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B. Hudson, J. Fragner (Eds.)
RESEARCHING THE TEACHING AND
LEARNING OF
MATHEMATICS II
2
The production of this publication has received fi nancial support
by the European Commision, DG XXII (Socrates, Erasmus) and
the Socrates National Agency in Austria
State College of Teacher Education Linz, Austria
Institute of Comparative Education (IVE)
Editors: B. Hudson, J. Fragner
Herausgeber: Pädagogische Akademie des Bundes OÖ,
Dir. Dr. Josef Fragner
All rights reserved
Linz 2005
TRAUNER VERLAG + BUCHSERVICE GmbH
ISBN 3-85499-188-6
ISBN 978-3-85499-188-5
Production:
Nucleus: Pädagogische Akademie des Bundes in Oberösterreich,
A-4020 Linz, Kaplanhofstraße 40
Cover:
Rudolf Trauner TRAUNER DRUCK GmbH & Co KG, Köglstraße
14, 4020 Linz
Layout: T. Peterseil
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CONTENTS
Acknowledgements
Introduction
Brian Hudson (Umeå University and Sheffi eld Hallam University) 7
The Mathematics Achievement of Slovene Students, Based on
TIMMS Data
Mojca Štraus (Educational Research Institute, Ljubljana) 9
Researching the Context in Mathematics
Zlatan Magajna (University of Ljubljana) 25
Can We Compare Like With Like in Comparative Educational
Research? – Methodological Considerations in Cross-Cultural
Studies in Mathematics Education
Birgit Pepin (The University of Manchester) 39
Exploring Children’s Conceptions of Zero in Relationship to Other
Numbers
Rona Ca erall (Sheffi eld Hallam University) 55
Transformation in a Series of Linear Images
Margaret Sangster (Canterbury Christ Church University College) 75
Designing a Learning Environment for Teaching Primary
Mathematics: ICT Use in Mathematics Education from Teachers’
Perspectives
Heidi Krzywacki (University of Helsinki) 97
Investigating Class Teachers’ Pedagogical Thinking and Action in
Mathematics Education: Theoretical and Methodological Overview
Sanna Patrikainen (University of Helsinki) 119
Concept Formation of the Concept of Function
Alena Kopáčková (Technical University of Liberec, Charles University of
Prague) 145
4
4
Tessellations by Polygons in Mathematics Education
Lucia Ilucová (Charles University of Prague) 161
Determining, Studying and Using Learning Styles – One of the
Aspects of the Technique of the Higher School
Ilmārs Kangro (Rēzeknes Augstskola, Rezekne Higher Education
Institution), Kaspars Politers (Liepājas Pedagoģij as Akadēmij a, University of
Latvia) 179
Teacher of Mathematics or Teacher Educator? Positionings and
Problematisations
Sigmund Ongstad (Oslo University College) 199
Validating as Positioning(s)? – A Discussion of Habermas’s Formal
Pragmatics
Hans Jørgen Braathe (Oslo University College) 213
Student’s Beliefs and A itudes towards Mathematics Teaching and
Learning – An Introduction to the Research
Kirsti Kislenko (Agder University College) 239
The Objects of Mathematics Education Research: Spo ing, and
Commenting on, Characteristics of the Mainstream
Dimitris Chassapis (Aristotle University of Thessaloniki) 253
Graph Comprehension of Primary School Students
Ioannis Michalis (Aristotle University of Thessaloniki) 271
Knowledge and Concepts of Rational Numbers Held by Elementary
School Teachers
Aristarchos Katsarkas (Aristotle University of Thessaloniki) 295
The Discourse of Primary School Teachers for Assessment in
Mathematics
Maria Vlachou (Aristotle University of Thessaloniki) 305
Mathematics Education as a Scientifi c Discipline: Implicit
Assumptions and Open Questions
Dimitris Chassapis (Aristotle University of Thessaloniki) 321
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5
ACKNOWLEDGEMENTS
I would like to acknowledge the contributions of colleagues who
acted as reviewers in the process of developing this publication:
Prof. Dr. Erkki Pehkonen, University of Helsinki
Prof. Dr. Sigmund Ongstad, Oslo University College
Dr. Birgit Pepin, University of Manchester
Dr. Zlatan Magajna, University of Ljubljana
Dr. Jarmila Novotna, Charles University of Prague
Assoc. Prof. Dr. Dimitris Chassapis, Aristotle University,
Thessaloniki
Also I wish to thank Miranda Barker of Wordwise Edit for all her
support with the editing of the fi nal texts.
The Intensive Programmes from which these papers arose were
supported under the European Commission under the Erasmus
Action (Higher Education) of the Socrates Programme. This activity
was also supported by my National Teaching Fellowship project
which was awarded by the Higher Education Academy in 2004.
Prof. Dr. Brian Hudson
Co-ordinator of MATHED 2004 and 2005 Intensive Programmes
Sheffi eld
July 2006
6
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7
Brian Hudson, Josef Fragner
7
INTRODUCTION
This is the second volume of papers arising from the MATHED
Intensive Programmes Researching the Teaching and Learning of
Mathematics. These papers arise from seminars and workshops
that took place during the summers of 2004 and 2005. The second
MATHED Intensive Programme was held in the Tolmin valley in
Slovenia from 11th to 22nd August 2004, hosted by the University of
Ljubljana. The third event was held in Vij landi in Estonia from 3rd to
13th July 2005, hosted by Tallinn University.
The partner institutions taking part were Sheffi eld Hallam University
(co-ordinator), Pädagogische Hochschule, Linz; Charles University
of Prague; University of Helsinki, University of Riga, Oslo University
College, Oxford Brookes University, Tallinn University and Aristotle
University of Thessaloniki. The MATHED Intensive Programmes
have been closely associated with the Socrates-Erasmus EUDORA
Project.
The intensive programmes enabled participants to examine issues of
policy and practice relevant to their work in mathematics education
within an international context. The programmes aimed to enable
participants to develop a critical focus on the nature of research into
the teaching and learning of mathematics in an international context
RESEARCHING THE TEACHING
AND LEARNING OF MATHEMATICS
II
Brian Hudson, Sheffi eld Hallam University and
Umeå University
Josef Fragner, Pädagogische Hochschule, Linz
8
INTRODUCTION
8
and to develop theoretical approaches and methods appropriate
to comparative research. This was intended to support the further
development on the part of the participants of their understandings
of the methodological complexities of research in mathematics
education, to identify current themes in mathematics education and
critically analyse their signifi cance and to relate these outcomes to
their own contexts.
The content of the intensive programme was based on the research
interests of the participants and involved active participation by
students and collaboration with peers from the outset. Each student
led a workshop and/or seminar based on their current work and
interests. In addition staff members led workshops on current issues
in mathematics education research.
Contact
Prof. Dr. Brian Hudson, Sheffi eld Hallam University, Division of
Education and Humanities
B.G.Hudson@shu.ac.uk
Umeå University, Department of Interactive Media and Learning
(IML)
brian.hudson@educ.umu.se
Prof. Dr. Josef Fragner, Direktor, Pädagogische Hochschule, Linz
Josef.Fragner@phlinz.at
9
Mojca Štraus
9
Abstract
In the last few decades, student achievement has played a central role
among the indicators used to evaluate the quality of education systems.
Accordingly, recent education reform in Slovenia included the achievement
of international standards of knowledge and skills as an important goal.
This chapter examines to what extent this goal has been achieved in the
non-reformed system. It describes the mathematics achievement of students
in the fi nal grade of the non-reformed compulsory education in Slovenia
based on the data from the Third International Mathematics and Science
Study (TIMSS). TIMSS data are also used to provide information about
achievements of students from other European countries, which are taken as
a point of reference for describing the Slovene achievement. Another point of
reference is derived from the a ainment targets in the reformed mathematics
curriculum. The goal of these comparisons is to provide information to
support the eff orts to successfully implement the reforms.
Keywords: Achievement, curriculum reform, TIMMS
Introduction
The political, social, and economic changes that occurred a er
1991 urged Slovenia to reform its education system. These reforms
encompassed the structure of the school system as well as the
curricula of all school subjects. The goals for reformed education
THE MATHEMATICS ACHIEVEMENT
OF SLOVENE STUDENTS, BASED
ON TIMSS DATA
Mojca Štraus, Educational Research Institute,
Ljubljana
10
THE MATHEMATICS ACHIEVEMENT OF SLOVENE STUDENTS, ...
10
explicitly stated that Slovene education: ‘… makes possible the
achievement of internationally comparable standards of knowledge
a er the completion of the primary education …’ (White Paper, 1996,
92)1. The reform is currently being implemented into the school
system through a 10-year process of stepwise transformations and it
is expected to be completed in the 2008/2009 school year.
Within this context the aim of curriculum development in Slovenia
was to reduce curriculum content and increase integration among
diff erent school subjects with more emphasis on inter-disciplinary
and conceptual knowledge. Throughout the political debate
surrounding the reform in Slovenia, comparisons with other
countries, and especially with those from the European Union,
were emphasised. Given contemporary worldwide a ention to
achievement, among the important questions to be answered were:
‘How well do Slovene students perform in comparison with students
from other countries? Do they reach expected levels of achievement?
What should be expected of the students?’ The answers to these
questions were sought through opinions of experts, experiences of
teachers and others involved in education and, where available, data
from empirical studies.
As part of the curriculum reform the a ainment targets, also called
the standards, were set for all school subjects at two levels in lower
grades and at three levels in the two fi nal grades, grades 8 and 9
(Učni načrt, 2002, translation: Curriculum Guide). When a student
masters the set of contents labelled minimum standards (called Level
1 standard in this chapter) he or she receives a pass grade. The second
level includes fundamental standards (called Level 2 standards).
They are defi ned as the knowledge and skills that it is expected that
the average student should a ain, and represent the knowledge and
skills that teachers should strive for their students to gain. The higher
level standards (Level 3) represent the knowledge and skills to be
expected of higher achieving students.
In the process of curriculum reform, two perspectives for describing
Slovene achievement are embedded. The fi rst is the intended
curriculum for mathematics in Slovenia (what teachers are supposed
to teach and students are supposed to learn). This chapter describes
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Mojca Štraus
11
the mathematics achievement of Slovene students in the non-
reformed system in the light of the newly set standards. Analysing
mathematics achievement of students just before the start of the
implementation of the new curriculum will enable insight into ‘the
starting point’ of the new curriculum. Future assessments carried
out in Slovenia may utilise this information to fi nd out whether
improvement measures introduced in the reformed curriculum have
had the desired eff ects on student achievement.
The second perspective from which Slovene mathematics
achievement will be examined is a comparison with achievements
of students from other European countries. The relevance of this
perspective is also clear from Slovene policy documents. In order to
identify areas of Slovene achievement in which improvements might
be desired, countries with similar or higher overall achievements
than Slovenia are considered relevant for comparison. The selection
of these countries is described in more detail in the section on
methodology. The relevance of comparisons of Slovenia with these
countries is underlined by Belgium Flemish and the Netherlands
being in the European Union for a number of years and by the very
recent accession, in May 2004, of Hungary and the Slovak Republic,
as well as Slovenia.
Corresponding to the two perspectives for describing Slovene
mathematics achievement, two research questions are posed:
1. How well did Slovene students at the end of compulsory education
in the non-reformed system in the late 1990s perform in mathematics
when compared to the a ainment targets in the reformed mathematics
curriculum?
2. How well did Slovene students at the end of compulsory education
in the non-reformed system in the late 1990s perform in mathematics
when compared to the performance of students in other European
countries?
There is no single ‘right’ perspective for describing student
achievement. Using two diff erent perspectives will provide more
insight into students’ achievements than a single one. Furthermore,
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THE MATHEMATICS ACHIEVEMENT OF SLOVENE STUDENTS, ...
12
the diff erences between the results of the analyses based on each
perspective can be examined. This is called convergence between
the two perspectives. The analysis of convergence of the results from
the two perspectives may highlight areas in Slovene achievement
in which improvements might be desired from both perspectives.
Furthermore, areas in which expectations in the intended curriculum
might be considered too high or too low, taking into account
achievements of students from other European countries, may also be
highlighted. Kellaghan (1996) describes this function of international
assessments as ‘enlightenment’.
Data on student mathematics achievement from the Third
International Mathematics and Science Study (TIMSS) will be
used. The next section briefl y presents the TIMSS database and the
methodology used. In the third section the results of the analyses are
presented and in the last section the main conclusions drawn from
these results are given.
Database and methodology
The Third International Mathematics and Science Study (TIMSS,
recently renamed Trends in International Mathematics and Science
Study; Robitaille, et al., 1993; Mullis, et al., 2003), is conducted
under the auspices of the International Association for Evaluation
of Educational Achievement (IEA; see e.g. IEA, 1998). Up to the
present, three TIMSS surveys have been carried out, in 1995, 1999,
and 2003. At the time of the present study, TIMSS 2003 data were not
yet available for analyses. To describe the performance of Slovene
students at the end of compulsory education in the non-reformed
system, the TIMSS 1999 data (IEA, 2001) will be used. The most
important feature of TIMSS for the present study is that it is based
on the curricula of participating countries. This enabled a link
between student achievement (the a ained curriculum) and the
curriculum as is prescribed in the offi cial documents (the intended
curriculum). Through this link, areas in student achievement in
which improvements might be desired can be identifi ed and possible
remedial actions developed.
TIMSS 1999 focused roughly on the end of compulsory education.
This was grade eight in almost all countries (Mullis, et al., 2000). In
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Mojca Štraus
13
Slovenia, grade eight students were taken as the target population
although they were, on average, slightly older than students in many
other countries (Mullis et al., 2000). The TIMSS design required that a
minimum of 150 randomly selected schools be sampled and in each
school one whole class of students in the target grade. In Slovenia and
the reference countries this yielded sample sizes of approximately
3000 students. The TIMSS mathematics achievement tests were
developed through an international consensus involving input from
experts in mathematics and measurement specialists (Garden, 1996).
The aim of TIMSS instrument development was to have items that
had maximum validity across participating countries and to test
as wide a range of school mathematics curricula as possible. The
items underwent an iterative development and review process,
including the pilot testing. Every eff ort was made to help ensure that
the tests represented the curricula of the participating countries and
that the items did not exhibit any bias towards or against particular
countries.
Measuring a ainment targets in the curriculum
Since the TIMSS achievement tests were not designed specifi cally
to describe the Slovene intended curriculum for mathematics, the
mutual coverage between the intended curriculum and the TIMSS
items needed to be examined in order to provide evidence of the
appropriateness of the test for measuring and linking the intended
and the a ained curricula in Slovenia. For each a ainment target
in the curriculum, a Slovene mathematics curriculum specialist
indicated whether it was covered by the items in the TIMSS tests.
The percentage of a ainment targets at Levels 1 and 2 that were
covered was 77 % for the TIMSS 1999 test. The main topics that
were not covered in the TIMSS tests were geometry topics about
circles, triangles, constructing angles and triangles, parts of three-
dimensional geometry including the Pythagorean theorem, and
simplifying symbolic expressions using properties of operations.
These topics are mainly covered in the fi nal grades of compulsory
education. Also, very few a ainment targets at Level 3 were covered.
The TIMSS items can be seen as covering the ‘general’ part of the
curriculum reasonably well, while they do not cover its ‘specialised’
part.
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THE MATHEMATICS ACHIEVEMENT OF SLOVENE STUDENTS, ...
14
The reverse issue was also addressed. For each TIMSS item, three
mathematics curriculum specialists indicated whether it was covered
by the a ainment targets in the reformed mathematics curriculum
and at which level. The percentage of TIMSS 1999 items that were
covered was 97 %. These percentages for the mutual coverage
between the test and the a ainment targets were deemed suffi ciently
large to enable meaningful examination of Slovene achievement on
the basis of TIMSS items, at least at Levels 1 and 2. The measures
for item coverage were given values ‘Level 1’, ‘Level 2’, and ‘Level
3’.For each item, the median of the three measures was taken as the
measure that was used in this study. The reliability of these expert
judgments was 0.72 [percent? And of what?] which was deemed
acceptable (e.g., Wolf, 1994) considering that a ainment targets are
a novelty in Slovene mathematics education.
In the process of allocation of items to the levels of the standards, it
was found that in addition to the fi ve items for which the standards
in the intended curriculum could not be determined, two items had
problems with translation and only six items were allocated at Level
3. These items were excluded from further analysis.
Selection of countries for reference points
The second reference point for describing Slovene mathematics
achievement was based on achievements of students from several
other European countries. As mentioned, in order to identify possible
areas for improvement in Slovene mathematics education, countries
with similar or higher overall achievement than Slovenia were
selected. Further, since Slovenia exhibited stability in achievement in
the period between 1995 and 1999, countries were selected that had
similar or higher overall achievement in both TIMSS measurements
in the late 1990s, indicating stability in their comparison with
Slovenia. The countries were selected based on their IRT scores for
average achievements reported in Beaton et al. (1996) and Mullis et
al. (2000). Four countries were selected in this way: Belgium Flemish,
the Netherlands, Hungary, and the Slovak Republic. The average
achievement in Belgium Flemish was signifi cantly higher than
in Slovenia on this scale in both TIMSS data collections, while the
average achievements in the other three countries were similar to
Slovenia.
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Mojca Štraus
15
Assessing the correspondence of achievement in Slovenia with the
selected reference points
In order to describe the correspondence of mathematics achievement
of Slovene students with the standards in the intended curriculum,
TIMSS items were used as the link between the two sets of curricula.
Similarly, the TIMSS items were used to link the curricula a ained
in Slovenia and in the reference countries. However, while the
achievements were directly comparable in the form of the percentages
correct the measurements on the intended curriculum were carried
out in a non-numeric form. To make them comparable with the
Slovene achievements, they were operationalised into the ‘intended
percentage correct’. This was done on the basis of general descriptions
of the diff erences between the three levels of the a ainment targets
as follows: for items allocated at Level 1, the expected percentage
correct was set to 75 %; and for items allocated at Level 2, the expected
percentage correct was set to 50 %.
This operationalisation is, of course, a very simple model for the
expected percentage correct (or intended diffi culty) for individual
items. Items that are used to measure student achievement may
and should vary in their intended diffi culty as well as in other
characteristics. However, in the absence of explicit guidelines of how
these levels of the a ainment targets should be operationalised in order
to assess whether they have been achieved, this operationalisation
was deemed a suffi cient approximation. It was also considered
plausible by the mathematics curriculum experts who allocated the
TIMSS items to the levels of the a ainment targets.
Once the target and reference measures were calculated in the form
of comparable scores, correspondence between the two was assessed.
When the estimate for Slovene achievement was signifi cantly higher
than the reference point, taking into account the standard errors of
the estimates, this was taken as an indication of a strength in Slovene
achievement. Generally, the level of signifi cance was taken at 0.05
using Bonferroni adjustment for multiple comparisons. In the cases
where it was signifi cantly lower, this was taken as an indication
of a weakness. In the remaining cases, Slovene achievement was
described as corresponding with the standards.
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THE MATHEMATICS ACHIEVEMENT OF SLOVENE STUDENTS, ...
16
When the reference point was constructed on the basis of the
achievements of students from the four reference countries, four
reference measures were computed as the average percentage correct
estimates of achievement in these countries and their standard
errors. In comparison with each of these countries it was indicated
whether Slovene achievement was signifi cantly higher or lower.
For this a usual test of signifi cance was used (t-test with Bonferroni
adjustment and signifi cance level of 0.05). In this way, comparisons
with each of the four reference countries were examined. However,
since no particular country was taken as ‘the most important’ for
comparisons with Slovenia, it was judged that there were indications
of strengths or weaknesses in Slovene achievement if signifi cantly
higher or lower achievement in Slovenia was observed in at least two
comparisons.
Results
Following the design of the research questions, this section is
organised into three subsections. The fi rst presents the results of
analysis of correspondence of Slovene mathematics achievement
with the a ainment targets in the reformed curriculum. The second
presents the results of analysis of correspondence of Slovene
mathematics achievement with achievements of students in other
European countries. The third discusses the convergence of the
results between the two reference points.
Correspondence of Slovene mathematics achievement with the
a ainment targets in the reformed curriculum
The results of analysis of correspondence with the a ainment targets
by these content areas are presented in Table 1. Average scores (in
the form of the average percentage correct) of Slovene students
across content areas in the curriculum and across the levels of the
standards are presented, as well as the score on the total TIMSS 1999
mathematics test. As shown, average scores of Slovene students
varied across content areas and across levels of the standards. In
one cell in Table 1 the results are not shown nor is the content area
‘probability’ due to insuffi cient numbers of items (a minimum of
fi ve items was deemed necessary for meaningful analysis). It can
be observed from Table 1 that correspondences of scores with the
standards varied across content areas. When looking at the overall
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Mojca Štraus
17
level, a satisfying correspondence was observed for most content
areas. On items about ‘natural numbers’, ‘algebraic expressions’,
and ‘data representation’, average scores of Slovene students were
signifi cantly higher than the standards. Average scores in the content
areas: ‘meaning of rational numbers’ and ‘geometrical shapes’ were
lower than the standards.
Table 1. Mathematics achievement of Slovene students in 1999 by
content areas and for the total test2
The correspondences in Table 1 diff ered between the two levels
of the standards for most content areas. As expected, scores were
higher at Level 1 than at Level 2 in most content areas (p<0.05)4 ,
except in ‘geometrical shapes’ where they were identical. However,
except for the content areas ‘natural numbers’, ‘measurement’, and
‘data representation’, students’ scores at Level 1 were lower than
the standards with the lowest diff erence of 20 percentage points
in ‘geometrical shapes’. As explained in the methodology section,
the intended achievement at Level 1 is 75 percentage correct and at
Level 2 50 percentage correct. In many content areas, average scores
of Slovene students at Level 2 were higher than the standards and in
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THE MATHEMATICS ACHIEVEMENT OF SLOVENE STUDENTS, ...
18
no content area at this level were they lower. In quite a few content
areas weaknesses at Level 1 seem to be compensated by strengths at
Level 2.
The average score of Slovene students on all items in the TIMSS 1999
achievement test was 61 percent which corresponded with the standards.
This could be interpreted as showing that overall achievement of Slovene
students in mathematics is satisfactory when compared to the a ainment
targets in the curriculum.
When considering the diff erent levels of the standards, it can be observed
that the average score at Level 2 was lower than at Level 1. The results
in Table. 1 indicate that while scores in Slovenia corresponded with
the standards at the level of the overall test, there seem to have been
weaknesses at Level 1 and strengths at Level 2.
Correspondence of Slovene mathematics achievement with the
achievements of students in other European countries
To assess the comparability of Slovene achievement with the achievement
of students in other European countries, average percentage correct
scores of these students on the overall mathematics test and in content
subdomains were compared. The results are presented in Table 2.
Table 2. Mathematics achievement of Slovene students in 1999 by content
areas and in total compared to the reference countries
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Mojca Štraus
19
When we ‘zoom in’ on the strengths and weaknesses in Slovene
achievement by comparison with other European countries, it can
be observed that Slovene scores corresponded with the scores in the
reference countries in most content areas. The greatest diff erence
can be found in comparison with Belgium Flemish. An overview of
Table 2 across countries and across content areas reveals that Slovene
achievement was signifi cantly lower than at least two reference countries
in the content areas: ‘natural numbers’, ‘meaning of rational numbers’,
‘measurement’, and ‘geometrical shapes’. It seems plausible to argue
that, especially in the content area: ‘meaning of rational numbers’, more
general weaknesses in Slovene achievement existed. Considering the
‘European dimension’ emphasised in the Slovene policy documents,
further improvements might be focused on these areas.
Convergence of the results when compared to the two reference
points
By using two reference points, two descriptions of Slovene achievement
in mathematics were obtained. Table 3 presents a summary of the results
for locating strengths and weaknesses in Slovene achievement in content
areas of the Slovene curriculum. As shown, there were similarities and
diff erences in the descriptions of Slovene achievement between the two
perspectives. From both perspectives weaknesses in Slovene achievement
were observed in the content areas: ‘meaning of rational numbers’ and
‘geometrical shapes’. Content areas ‘algebraic expressions’, ‘functions
and proportionality’, ‘operations with rational numbers’, and ‘data
representation’ can be described as satisfactory from both perspectives.
Table 3. Summary of correspondence of achievement with the two
reference points
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THE MATHEMATICS ACHIEVEMENT OF SLOVENE STUDENTS, ...
20
The diff erences between the two perspectives emerged in the
content area ‘measurement’, in which Slovene achievement seemed
to correspond with the standards but it was lower than in two
of the reference countries. Similarly, average scores in ‘natural
numbers’ were higher than the standards, but were lower than in
two of the reference countries. These results reveal the importance of
understanding the standards in the reformed Slovene mathematics
curriculum in the light of achievement of students from other
countries.
Conclusion
In this chapter, the problem of describing Slovene mathematics
achievement to serve the needs for comparative international
information as an input into the process of curriculum reform,
and its implementation, has been addressed. This adds to other
studies (e.g., Magajna, 2000), in shedding more light on to the areas
in which improvements might be desired and be possible. Two
research questions were formulated, giving two perspectives for the
description of achievement. The fi rst examined the correspondence
of Slovene achievement in mathematics with the a ainment targets
in the reformed curriculum and the second with the achievements
of students from four other European countries: Belgium
Flemish, the Netherlands, Hungary, and the Slovak Republic.
These correspondences were examined at the level of the overall
mathematics domain and in content subdomains of the curriculum.
The most general fi nding in this study is that mathematics achievement
of Slovene students in the non-reformed system corresponded with
the a ainment targets in the reformed curriculum. However, this
is at the level of overall mathematics achievement only. Detailed
analyses revealed variation in the correspondences. When looking
at achievements at Level 1, weaknesses in student achievement were
observed. These were largely compensated with strengths at Level 2.
There seem to be defi ciencies in the knowledge and skills of Slovene
students that are inappropriate from the perspective of the intended
reformed curriculum. In order to achieve the goals of the reform,
a ention must be focused on the knowledge and skills at Level 1, that
is on knowledge and skills that nearly all students should master.
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Mojca Štraus
21
Further analyses, carried out for diff erent content subdomains in
the Slovene curriculum, revealed that the contrast between Level
1 and Level 2 occurred in most of these areas. The complementary
comparisons with other European countries revealed also that
improvements might be desired in the content areas of: ‘meaning of
rational numbers’ and ‘geometrical shapes’.
This study was basically descriptive. It provided information
on the strengths and weaknesses observed in Slovene students’
mathematics achievement when compared to the relevant reference
points. By linking student achievement in the non-reformed system
to the reformed curriculum it provides information on the point from
which the reformed curriculum is starting. This information may be
also used in possible future studies to examine whether the desired
eff ects of the reforms can be observed.
The results in this chapter pointed out the importance of in-depth
analyses of the national results in an international context. For
example, the international TIMSS reports (Beaton et al., 1996; Mullis
et al., 2000) showed that Slovenia is among the higher achieving
European countries. However, in more detailed comparisons it was
found that there is room for improvement.
This study also showed the importance of describing student
achievement from several perspectives. While the intended
curriculum is an important perspective for student achievement, it
was shown that there might be diff erences between the intentions in
the national curriculum and what it is possible to achieve as observed
in other countries. This function of international assessments is termed
‘enlightenment’ (Kellaghan, 1996). It may be that the intentions in
the curriculum are too high, however, it may also be that they are
too low. Using the results of this study, Slovene educators are in a
position to understand be er student achievement and at the same
time the a ainment targets that were set in the reformed curriculum.
This study was not an evaluation of the a ainment targets but aimed
at providing information on what they mean and how can they be
used. Through measurements of student achievements and analyses
of their correspondence with the standards, as well as with other
reference points, these standards may be refi ned further in terms
of wording and content as well as in terms of intended levels of
22
THE MATHEMATICS ACHIEVEMENT OF SLOVENE STUDENTS, ...
22
achievement. Although the perspectives used in this study certainly
do not refl ect all possible views from which student achievement
could be described, they provide a wealth of information to help
Slovene educators in their eff orts to improve Slovene mathematics
education and its outcomes.
Footnotes:
1 The authors used the term primary education to describe the
compulsory part of the Slovene education system. Sometimes the
terms basic or elementary education are used. In this study, the term
compulsory education will be used.
2 In all tables in this chapter, rounded estimates are presented, while
tests of signifi cance were carried out using non-rounded values. This
may cause some inconsistencies in the tables.
4 The signifi cance of this diff erence is not indicated in Table 1 to avoid
confusion with indications of whether the scores correspond with
the standards. This signifi cance can be determined using t-test with
the standard error that is presented in Table 1. The same holds for
similar comparisons in other tables in the remainder of this chapter.
References
Beaton, A.E., Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Kelly, D.L.,
and Smith, T.A. (1996) Mathematics Achievement in the Middle
School Years: IEA’s Third International Mathematics and Science
Study (TIMSS). Chestnut Hill, Mass: Boston College.
Garden, R.A. (1996) Development of the TIMSS Achievement Items,
in D.F. Robitaille, and R.A. Garden (eds), TIMSS Monograph No.2:
Research Questions and Study Design, pp. 69–80. Vancouver, Canada:
Pacifi c Educational Press.
International Association for Evaluation of Educational Achievement
(IEA) (1998) IEA Guidebook 1998: Activities, Institutions and People.
Amsterdam: Author.
IEA (2001) TIMSS 1999 International Database. Chestnut Hill, Mass:
Boston College.
Kellaghan, T. (1996) IEA Studies and Educational Policy. Assessment
in Education, 3, pp.143–60.
Magajna, Z. (2000) Obravnava primanjkljajev matematičnega znanja
slovenskih učencev v učnem načrtu slovenske osnovnošolske
matematike [Addressing Slovene students’ lack of mathematical
23
Mojca Štraus
23
skills in the elementary school mathematics syllabus]. Sodobna
Pedagogika, 2, pp.162–80.
Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Gregory, K.D., Garden,
R.A., O’Connor, K.M., Chrostowski, S.J., and Smith, T.A. (2000)
TIMSS 1999 International Mathematics Report: Findings From IEA’s
Repeat of the Third International Mathematics Science Study at the
Eighth Grade. Chestnut Hill, Mass: Boston College.
Mullis, I.V.S., Martin, M.O., Smith, T.A., Garden, R.A., Gregory, K.D.,
Gonzalez, E.J. Crostowski, S.J., and O’Connor, K.M. (2003) TIMSS
Assessment Frameworks and Specifi cations (2nd edn.). Chestnut
Hill, Mass: Boston College.
Robitaille, D.F., Schmidt, W.H., Raizen, S.A., McKnight, C.C., Bri on,
E., and Nicol, C. (1993) TIMSS Monograph No.1: Curriculum
Frameworks for Mathematics and Science. Vancouver, Canada:
Pacifi c Educational Press.
Učni načrt: program osnovnošolskega izobraževanja: matematika.
[Curriculum guide: the program of elementary education:
mathematics] (2002) Ljubljana, Slovenia: Ministry of Education and
Sports.
White Paper on Education in the Republic of Slovenia (1996) Ljubljana,
Slovenia: Ministry of Education and Sports.
Wolf, R.M. (1994) The Validity and Reliability of Outcome Measures,
in A.C. Tuij nman, and T.N. Postlethwaite (eds), Monitoring the
Standards of Education: Papers in Honor of John P. Keeves, pp.121–
32. Oxford: Pergamon Press.
Contact
Dr Mojca Štraus
Educational Research Institute, Ljubljana
Slovenia
mojca straus@pei.si
24
THE MATHEMATICS ACHIEVEMENT OF SLOVENE STUDENTS, ...
24
25
Zlatan Magajna
25
Abstract
The chapter is an analytical account of conceptualisations of the context in
mathematics. It is not intended to comprise all possible conceptualisations;
it a empts to be pragmatic and to focus results that were found to be
relevant to mathematics class practices. These conceptualisations of context
are related to other subject domains, for example the cultural environment,
and to diff erent theories of connecting mathematics to the context: common
modelling, ethnomathematics, and activity theory. Furthermore they are
based on diff erent views of the nature of mathematics. To illustrate these
ideas, a small research topic comparing the treatment of the context in two
mathematics textbooks for Slovenian vocational schools is described.
Key words: activity theory, context, ethnomathematics, modelling,
word problems
Introduction
Context is a commonly used word in mathematics education, yet its
meaning is o en unclear. In this author’s experience most teachers
understand contextualisation of mathematics as ‘using words
from outside mathematics in a mathematics class’ and they believe
that contextualisation of mathematics is mainly important for two
reasons: 1. for motivational purposes, and 2. for learning to apply
mathematics to real-world situations. Such an understanding of
RESEARCHING THE CONTEXT IN
MATHEMATICS
Zlatan Magajna, University of Ljubljana
26
RESEARCHING THE CONTEXT IN MATHEMATICS
26
contextualisation appears rather narrow and perhaps naive, for the
role of context in mathematics is multifarious: the context is not just
something from which data are taken and to which mathematics
can be applied – the context may be essential in helping students to
build mathematical knowledge, to relate mathematical knowledge
properly to their beliefs, their values, to their cultural background
and foreground as well as to their professional activity.
This article is an analytic account of conceptualisations of the
context in mathematics. It is not intended to include all possible
conceptualisations of the context, rather it a empts to be pragmatic
and to focus on understandings that the author fi nds to be relevant to
practices in mathematics classes. Three areas of contextualisation of
mathematics will be covered: subject domains, culture, and activity.
Related to these are three diff erent ways of connecting mathematics
to the context: common modelling, ethnomathematics, and activity
theory. Note that the three areas of contextualisation also refl ect
diff erent views of the very nature of mathematics, but this issue will
not be pursued here.
Subject-domain as context
Greeno (1991, 174) defi nes a subject-domain as “a structure of facts,
concepts, principles, procedures and phenomena that provides
resources to the cognitive activity of knowing, understanding, and
reasoning”. Mathematics, seen as a formal body of knowledge,
is a subject-domain, and so are physics, economics, and perhaps
even kni ing. Usually, we think of mathematical knowledge as a
cognitive structure, consisting of representations and procedures to
manipulate them, which is (in a way) a subset of mathematics as a
subject-domain.
Context is o en interpreted as an environment in which the
notion of a concept is related to its meaning, i.e. as a general set of
conditions that organise the meaning of a notion at the linguistic
level. In mathematics education, however, the term ‘context’ is o en
used in a specifi c way in the sense of ‘applying words from outside
mathematics’ (Sträßer, et al., 1989). The underlying assumption is
that there is a formal, abstract, ‘decontextualised’ mathematical
subject-domain, in which the mathematical objects have meaning,
and a real-world environment or other subject-domains, which
27
Zlatan Magajna
27
are referred to as ‘the context’. Mathematics is thus understood as
a body of knowledge in which meanings are detached from the
experiential world. Pu ing mathematics in context, as in the case of
school-mathematics tasks given in context, is understood as relating
– o en just at a semantic level – the meanings of the mathematical
objects outside mathematics.
There is no doubt that students sometimes apply school-learnt
mathematical procedures or ideas to situations outside the
mathematics class. Yet there is plenty of evidence that children o en
do not activate real-world knowledge in solving school-mathematics
word problems. Many pupils, for example, claim that if we pour
together into a container 1 litre of water at 40C and 1 litre of water of
80C the resulting mixture will be 120C (Verschaff el, et al., 1994). In
fact, an analysis of typical school-mathematics tasks given in context
(usually as word problems) shows that the intended links between
the situation described and the supposed mathematical meanings
can be quite artifi cial, sometimes even whimsy. DeLange cites an
extreme example of a story problem quoted by Pollak:
Given that two bees can gather nectar from 100 hollyoak blossoms in
30 minutes and assuming that each bee works eight hours a day, fi ve
days a week, how many blossoms do these bees gather nectar from
in a summer season of fi een weeks? (DeLange, 1996, p. 67)
One cannot but agree with Lave (1992) that such problems (and indeed
most story problems in school-mathematics textbooks) are not aimed
at elaborating intrinsic connections between abstract mathematics
and real-world contexts but are a socioculturally established way
of grounding the abstract mathematical concepts in familiar terms
for pupils who have only a concrete grasp of mathematics. More
precisely, the context (in mathematics) sometimes serves as a general
motivator, and sometimes as a facilitator, by suggesting thinking
in terms of well known representations instead of abstract entities
(Boaler, 1993). Sierpinska (1995) claims that everyday contexts in
school-mathematics are not intended to be authentically everyday
so that pupils learn how to use mathematics in everyday life – their
function is to be authentically mathematical so that they help pupils
to learn abstract mathematical ideas. Boaler (1993), on the other
28
RESEARCHING THE CONTEXT IN MATHEMATICS
28
hand, believes that contexts in school mathematics promote the
perception of common underlying structures that enable pupils to
use mathematical knowledge in diff erent contexts.
The common way of applying mathematics (as a subject domain)
to other subject domains is mathematical modelling (see Figure
1). It presupposes two distinct systems of meanings: the system of
mathematical meanings and other systems of meanings, usually
referred to as real-world or some subject-domain. The two are, in
theory, autonomous, but there is some sort of parallelism between
them. Basically, given a real-world phenomenon the idea is to set
a ‘mathematical equivalent’ (e.g. a formula or a set of diff erential
equations), which is elaborated ‘mathematically’ without any
reference to the original model. Then the results obtained are
translated back into the real-world context and evaluated.
Such dualistic reasoning is part of the epistemological tradition
of the Western scientifi c community and has signifi cant didactical
implications. The most important one is perhaps the importance and
reasonableness of learning pure, ‘decontextualised’ mathematics.
Second, the mathematical reasoning in the second phase of modelling
is, in theory, done without reference to the original situation. Third,
the link between mathematics and the real-world context is quite
artifi cial, for what ma ers is that the appropriately interpreted
outcomes of the calculations fi t the experimental data. For students it
is, as a rule, disturbingly diffi cult to fi nd the links, through reasoning,
between a real-world phenomenon and its mathematical model.
Applying mathematics by modelling is defi nitely successful in
science and is deeply rooted in school practice. Perhaps the modelling
paradigm is most clearly expressed in the idea of the technique
curriculum, which is based on mathematical procedures, methods,
skills, rules and algorithms (Bishop, 1988). Since the ‘linking’ part of
the modelling is beyond the grasp of most students, they simply learn a
number of models (application-related formulas, equations, etc.) and
essentially all their eff ort is put into learning to apply the established
mathematical model. In other words, the majority of students are
supposed to learn pure (i.e. decontextualised) mathematics in
order to use the techniques they have learned in various modelling
29
Zlatan Magajna
29
situations. Yet nowadays there is plenty of evidence that when
solving mathematical problems in out-of-school situations people
only rarely resort to school learnt modelling, instead they use job-
specifi c techniques and methods (e.g. Noss and Hoyles, 1996). The
fact that mathematics students do a lot of exercises (usually stated as
word problems) on modelling does not necessarily mean that they
are able to ‘mathematise’ real-world situations. Very o en they just
‘“respond to word problems according to stereotyped procedures
assuming that the modelling situation is ‘clean’” (Greer, 1993). Thus,
a lot of eff ort is put into fi nding ways of enabling the students to
learn how to model.
Figure 1. Mathematical modelling
Culture as context
From this perspective mathematics is not ‘a subject domain of
universal validity’ but rather a culture dependent interpretational
means. Mathematics is still considered to be a body of knowledge,
but this knowledge is a social category, which is socially distributed,
transmi ed by social means, and is inextricably linked to other
elements of human culture. Bishop (1988) conceives mathematics
as a cultural product which has developed in societies as a result
of universal activities: counting, locating, measuring, designing,
playing, and explaining. Mathematics is, in this sense (a part of) the
symbolic technology of culture. In this sense mathematics diff ers
from culture to culture. Western mathematics is thus not universal, it
is perhaps just the most widespread and well known, but traditional
societies, non-Western cultures and even various cultural groups in
30
RESEARCHING THE CONTEXT IN MATHEMATICS
30
the Western world have their own mathematics, coherent with their
culture.
Thus mathematics is not something ‘decontextualised’ for it is
embedded in its cultural context. The links between mathematics and
various cultural constituents are o en hardly discernible. However,
they become apparent or even disturbing if the mathematics of one
culture is learnt in another cultural environment. The resulting
tensions may be simply ignored (e.g. assuming that mathematics is
a culture-free knowledge area) or may be taken into consideration at
various levels in the curriculum, teaching style, and language.
Boaler (1993) points out that a mathematical class is by itself a cultural
entity in its own right and thus that mathematical knowledge and
cognition cannot be separated from it. On the other hand ‘school
mathematics’ should acknowledge that (practical) solutions
developed by various cultural groups (outside school) to real-life
mathematical problems are also mathematical. According to Boaler,
both of these links should be considered if we want pupils and
students to learn mathematics that is both meaningful to them and
also useful in general situations.
If mathematics is understood as a cultural phenomenon, it is
worthwhile to take into account that culture, and mathematics
in particular, can be experienced on diff erent levels – a fact that
has been pointed to by Sierpinska (1994, 161). It is based on the
theoretical considerations of C. T. Hall, who has identifi ed three such
levels distinguished by specifi c ways of transmission of knowledge,
emotional relations, forms of communications (Sierpinska, 1994).
From this perspective the mathematics of mathematicians (as a
cultural product) is not all technical, that is explicit and rationally
justifi ed or explained. The way proofs are wri en, the way problems
are a acked, the expected rigour in argument, etc. belong to the
informal level. And there is also a level of mathematical beliefs,
unquestioned facts (not axioms) taken as self-evident.
Ethnomathematics is just another approach to relating mathematics to
its cultural milieu. Gerdes (1994) speaks of ethnomathematics as “the
cultural anthropology of mathematics and mathematical education”.
31
Zlatan Magajna
31
During the last two decades the concept of ethnomathematics has
diversifi ed into at least three directions: 1. as the mathematics
of various cultural groups, e.g. Australian aboriginal cultures or
the street vendors in Brasilia; 2. as an emancipatory movement in
former colonial societies, opposing world welfare injustices and the
supposed superiority of Western culture; 3. as an educational theory
(described below).
Ethnomathematics as an educational theory stresses the need to
base mathematics education on the students’ cultural background,
thus using their out-of-school experience, extracting mathematical
ideas from the cultural environment and embedding them into it.
It recognises that each cultural se ing, including underprivileged
minorities or non-Western populations, possesses knowledge for
coping with ‘mathematical’ challenges (counting, measuring, etc.).
Organised learning in school should take account not only of such
mathematical practices, but also consider the related jargons, codes
and styles of reasoning (D’Ambrosio, 1991). However, some authors
are opposed to such simplifi ed linking of (school) mathematics and
ethno-culture. Vithal and Skovsmose (1997), from the standpoint
of critical education, claim that school mathematics should train
students’ minds in order to enable them to cope with today’s
technological society, that school mathematics should give the
students a tool to improve the life of all of society – it should not be
seen as an essentially neutral ‘defrosting’ of mathematical ideas from
traditional artefacts.
Activity as context
A third view of mathematical knowledge focuses on (mathematical)
actions carried out in actual situations. Based on the theory of
activity originated by Vygotsky and developed by Leont’ev and
later by Engeström, Wertsch and others, actions are the constitutive
basis of a human mind (Wertsch, 1985). The actions people do,
according to this theory, are not the result of applying some sort
of abstract (decontextualised) knowledge to specifi c situations, but
are instead inextricably linked to the situations in which they are
learnt and executed. Adding scores while playing a card game and
adding numbers while practising in a mathematics class are perhaps
mathematically isomorphic problems but considered as actions
32
RESEARCHING THE CONTEXT IN MATHEMATICS
32
they are certainly not. O en the same person solves apparently
equivalent mathematical problems diff erently in various situations,
using diff erent methods, with diff erent success rates, etc. It this sense
the knowledge appears to be situated. Several studies confi rm that
there is a gap between school-learnt mathematical knowledge and
mathematical practices carried on elsewhere. Nunes, Schliemann,
and Carraher (1993), for example, studied the mathematical
behaviour of children selling candies or other goods on the streets
of Brazilian towns. They were, in general, successful in solving the
mathematical tasks in selling situations, less successful in (apparently)
isomorphic tasks given as word problems, and even less successful
in performing equivalent ‘decontextualised’ school-like calculations.
Another well known study that confi rmed the existence of a schism
between mathematical practices in various activities is the Adult
Mathematics Project (Lave, 1988). Here is an illustrative example. In
one of the tasks the participants (housewives) were asked to choose
the best value-for-money buy given several articles of the same type
and quality with respective weights and prices. When the problem
was given in the form of school-like exercise (performed in a home
se ing) the participants, on average, were able to identify the best
buy in 59% of cases while in a supermarket se ing in a real situation
the success score was a remarkable 98%.
Mathematical tasks may be given ‘in abstract’, but the actions
undertaken in order to solve them always occur in specifi c situations.
These indicate the connotations of the task, add many additional
restrictions, and explicit and implicit assumptions which imply
specifi c ways of acting. It therefore seems inappropriate to view
individuals’ mathematical actions as applications of formal, abstract
knowledge learnt at school to concrete situations. Instead, it appears
that mathematical actions undertaken in specifi c situations result
from situation-specifi c learnt knowledge, consisting of specifi c
strategies, methods, tool-related procedures (see, for example,
Carraher et al., 1987). Noss and Hoyles (1996) claim that the process
of abstraction that produces mathematical knowledge should be seen
more as synthesising diverse situational links than as disconnecting
or decontextualising from them. They use the expression situated
abstraction to underscore the idea that abstraction develops in an
activity through refl ection.
33
Zlatan Magajna
33
If knowledge is assumed to consist of actions, then context should
be considered as the environment in which the action occurs. This
environment is sometimes seen simply as ‘the situation’ in which
the action or learning happens; however, the proper environment
of an action is the activity in which the action occurs. Activity is a
socioculturally defi ned context in which human functioning occurs
(Wertsch, 1985). Examples of activities are school-learning, playing,
working-on-a-job, gambling, etc. Activity as a context, according
to the Vygotskian perspective, aff ects cognition on two levels. On
a sociocultural history level it provides tools, methods, schemes,
etc. for cognitive activity and for solving problems. On the level
of immediate social interactional context it structures the cognitive
activity. Being engaged in an activity means interacting with, or even
adopting, the socially shared cognitive tools, methods, strategies,
rules, ways of social interacting, and behaviour that are part of
the activity and are coherent with the aims of the activity – and all
this (context) situates the knowledge (actions). Work as activity is
a particularly important case of a context of mathematical actions,
for one of the aims of learning mathematics at school is its use in
pupils’ and students’ future professional life. Until a few decades
ago work mathematics was considered essentially an application of
the mathematics learnt at school, but nowadays we are aware of a
considerable discontinuity between school and work mathematics –
though there is a certain (not yet well understood) interplay between
them. Millroy (1992), for example, made an ethnographic study of a
group of carpenters, which nicely illustrates how the mathematical
methods and even concepts in work situations diff er from those
learnt at school. Working as an apprentice in a carpentry workshop
she discovered that mathematical cognition in such situations has
an important social connotation and is inextricably linked to other
job-related practices, conventions, tools and social relations in the
particular activity. For example, she was taught by her teacher about
perpendicularity in relation to drilling holes perpendicular to a given
plane, together with the technique of drilling the hole and ‘feeling’
the perpendicularity in the tool and so interactively adapting the
drilling direction.
34
RESEARCHING THE CONTEXT IN MATHEMATICS
34
An illustrative application
In school-mathematics, as we have seen, context may be used
with diff erent meanings (depending on the perception of what
mathematics is) and for diff erent purposes. In this section we shall
consider the role of context in mathematics education in vocational
schools in Slovenia. We shall illustrate this by reporting a small-scale
study, which formed the diploma thesis by Maja Lebar, a student
at the Faculty of Education in Ljubljana. In the study two textbooks
used in mathematical courses in Slovenian vocational schools were
considered. One of the textbooks (referred to as Textbook I) is more
‘mathematically oriented’, the author put a lot of emphasis on nicely
elaborated examples, on systematic elaboration, on developing
‘mathematical ways’ of thinking. The other textbook (referred to as
Textbook II) is more ‘student oriented’, the authors tried to simplify
elaborations and to gain the students’ a ention with appealing
illustrations related to (at least apparently) real-life situations.
Table 1. Categories used in the classifi cation of the task
The relation between mathematics and the context used is not considered in
the textbook (so that the students presumably did not relate their experience to
mathematics)
0 There is no non-mathematics context The exercise relates strictly to considered
(mathematics) topics. The exercise also
relates to other mathematics topics.
1a The context serves purely for
motivational purposes The context is mentioned but is not
related to the mathematical content.
1b The context is artifi cial, whimsy The context is far from students’
experience. The students presumably
just read the data and did not consider
the situation itself. The students
presumably had not and would not
encounter the situation described. The
context is clearly impossible in real life.
In real situations the described task
cannot or should not be solved with the
given data. The context used does not
help in understanding mathematics nor
does mathematics1b
35
Zlatan Magajna
35
1c The context is authentic but the
students presumably do not relate it to
mathematics
The students will presumably not relate
the context and mathematics.
The context relates mathematical knowledge and student’s experience The relation
between mathematics and the used context is important and is considered in the
textbook or is evident (thus the students presumably relate their experience and
mathematics)
2a The experience helps students to
build mathematical knowledge The context is authentic and the
students are encouraged to relate it to
the mathematics. The tasks considered
are well known to the student who can
solve them by common sense.
2b Mathematics enables a new
interpretation of students’ experience The students can meaningfully interpret
their experience with mathematical
concepts. The mathematics considered
gives a new dimension to students’
experience. Mathematics helps in
understanding a context that has not
been experienced by the student but is
nevertheless well known to them.
2c The context and the mathematics
under consideration complement each
other
The students’ experience and
mathematical thinking are intertwined.
A complex task has to be solved by
resorting to the students’ experience.
3 Mathematics is related to values,
emotions, tradition or other elements of
cultural identity
The text refers to values, emotions,
tradition or other elements of cultural
identity.
4 The context is related to a (professional)
activity The solution of a problem or the
interpretation of the result requires
knowledge about a (professional)
activity, e.g. conventions, norms,
social relations between participants.
In the exercise various elements of a
professional activity play a prominent
role. The solution of the task depends
on specifi c assumptions, suppositions or
circumstances related to a professional
activity.
Here is a brief description of the methodology that was used to
compare the role of the context in the two textbooks. First, the unit of
analysis had to be chosen. Since both textbooks are based on exercise-
like segments (e.g., exercises, examples), they were set as units of
analysis (note that precise rules for tackling compound exercises
36
RESEARCHING THE CONTEXT IN MATHEMATICS
36
had to be set). Second, possible roles of the context were specifi ed
with related criteria for classifi cation (Table 1). Third, the units of
analysis (i.e. exercises, examples) were categorised according to the
classifi cation developed (Table 2). Part of the units of analysis was
reviewed independently by two experts to check the appropriateness
of the classifi cations used.
Table 2. Numbers of types of exercises in the two textbooks
Textbook I Textbook II
No. % No. %
0 There is no non-mathematics
context 457 73.83 688 79.81
1a The context serves purely for
motivational purposes 3 23.91 5 19.03
1b The context is artifi cial,
whimsy 128 140
1c The context is authentic but is
not related to mathematics 17 19
2a The experience helps the
student to build mathematical
knowledge
3 1.29 3 0.81
2b Mathematics enables a
new interpretation of students’
experience
33
2c The context and the mathematics
complement each other 21
3 Mathematics is related to other
elements of cultural identity 0 0.00 0 0.00
4 The context is related to a
(professional) activity 1 0.16 2 0.23
Other 5 0.81 1 0.12
The fi gures in Table 2 speak for themselves. Though the textbooks
diff er in many respects they relate mathematics to the context in a
similar way. The contextualisation of mathematics is done, basically,
by ‘mentioning words from outside mathematics’. Both textbooks
very rarely explicitly use students’ everyday or professional
experience to help them build their mathematical knowledge or,
vice versa, use mathematics to gain be er understanding of the
world. The textbooks also do not relate mathematical knowledge to
37
Zlatan Magajna
37
cultural values or students’ emotive sphere, and essentially they also
do not consider mathematics as part of other professional activities.
Obviously, teachers in vocational schools may treat the examples
and exercises in the textbooks in a diff erent way – they may, for
example, emphasise and elaborate the links between mathematics
and the context mentioned in the exercises. The table merely shows
that this is not done in the textbooks themselves. As a ma er of
interest it is worth noting that the new curriculum for vocational
schools in Slovenia strongly emphasises the intrinsic links between
mathematics, students’ experience and cultural environment, and
professional activities.
References
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and Oral Mathematics. Journal for Research in Mathematics
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Learning Cultures and Computers. Dordrecht: Kluwer Academic
Publishers.
Nunes, T., Schliemann, A.D. and Carraher, D.W. (1993) Street
Mathematics and School Mathematics. Cambridge: Cambridge
University Press.
Sierpinska, A. (1994) Understanding in Mathematics. London: The
Falmer Press.
Sierpinska, A. (1995) Mathematics: “in Context”, “Pure”, or “with
Applications”? For the Learning of Mathematics, 15(1), pp. 2–15.
Sträßer, R., Barr, G., Evans, J. and Wolf, A. (1989) Skills Versus
Understanding. Zentralbla fur Didaktik der Mathematik, 21(6), pp.
158–68.
Verschaff el, L., DeCorte, E. and Lasure, S. (1994) Realistic
Considerations in Mathematical Modelling of School Arithmetic
Word Problems. Learning and Instruction, 4, pp. 273–94.
Vithal, R. and Skovsmose, O. (1997) The End of Innocence: A Critique
of ‘ethnomathematics’. Educational Studies in Mathematics, 34, pp.
131–57.
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Contact
Dr. Zlatan Magajna
Faculty of Education, University of Ljubljana
Slovenia
zlatan.magajna@uni-lj.si
39
Birgit Pepin
39
Abstract
The literature claims that one of the main problems that tends to arise in
cross-national comparative analysis is that of equivalence – how to study
the same problem in diff erent societies and cultures. The core issue is to
compare like with like, in order to develop a deeper understanding of the
phenomena under study and to be able to generalise. In this paper it is argued
that it is also necessary to study anomaly, recognising that anomalies can
contribute to a be er, albeit more complex, picture of the processes that are
being studied.
Keywords: Comparative education research, cross-cultural studies
Introduction
In recent decades, and at fi rst stimulated by the United Nations
and its agencies, international and interdisciplinary collaboration
has been encouraged, and international comparisons increasingly
used in the social sciences by researchers and policy makers. This
trend is particularly notable in the area of mathematics education.
Freudenthal argued in 1975 that:
In no branch of teaching have international contacts been closer and
more numerous than in mathematics; no subject ma er has known
CAN WE COMPARE LIKE WITH LIKE
IN COMPARATIVE EDUCATIONAL
RESEARCH – METHODOLOGICAL
CONSIDERATIONS IN CROSS-
CULTURAL STUDIES IN
MATHEMATICS EDUCATION
Birgit Pepin, School of Education, The University of
Manchester
40
CAN WE COMPARE LIKE WITH LIKE IN COMPARATIVE ...
40
as much international comparing and infl uential research than
mathematics. (Freudenthal, 1975, p. 130)
International activity continues to be evident in the numbers of
participants a ending the four-yearly International Congress
on Mathematical Education (ICME) which now brings together
mathematics educators from almost one hundred countries. Whilst
the seemingly ongoing internationalisation of mathematics education
off ers opportunities for international and comparative research, there
are also greater challenges in its undertaking. Clarke (2003) off ers the
following.
Challenges confronting the international research community
require the development of test instruments that can legitimately
measure the achievement of students who have participated in
diff erent mathematics curricula, research techniques by which the
practices, motivations, and beliefs of all classroom participants might
be studied and compared with sensitivity to cultural context, and
theoretical frameworks by which the structure and content of diverse
mathematics curricula, their enactment, and their consequences can
be analysed and compared. (Clarke, 2003, p. 144)
Critical ideas on comparative studies in mathematics education
There have been numerous international studies and comparisons
of a ainment in mathematics education over the past twenty years.
The First International Mathematics Study (FIMS) was carried out in
1964, with twelve countries participating (Husén, 1967). At the time
Freudenthal (1975) had already stated that cross-national comparisons
are not valid without considering curricular aspects. He argued that
a country’s success depended to a large extent on the degree to which
the test instrument (what is tested and how it is tested) was aligned
with the mathematics curriculum of the particular country.
The Second International Mathematics Study (SIMS) was carried out
between 1980 and 1982, and twenty countries participated. Taking
into consideration the curricular criticism, SIMS diff erentiated
between the intended, the implemented and the a ained curriculum
(for description see Travers and Westbury, 1989). In 1992 Westbury
suggested that ‘the lower achievement of the United States is the
41
Birgit Pepin
41
result of curricula that are not as well matched to the SIMS tests
as are the curricula of Japan’ (Westbury, 1992, 18). Thus he raised a
common concern, i.e. that such studies might – albeit unintentionally
– measure li le else than the alignment between the test instrument
and the curriculum of the particular country (Clarke, 2003).
In 1995 the Third International Mathematics and Science Study
(TIMSS) was carried out, with over 40 countries participating (for
description see, for example, NFER, 1997 or Beaton and Robitaille,
1999). Alerted to the problems of earlier studies, the researchers paid
particular a ention to curricular organisation.
Combining sequence data on the fl ow of mathematics curricula (that
is, the grades at which topics were typically covered) with document-
based data on the commonly intended topics at the three key grades,
revealed a picture of considerable cross-national diversity. This again
suggests caution in interpreting mathematics a ainment results.
(Schmidt, et al., 1997, p. 19)
Alongside TIMSS, a series of smaller accompanying studies
drew upon additional sources of data, amongst them a videotape
classroom study which analysed 8th grade lessons in the USA,
Germany and Japan (Stigler and Hiebert, 1997). Clarke highlights
what he regards as one of the most important results of conducting
the video studies.
One of the most powerful outcomes of large-scale video studies
such as this has been the demonstrated potential of the video data to
sustain multiple analyses. (Clarke, 2003, p. 165)
This was, for example, refl ected in the study of Kawanaka, Stigler and
Hiebert (Kawanaka, et al., 1999) who analysed teacher questioning.
They concluded that:
Teaching and learning, as cultural activities, fi t within a variety of
social, economic and political forces in our society. Every single
aspect of mathematics education, from a particular teacher behaviour
to national policy, must be considered and evaluated within a socio-
cultural context. (Kawanaka et al., 1999, p. 103)
42
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42
More recently, the OECD Programme for International Student
Assessment (PISA) included ‘mathematical literacy’ as one of its foci
(see for example, www.pisa.oecd.org).
Keitel and Kilpatrick (1999) give probably one of the most condemning
verdicts of large-scale comparative international studies, and TIMSS
in particular, when they criticise the ‘rationality and irrationality
of international comparative studies’. They refer to fundamental
problems in those studies, such as ‘comparing the incomparable’,
‘many visions, many aims, one test’, and ‘problems of comparing
curricula across educational systems’, already cited by other authors
(Bracey, 1997; Husen, 1983; Westbury, 1989). Their fi nal verdict is
that:
TIMSS threatens to poison for some time the waters of educational
policy, as politicians and researchers scramble to take advantage
of what TIMSS allegedly says about the teaching and learning of
mathematics in their country. … [International comparative studies]
rest on the shakiest of foundations – they assume that the mantle of
science can cover all weaknesses in design, incongruous data and
errors of interpretation. They not only compare the incomparable,
they rationalise the irrational.
(Keitel and Kilpatrick, 1999, p. 254)
Beside those large-scale studies (mainly conducted by the IEA), there
have been numerous small-scale, sometimes entirely qualitative,
perhaps ethnographic, or sometimes ‘mixed’, studies.
The growing interest in the use of qualitative research methods has
been a refl ection of epistemological critiques of the positivist traditions
of social science that have, until recently, dominated educational
research, in particular in mathematics education. Numerous authors
have emphasised the advantages of case studies to survey research
in comparative international research. Some even argue that those
studies can inform policy.
Studies such as Spindler (1987), Tobin et al. (1989), or Shimahara and
Sakai (1995) provide powerful insights into the way teacher practice
and belief is shaped within diff erent national contexts. Comparative
43
Birgit Pepin
43
studies that provide this level of ethnographic detail hold much
potential for educational policy. (Letendre, et al., 2002, p. 23)
Keitel and Kilpatrick state:
… the treatment of school mathematics curricula in international
comparative investigations is a story of increased eff orts to take
aspects of curriculum complexity into account. It is also, however,
a story of persistent failure to probe suffi ciently below the surface
of, and to challenge assumptions about, what is to be understood as
curriculum. (Keitel and Kilpatrick, 1999, p. 242)
Keitel and Kilpatrick argue that internationally equivalent
instruments need to be developed, that is instruments that have
‘internationally-comparable’ relations to the national curricula.
Otherwise international studies would continue, in the words of
Torsten Husén, to ‘compare the incomparable’. They criticise that the
present standard of measurement (a common set of tasks organised
by content) leaves no allowance for ‘diff erent aims, issues, history
and context across the mathematics curricula of the systems being
studied’, which leads to the situation, in their view, that ‘no one
really addresses how well the students in a system are learning the
mathematics curriculum that their system has off ered them’ (Husén,
1983, 243).
The critiques of positivism argue that there is a fundamental
diff erence between the study of natural objects and human beings,
in the sense that human beings interpret situations themselves and
give meanings to them. Some authors go so far as to argue that any
worthwhile sociological explanation must be related to the actual
ways in which people themselves interpret their situations. This has
major implications for the conduct of research in the sense that it
requires researchers to observe and interact/communicate with the
subjects of their research. Blumer asserts that:
… one would have to take the role of the actor and see his world from
his standpoint. This methodological approach stands in contrast to
the so-called objective approach …, namely that of viewing the actor
and his action from the perspective of an outside, detached observer
44
CAN WE COMPARE LIKE WITH LIKE IN COMPARATIVE ...
44
… the actor acts towards his world on the basis of how he sees it and
not on the basis of how that world appears to the outside observer.
(Blumer, 1971, p. 21)
It also refl ects the belief that curricular activities are ‘culturally
embedded’. Cogan and Schmidt (1999) suggest that ‘not only is
teaching an activity embedded in culture, but so is what is taught’
(p. 77). Kawanaka et al. (1999) also point to the ‘cultural’ side of
classroom practices and emphasise that:
Teaching and learning, as cultural activities, fi t within a variety of
social, economic and political forces in our society. Every single
aspect of mathematics education, from a particular teacher behaviour
to national policy, must be considered and evaluated within a socio-
cultural context. (Kawanaka et al., 1999, p. 103)
How do researchers understand diff erent pedagogies in the light
of diff erent educational ‘cultures’? If they believe that pedagogy
is ‘culturally embedded’, what are the cultural, intellectual and
philosophical underpinnings that infl uence teaching in diff erent
countries? These and more questions have to be researched and
answered, if the aim is to develop an understanding of curricular
processes in diff erent countries.
Furthermore, reducing methodology to a series of techniques,
carried out in specifi c stages, as is o en done in quantitative studies
(for example, conduct of interviews, design of questionnaires), plays
down the importance of social processes and the context of research
(Vulliamy et al., 1990). Burgess (1984) argues that this misrepresents
the nature of social scientifi c enquiry and the practice of research. He
suggests that:
Recent developments in research methodology indicate that
‘methodology’ involves a consideration of research design, data
collection, data analysis, and theorising together with social, ethical
and political concerns of the social researcher. In short, research is no
longer viewed as a linear model but as a social process. … Accordingly,
questions now need to be raised about the actual problems that
confront researchers in the course of their investigations and some
45
Birgit Pepin
45
consideration needs to be given to the ways in which techniques,
theories and processes are developed by the researcher in relation to
the experience of collecting, analysing and reporting data. (Burgess,
1984, p.2)
For qualitative researchers, the social nature of the research process is
likely to be more obvious from the outset, given the usually extended
communication periods between the researcher and the researched.
For example, educational ethnographers, such as Woods (1986), or
case-study researchers, such as Walker (1985), tend to draw more on
personal experiences than quantitative researchers. This refl ects the
extent to which research procedures diff er according to the context
of their use (Vulliamy et al., 1990).
The search for equivalence
Warwick (Warwick and Osherson, 1973) compares the choice of the
research method to an accessory lens for a camera, where it depends
on the type of picture that the researcher wants (broad panoramic
or intense concentration on detail). In this light the comparative
perspective serves as a systematic se ing in which to evaluate a
theory formulated in terms of common factors, and the comparative
focus becomes central when an a empt is made to come to terms
with social reality in a specifi c se ing and when diff erences are of
interest for their impact on society (Hantrais et al., 1985).
Warwick and Osherson (1973) consider the core issue in comparative
research methods that is equivalence – ‘how to study the same
problem in diff erent societies and cultures’. They regard this as the
‘central theoretical and methodological question raised by cross-
societal comparisons’ (p. vii). Furthermore, they explore the various
facets of equivalence. Conceptual equivalence refers to the question of
whether the concepts under study have equivalent, or any, meaning
in the cultures which are being considered. Some concepts have
meaning in many but not all cultures and se ings. A major challenge
of comparative research is to provide conceptual understandings
that have equivalent, though not necessarily identical, meanings in
the se ings under study. Probing of subjects/participants for their
meaning and long periods in the fi eld (in order to get to know the
46
CAN WE COMPARE LIKE WITH LIKE IN COMPARATIVE ...
46
context in which subjects/participants are working) are two of the
possible ‘solutions’ suggested by the literature.
Another problem is that of equivalence of measurement, which
involves the challenge of developing equivalent indicators for the
concept under study. Theoretically applicable concepts may diff er
in their salience for the culture as a whole, or respondents might
be unwilling or unable to discuss sensitive topics. Warwick and
Osherson (1973) suggest that the question format should be fl exible
(as in semi-structured interviews).
There is also the problem of linguistic equivalence. Linguistic
equivalence refers to the problem of translation itself. Warwick
and Osherson (1973) assert that the issue of linguistic equivalence
is inseparable from the theory and concepts guiding the study, the
problem chosen and the research design. They off er suggestions
with a view to se ing the problem within the broader framework of
conceptualisation and research design. Amongst them are: that the
research problem should be salient to the cultures involved; that the
primary emphasis in translation should be on conceptual equivalence
(‘comparability of ideas’); that many problems of translation could
be avoided by advance familiarity with the cultures under study
and that conceptual-linguistic equivalence can be improved through
extensive pre-testing in the local culture, in particular qualitative pre-
tests. Basically, as Warwick and Osherson (1973) point out, conceptual
equivalence, equivalence of measurement and linguistic equivalence
are ‘tightly interwoven and should be treated as a single fabric’.
We have argued elsewhere (Pepin, 2000 and 2002), and exemplifi ed in
other studies (Pepin, 1999a and b; Pepin and Haggarty, 2001; Haggarty
and Pepin, 2002), in which ways a qualitative approach can contribute
to comparative research design, and how to address the problems
of equivalence. In those papers it has been asserted that, whichever
methodology is employed, using research strategies cross-nationally
highlights problems of culture, language and communication, which
infuse all aspects of the research. Cross-national studies have to
grapple with language and communication problems intensively
at the stage of formulating problems and defi ning the meaning of
concepts and interpreting fi ndings. The advantage of qualitative
47
Birgit Pepin
47
studies, we have argued, lies in their in-built potential for establishing
conceptual equivalence by their underpinning epistemology which
has implications for the conduct of the research and the strategies
used. Researchers usually stay in the fi eld for long periods of time,
which gives them the opportunity to become ‘enculturated’ and
probe for meanings, which in turn is likely to help in understanding
the context under study, counter threats to the validity of the fi ndings
and help to establish conceptual equivalence.
A diff erent view of ‘comparing’
To develop a coherent and ‘complete’ picture of research design and
methodology in comparative international research education is a
diffi cult undertaking. It could be argued that this is due to various
factors and we suggest reasons why this might be the case (adjusted
from Banathy, 1991):
(1) a disconnected or ‘piecemeal’ approach to research design;
(2) the apparent failure to integrate new ideas from other research
areas (i.e. science);
(3) the fragmented discipline-by-discipline study of education;
(4) reductionist approaches to research design;
(5) thinking within existing boundaries.
The reader might usefully ask for a brief explanation of the above-
mentioned factors and why they seem valid reasons. However, we
would like to refer the reader to the ideas explained below which
relate to some of the above-mentioned reasons (and leave the
rationale for a later stage). We would like here to concentrate on
the main intention of this chapter, which is to look at the place of
studying ‘anomaly’ in comparative education research design.
The word ‘anomaly’ is described in the dictionary (Hanks, McLeod
and Urdang, 1979) as a ‘deviation from the normal or usual order’
or ‘irregularity’. The word ‘anomalous’ comes from the Late Latin
‘anomalus’ or Greek ‘anomalos’ which means ‘uneven, inconsistent’
(as ‘homos’ means ‘one and the same’). In statistical language it is
likely to mean the ‘outliers’, the ‘odd one out’ from the rule or law or
theory that was established.
48
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48
Whilst acknowledging the advantages of qualitative research
designs in cross-national comparative studies, the main purpose of
this chapter is to show that it is not enough to consider qualitative
approaches in our endeavour to achieve equivalence. In fact, we
argue that it is not suffi cient to consider comparing ‘like with like’,
indeed it might be impossible. We propose that it might be benefi cial
to study ‘anomalies’, that is cases that do not compare with others, in
order to ‘defi ne’ the boundaries of our developing theories and thus
help to deepen our understanding.
Discovery commences with the awareness of an anomaly, i.e. with
the recognition that nature has somehow violated the paradigm-
induced expectations that govern normal science. It then continues
with a more or less extended exploration of the area of anomaly. And
it closes only when the paradigm theory has been adjusted so that
the anomalous has become the expected. (Kuhn, 1970, pp. 52, 53)
Doll (1986) contends that a mechanistic view of education has
dominated western thought right through to the twentieth century,
and that this worldview underpinned a positivist view of knowledge.
Central to this approach, he asserts, is an emphasis on analysis to
identify independent elements or parts that equate to variables in
experiments. Only over the past 20–30 years has another paradigm,
namely postmodernism, become a popular way of thinking,
contradicting the notion of objective reality and promoting ‘pragmatic
doubt’ (Doll, 1993, 61). Thomas (1998) provides an interesting account
of the ‘myth of rational research’.
In its fl ight from ‘positivism’ educational enquiry still cleaves to a
faith in the ordered and the rational. Educationists continue to believe
in an order, accessible via rational inquiry and ordered refl ection,
governing human aff airs and thought. This belief has … unwelcome
consequences. First, it promotes the notion that certain rationalistic
ingredients are obligatory in research: a technology of inquiry is
thus constructed and maintained. Consequently, inquiry (even
interpretative inquiry) is formulaic; it follows predictable ruts and
leads o en to uninteresting fi ndings. Second, a belief in the ordered
mind leads to a faith in certain models of mind, and in ‘personal
49
Birgit Pepin
49
theory’ which can be developed via particular and orthodox methods
of fi nding out …(Thomas, 1998, p. 141)
It appears that the most dominant paradigm for educational thought
over the last century has been the mechanistic view and ways of
thinking, infl uencing ideas of teaching and learning (Hoban, 2002)
and theories of their measurement in research design.
The mechanistic worldview assumes that reality can be observed,
explained and predicted. According to Doll (1993) the legacy of
a mechanistic worldview is a narrow-minded methodology for
conducting research that produces prescriptive knowledge, and
subdivided and disconnected parts to be taught to learners.
It is Newton’s metaphysical and cosmological views – not his scientifi c
ones – that have dominated modern thought for so long, providing
a foundation in the social sciences for causative predictability, linear
ordering, and a closed (or discovery) methodology. These, in turn,
are the conceptual underpinnings of scientifi c (really scientistic)
curriculum making. (Doll, 1993, p. 34)
Schön (1983) called this way of thinking ‘Technical Rationality’
(p. 21). He claimed that it evolved from positivism and called it
‘positivist epistemology of practice’ (p. 31). Furthermore, he contends
that university researchers need to acknowledge the ‘complexity,
uncertainty, instability, uniqueness, and value-confl ict’ (p. 39) of work
se ings. ‘Discrete’ knowledge, ‘discretely’ researched and ‘packaged’
in separate boxes, presents a technical view of education and does not
encourage a way of thinking that considers how aspects of education
have an eff ect on one another. In addition, presenting educational
knowledge in discrete packages can lead to a misrepresentation of
the complexity evident in the phenomena that are being researched,
and the ways they can be understood.
Thus, it is proposed that the technical view of knowledge and
discovery appears to be an a empt to force phenomena of teaching
and learning, and their investigation, into conceptual and separate
boxes supplied by the research community. We argue that research
should also proceed without such boxes, i.e. by studying anomalies,
50
CAN WE COMPARE LIKE WITH LIKE IN COMPARATIVE ...
50
whatever the element of arbitrariness in their historic origins and,
occasionally, in their subsequent development.
This sense of arbitrariness, however, does not mean that the
researchers studying ‘without such boxes’ could practise without
some set of considered beliefs. Questions like the following have to
be answered when studying an anomaly:
1. What are the ‘fundamental entities’ of which this case is
composed?
2. What are the links/relationships between the entities within the
case?
3. How does this anomalous case interact with the other ‘rule-like’
cases?
4. In which ways does this case help to understand the boundaries of
the developing theory, and indeed the theory itself?
Feyerabend (1993) suggests that we may use ‘counterrules’:
hypotheses that contradict well-informed theories. He advises us
to proceed counterinductively. This, of course, gives rise to various
questions, in particular to those of whether and under what conditions
it is appropriate/useful/reasonable to use counter induction rather
than induction. He claims that:
The evidence that might refute a theory can o en be unearthed only
with the help of an incompatible alternative. … Also, some of the
most important formal properties of a theory are found by contrast,
and not by analysis. A scientist who wishes to maximise the empirical
content of the views he holds and who wants to understand them
as clearly as he possibly can must therefore introduce other views;
that is, he must adopt a pluralistic methodology. (Feyerabend, 1993,
pp.20, 21)
This rings true with much of what comparative education can off er
as potential. We can consider the very act of comparing, for example.
Rather than being a second-order activity, comparison is, arguably,
central to the very act of knowing and perceiving and therefore likely
to be essential to social scientifi c analysis (Hantrais and Mangen,
1996).
51
Birgit Pepin
51
Feyerabend (1993) contends that we must compare ideas with other
ideas (rather than with ‘experience’) and must try to improve rather
than discard the views that have failed in the competition.
Knowledge so conceived is not a series of self-consistent theories
that converges towards an ideal view; it is not a gradual approach to
truth. It is rather an ever increasing ocean of mutually incompatible
alternatives, each single theory, each fairy-tale, each myth that
is part of the collection forcing the others into greater articulation
and all of them contributing, via this process of competition, to the
development of our consciousness. (Feyerabend, 1993, pp. 21)
Conclusions
International and comparative research in education, in particular
in mathematics education, has been commonly wedded to large-
scale achievement studies that are, allegedly, generalisable and can
inform policy-makers. In terms of the general features of the research
design, it is claimed that one has to compare ‘like with like’, one has
to establish equivalence. These studies have been criticised by many
authors, in particular on the basis that the research design does not
take account of the cultural embeddedness of the phenomena under
study. This has been addressed by researchers who have implemented
more qualitative approaches into their design.
In this paper it has been argued that it is not enough to implement
qualitative approaches in comparative research design. Indeed,
we believe that an important aspect of developing ‘comparative’
theory is to consider the study of ‘anomalies’, in order to help us to
develop a deeper understanding of the phenomena and delineate the
boundaries of our theories.
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Pepin, B. (2000) Reconceptualising Comparative Education: The
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their Use in English, French and German Classrooms: A Way to
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Curriculum Making Processes (in the series “Explorations”), Zurich:
Peter Lang.
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Classroom Mathematics Instruction: An Overview of The TIMSS
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Teaching and Learning: Recent Findings and New Directions, in.
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Contact
Dr Birgit Pepin
The University of Manchester
School of Education
Manchester, UK
birgit.pepin@manchester.ac.uk
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Rona Catterall
55
The paradoxes posed by an innocent number (zero), ra ling even
this century’s brightest minds and threatening to unravel the whole
framework of scientifi c thought. … It (zero) provides a glimpse of the
ineff able and the infi nite. That is why it has been feared and hated
– and outlawed. (Seife, 2000, p. 2)
Abstract
This paper is part of a larger project to investigate children’s conceptions
of zero. It arose from the researcher’s specifi c interest in noting, during
many years in primary education, that zero created its own mathematical
problems for children and for teachers. Examples, provided by teaching staff ,
illustrated where zero created cause for concern in various areas of children’s
mathematics. These examples mirrored this researcher’s experiences. When
it is admi ed that researchers tend to leave out the zeros knowing that
they cause problems (Guedj, 1996; Suydam and Dessart, 1976) then it
is not surprising that there is a noticeable lack of research on children’s
understanding of zero. This paper concentrates on one aspect of the research
fi ndings; children’s reaction to zero in relationship to other numbers.
Keywords: conceptions of zero, primary education
EXPLORING CHILDREN’S
CONCEPTIONS OF ZERO IN
RELATIONSHIP TO OTHER
NUMBERS
Rona Ca erall, Sheffi eld Hallam University
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EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
56
Introduction
Schools for this study were selected on the basis that the children
would be aff ected by internal and external factors. Within the school
there would be a variety of teaching styles and mathematical activities
infl uenced by a range of professional interpretations by teachers
with their own mathematical ability and philosophical viewpoint. In
their editorial article, on ‘Eff ective Educational Research’, Riding and
Wheldall saw the value of pure research carried out with atypical
subjects; however, they were ‘pressing for more research with normal
children in normal schools’ (Riding and Wheldall, 1981, 6).
Five ‘normal’ primary schools were selected. The schools were
situated within a square mile, in northern England. The questionnaire,
previously piloted in a diff erent school, was given to the Year 6 (Y6)
children (aged 10–11 years) in the fi ve selected schools, in July 2002.
In total there were 100 returns. On analysis no one school showed
any signifi cant, singular result, indeed quite the opposite in that
there were many common features.
In his work on children’s errors in algorithms, Engelhardt
acknowledges his research shortcomings in that he did not include
talking to the child. He felt that examining only wri en performance
without the opportunity to investigate a given error further, greatly
increased the possibility of misjudging a pupil’s erroneous approach
(Engelhardt, 1977). While the questionnaire produced valuable
quantitative data its limitations as a method of data collection needed
to be addressed. The opportunity to gain insight into children’s
thinking came from the use of interview-tasks undertaken in one of
the original fi ve schools; Research School A in July 2003, with twenty
children in Y6 (10–11 year olds). Throughout the tasks the children
were encouraged to explain and illustrate their answers. In turn this
provided an opportunity for the researcher to observe each child and
to ask probing questions. Experience has led the researcher to be in
full agreement with Ginsberg that errors are seldom capricious or
random. He goes on to say:
Typical children’s errors are based on systematic rules … Children’s
faulty rules have sensible origins. Usually they are distortions or
57
Rona Catterall
57
misinterpretation of sound procedures (Ginsberg, in Byers and
Erlwanger, 1977, p. 274).
Here Ginsberg is speaking of errors; this researcher also believes
that the correct answer can also be the result of distortion and
misinterpretation. The children were encouraged to explain all
answers, whether or not the researcher knew them to be correct.
Exploring children’s number ordering – including zero
The introduction of The National Numeracy Strategy (NNS),
(DfEE, 1999) is said to have led to signifi cant changes in primary
mathematics teaching in England (Earl et al., 2003). One of the NNS
recommendations, pertinent to this research paper; is the display
and use of a number line, including the zero number symbol, in each
classroom.
The reasons for including ordering numbers were to gain insight
into how the children viewed zero within the number order and to
explore their understanding of zero’s relationship to other numbers,
but without the involvement of any calculations or algorithms.
The sets of numbers used were:
• small single digits
• simple fractions plus zero
• decimals plus zero
The same series of numbers, in the same order as in the questionnaire,
were presented to the children in the Y6 task-interview situation. This
meant there was a comparison between the raw and the collated data
from the questionnaire and that collected from the interview-task.
In the interview-task each of the numbers was wri en on a piece
of card and the child was asked to read each symbol. If the child
could not read the symbols and did not appear to be familiar with
fractions or decimals then the researcher did not continue with the
relevant question. The interview-task allowed participants to expand
on their answer, to provide an explanation as to how the answer was
reached. It also allowed the researcher to ask questions in order to
gain clarifi cation. Whatever the answer, correct or not, the child was
asked for an explanation.
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EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
58
The numbers and the order presented in the questionnaire and in the
interview-tasks were:
Table 1. Interview tasks
(a) 1/4 1/2 2 1 0
(b) .3 .4 0 .5 .1
(c) 1/4 3/4 1/4 0 1/2
(d) 0.4 5 1.2 8 0
(e)30547
(f) 8 5 7 1 0 4 3 2 9 6
Collating and classifying the data
When classifying the answers there was an acute awareness that the
results contained the child’s depth of understanding of fractions and
decimals as well as that of zero. An example of this could be seen in
some of the answers given to question (c) ordering:
¼ ¾ ¼ 0 ½
Table 2. Options
option 31…... 0 ¼ ½ ¾
option 32….. 0 ½ ¼ ¾
option 33….. 0 ¼ ¾ ½
option 34…... 0 ¾ ¼ ½
option 35….. 0 ¾ ½ ¼
The mis-ordering of the fractions and decimals had to be taken into
consideration when isolating the child’s positioning of zero from the
other factors. Options 31 to 35 (Table 2) place zero at the beginning of
the order sequence. This appeared to be the important element as far
as this study was concerned. The answers in Table 2 were classifi ed
as ‘zero then fractions’; it was this element, regardless of the fraction
order, which was noted as signifi cant.
59
Rona Catterall
59
In the process of the explanation some children changed their mind
and re-ordered the cards. The numbers shown in brackets are the fi rst
answers. There was no apparent pa ern in the change, either with
individual children or with specifi c explanations. Some changes were
from correct to incorrect answers, some from incorrect to correct. In
many of the sections there is strong connection between the percentage
of answers from the questionnaire and the fi rst answer given in the
task-interview. This is especially noted in the fi rst fraction and the fi rst
decimal ordering: Table 4 and Table 8 In the second fraction and second
decimal question: Table 5 and Table 9 the connection is not as close. The
researcher felt that this was due to the task interview children explaining
their reasons and o en noting their error. This then had an eff ect on their
approach, later in the interview, to the second, related question. In this
second question some children made reference to the fi rst question and
gave the same rationale. While the comparison of questionnaire and
interview-task results in the second question are not as strong they are
still noteworthy.
Data analysis of the results
The analysis was undertaken in three sections:
a) Fractions and Zero
b) Decimals and Zero
c) Single Digit Numbers and Zero
In relation to Fractions and Zero two ordering questions contained
fractions.
Question A – fractions and whole numbers plus zero
Question C – simple fractions plus zero
Question A ½ ¼ 2 1 0
Two frequent answers, in both the questionnaire and the task-
interview were:
Table 3. Frequent answers to Question A
Questionnaire Task-interview
(zero/fractions/whole
numbers) 43% (45%) 60%
(fractions/zero/whole
numbers) 48% (40%) 25%
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EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
60
The strong link between the questionnaire and task-interview fi rst
answers is reassuring from the aspect of methodology selection.
However, it is not possible to know the rationale for the questionnaire
answers. It cannot be assumed that there is a connection between
their reasoning and the reasons given by the children in the task-
interview group.
Reviewing the rationale for the fi rst set of answers:
Table 4. Rationale for fi rst set of answers to question A
Questionnaire Task-interview
(zero/fractions/whole
numbers) 43% (45%) 60%
(fractions/zero/whole
numbers) 48% (40%) 25%
Of the 60% of children in the task-interview who gave the answer
as zero/fractions/whole numbers the reasons given by 70% of
these children involved comparing the size of zero with the other
numbers.
• zero is lower than 1, zero is a whole one lower than one
• ¼ and ½ are more than zero
• 0 is nothing, ¼ is something, zero isn’t anything
• ¼ is more than nothing
Some of the explanations given during the task-interview were
similar but the outcome, the number order, diff ered. One rationale,
that zero is a whole number, was used by two children to explain
a) why 0 went before the fractions 0 ¼ ½ 12, zero is a whole
number, zero has no parts to it
b) why 0 went a er the fractions ¼ ½ 0 1 2, these ¼ ½ are parts
of a whole, 0 is a whole number
Analysis of the rationale for the second set of answers:
Table 5. Rationale for second set of answers to question A
61
Rona Catterall
61
Questionnaire Task-interview
(zero/ fractions/whole
numbers)
43% (45%) 60%
(fractions /zero/whole
numbers) 48% (40%) 25%
Observing the children ordering the cards the initial reaction of
many children was to place fractions/zero/whole numbers. The
explanations were not that children were pu ing zero a er the
fractions but that they saw 0 as being before 1. Pu ing zero next to
one was o en the initial reaction but, as seen in the chart, on refl ection
a number of children changed their order to zero then fractions.
• you have to have 0 in front of 1
• it goes 0 then 1, then 2
• because 0 is more than ¼, zero comes before one
• zero is a er ½ and zero is the next smallest, then you get 1, like
with 0,1,2,3
The three children who gave answers other than option 1 and option
3 (see Table 5) also used the zero before one rationale.
• 0 1 2 ½ ¼ 0 always goes fi rst, before 1 and then 2
• 0 ¼ 1 ½ 2, zero is before 1, but one quarter ¼ comes in
between.
• 0 1 ¼ ½ 2 zero is before 1 then ..reads ¼ as one and a
quarter, ½ as one and a half, then 2
Question C ¼ ¾ ¼ 0 ½
The frequent answers, in both the questionnaire and the task-
interview were classifi ed as:
Table 6. Children’s explanations for question C
Questionnaire Task-interview
zero then fractions 52.5% (75%) 80%
fractions then zero 44.4% (25%) 20%
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EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
62
Looking at the children’s explanations for the fi rst of these
classifi cations, zero then fractions, 81% of the reasons given by the
children involved comparing the size of zero to the other numbers:
• zero is nothing [child draws the fractions as fractions of a circle]
you’ve got none of the circle there
• because these ¼ ½ ¾ are more than zero
• zero has no parts to it
The other reasons for pu ing zero then the fractions were:
• zero always goes fi rst
• because 0 is not a number. I’m pu ing it at the front but it could
go at the end
• Not sure where the zero goes
The children’s explanations for the second classifi cation were most
revealing.
Table 7. Children’s explanations for question C
Questionnaire Task-interview
zero then fractions 52.5% (75%) 80%
fractions then zero 44.4% (25%) 20%
While question A included fractions and single digits (½ ¼ 2 1 0)
question C contained only simple fractions plus zero (¼ ¾ ¼ 0 ½).
The reason for this was to remove the temptation for the child to put
0 and then to put 1. Nevertheless the most common reason given for
pu ing the 0 a er the fractions was linked to the number line order
of zero, one, two, three, etc.
• zero is more than the fractions, it starts the whole numbers, zero,
one, two, three ….
• ¾ is the smallest, then we go bigger up to zero, then one, two and
we count up.
• zero always goes fi rst. When you have loads of numbers then 0
(pointed to the symbol on the card), you always write this fi rst.
It’s like zero, one, two, three, four, fi ve, six, seven, eight, nine, ten.
You always say zero fi rst.
63
Rona Catterall
63
The other explanations for pu ing the fractions then zero made
reference to the ‘nothingness’ of zero:
• Nought is just nothing (puts the zero a er the fractions) – it could
go anywhere. (Why?) because 0 is not a number. I’m pu ing it
at the front but it could go at the end (Why?) – because it is
nothing, so it can go anywhere. It doesn’t mean anything so it
doesn’t ma er where it goes.
• Not sure where the zero goes. (Why not?) – I could put it there
(Moves the card to the end of the sequence). (Could you put
it anywhere else) Yes, there (points to the space between two
fractions), or there (points to the space between two other
fractions).(Why could it go in all these diff erent places?). It’s
nothing so it can go anywhere – it’s nothing so it makes no
diff erence where you put it.
• Nought is just nothing (puts the zero a er the fractions) – it could
go anywhere. (Why?) Because it is nothing. (Here the ‘nothing’
was said with great emphasis.
These children saw zero, the symbol 0, as being nothing, of no
importance. Thus it does not ma er where it goes or even if it is le
out.
The following questions relate to Decimals and Zero:
Question B .3 .4 0 .5 .1
The frequent answers, in both the questionnaire and the task-
interview were classifi ed as:
Table 8. Frequent answers to question B
Questionnaire Task-interview
Zero then decimals 49.5% (45%) 55%
While explanations of the decimal and zero sequence sometimes
threw light on a child’s understanding of zero the also linked closely
with the child’s understanding of decimals. It proved diffi cult to
diff erentiate between the two. What was of interest was the reference
to zero being a whole number, particularly as this was used as the
main reason for thinking the decimals were smaller than zero:
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EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
64
• .1 .3 .4 .5 0 because zero is a whole number
• 0 .1 .3 .4 .5 zero comes fi rst then you have zero point
1 (0.1) then you go up to 0 on its own. The
number before the point is a whole one. A
whole one, what do you mean? A whole
number like one and two.
Question D 0.4 5 1.2 8 0
The top three order of frequency of answers, in both the questionnaire
and the task-interview, were:
Table 9. Frequency of answers to question D
Questionnaire Task-interview
zero then decimals 60.8% (60%) 55%
Zero amongst the decimals
and whole numbers
28.9% (30%) 35%
decimals then zero 2% 5%
It was surprising to note the high number of responses which placed
zero amongst decimals (0.4, 0, 1.2, 5, 8) with 0 coming a er 0.4. Here
the question of whether zero was a whole number was again raised;
these were not the same children who had voiced their thoughts on
this issue in the fractions sequencing.
• puts zero then 0.4, changes to 0.4 then zero. (You don’t seem
too happy with the answer?) I don’t know if the zero is a whole
number or not. (Points to the 0 card.)
• 0.4 then zero, changes to put zero then 0.4, (Are you unsure?) I
think that’s right, it depends if zero is whole one. (A whole one?
Can zero be one? – child laughs.) I mean a whole number.
• zero then 0.4, then changes to 0.4 then zero, says 0.4 (nought
point four) is smaller than zero. Why did you change? Well,
I thought that zero is a whole number so a decimal must be
smaller.
The following questions relate to Single Digit Numbers and Zero
65
Rona Catterall
65
Question E 3 0 5 4 7
Table 10. Responses to question E
Results of Ordering Numbers –
Question E
Question E 3 0 5 4 7 Questionnaire Task-interview
Total number of responses 96 20
option 56…..0 3 4 5 7 92 20
Spoiled answers 4
Question F 8 5 7 1 0 4 3 2 9 6
Table 11. Responses to question F
Results of Ordering Numbers 7 Question
F
Question F 8 5 7 1 0 4 3 2 9 6 Questionnaire Task-interview
Total number of responses 96 20
option 62…..0 1 2 3 4 5 6 7 8 9 92 20
Spoiled answers 4
The ordering of the single digits produced 100% correct answers
(excluding the spoiled answers). When the children were asked for
their reasons for placing zero 0 3 4 5 7 and 0 1 2 3 4 5 6 7 8 9 the
answers were all connected with the number order:
• You say zero, one, two, three (etc)
• It’s that order on the number line
In the interview-task sessions each child was asked “Could you put
the zero card in any other place?” In question E (Table 10) the answer
was always “no”. However, in question F (Table 11) where all the
single digits were involved, three children said “yes”.
Two children said that the zero (0) could go a er the 9. This gave
1234567890, the two explanations were:
• because when we were in Mrs ~ class we had a number line on
the wall that had 0 at that end.
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EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
66
• I’ve seen 1234567890 in lots of places – it’s on the computer
keyboard.
One child said that
• You could leave it out [child removed the 0 card] or it could go
anywhere [child moved the 0 card to diff erent places between the
other digits. Asked why the child explained that] because zero is
worth nothing it doesn’t ma er where it goes or whether you put it
in or not. [Asked if you could do this with other numbers the child
said] No, because they’re something and zero is nothing.
Summary and discussion of the main points
The teaching and learning of early number is a complicated,
multifaceted task. Part of this is the learning of the number order,
that is to recite the number words (zero), one, two, three, four, etc.
and the knowing of the symbol order (0) 1, 2, 3, 4, etc.
Most mathematical educators are reluctant to discuss memory when
considering mathematical learning and understanding, Gagne (1970)
being one of the few exceptions. This researcher agrees with Morris
(1981) that reliance upon memory can have deleterious eff ects, as
reliance upon memory adds signifi cantly to ‘mathematics anxiety’,
especially when memorisation replaces understanding as this o en
results in confusion. Freedmont (in Byers and Erlwanger, 1971) goes
to one extreme when he described rote learning as one of the time-
honoured enemies of eff ective mathematics learning. At the other
extreme is Krutetskii (1976) who considers mathematical memory
to be one of the abilities which distinguish mathematically capable
from mathematically incapable students.
This researcher’s experience is in keeping with Byers and Erwanger
(1985) as they see the discrepancies between theory and practice, in
that teachers in mathematics, far from ignoring memory, are very
cognisant with the problems it presents. Classroom teaching and
learning of mathematics departs signifi cantly from what theorists
have proposed. The main diff erences may be summed up in two
words: repetition and practice.
67
Rona Catterall
67
The importance of memory for doing mathematics, from the lowest
computations to the more sophisticated proofs is almost self-evident.
The most crucial question is not whether memory plays a role in
understanding mathematics but what it is that is remembered and how
it is remembered by those who understand it – as well as those who do
not. (Byers and Erwanger, 1985, p. 261)
When learning the number order and the number symbol sequence,
memory is of paramount importance. The 1 to 10 range of number
words is learned by rote with exposure to the sequence of number
names and the experience of moderate amounts of sequence production
activities (Fuson and Hall in Byers and Erwanger, 1985; Maclellan in
Thompson, 1997). As a result many children can recite the number
sequence to 100 by the time they are about six years of age (Maclellan,
1997). The learning of the number and order of the number names is
dependent upon aural and verbal memory while the learning of the
number sequence of the number symbols, the number line order, relies
upon visual memory. The marrying of the number names and number
symbols does require considerable eff ort ‘because nine words and
symbols have to be associated, and none of them is predictable from
any of the others’ (Wigley 1997, 116). Interesting Wigley speaks of nine
symbols, one can only assume that he was not including zero.
As far back as 1883 Galton referred to the visual number line, ‘persons
who are imaginative almost invariably think of numerals in some
form of visual imagery’ (Thompson, 1990, 116). A hundred years
later, Ernest (1983) found that, as a result of a questionnaire given
to teacher training college staff , 65% had an internalised number
line. All but 5% of these were straight-line number forms which he
deduced were ‘… possibly stimulated by the greater use of graded
rulers and physical number lines since Galton’s time’. (Thompson,
1990, 116)
Two decades later many schools are following the NNS
recommendation that each classroom displays a number line,
including the zero number symbol. While the interview-tasks were
conducted in a room other than the classroom, each of the research
classrooms did have a number line on the wall. All the number lines
began with 0 and there was a le to right ordering:
0 1 2 3 4 5 6 7 8 9 …
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EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
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As has been seen in the ordering of fractions and zero a number of
children placed the fractions in such a way as to preserve the 0, 1, 2,
number order (as with, ½ ¼ 1 2 3). Zero was seen as a symbol with
a specifi c place on the number line, next to the number one. An area
of tension was created with the children who wanted to preserve this
number order relationship but at the same time they used reasoning
which would put other numbers between 0 and 1. This reasoning
o en contained a reference to the size of zero; its nothingness or to
whether zero was ‘a whole number’. These two aspects are discussed
in the following sections.
Children, throughout the ordering tasks, raised the question as to
whether or not zero was a whole number. To try to gain further insight
into the children’s understanding the Y6 task-interview children
were put into random groupings of four/fi ve children for an informal
discussion. The debate centred round how one recognised whole
numbers, where they were to be found and what their a ributes
were. The following statements represented the ‘commonly agreed’
opinions of the children.
What are whole numbers?
• Full ones, not part of something like fractions.
• Like the numbers on a number line …but negative numbers are
not whole numbers… because the minus sign meant that they
are less than zero.
The discussion moved on to the topic of fractions and whole
numbers.
• Whole numbers are full numbers not bits like fractions.
• Whole numbers appear in front of fractions.
• You write it big in front of the fraction like one and a half, or
three and a half. (The child wrote 1½ or 3½ ). But you don’t
write zero and a half. (The child wrote 0½, the other children
laughed.)
The children were asked why you didn’t write zero and a half? A er
a pause the responses were:
• You just don’t. It’s silly.
• You can cut whole numbers into pieces but you can’t do that
with zero. You can’t cut nothing.
69
Rona Catterall
69
• You can’t have half of nothing.
The children were undecided as to which aspects of zero to ‘stress’
and which to ‘ignore’ in which situation. Their explanations were
logical within the limits of their knowledge and allowing for their
ideas of keeping to the conventions they had learned, including the
way to say, to do, to write and to order numbers. By association, if
zero were a ‘whole number’ then ‘0’, would be expected to behave like
the other single digit number symbols. But they found areas where
the ‘0’ sign was not used while 1, 2, 3 … were. Within each group
the children’s reasoning moved from the abstract consideration of
number symbols to the practical, concrete consideration of ‘zero’s
worth’. The ‘nothingness of zero’ is considered in the fi nal section.
This paper has concentrated on the children’s ordering of number
symbols and in their explanations of their number orders the term,
‘worth nothing’ was frequently used. The train of thought of a few
children was reminiscent of the riddle of assumptions – a bird has
wings, a bird can fl y, a penguin has wings so a penguin can fl y. This
transferred to zero-as-zero is worth nothing, nothing is worthless
and of no value, if it is worthless it is of no signifi cance, if it is of no
signifi cance it has no eff ect, if it has no eff ect then it can be ignored.
This concept of insignifi cance is expressed in the ordering of single
digit numbers and with ordering fractions and zero. There is a tension
between the children’s conceptions and use of the symbol for zero and
the ‘nothingness’ of the empty set which is seen when the abstract ‘0’
symbol is explained in concrete terms using the problematic‘ zero
language’. To quote Rotman, zero serves as:
… the site of an ambiguity between an empty character . . . and a
character for emptiness: a symbol that signifi es nothing. (Rotman,
1993, p. 26)
There is no intention, within this study, to assume generalisations
beyond the area of data collection contained in the research. The
aim is not to test a hypothesis but to undertake an exploratory
investigation of an as-yet-uncharted area of student experience; the
data being used to look at the nature of a itudes and trends. This
paper reports on part of an ongoing research project containing other
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EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
70
areas of inquiry, to be reported elsewhere where. Early analysis of
the data, from children aged 3 to 10, suggest that the notions of zero
as ‘empty’, ‘worthless’ and ‘insignifi cant’ are high profi le features.
References
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Years of Schooling. London: The Falmer Press.
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Dickinson, L., Brown, M. and Gibson, O. (1984) Children Learning
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Holt, Rinehart and Winston.
Department of Education and Employment (DfEE) (1999) The
National Numeracy Strategy for Teaching Mathematics Framework
for Teaching Mathematics from Reception to Year 6. London: DfEE.
Dolan, C. (ed.) (1998) The Development of Mathematical Skills. Hove,
East Sussex: Psychology Press.
Donaldson, M. (1987) Children’s Minds. London: Fontana Press.
71
Rona Catterall
71
Earl, L., Levin, B., Leithwood, K., Fullan, M. and Watson, N. (2001)
Watching and Learning 2; OISE/UT Evaluation of the Implementation
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UT.
Engelhart, J. (1977) British Journal of Educational Psychology, 47, pp.
149–54.
Ernest, P. (1983) Mathematical Education for Teaching, 4, pp. 51–53.
Geldman, R. and Gallistel, C.R. (1978) The Child’s Understanding of
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Guedi, D. (1998) Numbers: The Universal Language (London:
Thames and Hudson).
Hughes, M. (1986a) Children and Number: Diffi culties in Learning
Mathematics, Oxford: Basil Blackwell.
Hughes, M., Desforges, C., Mitchell, C. and Carre, C. (2000b)
Numeracy and Beyond: Applying Mathematics in the Primary
School. Buckingham: Open University Press.
Inhelder, B. and Piaget, J. (1967) The Early Growth of Logic. New
York: Norton.
Kaplan, R. (1999) The Nothing That Is: A Natural History of Zero,.
New York: Penguin.
Landau, E. (1954) Foundation of Analysis, The Arithmetic of Whole,
Rational, Irrational and Complex Numbers. New York: Chelsea
Publication Company.
Lappan G. (ed.), (1987) Arithmetic Teacher, 35, pp. 42–44.
Lee, K. (ed.) (2000) Childhood Cognitive Development – Essential
Reading. Oxford: Blackwell.
Maclellan, E (1997) The Importance of Counting, in Thompson, I.
(ed.) (1997) Teaching and Learning Early Number. Milton Keynes:
Open University Press.
Mant, A. (1983) Leaders We Deserve. Martin Robinson, London.
Menninger, K. (1992) Number Words and Number Symbols, New
York: Dover Publications, Inc.
Oesterle, R.A. (1959) The Arithmetic Teacher, 6, pp. 109–11.
Orton, A. (1987) Learning Mathematics: Issues, Theory and Classroom
Practice. London: Cassell.
Pogliani, L., Randic, M. and Trinajstic, N. (1998) International Journal
of Mathematical Education in Science and Technology, 29, pp. 729–
44.
72
EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
72
Riding, R. and Wheldhall, K. (1981) Educational Psychology, 1, pp.
6–11.
Rotman, B. (1993) Signifying Nothing: The Semiotics of Zero.
Stanford, California: Stanford University Press.
Seife, C. (2000) Zero: The Biography of a Dangerous Idea. London:
Souvenir Press.
Shuard, H. and Rothery, A. (eds) (1984) Children Reading
Mathematics. Oxford: The Alden Press.
Suydam, M.N. and Dessart, D.J. (1976) Classroom Ideas from
Research on Computational Skills. Reston, Virginia: National Council
of Teachers of Mathematic
Thompson, I. (ed.) (1997a) Teaching and Learning Early Number.
Milton Keynes: Open University Press.
Thompson, I. (ed.) (1999b) Issues in Teaching Numeracy in Primary
Schools. Buckingham: Open University Press.
Thompson, S.V. (1990) Educational Psychology, 10, pp. 140–49.
Wigley, A. (1997) Approaching Number Through Language in
Thompson, I. (ed.) Teaching and Learning Early Numbe. Milton
Keynes: Open University Press.
Wheeler, M.M. and Feghali, I. (1983) Journal for Research in
Mathematics Education, 14, pp. 147–55.
Contact
Dr Rona Ca erall
Sheffi eld Hallam University
Sheffi eld, UK
rona.ca erall@btinternet.com
full agreement with Ginsberg that errors are seldom capricious or
random. He goes on to say:
Ca erall, R. (1999) The Number Pack – Practical Pre-School.
Leamington Spa: Step Forward Publishing Limited.
Clark, A. and Moss, P. (2001) Listening to Young Children; The
Mosaic Approach. London: National Children’s Bureau for the
Joseph Rowntree Foundation.
Cohen, L. and Manlon, L. (1994) Research Methods in Education.
London: Routledge.
Cook, T.D. and Reichardt, C.S. (eds) (1979) Qualitative and
Quantitative Methods in Evaluation Research. London: Sage.
73
Rona Catterall
73
Dickinson, L., Brown, M. and Gibson, O. (1984) Children Learning
Mathematics: A Teacher’s Guide to Recent Research. East Sussex:
Holt, Rinehart and Winston.
Department of Education and Employment (DfEE) (1999) The
National Numeracy Strategy for Teaching Mathematics Framework
for Teaching Mathematics from Reception to Year 6. London: DfEE.
Dolan, C. (ed.) (1998) The Development of Mathematical Skills. Hove,
East Sussex: Psychology Press.
Donaldson, M. (1987) Children’s Minds. London: Fontana Press.
Earl, L., Levin, B., Leithwood, K., Fullan, M. and Watson, N. (2001)
Watching and Learning 2; OISE/UT Evaluation of the Implementation
of the National Literacy and Numeracy Strategies. Toronto: OISE/
UT.
Engelhart, J. (1977) British Journal of Educational Psychology, 47, pp.
149–54.
Ernest, P. (1983) Mathematical Education for Teaching, 4, pp. 51–53.
Geldman, R. and Gallistel, C.R. (1978) The Child’s Understanding of
Number. Cambridge, MA: Harvard University Press.
Guedi, D. (1998) Numbers: The Universal Language (London:
Thames and Hudson).
Hughes, M. (1986a) Children and Number: Diffi culties in Learning
Mathematics, Oxford: Basil Blackwell.
Hughes, M., Desforges, C., Mitchell, C. and Carre, C. (2000b)
Numeracy and Beyond: Applying Mathematics in the Primary
School. Buckingham: Open University Press.
Inhelder, B. and Piaget, J. (1967) The Early Growth of Logic. New
York: Norton.
Kaplan, R. (1999) The Nothing That Is: A Natural History of Zero,.
New York: Penguin.
Landau, E. (1954) Foundation of Analysis, The Arithmetic of Whole,
Rational, Irrational and Complex Numbers. New York: Chelsea
Publication Company.
Lappan G. (ed.), (1987) Arithmetic Teacher, 35, pp. 42–44.
Lee, K. (ed.) (2000) Childhood Cognitive Development – Essential
Reading. Oxford: Blackwell.
Maclellan, E (1997) The Importance of Counting, in Thompson, I.
(ed.) (1997) Teaching and Learning Early Number. Milton Keynes:
Open University Press.
Mant, A. (1983) Leaders We Deserve. Martin Robinson, London.
74
EXPLORING CHILDREN’S CONCEPTIONS OF ZERO IN ...
74
Menninger, K. (1992) Number Words and Number Symbols, New
York: Dover Publications, Inc.
Oesterle, R.A. (1959) The Arithmetic Teacher, 6, pp. 109–11.
Orton, A. (1987) Learning Mathematics: Issues, Theory and Classroom
Practice. London: Cassell.
Pogliani, L., Randic, M. and Trinajstic, N. (1998) International Journal
of Mathematical Education in Science and Technology, 29, pp. 729–
44.
Riding, R. and Wheldhall, K. (1981) Educational Psychology, 1, pp.
6–11.
Rotman, B. (1993) Signifying Nothing: The Semiotics of Zero.
Stanford, California: Stanford University Press.
Seife, C. (2000) Zero: The Biography of a Dangerous Idea. London:
Souvenir Press.
Shuard, H. and Rothery, A. (eds) (1984) Children Reading
Mathematics. Oxford: The Alden Press.
Suydam, M.N. and Dessart, D.J. (1976) Classroom Ideas from
Research on Computational Skills. Reston, Virginia: National Council
of Teachers of Mathematic
Thompson, I. (ed.) (1997a) Teaching and Learning Early Number.
Milton Keynes: Open University Press.
Thompson, I. (ed.) (1999b) Issues in Teaching Numeracy in Primary
Schools. Buckingham: Open University Press.
Thompson, S.V. (1990) Educational Psychology, 10, pp. 140–49.
Wigley, A. (1997) Approaching Number Through Language in
Thompson, I. (ed.) Teaching and Learning Early Numbe. Milton
Keynes: Open University Press.
Wheeler, M.M. and Feghali, I. (1983) Journal for Research in
Mathematics Education, 14, pp. 147–55.
Contact
Dr Rona Ca erall
Sheffi eld Hallam University
Sheffi eld, UK
rona.ca erall@btinternet.com
75
Margaret Sangster
75
Abstract
The purpose of the study was to see if children could complete linear pa erns
which had been translated, refl ected or rotated, to consider which they found
easier and whether abstract or picture images generated a diff erent response.
The study was carried out with 100 children between the ages of 7.6 and
11.6 years in one junior school 7–11 year olds). The results of the study
showed that translation pa erns were most successfully completed, followed
by refl ection and then rotation. Picture images were more successfully
completed than abstract design images.
Keywords: Linear pa erns, primary education, refl ection, rotation,
translation
Introduction
In mathematics there have been advocates of pa ern (Whitehead,
1925; Sawyer, 1955; Devlin, 1997) but it appears there is more than
one interpretation of the term. These three mathematicians refer to
the fundamental structures of mathematics when they talk about
pa ern. They see pa ern underpinning mathematics in a similar
way to the notion of ‘equivalence’. Without pa ern, it would not be
possible to make sense of situations, extrapolate or predict outcomes.
Parallel to this is a more limited use of the term pa ern in mathematics,
referring to various strands which are diff erent types of pa ern, such
TRANSFORMATION IN A SERIES OF
LINEAR IMAGES
Margaret Sangster, Canterbury Christ Church
University College
76
TRANSFORMATION IN A SERIES OF LINEAR IMAGES
76
as refl ective symmetry, or number sequences. It is possible that the
primary mathematics curriculum has become very fragmented into
such topics which involve pa ern. It is unlikely that the teacher has
a sense of pa ern underpinning mathematics and approaches much
mathematics as unconnected topics (Askew et al. 1997).
Of the two elements operating in the generation of a mathematical
pa ern, the fi rst could be called the ‘image’. This is the visual unit of
the pa ern. It might be a series of beads:
red, green, blue, red, green, blue, red, green, blue……
Here the image or unit is a series of three beads, ‘red, green, blue’. In
this case the image is repeated in a linear fashion. The second element
of a pa ern in mathematics is to describe the action or ‘function’ of
the pa ern. This could be referred to as the rule. In the case of the
beads the function is a linear translation of three places to the right.
Sometimes the pa ern is not so obvious because the image does not
repeat. The function is the pa ern and as such it changes the image:
2, 4, 6, 8, 10, 12, 14………
Here the function is to add 2 (+2) but in doing so we get a diff erent
image. It is still a pa ern because there is a constant relationship and
the sequence can be extended and predictions made. Children need
to understand the regularity of the number system to accept that this
is a pa ern.
Both these examples are linear. It is possible to develop pa erns in
two and three dimensions. It is possible to make interesting 2D and
3D structures, but they must be classifi ed as designs unless there is a
predictable element of repeat or growth present.
Refl ective symmetry pa erns are easy to see and create (Bruce and
Morgan, 1975) but they are probably the hardest to describe. One
could consider they grow outwards from a point, line or plane:
12, 11, 10, 9, 8, 7, 8, 9, 10, 11, 12
77
Margaret Sangster
77
in this case starting at 7 and using a +1 function/rule and growing in
two directions.
To meet the situation where more than one image is presented and
there is a relationship or regularity present which can be perceived
then the pa ern can be extended or described in mathematical terms.
A simple defi nition of pa ern could be:
A sequence of elements with a perceived rule for development.
It is very important that an element of repeat is involved either by
applying the rule or in the use of the image generated by the rule.
This allows for prediction to occur and allows mathematicians to
express the situation in terms of a mathematical relationship.
Pa ern Seeking in Mathematics
For pa ern seeking to be successful it is necessary to recognise the
structure within a situation. For example, in a pa ern of beads; red,
red, blue, blue, red, red, blue, blue, it is important to realise that two
colours, red and blue, are being used and that there are pairs of them
alternating. From this information one could reproduce the pa ern
or construct one with a similar structure and diff erent images.
All mathematical pa erns have structure, but so too have many
designs. The diff erence is that a mathematical pa ern can have
a rule applied to it which absorbs the structure and allows one to
continue the pa ern. 112233 is a structure. If one applies a rule it then
becomes a mathematical pa ern. For example, one rule might be:
create a new image, double it and continue in a linear fashion when
applied, a mathematical pa ern of 112233‡‡ might acceptably occur.
Another rule might be: repeat the unit of 112233, which would then
generate 112233112233. A third alternative rule might be: add one
to the previous number and then repeat it, which would generate
11223344 …
To carry out the above actions a child would need to apply pa ern
seeking strategies to establish both the structure and the rule.
This could involve counting or use of operations, or use of verbal
cues. Prior knowledge of possibilities would enable a rule to be
78
TRANSFORMATION IN A SERIES OF LINEAR IMAGES
78
created. Creating a rule in a new situation would require some
transfer of mathematical knowledge. Seeing structure is noticing
something which holds a design or pa ern together. To continue the
mathematical pa ern requires recognition and sometimes statement
of the rule. It is possible that children could recognise structure and
continue a rule without verbalising the situation.
Pa ern in image sequences
In this study the children were expected to work out the pa ern
contained in a linear sequence of three images and then select the
next correct image and manipulate it into a fourth space. In contrast
to the standard spatial IQ tests using pencil and paper the children
were able to select a card and physically match and manoeuvre it
before placing it at the end of a series of similar images. There was
an expectation that the children would recognise the structure of
the pa ern. In addition, the children were required to continue the
pa ern. The images repeated but the rule was not overtly stated. This
was the key to the task, seeing the hidden regularity and continuing
the rule. Presmeg (1992, 605) refers to this as the ‘regularities and
commonalities’ found in pa ern imagery. Gibson (1968, 284–86) in
his modifi ed theory of perception recognises the need to respond
by:
• isolating external invariants
• learning the aff ordance of objects
• detecting the invariance of events
• the development of selective a ention
These factors would be strongly relevant to a child selecting cards
and deciding the appropriate orientation. Focusing on the image,
having a perception of the orientation of the image, recognising the
rule by which the image moves and remaining concentrated on the
relevant factors.
Three types of transformation were used in this study, refl ection,
translation and rotation. Sheppard and Cooper (1982) found when
working with computer image transformation, that inverted images
(refl ections about a horizontal axis) were easily matched. On their
computer screen the 3D images would distort to maintain perspective
79
Margaret Sangster
79
but inversion retained its points of reference. In this study none of the
images distorted so all images would retain their points of reference
but children might confuse inversion with 180 degree rotation. Bruce
and Morgan (1975) found that refl ective symmetry in repetition is
more easily recognised than non-refl ective repetition (translation,
rotation) although Sheppard and Cooper (1982) found with the
computer images that refl ection took longer to match than rotation. In
contrast Mach (1959) noticed the immediate recognition of translated
shapes. Corcoran (1971) suggests that rotating a symmetrical image
is easier than rotating an asymmetrical image. He also suggests
some people fi nd rotation diffi cult because they visualise the image
strongly in a vertical position. This indicates that children might
fi nd translation pa erns easy to transform and rotating symmetrical
images easier than asymmetrical.
A second aspect of the study was to see if children manipulated
picture or abstract images more easily. This question developed from
the Orton presentation (1993) where one image seemed impossible
to mentally rotate until it gained the ‘identity’ of being rather like
a duck. This allowed it to be moved as a whole. Corcoran (1971)
suggested that responses will vary depending on whether the
image is in outline or blocked in, also whether it fi ts with previous
experience. Haber (1971) also supported feature detecting, as did
Gibson (1950). Corcoran (1971) is a supporter of the theory of the
mind extracting features from a situation rather than recording the
complete image. He suggests that responses will vary depending on
whether the image is in outline or blocked in. There is the possibility
that images might contain cues or features which would enable the
image to be more easily rotated. The children in this study will be
asked which image they found the most diffi cult and which the
easiest to transform.
Nola (1997) off ers six types of knowing which could prompt thinking
about children’s depth of understanding of pa ern. Paraphrased and
applied to pa ern they are:
1. A person knows a direct object (e.g. that is a pa ern)
2. A person knows how to do something (e.g. draw a refl ection,
a skill)
3. A person knows how to explain (how a pa ern works)
80
TRANSFORMATION IN A SERIES OF LINEAR IMAGES
80
4. A person knows why something works (e.g. can explain the
rule behind the images)
5. A person knows that something happens (e.g. that it can be a
refl ection, etc…)
6. A person knows what something is (e.g. it is a refl ection
pa ern), (Nola, 1997, p. 62)
Types 2, 3 and 4 appear to require far more intricate and active
knowledge. Knowing the name of something allows identifi cation
only. For viable use to occur when learning about pa ern there is a
need to reach the knowing of 2, 3 and 4.
Method of data collection and rationale
The study was carried out with 100 children between the ages of
7.6 and 11.6 years in one junior school (7–11 year olds). This was a
one-to-one task with a book containing the pa erns and cards which
had to be selected and placed correctly on the book (Figure1). Each
child was asked to spread out 20 cards. There were pairs of cards
which were similar but not identical. The children were then shown
10 linear series of three cards and a space. They were presented with
these one at a time and they had to select one card and place it in
the space with the correct orientation. Only one card was correct
and that had to be place in one of four rotational positions. The
researcher recorded correct and incorrect responses. At the end the
child was asked which they considered the hardest and which the
easiest pa ern to complete. This was also recorded. This, therefore,
did not require children to mentally hold and move images from one
location to another. It also allowed for some trialling to take place
when doing the matching. This practical approach meant that the
skill level was lower but the understanding of the pa ern sequence
was captured.
Individual interviews were selected as the best method of collecting
data as this allowed for a hands on approach. A task based, practical
data collection was selected as this obviated the need for wri en
responses or reading and therefore allowed the children to focus more
strongly on the problem presented to them and demonstrate their
understanding without needing to read or explain their reasoning.
The ability to apply knowledge indicates a certain degree of security
81
Margaret Sangster
81
in that knowledge. It also meant that the data collection could be
extended further down the primary age group.
A small scale survey/interview style was chosen because this was
considered to be a particularly appropriate method to use with
young children. The material was visual. There was no writing
required. The children were in a one-to-one situation or a two-to-
one situation with the researcher. It was diffi cult for them to copy.
They had the opportunity to talk and ask questions if they wished.
The presence of other children nearby could have been considered
reassuring although none of the children were shy to the point of not
participating which was a positive factor. It is possible that this was
due to the presence of practical materials and tasks.
The additional advantage of seeing individuals or pairs was that
it was possible as a researcher to observe closely the children’s
responses and ask questions to ‘probe’ as Brown and Dowling (1998)
would say. As one of the purposes is to try and build a picture of
developmental understanding, opportunities to prompt and probe
rendered the data more useful. ‘How a person reasons is not open to
direct inspection’ (Brown and Dowling, 1998, 61), so one is therefore
forced to make assumptions. This would place any such responses
in the realm of qualitative data. The tabulated results could be
regarded as a ‘small-scale survey’ and have been treated as such. It is
recognised that the size of the survey can only allow for conclusions
of an indicative nature to be drawn.
The data is considered to be reliable as similar results are likely to be
achieved with other groups of children using the same data collection
methods. The validity of the study in terms of ge ing children to
continue a pa ern, indicating whether abstract or picture images
are more easily managed and discerning the diff erent responses for
translation, refl ection and rotation were considered to be secure.
The perception of the particular rule in rotational situations was
questionable, as results will indicate.
82
TRANSFORMATION IN A SERIES OF LINEAR IMAGES
82
83
Margaret Sangster
83
84
TRANSFORMATION IN A SERIES OF LINEAR IMAGES
84
The instructions were:
Turn over all 20 cards and spread them out.
There is one card that fi ts a particular way round.
You have to look at the others (pointing to the sequence) to see which
way round it goes.
Look closely at what is happening.
Occasionally a second comment was made:
You can try them and see.
For each of the ten series there were two cards which had designs
similar to the given images. One only was correct and this needed to
be placed the right way round. Given that the children were going
to select one of the two probable cards as each pa ern was very
diff erent, there were eight possible placements from the two cards. A
tick in the recording table indicated a correct placement. The children
were then asked which they thought was the hardest and which the
easiest pa ern. The linear pa erns off ered included translations,
refl ections and rotations. Some of the images off ered were simple
pictures and others were abstract designs. One of the designs was
symmetrical, four were asymmetrical.
Table 3.To show details of the linear transformation pa erns
Series 1 A picture image of a cup and saucer with a single fl ower motif. In
the series the cup is seen the right way up and then rotated through
180 degrees. The third image is the cup in the original position. It
is anticipated the fourth cup will be rotated 180 degrees again. The
selection is likely to be made between the image and its refl ection.
Series 2 A straight lateral translation of an abstract image of rectangles.
Selection is likely to be made between the image and its refl ection.
Series 3 A picture image of a cartoon face which has been rotated 90 degrees
anticlockwise in each position. A further rotation of 90 degrees is
anticipated. Selection is likely to be made between a face set square
on a card and one place diagonally (45 degrees).
Series 4 A symmetrical abstract crown image. The image is rotated 90 degrees
anticlockwise and then in the third space returned to the original.
It was expected that the fi nal space would be a repeat of the second
space, a rotation of 90 degrees anticlockwise. Selection is likely to be
made between a crown with a blue band and a crown with a purple
band.
85
Margaret Sangster
85
Series 5 Exactly the same rule as series 4 but the image is a teddy bear. The
selection is likely to be made between a teddy bear with a red bow tie
and a teddy bear with a blue bow tie.
Series 6 An abstract asymmetrical image. The second image is a refl ection,
the third a refl ection again which returns it to the original image.
The selection is likely to be made between the original image or its
refl ection.
Series 7 A lateral translation of a sailing boat image. The selection is likely to
be between the original image and a refl ection.
Series 8 An abstract image of triangles set on the diagonal with a rotation
of 180 degrees each time. The selection is likely to be between the
original image and its refl ection.
Series 9 A picture image of a lorry with pop-ice inscribed on it. The series is
refl ections. The selection made is likely to be between the original
image and a refl ection.
Series 10 An abstract ‘blob’ rotated 90 degrees anticlockwise. The selection
made is likely to be between an original card and the image placed
diagonally on the card (45 degrees).
Results
It was observed that children responded to the task with varying
degrees of speed. It was noticeable that many of the oldest group
(age 10.6–11.6 years) selected the two relevant cards and compared
them side by side. Here we see the comparison strategy (Fellows,
1968).This was not something the younger children did. In fact, it
was observed that a few of the younger children failed to scan all the
cards when they were looking for a match.
Several of the children described the rotation in terms of the cards
facing in all four directions. One child used the four compass points
to explain what was happening. When asked, the child said they had
recently done work on compass points in the class. This is an example
of transferring previous experience to the new situation (Corcoran,
1971). Experience of symmetry and pa ern work was mixed which is
fairly normal for a junior school which followed the Peak published
mathematics scheme (Nelson).
As soon as the children completed the task they were asked which
they thought was the easiest and which the hardest series to do. They
could look through the cards if they wished. Correct and incorrect
86
TRANSFORMATION IN A SERIES OF LINEAR IMAGES
86
responses were recorded (Table 2) as well as the children’s fi nal
decision on easy and hard pa ern (Table 5).
Table 2. To show correct responses to linear transformation pa erns
by age group
Year group
out of s1 s2 s3 s4 s5 s6 s7 s8 s9 s10
6 (10.6 – 11.6)
25 25 25 13 16 22 17 25 17 25 14
5 (9.6 – 10.5)
25 25 25 12 18 21 21 25 20 25 3
4 (8.7 – 9.5)
25 24 25 5 11 17 15 25 15 24 6
3 (7.6 – 8.6)
25 24 22 5 13 19 16 25 12 22 1
Total (out of
100) 98 97 37 58 79 69 100 64 96 24
This would indicate an overall order of diffi cult using correct
responses (out of 100):
Table 3. To show overall correct responses to linear transformation
pa erns
series correct responses
7 100
198
297
996
579
669
864
458
335
10 24
87
Margaret Sangster
87
Examining the results in terms of overall success rate, this is very
high for some of the series. Image 7, the boats, a simple translation
scored 100% correct response. This would support Mach’s view that
translation is the easiest of symmetries to work with (Sheppard and
Cooper, 1982). But a 98% correct response is given to series 1, a tea cup
picture with a 180 degree rotation. High in the performance is also
series 2, a translation of abstract rectangles and series 9, a refl ection
of vans. Commonality is not clear. In the top four results there are
both pictures and abstract designs and all three forms of symmetry.
Poor performance occurred in series 3, a rotating picture of a cartoon
face and series 10 a rotating abstract splodge.
If the order of successful performance data is placed against the type
of response required and whether the design is a picture image or
abstract image the following table is generated:
Table. 4 To show type of design and nature of transformation
matched to number of correct responses
series picture/abstract transformation
7 picture translation
1 picture rotation 180 degrees
2 design translation
9 picture refl ection
5 picture rotation 90 degrees & return
6 design refl ection
8 design rotation 180 degrees
4 design rotation 90 degrees & return
3 picture rotation 90 degrees
10 design rotation 90 degrees
From this it can be seen that two factors appear to be operating here.
The results would seem to suggest that using a picture to generate a
pa ern is very helpful. At the same time series 3 is a picture and had
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88
a poorer success rate. The children found it hard to continue a series
which required a 90 degree rotation in one direction, as these three
series (3, 4 and 10) are at the bo om of the table. Even the picture did
not help. When considering rotations as a 180 degree turn this seems
to be easier. The two translations are high scorers, confi rming Mach’s
view (1982). The refl ection does seem to be easier than most rotations
(Bruce and Morgan, 1975) but is not as well completed as the picture
rotations. Series 5 initially seems to score higher than one would
expect for a 90 degree rotation. It is the image of the teddy bears.
There is the possibility of a third infl uence entering the equation, that
of Corcoran (1971) familiarity.
The series 5 has interesting results when the table for children’s
opinion of ‘easiest’ and ‘hardest’ is examined below.
Table 5. To show which series were thought to be hardest and easiest
by the children
series hardest easiest
11 41
21 1
33 5
47 0
50 13
628 1
70 16
82 1
92 21
10 51 0
no opinion 5 1
The series identifi ed as easiest were 1, 5, 7 and 9 which are all picture
images. This included translation, 180 and 90 degree rotation and
refl ection. This indicates that pictures were considered easier to
manipulate than designs. The unexpected success in performance
is series 2 which is a translation/design which children did not
considered easiest but performed well on. It is possible that this was
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because it was a translation. Both translations have high success
rates.
Looking at the series which children judged the hardest, series 10
and 6 dominate (Table 5). 10 is a design/90 degree rotation and 6 is a
design/refl ection. Children’s judgement matches their performance
with series 10. The series 6 refl ection lies in the middle order of
success. If translations are relatively easy and rotations are relatively
hard, one would expect refl ections to lie in the middle order. Children
appear to be consistent when selecting pictures as easier. They have
also identifi ed designs as the hardest.
From observation of the children’s approach to series 3 and 10, which
gave the poorest results many placed the correct card in a similar
position to space two so that the series read:
original, 90 degrees anticlockwise, 180 degrees anticlockwise, and
the return to 90 degrees clockwise. At fi rst this seems incorrect,
especially if one is anticipating a continuing anticlockwise rotation.
But earlier in series 4 and 5, the image had rotated 90 degrees and
returned 90 degrees. This series could be viewed in a similar way;
rotate 90, return 90, rotate 90, return 90 to original. This would give a
four image unit of repeat too.
This then calls into question the validity of the results on the rotation
pa erns. From this we can see that three rotations are not suffi cient
to be unambiguous.
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90
Discussion
The performance of the children throughout the age groups was high.
Some of the youngest children tailed off on the rotation situations,
but there were good performances in all age groups. The top two age
groups have met rotation on at least one occasion in their published
mathematics scheme (Peak, Nelson).
From observation, most children appeared more confi dent in the
picture situations and very confi dent in the translation situations.
When viewing the abstract images they did not speak, they seemed
to be searching for reference points in the designs as suggested by
Gibson (1968) and Haber (1971). This was particularly evident with
series 6. Many seemed to arrive at a satisfactory classifi cation and
then rotate the card. Far more problems were experienced with series
10 where it seemed many of the children did not fi nd a satisfactory
cue and were le hazarding a guess. Those who seemed to use the
red dot as a cue appeared to have more success. This could be an
indication of using a distinctive feature within the design. These
statements are speculation based on observed response and would
be worth further investigation.
Corcoran (1971) suggested that blocked images would be easier to
manipulate than outlines. Of the abstract images in the study they
were mostly blocked with colour. The exception was series 6 which
had a line infi lled lightly with pa ern. This was one with which
children had poorer results than one would expect from a refl ection.
Corcoran also suggests it is easier to rotate symmetrical shapes. The
only image that fell into this category was series 4 with the abstract
‘crown’ image. Results indicate children were reasonably successful
with this series but their opinion of the series was ‘diffi cult’.
None of the children showed uncertainty about the task or confusion
in placing the cards. Only a few children selected the wrong card.
On most occasions this appeared to be because they had failed to
spot an alternative card or they had not registered the detail such
as the colour of the bow tie or the crown. This would suggest that
one or two had low scanning or low observation skills or even poor
colour discrimination. This would support Fellow’s view of failure
strategies (1968).
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Four strategies observed which seemed to facilitate correct solutions
were: scanning the whole selection of choices, observing detail within
a design, holding the options next to each other before selection of
the card and placing the choice in position and ‘seeing’ if it fi ed.
In the context of continuing a linear pa ern of spatial images the
study showed several trends. Picture images were more successfully
manipulated than abstract design images. The children also identifi ed
the series with the picture images as the easier tasks. Results showed
that the abstract designs were harder to continue unless it was in the
context of a translation pa ern. The children themselves convincingly
chose two of the abstract designs as the hardest series to complete.
The images of spatial pa erns might have an eff ect on the success rate
of continuing pa erns. If the fi ndings of the Gestalt school (Haber,
1971, Bruce and Green, 1986) are applicable then certain images will
be easier to manipulate than others. In a simple form, this could be
pictorial images compared with design images. If the features in
the design are not distinctive they might be diffi cult to manipulate
(Orton, 1993).
In terms of symmetry, translation pa erns were the most successfully
completed pa erns. Refl ection pa erns fell between translation and
rotation apart from a 180 degree rotation of a picture (Figure 1,
series 1: teacups). The 90 degree rotations were the least successfully
completed series. Responses were more varied and it would have
been useful to have collected exactly what the incorrect responses
were although observational data has been discussed earlier. There
are strong indications that children had not developed a strong sense
of rotational rules within linear pa erns and were therefore not
recognising it as a linear pa ern. But also the task, by off ering only
three images, allowed for an interpretation of inverting positions
1 and 3 and repeating positions 2 and 4. There is a need to revisit
rotation contexts to establish what rule children are applying.
Observation of the strategies employed by the children off ered
possible ways of improving the success rate of the children. These
included scanning all the materials, paying a ention to detail, using
matching strategies before making decisions and trialling solutions.
Pa ern seeking requires the skills of: recognition of the same pa ern,
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recognition of similar structures, categorising pa erns by property,
generalising the properties suffi ciently to enable transfer to new
situations, and being able to conjecture or predict what might happen
next by identifying a rule and applying it. When Bird (1991) observed
5 and 6 year-olds at work on problem solving, she amassed a list of
96 skills and strategies which she considered the children had used.
From this list we can see some of the skills mentioned here: searching
for/fi nding pa erns and relationships, conjecturing, extrapolating,
manipulating, deciding on rules, recognising equivalence, pa erning,
generalising, using rules, transforming, classifying, structuring and
comparing.
These are skills which need to be learned and refi ned. For example
checking is a strategy which leads to more consistent results. Closely
aligned to checking is a child’s perception of accuracy. As Gibson and
Gibson (1950) observed, ‘similar’ was acceptable to the children when
asked to identify the ‘same’ pa ern. There might be a need to clarify
that ‘same’ means ‘identical’ in mathematics. Fellows (1968) found
children failing to match images because they would not orientate
them, or not take suffi cient notice of the features, or not compare
carefully enough, or not carry through the task. The positive form
of any of these might be considered strategies which will support
successful work with pa ern images. Vurpillot (1968) found that the
younger children were weaker at scanning and comparison.
Mason (1989), indicates it is important to direct students’ a ention
to the signifi cant features that allow for abstraction to take place.
This is particularly important at a later stage of pa ern work when
formulae are being sought to generate any stage of the pa ern
(Orton and Frobisher, 1994). If pa erns remained as a collection of
diff erent images in the mind, eventually a large collection would
amass. These could be drawn out of memory for matching purposes.
However, not many pa erns are the same. This bears some of the
same characteristics as investigations and problem solving and
the diffi culty of transfer. It has been suggested that recognition of
properties and generalisation enable transfer. In research, it has been
diffi cult to establish that regular transfer takes place in problem
solving because problems can be so very diff erent mathematically
(Sco , 1977; CASE, 1988; CAME, 1997). These projects have met with
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varying degrees of success. Pa erns too, can be very varied. Transfer
will probably depend on a range of mathematical knowledge as well
as an awareness of structures and rules. For example, knowing that
rotation is about the amount of turn allows the child to focus on this
property when working with any rotation pa ern.
Mason, Burton and Stacey (1986) consider that when students
generalise, they focus on aspects common to many examples and
ignore other features. Generalisation is the action of drawing out
information from a situation which can be used elsewhere. For
example, the fact that there is a repeat of the unit in a particular
linear translation could be a generalisation made about other linear
situations using diff erent materials. For the generalisation to occur the
structure and/or the rule of the pa ern has to be recognised and stored
in memory. This implies a more sophisticated level of understanding
(Nola, 1997) based on the recognition of the mathematical properties
of a situation.
It is possible to continue some pa erns with only a limited amount
of understanding about their structure and rule. Refl ecting on the six
types of knowing proff ered by Nola (1997) it might be that a child
could say, ‘It’s a mirror pa ern’ (refl ection). Here they are showing
they know what something is (number 6 in Nola’s list), but it does
not necessarily mean they know how to explain the refl ection pa ern
(number 3), or why it works (number 4). Teachers need to be aware
of what depth of understanding a child has and what depth of
understanding they expect from the child and their ability or inability
to express their understanding of the situation in words.
References
Adhami, M., Johnson, D. and Shayer, M. (1997) Does ‘CAME’
work? Summary Report on Phase 2 of the Cognitive Acceleration
in Mathematics Education, CAME, Project, in Proceedings of the
Day Conference of the British Society for Research into Learning
Mathematics (Bristol, November 1997).
Askew, M., Brown, M., Rhodes, V., Johnson, D. and Wiliam, D. (1997)
Eff ective Teachers of Numeracy (Final report) London: Kings.
Bird, M. (1991) Mathematics for Young Children – An Active Thinking
Approach, London: Routledge.
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94
Brighouse, A., Godber, D. and Patilla, P. (1981–5) Peak Mathematics,
Walton-on-Thames: Nelson.
Brown, A., Dowling, P. (1998) Doing Research/Reading Research,
London: The Falmer Press.
Corcoran, D. (1971) Pa ern Recognition, Harmondsworth: Penguin.
Devlin, K. (1997) Mathematics: the Science of Pa ern, New York:
Scientifi c American Library.
Fellows, B. (1968) The Discrimination Process and Development,
Oxford: Pergamon.
Gibson, J. (1950) The Perception of the Visual World, Cambridge,
Mass: Riverside.
Gibson, J. (1968) The Senses Considered as Perceptual Systems,
London: Allen and Unwin.
Haber, R. (1971) Where are the Visions in Visual Perception. Chapter
3 in Imagery: Current Cognitive Approaches (S. Segal ed.), New
York: Academic Press.
Mach, E. (1959) in The Analysis of Sensation (107) quoted in Mental
Images and Their Transformation Sheppard, R. and Cooper, L.,
Cambridge, Mass: MIT Press.
Mason, J. Burton, L. Stacey, K. 1986 Thinking Mathematically,
London: Addison-Wesley.
Mason, J. (1989) Mathematical Abstraction as the Result of a Delicate
Shi of A ention For the Learning of Mathematics 9, 2 June 1989,
pp. 2–8.
Nola, R. (1997) Constructivism in Science and Science Education: A
Philosophical Critique, Science and Education 6, 1–2, pp. 55–83.
Orton, A. Frobisher (1996) Insights into Teaching Mathematics,
London: Cassell.
Orton, J. (1993) Pa ern in Relation to Shape The Proceedings of
the British Congress on Mathematical Education (G. Wain ed.),
University of Leeds, July 1993.
Presmeg, N. (1992) Prototypes, Metaphors, Metonymes and
Imaginative Rationality in High School Mathematics Educational
Studies in Mathematics, Vol 23, 6, pp. 595–610.
Sawyer, W. (1955) Prelude to Mathematics (Pelican series),
Harmondsworth: Penguin.
Sco , N. (1977) Enquiry Strategy and Mathematics Achievement,
Journal for Research in Mathematics Education 8, pp. 132–43.
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Sheppard, R., Cooper, L. (1982) Mental Images and Their
Transformation, Cambridge, Mass: MIT Press.
Whitehead, A. (1925) An Enquiry Concerning the Principles of
Natural Knowledge, Cambridge: Cambridge University Press.
Vurpillot, E. (1968) The Development of Scanning Strategies and
their Relation to Visual Diff erentiation, Journal of Experimental
Child Psychology 6 5, pp. 622–50.
Contact
Dr Margaret Sangster
Faculty of Education
Canterbury Christ Church University,
Canterbury, UK
m.sangster@canterbury.ac.uk
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Abstract
This chapter is based on a plan of a research in mathematics education,
especially in ICT use in primary mathematics from the point of view of
teachers. It outlines the way in which design research can be used as a form
of educational research focused especially on teaching mathematics and
using ICT in mathematics education. Many computer tools and resources
have been developed for school activities generally, as well as especially
for mathematics education. However, although Finnish schools have high
standards of equipment applications in use, ICT is not as common as
some authorities would like it to be. One aim of this research is toestablish
teachers’ a itudes and perceptions towards technology and the use of
ICT in mathematics education. Experiences and opinions of teachers are
central. Another aim is to develop a model for a learning environment for
teaching mathematics that combines e computer resources with elements
of the traditional learning environment, and also takes into consideration
teachers’ opinions. The focus is on the background and methodology of
the research. The current state of research is also described by focusing on
problem analysis, which includes need assessment and the clarifi cation of
constraints.
Keywords: design research, ICT use, mathematics education, teacher
a itude and perception, usability
DESIGNING A LEARNING
ENVIRONMENT FOR TEACHING
PRIMARY MATHEMATICS: ICT USE
IN MATHEMATICS EDUCATION
FROM TEACHERS’ PERSPECTIVES
Heidi Krzywacki, University of Helsinki
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DESIGNING A LEARNING ENVIRONMENT FOR TEACHING PRIMARY ...
98
Introduction
Over several years, development of technology has been enormous and
it has become more common to use information and communication
technology (ICT) in education. Many kinds of applications, digital
learning materials and digital learning environments have been
developed for school activities generally and also especially for
mathematics education. The revision of the framework curriculum
for comprehensive schools in Finland was published in January 2004.
One of the objectives emphasised in mathematics education was the
development of mathematical thinking and ICT use in supporting
learning processes (NBE 2004). Information technology has an
important role in modern learning environments.
Although Finnish schools have high standards of equipment and
use several educational applications, ICT is not as common as some
authorities would like it to be. For example, it might be surprising
that the students representing schools known to use ICT intensively
were not using it as an integrated part of their everyday schoolwork.
(Hakkarainen et al. 2000) On the other hand, using ICT in a particular
subject should not be emphasised just for its own sake. Firstly, it is
important to think carefully about what ICT can or cannot off er.
Secondly, it is central to consider how children learn, according
to educational research. Thirdly, relationships between ICT, the
discipline of the subject domain and the aims of the particular school
subject, should all be examined carefully when applying ICT in
education. (Haydn 2003)
An important part of teaching primary mathematics is to concretise
contents and to use manipulatives in order to help in understanding
and learning mathematical concepts. This issue has to be taken into
consideration while considering what kind of possibilities digital
learning materials and environments off er for teaching mathematics.
In this research one aim is to develop a learning environment for
teaching mathematics which combines the elements of real and virtual
learning in a suitable way for teaching primary mathematics.
This research also focuses on teachers’ a itudes, skills and perceptions
of ICT use in mathematics education. Besides the need for diff erent
educational technology applications, it is important that teachers
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Heidi Krzywacki
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and students master ICT well and are also willing to use ICT in
school activities. The teacher has a central role in school activities
as a decision maker and has a great impact on implementations
of mathematics education. The teacher should be able to create a
learning environment with his/her students which provides a suitable
se ing for mathematical learning and in which ICT use is integrated
with mathematics in meaningful ways. There are several studies on
the use of technology in education, examining for example, teachers’
and students’ technological skills (Baylor and Ritchie 2002), a itudes
and perceptions toward technology (Carey et al. 2002; Christensen
2002; Ruthven et al. 2004) and also fi nding out the ways technology
has been used in education (Hakkarainen et al. 2000; Ruthven and
Hennessy 2002). Here the focus is on teachers’ perceptions and
a itudes toward ICT, as well as on technological skills evaluated by
the teachers themselves. Those themes comprise the basis for a model
of a learning environment of teaching and learning mathematics
using ICT meaningfully and with suitable support.
This chapter focuses on the background concepts of the research, on
methodology and on the present state of research. Four frames of
reference are introduced: computer-based education and ICT use in
education; the concept of a learning environment; usability of ICT;
and the teacher’s role in teaching mathematics especially with the
help of computer tools and resources. The design research approach
is one way to discover the characteristics of ICT use in primary
mathematics and to take teachers’ a itudes and perceptions into
consideration as users’ opinions in the development of a model for
a technological learning environment (see Henneman 1999; Sugar
2001; Edelson 2002). At the beginning of the research the fi rst two
phases: design procedure and problem analysis, are central, so that
the review of the background concepts and of the methodological
se ing is emphasised (see Edelson 2002).
Theoretical Background
The background to the research is based on four frames of reference
that aff ect the research and the problem analysis. The fi rst is the
concept of computer-based education and use of information and
communication technology (ICT) in education. The focus must be
clarifi ed carefully as defi nitions of computer-based education vary
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DESIGNING A LEARNING ENVIRONMENT FOR TEACHING PRIMARY ...
100
greatly. Secondly, as the aim is to develop a learning environment
which integrates technological solutions as part of teaching and
learning mathematics, the concept of the learning environment
is central. The third concept is usability of ICT in education.
Pedagogical usability is a most important criterion for teachers in
evaluating educational solutions from technology in the context of
mathematics. The fourth concept is teachers’ pedagogical thinking
and teachers’ roles as decision makers in school activities. Teachers’
a itudes towards and perceptions of technology as well as teachers’
technological competence, impact on implementations in mathematics
education. On the other hand, conceptions of and a itudes towards
mathematics rather than technology are emphasised in practice
(Pietilä 2002).
Computer-based education and the use of ICT
Defi nitions of ICT in education vary greatly. As this research focuses
on using computer tools and resources in mathematics education in
general, and also on a itudes and perceptions towards technology,
the defi nition of ICT use in education needs to be clarifi ed. The uses
of technology and ICT in education may be single digital learning
units or virtual learning environments with many technological
tools. Digital learning material can be defi ned as digitally-published,
computer-based learning material (e.g. CD-ROM or learning tasks
involving the use of ICT). The common feature of digital learning
material is that it aims at particular objectives of learning and
can be used through its own operating system. A virtual learning
environment is an application in which unit(s) of learning material
can be placed, and which has tools for studying and learning
processes. There are two kinds of environments, those with units of
digital learning material and those with characteristics and tools for
learners and teachers (Nokelainen 2004). There are also computer-
based, technological applications and tools which cannot be included
in the defi nitions above. Some were produced initially for other
activities, but are now used in education. In this chapter the focus
is on all kinds of technological applications which are in some way
computer-based and both known to, and available for, teachers.
Reeves describes and evaluates diff erent forms of computer-based
education (CBE) according to fourteen pedagogical dimensions (Reeves
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1997, 2). This model suggests that these pedagogical dimensions
can possibly be used as criteria for comparing, understanding and
evaluating diff erent implementations of computer-based education.
According to Reeves (1997, 2) the dimensions are concerned with
those aspects of the design and implementation of CBE that directly
aff ect learning. These fourteen pedagogical aspects are:
1) epistemology: theories about the nature of knowledge ranging
from objectivism to constructivism;
2) pedagogical philosophy: approaches to teaching and learning
ranging from a strict instructivist philosophy to a radical
constructivist one;
3) underlying psychology: dimensions related to the basic
psychology underlying CBE, ranging from behavioural to
cognitive psychology;
4) goal orientation: dimensions related to the degree of focus
represented by a programme ranging from sharply-focused to
unfocused;
5) experiential value: a continuum ranging from abstract to
concrete;
6) the teacher’s role: pedagogical roles ranging from didactic to
facilitative;
7) fl exibility: a continuum of programme fl exibility ranging from
unchangeable to easily modifi able;
8) the value of errors: a continuum ranging from errorless learning
to learning from “trial and error”;
9) origin of motivation: a dimension ranging from intrinsic to
extrinsic motivation;
10) accommodation of individual diff erences: a continuum that
ranges from non-existent to multi-faceted;
11) learner control: a dimension of control ranging from complete
program control to unrestricted learner control;
12) user activity: related to learning environments ranging from
mathemagenic (access to various representations of content)
to generative (engaging learners in the process of creating,
elaborating and representing knowledge);
13) co-operative learning: ranging from a complete lack of support
for cooperative learning to inclusion of cooperative learning;
and
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102
14) cultural sensitivity: ranging from non-existent to integral (Reeves
1997).
As these are all criteria for evaluating and describing the particular
technological application, they seem to be useful for this research. The
most central described by Reeves (1997) are pedagogical philosophy;
goal orientation; experimental value; teacher role; fl exibility; and
user activity. It is important to notice that the focus is not only on
computer-based education (CBE), as important as CBE is in teaching
and learning mathematics in a learning environment with computer
tools and resources in a traditional way. All the points described are
related to the characteristics of the traditional learning environment,
which also happen to defi ne the characteristics of the technology
used in mathematics education.
Learning environment with ICT use
As the aim of the research is to develop a model of a learning
environment which integrates computer tools and resources into
teaching and learning mathematics, the concept of a learning
environment is central. The term learning environment related to
technological applications is usually linked to computer networks and
the WWW. In this research a learning environment is understood as a
combination of traditional real classroom practices and technological
(computer based) applications used especially for teaching and
learning mathematics. The learning environment can be understood
as those factors that defi ne the context of the learner’s studying and
learning (La u 2003, 23). It seems to be external to, and something to
be absorbed by, the learner before it can become internal to a learner.
The educator’s role in creating the learning environment is however
emphasised. The teacher acts as a supervisor to motivate learners
to use the possibilities off ered and also controls the activities by
reshaping the environment during the studying and learning process.
(La u 2003, 24–25)
The development of technology has had an eff ect on educational
aims and on learning environments (Haydn 2003; NBE 2004). ICT use
has become one of the central characteristics of modern education.
Now there are diff erent kinds of technological solutions in education
and new ways of supporting learning in schools. Finland like other
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Heidi Krzywacki
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industrialised countries has invested in ICT and therefore the standard
of equipment is high. Yet despite the high profi le of ICT in education,
it seems to be quite demanding to fi nd practical and meaningful
ways to integrate ICT into school practice. Certainly not all teachers
are using ICT as a routine part of their teaching (Hakkarainen et al.
2000; Ruthven and Hennessy 2002; Haydn 2003). There are several
reasons which either encourage or discourage teachers in using ICT
in mathematics lessons. Forgasz (2002) has found four critical factors:
computer access; teachers’ confi dence and skill levels; availability
of appropriate so ware; and technical support. One issue here is
that even those teachers who do integrate ICT into their lessons are
sceptical of its advantages for enhancing learning.
Besides the exigencies of ICT use, mathematics as a school subject
has specifi c characteristics to be taken into consideration. According
to Haydn (2003, 4–6), three propositions can be made for integrating
ICT be er into teaching activities within the real world of teaching
and learning. Those can also be applied in mathematics education.
The fi rst proposition is the need to think carefully about exactly what
ICT can or cannot off er. ICT shouldn’t be seen as an educational
miracle. As it is mathematics education under discussion, the
benefi ts and dangers of ICT in teaching mathematics and for those
who are teaching mathematics in school must be carefully examined.
Designing ICT applications with collaboration between teachers and
producers is one solution. The second proposition is the need to think
about technology in the context of how children learn in general. It is
also central to fi nd out how the characteristics of the new technology
fi t with the nature of mathematics the kind of learning strategies that
will be emphasised. The third proposition is the need to think about
the relationship between ICT, the discipline of mathematics and the
purposes of school mathematics. Much of the discussion about ICT
and education has been non-subject-specifi c. According to Haydn
(2003, 5), even if there are applications that can have possible uses
in all subjects, there are also those which have more potential for
enhancing learning in some subjects than in others.
Teachers’ a itudes should be explored in discovering the
characteristics of suitable learning environments with ICT. Teachers
are those who have the greatest impact on what takes place in
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104
classrooms. Ruthven and Hennessy (2002) have developed a model of
computer use in education. The study hints at a gradual mechanism
whereby teachers initially view technology through the lens of their
established practice, and employ it accordingly. There were several
themes (according to the teachers’ views), e.g. productivity, progress
and participation as well as suffi ciency, which should be considered
while creating the learning environment with computer tools and
resources.
Usability of ICT
The third concept that is part of the framework behind the research
design is usability of information and communication technologies
in the educational fi eld. There are several systems for defi ning
usability. (Nielsen 1993; Henneman 1999; Nokelainen 2004) It can
be measured by, e.g. usability evaluations, questionnaires and focus
groups. Usability, or ease of use, may be defi ned as the eff ectiveness,
effi ciency and satisfaction with which specifi ed users achieve
specifi ed goals in particular environments. Usability goals should be
set in these three areas. Aspects of may include ease of installation,
ease of learning, user support, productivity, user errors, and ease of
customisation. (Henneman 1999, 137–38) Henneman (1999) defi nes
usability in general, not particularly in an educational context. As this
research is aimed at fi nding out teachers’ technological perceptions
and also a itudes toward computers, their professional knowledge
in the use and evaluation of ICT in mathematics education should
be emphasised (Da Ponte et al. 2002). Criteria of usability can also
be divided into the pedagogical and technical criteria of usability
(Nokelainen 2004). As the focus of the research is on teaching and
learning mathematics, the characteristics of pedagogical usability
can be used for evaluating the educational solutions of technology.
Pedagogical usability can be defi ned according to ten diff erent
themes (Nokelainen 2004; see Reeves 1997):
• i) learner control: Does a learner have a feeling that he/she is
controlling the learning situation? Is he/she able to focus on
essential things according to objectives of learning?
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• ii) learner activity: How much do the ICT tools activate
learners? How problem-centred is the computer-aided learning
environment?
• iii) cooperative learning: How do ICT tools support the
learners and give opportunities to cooperate?
• iv) goal orientation: What are the objectives/goals for learning?
• v) applicability: In what way are the contents of the material
related to real-life situations? How are the skills for real life
enhanced?
• vi) added value: What kind of new applications or new
approaches do ICT tools off er for mathematics education?
• vii) motivation: What kinds of motivation arise from the use of
ICT ? How does the use of ICT motivate learners?
• viii) valuation of previous knowledge: How are the skills and
previous knowledge of learners taken into account)
• ix) fl exibility: How fl exible are the ICT tools which are used? Is
there a opportunity of varying tasks individually?
• x) feedback: What kind of feedback do learners get?
As Haydn (2003) emphasises, the subject domain and content of
mathematics should be taken into consideration while evaluating
the use of ICT in mathematics education. However, the criteria
for usability defi ned by Nokelainen (2004) and the characteristics
of computer-based education described by Reeves (1997) are on a
relatively general level. It is important to note that the characteristics
of usability should not be used for judging computer tools and
resources without taking a particular content and learning situation
into account. The general features of mathematics education should
be considered in the light of the criteria listed above. Only then
should mathematics teachers use those criteria as tools for planning
their lessons and for evaluating technological applications..
Teachers’ pedagogical thinking and conceptions of computers
The fourth viewpoint is teachers’ pedagogical thinking and their role
as decision makers in school activities. This can be considered as
a main tool of evaluation and creation of the learning environment
suitable for mathematics education. In the Finnish school system
teachers are decision makers who have a great impact on the
implementation of mathematics education within the curriculum.
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106
Teachers have the responsibility for choosing educational materials,
in this case computer tools and resources. Other specifi c issues related
to teachers’ roles, are their view of mathematics and their conceptions
of technology particularly as part of mathematics education.
The pedagogical thinking of teachers in general has been examined
in diff erent contexts and in relation to diff erent goals. Research has
varied in terms of context and research subjects (e.g. Alexandersson
1994; Alexandersson 1995; Oser and Baeriswyl 2001). The pedagogical
thinking of teachers has been analysed in several models (e.g. Van
Manen 1977; Patrikainen 1997; Kansanen et al. 2000). The concept of
interaction is central to teaching. Interaction is part of the educational
process and there is no educational process without interaction or
values (Kansanen 1993, 53–54; Kansanen et al. 2000, 27–28). In order
to understand teachers’ actions, it is crucial to examine the concepts
guiding the teacher, for example consciousness and reasoning.
Pedagogical thinking can be defi ned as the making of decisions
according to the situation and there is always reasoning behind the
decision. The decision-making is refl ective; there is time to think
through diff erent alternatives; the teacher does not have to put
the decision into practice immediately. In the light of this research
teachers will have time to evaluate the source material. (Kansanen
1993, 51) The teacher is a decision maker in his/her work and s/
he gives reasons for acting in a certain way through pedagogical
thinking. The degree of consciousness in the teacher’s thinking about
purposiveness can vary: from being a total technician to being an
independent decision maker. A teacher can develop his/her work
through pedagogical thinking. (Kansanen 1993, 61)
On the other hand, conceptions of and a itudes toward mathematics
are emphasised in practices (Pietilä 2002). Teachers’ thinking refl ects
a view of mathematics, that is a set of beliefs and a itudes towards
mathematics that develops with exposure to diff erent experiences.
Their own views of mathematics consist of their knowledge,
beliefs, conceptions, a itudes and emotions about 1) themselves as
learners and teachers of mathematics and 2) mathematics and its
teaching and learning. (Pietilä 2002, 23–24) Motivational factors and
meaningfulness in the learning process are central to both students’
and teachers’ opinions about learning materials. Beliefs, a itudes
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and practices in teaching and learning mathematics are signifi cant
determinants of the way teachers view their role as educators as
well as the way in which students view their role as learners. On
the other hand the method of teaching and learning aff ects a itudes
towards mathematics as it also does those of student teachers
(e.g. Utsumi and Mendes 2000, 241; Macnab and Payne 2003, 55).
Especially interesting is to discover the conceptions teachers have of
mathematics education, of ICT use and of ICT use especially as a part
of mathematics education.
A itudes and perceptions of teachers toward computer tools
and resources have been highlighted in research. These, together
with their technological competence, will have an impact on
implementation in the classroom. Naturally willingness to try,
experience in, and knowledge of, diff erent approaches will aff ect
whether the technological applications will be chosen or not and also
how they will be used. Positive a itudes and teachers’ willingness
to integrate ICT in teaching and learning mathematics are crucial
to bring about the educational change related to technological
development. According to Carey (2002, 224), the extent to which
nations can exploit the potential economic, social and educational
benefi ts of ICT is dependent upon individual citizens’ perceptions of
and a itudes toward technology; their access to computers; and their
experiences in using those technologies. Research suggests that the
greater the access and usage, the more positive the a itudes towards
the technology (Christensen 2002; Hong and Koh 2002).
The fi ndings described above reveal only one mechanism related
to a itudes to technology. As the teacher’s role is central, a itudes
towards technology must also be emphasised in teacher education
(e.g. Da Ponte et al. 2002; Hazzan 2002). Teachers should be aware
of their a itudes and also of arguments for and against the use of
computers that might infl uence their choosing to employ them in
mathematics lessons. A further point is also that student teachers
might support positive changes in schools if they are motivated
enough and can resist discouragement from veteran teachers.
(Hazzan 2002) A itudes of teachers and of students aff ect each other
so positive a itudes from students might also aff ect school practices
(Christensen 2002). Som student teachers are suspicious and fearful
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108
of ICT. It is therefore important to make them more confi dent
users of ICT and help them to develop a positive relationship with
technology. Future teachers should also take a critical a itude towards
technology. Professional knowledge of using ICT and technological
skills are also crucial for teachers. They should be able to integrate
ICT within their goals and objectives for mathematics teaching. They
should certainly be confi dent of their technological competence. (Da
Ponte et al. 2002) In this research, there are examples of models to
enhance and support use of ICT in education in the research fi eld of
teacher education. The studies described above reveal mechanisms
that could be considered while creating models in other research
fi elds.
Baylor and Ritchie (2002) have examined the factors facilitating
teacher skill, teacher morale and perceived student learning in a
technology-using classroom. They found that the degree of teacher
openness to change is a critical variable; teachers who are open to
change appear more easily to adopt technologies and thus their
technical competence increases. Unfortunately, it is diffi cult to
infl uence teachers’ openness to change. The other predictive feature
is the level of technology leadership and support for professional
development. The support of teachers, rather than of policy-makers,
seems to be of greater importance.
Method
Besides an overview of the theoretical background, a further aim of
this paper is to describe the design research approach as a method
especially in this educational research. Design research explicitly
exploits the design process as an opportunity to advance the
researcher’s understanding of teaching, learning and educational
systems. (Edelson 2002, 106–7) Here I hope to clarify this procedure
research, indicating especially the present juncture of the research.
Design research approach as an educational research method
Educational researchers are increasingly incorporating design into
their research activities as one form of educational research. However,
what is meant by design research varies between researchers just
as does the name for this particular approach (design experiment,
design research, design-based research, design studies or user-
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design research). The focus of these design eff orts also varies quite
dramatically; for example, from medical to technological research.
(Henneman 1999; Kim et al. 2001; Collings and Pearce 2002; Edelson
2002; Kelly 2003) The research described here is focused on teachers
and especially on teaching primary mathematics with computer tools
and resources. As one aim is to investigate a itudes and perceptions
toward computers as well as the technological skills of teachers,
and another is to develop a model for a learning environment
suitable for teaching mathematics with ICT, the design research
approach seems to be a good methodological choice (cf. Collings and
Pearce 2002). This kind of design research is both descriptive and
prescriptive, with the intention both to improve as well as describe
the use of ICT in mathematics education from the teachers’ points of
view. An important characteristic of design research is to eliminate
the boundary between design, development and research (Edelson
2002; The Design-Based Research Collective 2003).
Five features diff erentiate design experiments (design research) from
other methodologies. (Cobb et al. 2003; the Design-Based Research
Collective 2003) First, the purpose of design research is to develop
a class of theories about the process of learning and the means
designed to support that learning. The central goals of designing
learning environments and developing theories or ‘prototheories’
of learning are intertwined. Second, a design research process
is essentially iterative. Research takes place through continuous
cycles of design, enactment, analysis, and redesign. Third, designs
must lead to shareable theories that help communication between
educators and designers. Fourth, research must indicate how designs
function in authentic se ings. Fi h, design research should produce
solutions directly applicable in ordinary teachers’ classrooms. These
last features are related to the methods used in the research process.
They must not only document the success and failure but also focus
on interactions that refi ne our understanding of the learning issues
involved.
The process of design might seem complex. There are several
descriptions of design research (cf. Henneman 1999; Nelson et al.
2000; Sugar 2001) but in this paper we will use the one that Edelson
(2002) has described. Edelson’s model is quite simple and is a good
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110
basis for a methodological overview. According to Edelson (2002),
within any design a researcher or a research team has to make three
sets of decisions iteratively that determine the results of the process.
The fi rst set of decisions is the design procedure, the second is the
problem analysis and the third are the design solutions. The process
of design research is not, however, meant to be straightforward so
decisions will not be specifi ed all at once (Edelson 2002; cf. Henneman
1999). The design procedure specifi es the processes and the people
involved in the development of a design. In other words, these
solutions answer the question of how the design process will proceed.
When specifying the expertise and process required, the goals and
constraints of the design determine the decisions. When considering
the needs and opportunities the design will address, the problem
analysis will be characterised. The problem analysis specifi es the
goals or needs that the design is intended to address, together with
challenges, constraints and opportunities presented by the design
context. This typically evolves over the course of a design process
incorporating information from a variety of sources. In the design
solution the resulting design is described. The question is what form
the resulting design will take. In solution construction, designers
o en decompose a complex design problem into manageable
components. A researcher has to take on considerable challenges and
constraints and exploit the opportunities (Edelson 2002, 108–9).
In this research project, the user-centred approach is one of the
approaches (Henneman 1999, 135–36; Sugar 2001). Teachers’
a itudes and perceptions towards computers, as well as their
computational skills, are the central part of problem analysis in the
learning environment with technological applications (cf. Da Ponte
et al. 2002). Design research with three sets of decisions to be made
is the structure of the research. Teachers’ opinions will be taken into
consideration especially within the phases of design procedure and
problem analysis. Constraints and resources are described in the
initial design procedure. Firstly the relevant forms of expertise are
considered. The researcher and teachers as educators represent this.
Secondly, the processes (e.g. data collection and testing) to be used
during the research process have to be decided. Two methods are
used: a survey and interviews augmenting the survey data. While
developing the model of a learning environment to enrich the use
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of ICT observations will also be made. Thirdly the study se ing is
planned and the schedule of the project thought out. The researcher’s
role is thus an active one whilst observing and interviewing the
teachers.
Another set of decisions to be made is related to problem analysis. The
research problems as well as the goals of the research are described
below. The design of the learning environment for improving the
use of ICT is intended to be developed partly based on teachers’
conceptions. As a preliminary study has already taken place the
research problems have already been specifi ed. Accordingly teachers’
own conceptions and a itudes have a central role. Teachers’ opinions
about, and conceptions of, ICT use in mathematics vary greatly.
This results from their own technological skills and experiences in
using ICT. Teachers were not thinking about ICT directly related to
mathematics education but more generally and quite subjectively.
As the research is still new the outlines of the design solution
that will take place are not clear yet. The design solutions will be
a model of the learning environment using ICT in mathematics
education. Teachers’ opinions in the preliminary study suggest that
one of the most diffi cult problems preventing their use of ICT is
the lack of pedagogical support. Another problem is that teachers
are still not familiar with the use of ICT in their classrooms and
technological applications are not a part of the traditional primary
mathematics classroom. The model that will be developed here
should be prescriptive and it is planned to test it with teachers in
their classrooms.
Research focus
The aim of this research is the be er understanding of teachers’
technological skills, a itudes and perceptions towards technology and
using ICT in mathematics education. The experiences and opinions
of teachers are the central. Data will be collected by questionnaires
and supplementary interviews. A further aim is to develop a model
of the learning environment in a way that combines computer tools
and resources with elements of the traditional learning environment.
The supporting system for teachers is one part of the model. The
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DESIGNING A LEARNING ENVIRONMENT FOR TEACHING PRIMARY ...
112
development process will be carried out in response to the viewpoints
of teachers.
At the outset the problem analysis has to be made. The problems are
divided into two categories according to the aims of the research.
They will be specifi ed step by step during the research process; a
typical procedure for this kind of design research (Edelson 2002;
Carr-Chellman and Savoy 2004). The problems are based on two
themes: teachers’ using ICT in mathematics education and the
development of a model of the suitable learning environment for
primary mathematics education with ICT use.
How do teachers use ICT in their mathematics lessons?
• What kind of a itudes and perceptions do teachers have toward
computers and ICT use in general?
• What are their reasons to use or not to use ICT?
• How do teachers see the role of ICT in primary mathematics
education?
• How do teachers evaluate their confi dence in using ICT?
What are the features of the suitable learning environment for
primary mathematics with ICT use?
• What are the factors of the learning environment emphasised by
teachers which are relevant in order to improve the integration
of ICT use in mathematics education?
• How is it possible to give teachers the support they need with the
help of the model?
Research subjects
As this research is focusing on teachers and teaching primary
mathematics, the role of teachers will be central. The teacher is
a decision maker who strongly infl uences the way the learning
environment of mathematics teaching will be designed. His or her
a itudes towards, and perceptions of, both mathematics and ICT
use as well as their experiences in teaching mathematics, aff ect the
implementation of mathematics education. During the research
process a survey examining the a itudes, perceptions and skills of
teachers evaluated by themselves will be completed by a sample of
teachers (N=200) (see Francis et al. 2000). All research subjects will
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be elementary school teachers with diff erent kind of a itudes, skills
and experiences. It is also important to be aware that the schools will
not be chosen according to standards of equipment or technological
culture. As the aim is gain information about the mechanisms of,
and understand the reasons behind, teachers’ actions, the interviews
will take place with those key participants (N=10) who are willing to
engage intensively in the process of development.
Validity and reliability of research
In assessing research methods it is important to uphold the concepts
of qualitative analysis: credibility, transferability, dependability and
confi rmability (Lincoln and Guba 1985; Tynjälä 1991). In the design
process of this research, issues related to validity and reliability
should be taken into consideration. Methodological triangulation is
one way to solve the problem of credibility. Triangulation is based
on the threefold aspect of observations, interviews and quantitative
data based on a survey. Data collection must be suffi ciently extended
to give a credible picture of the phenomena and to ensure that the
researcher is familiar with the research area. However, the researcher
has an active role in the process and s/he may have an eff ect on action
in the classroom which could decrease the credibility of the research.
Transferability depends on how similar the researched context and
the application context are. A researcher has to describe and report
the process carefully. By making assumptions explicit the researcher
makes it easier to evaluate the research. Dependability relates to
elements which aff ect the circumstances of the research. Few elements
may be standardised well enough to aff ect the process. However, it
is hard to standardise the way the researcher’s approach will aff ect
the a itudes and opinions of the research subjects. Confi rmability
is a characteristic of the research data as well as the researcher. The
problem with qualitative research is that there is not just one true
reality but many possible points of view. Subjectivity is a part of
qualitative research and that is why the researcher’s preconceptions
and pre-understanding should be reported carefully.
Special features related to design research must also be considered.
It is important to defi ne the research subject clearly. It has to be
clear to the reader what the real research object is. The researcher
must describe the process of data analysis and its steps delicately
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114
enough for the reader to consider the transferability of the results
to other contexts (cf. Stake 1997). Ordinary design eff orts can be
augmented to yield useful research results in four ways. Firstly, the
work has to be connected to both research fi ndings and perspectives.
Secondly, the elements of design that typically remain implicit must
be made explicit. Thirdly, formative evaluation is critical because it
can identify inadequacies in the problem analysis. Fourthly, through
the process of generalisation a design researcher develops domain
theories, design frameworks, and design methodologies (Edelson
2002, 116–17).
Present juncture of research
As the research process is in its early stages, preliminary interviews
and observations (N=3) have only just been completed. The initial
research problems and study design have been specifi ed according
to the preliminary study. The research is now more focused on
teachers’ roles in using ICT in mathematics education and on their
needs related to ICT use. The teachers, who have all been informally
interviewed and observed for two lessons, wanted to emphasise the
importance of teachers’ professional knowledge and also the need for
technological support in schools. They talked especially about a need
for pedagogical support and about the motivational problems many
teachers have. All those teachers who took part in the preliminary
interviews were motivated to use ICT in their lessons and were also
quite innovative. The main reason for this was that the teachers were
willing to engage in the research process and they believed that
improving ICT use in schools was important. As the research focus
is on teachers, it is important to complete the survey with teachers
with diff erent kind of a itudes and professional knowledge.
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Contact
Heidi Krzywacki
Department of Applied Sciences of Education
University of Helsinki
heidi.krzywacki@helsinki.fi
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Sanna Patrikainen
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Abstract
Finnish students have been successful in international comparative research
e.g. in mathematics. But mathematics education is criticised because of the
behavioristic teaching tradition. A socioconstructivistic curriculum and
the new school system for all grades from 1 – 9 also off ers new challenges
to Finnish teachers. The purpose of this study is to describe the quality of
teachers’ pedagogical thinking and action. In this research I will investigate
how teachers teach mathematics in practice and how they give reasons
for their pedagogical decisions. The aim is to make visible the didactics of
mathematics through the thinking of class teachers and to conceptualise
it. The theoretical framework of this study consists of two parts: teachers’
own pedagogical thinking and the relationship between general and subject
didactics. The six participants in this study are class teachers from the
fi rst to the sixth grades. The data has been gathered by observing and by
using stimulated recall interviews. This chapter is a short introduction
to and overview of my doctoral dissertation. First the objectives of this
study, reasons for them and the connection between the objectives and the
theoretical framework are presented. Then the methodological solutions from
this study will be discussed. The opportunities of describing teachers’ actions
and understanding and the thinking of teachers in classroom situations are
considered and presented with an example of data.
INVESTIGATING CLASS TEACHERS’
PEDAGOGICAL THINKING AND
ACTION IN MATHEMATICS
EDUCATION: THEORETICAL AND
METHODOLOGICAL OVERVIEW
Sanna Patrikainen, University of Helsinki
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Key words: classroom research, mathematical beliefs mathematics
education, pedagogical thinking, stimulated recall method video
observation
Introduction
Finnish students have qualifi ed highly in recent international
comparative research, e.g. in mathematics learning (TIMMS
1999, PISA 2000). On the other hand, mathematics education is
still criticised because of a behavioristic teaching tradition which
fosters mechanical practices instead of facilitating the development
of diverse mathematical thinking. These two extreme views are
contradictory about the quality of mathematics teaching and learning.
In consequence, learning how mathematics is taught at schools is of
great importance.
One reason for the criticism of mathematics education could be the
beliefs and conceptions within the fi eld, which have an eff ect on the
teacher’s pedagogical thinking. The current Finnish curriculum is
based on a socioconstructivistic conception of teaching and learning.
The change in curricular thinking from a behavioristic approach
to a socioconstructivistic model should also initiate change in
mathematics teaching practices which will develop along the same
lines as a teacher’s pedagogical thinking.
This chapter is based on my master’s thesis of the same name
(Patrikainen, 2001, 2003). In that thesis the quality of the class
teacher’s pedagogical thinking and action in mathematics education
was compared with the goals of the Finnish curriculum. A ention
was especially drawn to how teachers taught in practice and on what
kind of mathematical beliefs the teacher’s pedagogical thinking is
based.
According to the results of the study, all teachers shared common
traits of pedagogical action. One was to divide the mathematics
lesson into three phases: orientating, teaching and practising. There
were also some diff erences in action related to the use of teaching
methods and to the speed of progress in the teaching-studying-
learning process.
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Another important result noticed was the division of the teacher’s
pedagogical thinking into general didactical, and subject didactical,
thinking. General didactical thinking included ideas related to
maturing as a person and to the organisation of the context of the
teaching-studying-learning process. However the subject didactical
thinking model focused on the subject, and its aspects, important
skills, teaching, learning, and evaluation. The mechanistic practice
of basic counting skills and the importance of understanding were
emphasised in the teachers’ mathematical beliefs. In practice, these
beliefs occurred in partial inconsistency with each other.
Objectives of the research
The purpose of this study is to describe the quality of teachers’
pedagogical thinking and action in mathematics education. It will
investigate how teachers teach mathematics in practice and what
reasons they give for their pedagogical decisions. The aim is to
depict the didactics of mathematics through the class teacher’s own
thinking and to conceptualise it.
Class teachers are expected to have a thorough knowledge of their
subject, in addition to pedagogical skills and theoretical knowledge of
teaching and learning. A socio-constructivistic learning environment
demands closer control of discipline and its methods than a
behavioristic learning environment. In a new Finnish school system
for all classes from fi rst to ninth it is possible that subject teachers
could work as class teachers.
In the light of these new challenges it is important to investigate the
class teacher’s thinking, especially in specialised subject disciplines.
The teacher’s pedagogical thinking in mathematics itself is seldom
investigated. The way that teachers think in this area is part of what
this study tries to illuminate. It should be possible to reveal, in part,
what kind of mathematics teachers class teachers are.
Research problems
The goals of this study are approached in the form of the following
questions:
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1. How is the class teacher’s pedagogical action represented in
practice in mathematics education?
1.1 Into what kind of pedagogical episodes are the mathematics
lessons divided?
1.2 What kind of structure does the action of the individual teacher
take?
1.3 What kind of common traits and diff erences are represented in
individual teacher’s actions?
2. How is the class teacher’s pedagogical thinking represented in
practice in mathematics education?
2.1 Into what categories is the class teacher’s pedagogical thinking
divided?
2.2 In which ways are the didactical reasons and beliefs of the
subject represented in the teacher’s pedagogical thinking?
3. What set of concepts could be used in describing pedagogical
thinking and action in mathematics?
4. How does the class teacher’s pedagogical thinking and action in
mathematics compare to the thought behind socioconstructivistic
curricular thinking?
Theoretical framework
The theoretical framework of this study consists, on the one hand,
of the teacher’s pedagogical thinking and, on the other hand, the
relationship between general didactics and subject didactics. The
main concepts are discussed both at a general educational level and
from the viewpoint of mathematics education.
The teacher’s pedagogical thinking
The starting point is that a teacher’s pedagogical action is guided
by his/her pedagogical thinking which is based on a personal set of
beliefs and practical theories (see Clark and Peterson, 1986; Kansanen,
1995, 1999; Kansanen, et al. 2000; Kosunen, 1994; Thompson, 1992).
A ention is drawn especially to the mathematical beliefs of the
teacher’s thinking (see Pehkonen, 1995, 1998, 1999) and to their eff ect
on teaching situations (see Pehkonen and Törner, 1996; Lindgren,
1998; Kupari, 1999; Thompson, 1992). The teacher’s personal beliefs
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are parallel to the behavioristic and constructivistic teaching and
learning theories which also have a background eff ect on mathematics
education.
The general and subject didactical goals of the Finnish curriculum,
based on current socio-constructivistic curricular thinking, are
considered to be the ideal model of mathematics education. Changes
in beliefs and teaching practice are seen as part of a teacher’s
professional development (see Pehkonen, 1994; Patrikainen, 1997).
The relation between general didactics and subject didactics
According to Kansanen and Meri (1999), general didactics and
subject didactics are usually in opposition to each other, along with
their respective background disciplines. These writers refer to Kla i
(1994), who has wri en that the relation between general didactics and
subject didactics is not hierarchical by nature, but rather reciprocal.
Both deal with the same problems in which a certain subject has
its own characteristics. The main diff erence between them is in the
extent to which their solutions and discussions can be generalised.
In this study, teaching is understood broadly as a teaching-studying-
learning process. Kansanen and Meri (1999) stated that the elements
of the teaching-studying-learning process could be described,
according to Herbart, with the didactic triangle, in which the most
usual approach is to view the relationship between the teacher and
the students: a pedagogical relationship (see Figure 1). In a didactical
sense the student’s relationship to the content is the most essential
because the whole instructional process aims at achieving the aims
and goals stated in the curriculum. The teacher’s task is to guide this
relationship, called a didactic relationship by Kansanen and Meri,
according to Klingberg (1995). Therefore, the didactic relationship
means a relationship to another relationship. First, there is a
relationship between the student and the content, which represents
itself as visible studying and invisible learning. Secondly, the teacher
has a relationship to the relationship between the student and the
content. Kansanen and Meri (1999) emphasise that concentrating on
the relationship between the student and the content is the core of a
teacher’s profession.
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Figure 1. The pedagogical relationship and the didactical relationship
in the didactic triangle
(Kansanen and Meri 1999)
Kansanen and Meri (1999) state that it is not possible to organise
the didactic relationship universally nor to defi ne technical rules for
it. Every teacher is supposed to consider his/her own actions and
decide how to cope with the relationship to the students’ studies.
This means that every teacher has a didactics of his/her own, and,
according to Kansanen and Meri (1999), this comes close to Elbaz’s
(1983) concept of the practical theories of a teacher or Kansanen’s
(1999) teacher’s pedagogical thinking.
The research process
This study is qualitative because the aim is to fi nd out how teachers
teach, what they think about mathematics in practice, and what kinds
of reasons are given for their pedagogical decisions. The starting
point is to describe a real life situation, where the research subject is
depicted as comprehensively as possible. The purpose is to discover,
or uncover, reality, not to verify statements already known.
At least two fundamentally diff erent approaches are used in making
qualitative analysis and interpretation. The fi rst is the inductive
approach, in which the data is analysed without theoretical pre-
assumptions, and the second is the deductive approach, in which
previous theories are exploited and proved. In this study the abductive
approach is used, instead of pure induction and deduction, in which
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the researcher has a basic principle, which guides the observation.
The theory can be used as a support, not for verifi cation, but as a
source of ideas. However, this research will proceed mainly based
on the collected data, and the fi nal theory and research questions will
develop alongside the process and during its fi nal stage.
The teachers participating in the research
In qualitative research it is important to fi nd informants who clearly
display the phenomenon being researched. At times it is benefi cial to
choose only extreme examples, but ordinary cases can be investigated
as well. Above all, the researcher should fi nd the best informants for
his/her specifi ed study. In my thesis mentioned earlier the teachers
could be characterised as ‘ordinary teachers’. Only one in six had
specialised in mathematics.
In the primary research here the subjects (6) are class teachers from
all grades from fi rst to sixth. They have proved to be teachers who
actively want to discuss issues and develop themselves in their
profession. They also want to apply the knowledge they have
studied in their training as teachers to practical work and to train
themselves further as well. Through studying these teachers the aim
is to illustrate versatile mathematics education and the pedagogical
thinking behind it.
Study design
The focus of this research, mathematics education, is a qualitative
phenomenon by nature and is therefore examined mostly with
qualitative methods (see Figure 2). The research material has been
gathered by observing and by using stimulated recall interview. By
using several methods the comparability of the research tends to
improve.
One of the characteristics of qualitative research is participation,
which means that both the subjects and the researcher him/herself
are involved in the research process. When the researcher tries to
preserve the phenomenon for research as it is, without manipulation,
the intention is said to be to reach the subject’s own point of view.
In this study, the research material will be gathered in natural, real,
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teaching situations which highlight, with the methods used, the
teachers’ own points of view.
Every teacher participating in the study will be observed by
videotaping the mathematics lessons during one teaching process,
which is estimated to last two to three weeks per teacher. Diff ering
from previous research studies, mathematics education itself is
investigated in real classroom situations in which the straight and
direct information about the teachers’ actions are received. Because
the aim here is to portray the teachers’ pedagogical thinking as it
relates to teaching and learning situations and to fi nd out the reasons
teachers themselves give for their pedagogical decisions, the teachers
are interviewed a er the videotaping using the stimulated recall
method.
The data gathered will be analysed by using the method most suitable
for qualitative research. In qualitative analysis, the researcher must
consider what the best method is for her/his own study. In this
study, videotapes, which have captured the mathematics lessons,
are transcribed into wri en form as descriptions of the events in
the lessons. A set of concepts is constructed from the theory which
describe the teaching-studying-learning process. With this set of
concepts the action of each teacher is described in the form of lesson
profi les. These lesson profi les are compared with each other to
highlight the similarities and diff erences between individual teachers
and her/his peers.
Stimulated recall interviews, which represent the teacher’s
pedagogical thinking, are also transcribed into wri en form. A er
that, they are categorised and analysed from the data bases. Through
analysis, the thinking underlying the didactics of mathematics is
portrayed. Through conceptualisation, a view is constructed of how
subject didactics in mathematics are represented in the class teacher’s
pedagogical thinking.
Finally, the connection between thinking and action is analysed and
compared to the thought behind the socio-constructivistic curricular
thinking.
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Figure 2. Study design
Video observation
Observation is one of the basic data gathering methods in scientifi c
research. By observing it is possible to investigate both human
activities and the context in which these activities take place. The
other salient methods, interviewing and survey, give us information
about what people think, feel, and perceive around them. But by
observing we can make sure what really happens and discover if
people are really acting as they say they do.
Since the 1960s, the nature of observational studies has developed
from quantitative, strictly structured procedures and analyses to
relatively undetermined data gathering and analysis processes
typical of qualitative research. Over the past few decades observation
methods have improved even more because of the development of
technology. The use of video and computers has enabled investigation
of complex and layered teaching and learning situations in great
detail, along with other additional methods like stimulated recall
interview.
Characteristics of video observation
Video is a suitable tool for gathering visual and aural information
because it easily captures diverse behaviour and complex interactions
in detail. The use of video recordings also enables researchers to
re-examine data again and again and from diff erent point of view.
Pirie (2001, 346) refers to Erickson (1992, 205) who notes that video
recordings are used especially when ‘the distinctive shape and
character of events unfolds moment by moment, during which it
is important to have accurate information on the speech and non-
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verbal behaviour of particular participants in the scene’. Cobb and
Whitenack (1996) emphasise the importance of social context where
the action being observed occurs. Videotaping produces data where
the context is in evidence all the time.
Paterson et al. (2003, 6–7) mention that some researchers have
designated the use of video recordings as ‘participant observation’
because subjectivity is a characteristic aspect of videotaped data.
Video observing requires constant decision making about what
to observe and record and it produces a selective description of
the phenomenon being studied which is based on the researcher’s
preconceptions and the objectives of the study. Pirie (1996, 553) points
this out. She writes that: ‘all research is to some degree subjective. We
see what interests us; we look with a purpose. The fi eld notes we
take are already an interpretation of the phenomenon that we study.
We rationalise as best we can the value of the data we gather and the
worthlessness or irrelevance of that which we do not.’
As mentioned above one of the features of videotaped data is its
selective nature. When the data is gathered the researcher has to make
another signifi cant decision concerning the form in which the data
will be analysed. Some researchers prefer working with videotapes
and others transcribe video into wri en form. According to Pirie
(1996, 555; 2001, 349) working with either videotapes or transcripts
is not intrinsically be er or worse, but it is diff erent and each has its
merits and demerits.
Powell et a.l (2003, 410–11) remind us that it is impossible to write
an exact transcript of verbal and non-verbal interactions captured
on videotape. However it is possible to produce wri en descriptions
which are close approximations and representative enough for
particular research purposes. Powell et al. (2003, 422–23) prefer the
use of transcripts in their research because they are a permanent
record and the researcher can consider more permanently the
meaning of specifi c u erances. When the research is reported,
transcripts can also provide evidence of fi ndings in the participants’
own words. Unlike Powell et al., Pirie (1996, 555–56; 2001, 349) works
exclusively with the videos when analysing the data. In her opinion
the wri en word is less conducive to discovering new insights than
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by re-viewing videos, and once the text is categorised and analysed
it is rarely read from a completely fresh perspective. Early transcripts
can ‘cement’ the hearing of speech.
Advantages and disadvantages of video observing
According to Hirsjärvi et al. (1998, 209–10) the most important
advantage of observation in general is the opportunity to gather
immediate information about the behaviour of individuals and groups
in natural situations. Observation is a useful way to investigate, for
example, interaction or situations in which the activities are diffi cult
to foresee or rapidly changing. Paterson et al. (2003, 3–4) state that an
observer’s accounts of naturally occurring events can off er a deeper
and less limited understanding of phenomenon being studied than
relying only on participants’ explanations.
Bo orff (1994, 245–47) refers to Grimshaw (1982) who determines
the two principal advantages of using video recordings: density
and permanence. Density relates to the advantage of videotaping
over the human observer. Rosenstain (2002, 6–7) and Bo orff (1994,
245–46) state that even the video recordings are to some extent
selective; the advantage is that the camera records everything which
is within its view. Traditional observation techniques are far more
selective from the start because it is impossible for a human observer
to simultaneously perceive all the relevant cues, or to remember
most of them when the particular moment has passed. According
to Powell et al. (2003, 407) density refers also to the capability of
video to capture both audio and visual streams in real time. Quoting
Collier and Collier (1986) Rosenstein (2002, 3, 6) adds that video can
record both context and action and it is possible to study ‘what’ is
happening but also ‘how’ something is happening. As mentioned
above the data gathered can be considered more exact, reliable and
objective than notes taken by the researcher.
The other main advantage of video recordings is their permanence
(Bo orff 1994, 245–47). Permanence enables us to revisit and analyse
gathered data as many times as necessary in various ways and
from diff erent perspectives. Powell et al. (2003, 410) refer to Martin
(1999) who notes that video recordings provide the opportunity for
considered interpretations. Pirie (1996, 555) considers it an advantage
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to have other researchers watch the video as well and thereby assess
the reliability of fi ndings. Because the video recordings are permanent
data it is also possible to combine them with additional methods, for
example stimulated recall interviews, to construct a shared meaning
of the phenomena under study together with those being observed.
There are also some problems in the use of observation and video
recordings and the researcher has to be aware of them. They can be
technical or they can relate to the nature and the reliability of the
data. Problems concerning law and ethics are also characteristic of
research in which video recordings are used.
The most obvious technical problem is in the use of video camera.
According to Hirsjärvi et al. (1998, 210) observational studies are
o en criticised because the presence of the researcher and the camera
can disturb, or even change, the course of events. Eskelinen (1993)
states that the teacher tends to consider the situation as more of a
performance than usual and feel him/herself embarrassed. However,
some researchers claim that being embarrassed and not in control of
your own behaviour is only momentary (Engeström et al. 1988; Lovén
1991; Prosser 1998). The tension subsides when the videotaping lasts
long enough.
The other signifi cant issue to be considered is the selective nature
of videotaped data, which is induced by both technical reasons and
those resulting from the researcher him/herself. Bo orff (1994, 246)
lists three reasons for the notion that no record is ever complete:
First, mechanical limitations are typical of all technical equipment.
Second, with video observation it is impossible to discern the
subjective content of the behaviour being observed. And third, when
the videotaped data is used there is no sensitising awareness of the
historical context of the observed behaviours. Powell et al. (2003, 408)
refer to Hall (2000) who states, likewise Bo orff (1994), that video
data is technology laden but also theory laden. During the data
gathering process, selections concerning the phenomenon under
study are made on the basis of the researcher’s pre-understanding
and other theoretical interests. These points must also be kept in
mind while analysing the videotaped data. As Powell et al. (2003,
411) note videotaping produces a lot of data and it may be diffi cult
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to distinguish the relevant from the irrelevant. Yet, quoting Roschelle
(2000), Powell et al. (2003, 408) state that the researcher can diminish
the technical and theoretical biases of video observation with
conscious decisions based on the research questions.
The laws and ethics of video methods are also important issues to
consider here. Citing Albrecht (1995) and Rosenstain (2002, 9) reminds
us of ownership, exposure and availability of data. Videotaped data
is vulnerable and easily abused because the video records the faces,
context and interactions of those being observed. Powell et al. l (2003,
408–409) emphasise the importance of participants under observation
and being videotaped to be fully informed and to understand what
it means to participate. Further, that they realise the implications of
having their actions recorded and that they consent to the uses of the
videotaped images.
Stimulated recall method
It is not easy to reach a teacher’s thoughts by using the traditional
observation methods. It is almost impossible to believe that a teacher
could think aloud his/her thoughts while he/she is in the middle of
the teaching situation. The stimulated recall interview method is
suitable for gathering information on thinking processes during the
action, without disturbing the situation itself and for investigating
these thinking processes, which would be invisible when observed
(see Patrikainen and Toom 2004).
The basic idea of the stimulated recall method
In educational research the stimulated recall method became more
widely used during the 70s, especially the teacher-thinking research
tradition. The teacher-thinking tradition was developed as a reaction
to behaviouristic interaction analyses. In this tradition a ention is
drawn to a teacher’s cognitive thinking and his/her decision-making
processes
Bloom (1953) is the one of the fi rst users of the stimulated recall
and he describes the method as follows: ‘The basic idea underlying
the method of stimulated recall is that the subject may be enabled
to relive an original situation with vividness and accuracy if he is
presented with a large number of the cues or stimuli which occurred
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during the original situation’ Bloom himself used an audiotape when
he investigated college students’ thinking processes in two diff erent
teaching situations. Nowadays it is more common to use videotape
as stimuli. Jokinen and Pelkonen (1996) mention that stimuli can, for
example, also be photographs or drawings.
Patrikainen and Toom (2004) while investigating teachers’ thinking
and action, suggest that the data gathering process begins with
recording the particular teaching situation and the actual stimulated
recall interview is carried out a erwards. In the interview the aim
is that the teacher describes his/her thoughts and actions during the
lesson and gives reasons for them. The teacher watches the video
together with the researcher and explains what he/she is doing and
why. The researcher can also ask some corrective questions.
Advantages and disadvantages of the stimulated recall method
The stimulated recall method enables the teacher to describe his/
her own interactive thoughts as authentically as possible in natural
situations and without disturbing the action itself. The method has
several other advantages. According to Eskelinen (1991), the data
gathering is fl exible, because both the researcher and the teacher
can choose the episodes to be taken into consideration. The method
allows the teacher to make spontaneous comments and the researcher
to ask specifi ed questions and gain additional information which
might be diffi cult to elucidate from text analysis. Eskelinen (1991)
also notes that the stimulated recall method decreases the problem of
forge ing, because the teacher’s actions are seen on the video which
has recorded his or her actions. Jokinen and Pelkonen (1996) state
that the teacher cannot draw upon educational theories in reporting
his/her own thinking and action, and the video helps the teacher to
keep focussed. Therefore there will not be a heavy inclusion of data as
o en happens in qualitative research. With the help of the stimulated
recall method it is also possible to investigate the thinking and action
of a number of people in the same situation and at the same time.
Gass and Mackey (2000) state that verbal reports enhancethe analysis
of similarities and diff erences between diff erent people’s thinking
processes.
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The disadvantages of the stimulated recall method can be technical
or in the nature and the reliability of the data. The main technical
problem is the disturbance caused by the presence of the video camera.
The researcher can diminish the negative eff ects of the videotaping
by bringing the camera into the classroom so that the teacher and the
pupils can get to used to it before the real data gathering situations
occur (see Marland 1986, Alexandersson 1994).
It is more diffi cult to deal with the methodological problems
concerning the nature and the reliability of the data. In a stimulated
recall interview the teacher watches the lesson from a diff erent
perspective from that of a real teaching situation. The teacher may
be embarrassed by his/her habits and behaviour, and the ability to
refl ect on one’s own thinking and actions requires metacognitive
skills. The extent to which the teacher is aware of his/her own
thinking and has the ability to verbalise it may vary. Calderhead
(1981) states that there are some areas of knowledge, which have
become automatic, are unconscious and have never been verbalised.
According to Eskelinen (1991) it is diffi cult to separate the thought
processes stimulated by the video from real interactive thinking.
The nature of gathered data has also aroused a lot of discussion.
One criticism is that stimulated recall does not produce information
about the interactive thinking processes in the particular situation,
but displays the teacher’s general beliefs or a empts to rationalise
his/her actions and the principles of teaching and learning. Gass and
Mackey (2000) note that persons are ‘sense-making’ beings who tend
to create explanations whether or not they are reasonable. Eskelinen
(1993) adds that instead of remembering the particular thought,
an individual may just give a view about his/her general ways of
thinking and action. In addition Eskelinen (1993) states that the
problem of analysing thinking processes a er the event, is that the
la er phases of the situation have an eff ect on the recall of the earlier
ones.
Where this method has met criticism Patrikainen and Toom (2004)
see these as strengths nowadays. The criticism of the nature of the
data can be considered as a strength of the method. That the data
may include thoughts from diff erent phases of the teaching situation,
and the data gathering process, is by no means a disadvantage but a
factor in enabling a wide spread of data and its varied use.
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Video observing and the stimulated recall method and its application
to research on mathematics education
As mentioned earlier in this chapter, the main interest of this research
is to describe the quality of teachers’ pedagogical thinking and action
in mathematics education. In other words the aim is to investigate
how teachers teach mathematics in practice, and what kind of
reasons they give for their pedagogical decisions. The data will be
gathered by applying two methods: observation and interview. The
observations are conducted by using video recordings which also
are the bases for stimulated recall interviews. These methods are
chosen because the purpose is to portray the teachers’ pedagogical
thinking as it relates to concrete teaching and learning situations. By
using both observation and interview it is possible to investigate this
phenomenon from two diff erent perspectives: The researcher is an
external observer who sorts out the behaviour which takes place in
the classroom. In the stimulated recall interview the teacher has an
opportunity to uncover the situation from the participant’s internal
point of view.
The data gathering process
In this research every teacher participating in the study is observed
by videotaping the mathematics lessons during a teaching period
of around two weeks, or about ten lessons. The videotaping of one
teacher is much longer than usual in this kind of study. Because of
the longer videotaped period, there are also more stimulated recall
interviews (see Clark and Peterson 1986). Stimulated recall interviews
are carried out twice a week, so that two lessons are discussed each
time. The episodes which are used as a stimulus are not especially
chosen, but the lessons are watched in full.
It is not simple to observe such a long teaching process. It requires
quite a lot of time from both the researcher and the teacher. There is
a considerable amount of data, and a heavy workload, in managing
and analysing the data. On the other hand there are advantages in
observing the whole teaching process. The aim of this study is to
give as versatile an insight into mathematics education as possible. I
have tried to achieve this by choosing diff erent informants and also
by observing diff erent teaching situations. That is the reason why
the whole teaching process in its various phases will be observed. In
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the long data gathering process there may appear to be saturation.
The same themes will emerge again and again in the interviews and
it can be assumed that the extra data will not yield much more new
information.
The researcher’s role
The researcher’s role as an information gatherer and analyser has to be
determined by the objectives of the study and the research questions.
In addition the researcher has to be aware of his/her preconceptions:
personal values, beliefs and theoretical knowledge, all of which are
present during the research process. To ensure validity and reliability
of the research the researcher must become conscious of his/her own
role and report its eff ects in the research process.
Observation is more than just recording data from the environment.
According to Fox (1998, 2) a researcher is active, not simply a passive
collector of data. Observation does not involve only vision but all
our senses and certainly the interpretation of that sense data. An ila
(2003) reminds us that it is important to be aware of all of the channels
through which observations can be made, as there is a danger of
missing some essential information or misinterpreting the data. She
states that observation is always selective and can lead to certain
interactions or negotiation processes where external perceptions
could infl uence our already existing concepts and thinking structures.
Therefore an individual researcher experiences things in a way
which might diff er from those of other researcher. When observing,
there is always a need for analysis and understanding. A researcher
has to have some expert knowledge and background information,
which the observation itself does not directly point out. Pirie (1996,
553; 2001, 346) mentions that video recording has been claimed as
a way to capture everything which is occurring in the classroom
so allowing a researcher to postpone that moment of focusing and
decision taking. She contends that ‘who we are, where we place the
cameras, even the type of microphone that we use governs which
data we will gather and which we will lose’. In this study the main
interest is to observe a teacher’s actions which is why the video is
focused mainly on the teacher, not the students.
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INVESTIGATING CLASS TEACHERS’ PEDAGOGICAL THINKING ...
136
In the stimulated recall interview the researcher has a certain role
in an interview situation, because s/he constructs the interpretation
of a mathematics teaching-studying-learning process together with
the teacher. Patrikainen and Toom (2004) state that it is important to
consider the researcher’s behaviour in the interview, especially from
the viewpoint of the objectives of the research. The researcher needs
certain data to fulfi l the aims of the study and s/he has to defi ne
his/her role and the form of the interview so that the relevant data
can be gathered. According to Calderhead (1981), the factors which
infl uence the nature of the data are the ways in which the teacher is
prepared for the interview and how he/she is instructed to comment.
The questions that the researcher asks while watching the video also
aff ects the data.
As mentioned before, the primary aim is to look for the basis of the
mathematics teaching and to describe the didactics of mathematics
together with the concepts, which have arisen from the class teacher’s
pedagogical thinking. In this kind of research, where the contents
of thinking are defi ned, the role of the researcher might be quite
dominant in stimulated recall interview. It is not so relevant to address
the thinking processes and events a teacher him/herself points out.
More important is the discussion about the mathematics teaching-
studying-learning process in as diverse a manner as possible with
the help of the videotapes. Consequently it is necessary to be aware
which type of thinking the data contains – interactive or post-active
thoughts – or general beliefs and principles of teaching and learning.
The most important discussion in stimulated recall interviews that
about the reasons for actions. Although the teacher is asked to explain
what he/she is doing and thinking at the moment and especially to
give reasons for his/her actions, experience shows that the teacher
mostly simply describes his/her action so the researcher’s job is to
pose those important why-questions.
The data example
An example of data generated during the data gathering process
is presented below (see Table 1). The purpose is to illustrate how
the teacher describes his/her own thinking and action during the
lesson and to show how the the phenomenon under research, the
137
Sanna Patrikainen
137
mathematics teaching-studying-learning process, appears in the data
gathered with the help of stimulated recall.
In this sample the teacher discusses decimal numbers with her ten-
year-old pupils. The mathematics lesson has started with the pair
exercise. The pupils’ task was to think about which one of the two
numbers is greater and to explain their opinion. A er this the teacher
gave another exercise, where the pupils had to consider which of the
numbers on the blackboard was greater than one. Then the teacher
asked two girls to come and circle those numbers.
On the grounds of a tentative examination of the data sample, it
can be seen that the teacher gives several reasons for her actions in
this particular episode. She explains the choice of exercise from the
viewpoint of mathematics teaching and learning: In her opinion the
same mathematical content has to be considered in many diff erent
ways. The way to study this content is argued also in the light of the
motivation of the pupils. It is interesting for the pupils to follow each
other’s working whilst thinking about the exercise. In addition, this
example shows that the teacher sets great store by the safe learning
environment. This can be noticed in several other episodes and
therefore it can be considered as relating strongly to this teacher’s
conception of the mathematics teaching-studying-learning process.
Table 1. Data from research on a class teacher’s thinking and action
(Patrikainen, 2003)
Teacher’s action during
the lesson Thinking and reasons
connected with the teacher’s
action
Examination of the
data
A = teacher A
P = pupil R = researcher
A = teacher A
The varying of
the teaching
and studying
methods
Teacher A writes more
numbers for the exercise
on the blackboard.
R: And then there was another
exercise. You wrote a group
of numbers on the blackboard
and asked which of them was
greater than one. What was
your aim here?
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INVESTIGATING CLASS TEACHERS’ PEDAGOGICAL THINKING ...
138
A: Well, then I write more
decimal numbers on the
blackboard and your
task is to consider which
numbers are greater than
one.
A: Well, my aim was that the
pupils think about that kind
of group of numbers and then
they consider… they consider
the greatness or smallness
of the numbers in relation to
something. Or the same thing
but in a diff erent way.
A: Which of those
numbers are greater than
one? Greater than one.
R: Why did you ask pupils to
come and circle the numbers on
the blackboard?
P: One whole? A: Those pupils are the ones
who really study the question.
And the others, who do not
come to the blackboard, always
follow more intensively while
they think about whether the
pupils in front are doing the
task right. So, if the teacher
does the exercise, they assume
that of course the teacher circles
the right numbers.
The motivation
of the pupils
A: Yes, one whole. Which
are greater?
A: Would you come with
your partner to circle the
numbers that are greater
than one?
A: Anna? Come, please.
And bring your partner
with you.
R: Ok. And then you again
asked two pupils to circle the
numbers. Come with your pair.
Two girls come to circle
the numbers on the
blackboard, which are
greater than one. The
teacher asks the rest of
the class what they think
about the answer.
A: Yes. It is easier that way.
You can always ask your friend
and you are not suddenly there
alone, facing the blackboard
and in panic or something… or
frantic about… So if something
happens, you have always a
friend to ask ‘do you agree with
me?’
The creation of
the safe learning
environment
Conclusions
A class teacher’s pedagogical thinking and action concerning
mathematics education could certainly be investigated in other
ways from those in which it has been described in this theoretical
and methodological overview to the broader programme of
research. In spite of the disadvantages of the video observing and
the stimulated recall methods these have been chosen because they
allow investigation of a teacher’s thinking in continuous connection
139
Sanna Patrikainen
139
with the real and natural teaching-studying-learning process.
These methods also emphasise a teacher’s own perspectives and
experiences in a particular situation, which is the central focus in
the investigation of the basis of mathematics education. When data
gathering is done in a natural classroom situation, it can be assumed
that it is possible to a ain more reliable research data . The next step
then, in this research process, is to consider reliable ways to analyse
a teacher’s thinking and action and to fi nd the relevant concepts to
describe a class teacher’s understanding of the mathematics teaching-
studying-learning process.
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Contact
Sanna Patrikainen
Department of Applied Sciences of Education
University of Helsinki, Finland
sanna.patrikainen@helsinki.fi
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144
145
Alena Kopáčková
145
Abstract
Three points of view of the concept of function are discussed: 1) a
phylogenetic view based on the history of mathematics, 2) a view of school
mathematics, 3) an ontogenetic point of view based on cognitive psychology
and some results of our own research. The parallels between the phylogeny
and ontogeny of concept formation processes in relation to the concept of
function are stressed here and a ention is paid to the role of the defi nition in
concept formation. One research task (a fragment of the research on the fi eld
of functional thinking) is described. The work of some students illustrates
the conclusions.
Key words: function, defi nition, concept formation, separated
(universal) model, representation of a concept, ontogeny,
phylogeny.
Introduction
The analogy between phylogeny and ontogeny of many phenomena
and processes is very well known in biology and psychology. The
idea that some parallels between ontogeny and phylogeny of concept
formation processes in the didactics of mathematics also exist, is not
old. It was noted in the 70s by P.M. Erdnij ev for the fi rst time, and
was brought into Czech and Slovak didactics of mathematics by Vít
and Milan Hejný. They called this principle onto-phylo parallelism
CONCEPT FORMATION OF THE
CONCEPT OF FUNCTION
Alena Kopáčková, Technical University of Liberec,
Charles University of Prague
146
CONCEPT FORMATION OF THE CONCEPT OF FUNCTION
146
(Hejný and Kuřina 2001, 77). Our research into concept formation of
function with Czech students (especially those over 15 years) arose
from this belief and fully justifi ed this idea.
Phylogeny of function
Development of functional thinking
The fi rst evidence of functional feeling and the mathematical
expression of dependencies came from Babylonia in the 2nd to 1st
millennia B.C. The tables of functions, which the old Babylonians
had used in astronomy, agriculture, building and trade (e.g. functions
like,
), survived. The Greeks in the sixth to the fi h centuries realized the
diff erence between the discrete and continuous variable, cultivated
infi nitesimal speculations, studied the concrete changes of the
variable quantity (especially motion) whilst geometers dealt with
general curves and tried to describe the kinematical rules of their
genesis. In spite of this, the idea of a general function or a general
variable occurred nowhere.
The idea of function only arose in the middle ages. The scholastics
Robert Grosseteste (about 1168–1253), Thomas Bradwardinus
(about 1290–1349), Nicole Oresme (about 1323–1382) and Richard
Swineshead (14th century) at the universities of Oxford and Paris,
studied and described the natural processes and motions and
speculated about the continuum. Functional dependency was
described by means of some rule given by words or in a graphical
way. The conception of natural laws like functional dependencies
gradually grew among scholars and diff erent theories of the change
of variable as the function depending on time, emerged.
In the 17th century, the concept of natural causality developed into the
concept of functional dependency. Newly created analytic geometry
(Pierre de Fermat 1601–1665, René Descartes 1596–1650) enabled the
description of dependency between two variables by means of the
equation with x and y on the base of the coordinate system. Isaac
Newton (1643–1727) and Go fried Wilhelm Leibniz (1646–1716)
147
Alena Kopáčková
147
created the basis of diff erential and integral calculus without any
exact delimitation of the concept of function. Newton and Leibniz
used diff erent contexts for their considerations; Newton’s ideas arose
from the description of the motion of a mass point, while Leibniz
was inspired by looking for the tangent line to a curve.
Leibniz fi rst used the word function in 1673 (in a diff erent meaning
from that usually used – function was closely tied to a tangent line of
a curve and it was represented by an abscissa, a segment of a tangent
line (Juškevič 1970, 144–45). Leibniz’s understanding of the new
mathematical word corresponded well with the Latin sense of the
word: fungor means to realize, to execute or to be active and functio
means activity, practising.
The concept of function
Johann Bernoulli defi ned the concept of function for the fi rst time in
Mémoires de l´Académie des Sciences de Paris as follows:
Function of the variable quantity is a quantity composed any way
from this variable quantity and constants (“On apelle ici fonction
d’une grandeur variable, une quantité composée de quelque maniére
que ce soit de ce e grandeur variable et de constants.”)
(Cantor 1901, 457; Youschkevitch 1976, 60; Bos 1975, 10.)
This defi nition represented the fi rst a empt to introduce the new
concept into mathematics. What functions did mathematicians in
the 18th century have in mind? Function was mostly understood
geometrically as a curve (usually one which can be sketched by one
uninterrupted motion of a hand) or geometric-kinematically as a
trajectory of a moving point.
Let us summarize briefl y how the concept of function developed
further. Leonhard Euler (1707–1783) defi ned function as an analytical
expression in 1748 and as a dependence of some quantity on the
other(s) in 1755. Sylvestre Lacroix (1765–1843) took the same view
in 1797 and Jean Antoine de Condorcet (1743–1794) in 1778. Joseph
Fourier (1768–1830) saw function as a more general correspondence
between quantities in 1822. Let us remark that Euler took the function
148
CONCEPT FORMATION OF THE CONCEPT OF FUNCTION
148
x
x1
→ for continuous everywhere because it can be described by only
one formula, whilst the function , which is described by means
of two formulas (if x > 0) and (if ), was for him
discontinuous. Many mathematicians a priori understood function
in the 18th century as a changing variable (e.g., Euler explicitly
rejected taking a constant for a function). Function was an explicit
(unique) assignment for Nikolaj Ivanovic Lobacevskij (1793–1856) in
1834 and for Peter Gustav Dirichlet (1805–1859) in 1837. The term
mapping was used by Richard Dedekind (1831–1916) in 1887, by
Georg Cantor (1845–1918) in 1895 and by Constantin Carathéodory
(1873–1950) from 1917 without no connection with the concept
of function defi ned before. The structuralistic conception of the
function appeared at the turn of the 19th and 20th centuries with
Giuseppe Peano (1858–1932) and the Nicolas Bourbaki group (1939):
function was defi ned as a set of (ordered) pairs or a special binary
relation. Finally, it is interesting to note the suggestions of logicians
to introduce function as a primitive concept (for example, Go lob
Frege (1848–1925), Alonzo Church (1903–1995), Friedrich Wilhelm
Schröder (1841–1902), Bertrand William Russell (1872–1970))
without defi nition through examples and historical excurses, in the
fi rst half of the 20th century. Logicians censured the large numbers of
undetermined terms in the classical defi nitions of function (such as
rule, formula, assignment, correspondence, relationship, operation,
quantity, variable, set, etc.) and tried to reduce them or to give up
the a empt to create pure mathematical defi nitions consistent with
mathematical theory.
Let us summarise the features of the phylogeny of the concept of
function to clarify the ontogeny of the concept:
• The concept of function developed gradually from natural to
abstraction, the formulation of the defi nition crowned a long
period of hard work by mathematicians on many concrete
examples of functions.
• Until the beginning of the 19th century, mathematicians usually
considered a function as a smooth continuous curve, which
was not constant. The function was expressed by means of an
analytic expression (in a more general sense than in school
mathematics today).
149
Alena Kopáčková
149
• The concept of function initially always corresponded with the
requirements of concrete real-life or scientifi c problems. Let
us underline the procedural character of the initial concept
of function and its connection with a change, especially in its
dependency on time. It implied the continuity and smoothness
of a function and usually also the no negativity of the argument.
The phenomenon of dependency and variability dominated at
fi rst, the uniqueness of the dependent variable emerged in later
defi nitions of function.
Function in school mathematics
School mathematics took much longer to accept the idea of function
in comparison with mathematics-science. The idea of function
spread in European schools from the beginning of the 20th century,
thanks to Felix Klein (1849–1925). Nowadays there are two essential
concepts of function in Czech schools:
• The classical approach to the concept, where the function is
defi ned as a rule or formula (in Euler’s view) or as a variable
quantity (in Bernoulli’s view) or as an assignment (according to
Lobacevskij or Dirichlet). The general concept is fi rst introduced
to Czech students at about the age of 15.
• The structuralistic approach to the concept, where the function is
defi ned as a set of ordered pairs, a mapping (from) one set into
another set (o en in the hierarchy of concepts: Cartesian product
of two sets – binary relation – mapping – function) by Peano,
Bourbaki, Dedekind, Cantor. This is how function is defi ned
mostly at universities and high schools.
The Czech curriculum follows the spiral principle of the arrangement
of topics, so the defi nition of function is repeated again at most
secondary schools a er being taught to students at secondary school
at age 15 for the fi rst time. Our research classifi ed about thirty
defi nitions of function. The system of phenomena relates not only to
Czech mathematical terminology, but also to the particular language
of Czech thus we will not discuss this here.
150
CONCEPT FORMATION OF THE CONCEPT OF FUNCTION
150
Ontogeny of the concept of function
The cognitive psychology of Piaget and Smith, the constructivism
in didactics based on the works of Tall and Vinner, 1981, Tall and
Grey 1985, Hejný and Kuřina 2001 and onto-phylo parallelism (see
earlier) are the theoretical bases of our reasoning. The concept is the
building block of cognitive structures. It is the essential unit of any
exact science, particularly mathematics. Milan Hejný (Hejný, 1990)
introduced the following model describing the concept formation of
an individual into Czech and Slovak didactics
motivation (syncretic stage)
separated models
universal model
discovery (understanding) of the concept
The constructivist stream in didactics is based on a belief that the
concept is not copying an ideal ( indeed constructivists believe that
no ideal concept exists). The understanding consists for them in
representations and individual constructing of the concept in the
mind of any student. From an ontogenetic point of view, we can
assume that a student reaches the stage of motivation (syncretic
stage) in the fi rst school years and even earlier. The feeling for
dependencies or functional thinking is developing at this stage. The
diff erent concrete examples of functions (e.g. linear, quadratic, direct
and indirect proportionality, goniometric functions in a triangle) with
their properties, relations and graphs are being taught in the early
years of secondary school and students begin to create the separated
models of the concept of function in their minds. The students
probably should have started creating the universal model of the
concept of function as they are learning the fi rst general defi nition.
Research
Our research in 1998–2003 included almost 700 Czech students (from
school and university, aged 15–22). The research combined both
quantitative and qualitative approaches. The quantitative approach
was used more in the fi rst phases of the research, when we discovered
the diff erent phenomena related to the ontogeny of the concept of
function. The qualitative approach was used in detailed analysis of
the work and responses of some individuals. The main methods of
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Alena Kopáčková
151
investigation were qualitative analysis of individual works, and
comparison and classifi cation based on our own phenomena We
used wri en tests, interviews, group discussions, and observations
as the essential research tools.
Research questions
• Do students create the universal model of representation of the
concept of function in their minds a er they meet the concept of
function in school? What position does the defi nition of function
have in concept formation?
• What kind of phenomena can we observe in connection with the
general concept of function of the students? What kind of obstacles
do students have?
• What causes problems in students’ understanding of the concept
of function? How do the diff erent representations of function
infl uence the understanding of the concept?
• Is it possible to re-educate students who meet obstacles in
understanding functions?
• Is the present way of introducing functions in school acceptable
and consistent with the conclusions of cognitive psychology and
phylogeny of the concept?
Results
Let us conclude the essential results of the research and then describe
one concrete research tool.
• The defi nition of function is a formality for the majority of students;
the students do not derive their images of function from the
defi nition; the mental representations of the concept of function
are independent of the defi nition.
• Students very o en take the concept of function for a set of concrete
examples more than a general category.
• The phenomena closely connected with function for students
are continuity, symmetry and regularity of some kind (given
for example by periodicity), and the existence of one analytical
expression, or sometimes smoothness of the graph. Many students
make a close connection between the concepts of function and
some noticeable change and dependency on time; many of them
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CONCEPT FORMATION OF THE CONCEPT OF FUNCTION
152
classify the examples by visible similarity to a well-known
model.
• The requirement of the explicitness of the dependent variable of a
function does not seem cardinal for many students.
• The phenomena very o en closely connected with “nonfunction”
are discontinuity, discreteness (even some university students
do not regard a real sequence as a function) and constantness. In
addition, irregularity in any sense (e.g. untypical location of the
graph and incompleteness) leads to the decision: “The relation is
not a function”.
• Some properties are taken as necessary (e.g. continuity, regularity,
variability), some as suffi cient (e.g. similarity with the well-
known model). Some properties are necessary and suffi cient
– e.g. existence of the analytical expression, existence of a graph:
“No, it is not a function because it is not possible to express it by
some formula” (necessity) or: “Yes, it is a function because it is
possible to express it by a formula” (suffi ciency).
Role of the defi nition
Let us direct our a ention to a concrete problem that we were
investigating. Two instructions were given with two coordinate
systems in the task: “Defi nition of the concept. Graph of the
(non)function”. More than 250 students solved the following:
1) Defi ne (describe in your own words) a function.
2) Sketch a graph of some function in the fi rst coordinate system and
“something” which is not a function in the second system.
The objectives of this task were to identify the stage of the
concept formation of the function and to fi nd out if there exists a
correspondence between the concept and its graphical representation.
In the terminology of Tall and Vinner it is the relationship between
concept image and concept defi nition (Tall and Vinner, 1981).
Analysis of the fi rst problem – defi nition of the concept
The analysis split into two main parts. Our initial intent to divide
the defi nitions of function by classifying them as: correct/incorrect
quickly became impracticable. First, there were only a few correct
defi nitions and the formulations were very diff erent, incomplete and
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Alena Kopáčková
153
confused both grammatically and semantically. Next, we recognised
the absurdity of such a classifi cation. A student’s defi nition that was
almost correct at fi rst sight was o en immediately followed by an
example that contradicted it. The diff erence between knowing and
understanding the defi nition was obvious. We decided to look for the
substance of the response, to try to guess what the author had wanted
to say. As the result of the lengthy comparative analysis, we created
a categorization based on twelve categories. The categorization
(abbreviated in comparison with the original Czech version) appears
in Table 1, below.
Category Description
(main features of the
category)
Concrete examples
(students’ responses)
Rule
Formula Function is defi ned as
a formula where the
existence of the analytic
expression is not required
explicitly.
The approach to the
concept is mostly
conceptual; however, it
can also have procedural
a ributes.
„Function – is a rule
which to any x from the
domain assigns one y.”
Mapping Function is defi ned as a
mapping.
This defi nition is
conceptual.
It occurred only withat
some university students
„Function is a mapping in
the set of real numbers.”
Relationship, dependence
between variables The relationship or the
dependence need not
be specifi ed; sometimes
the assumption of the
analytic expression is
obvious.
„Function = the
relationship between
variables x and y, e.g.: y =
x3, y = 2x + 1.”
Variable
Value
Set of values
Function is taken for
a variable quantity;
however, the term
quantity alone did not
occur.
„Function is some
variable that consists
from the ordered pairs
x, y, where for any x is
ordered one y.”
„Function is a set of
values…H and R.”
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CONCEPT FORMATION OF THE CONCEPT OF FUNCTION
154
Equation The word equation is here
a synonym of the word
(analytic) expression. This
defi nition has obviously
conceptual character.
“„Function is an equation
by thatwhich the
points of a graph are
calculated.”
„Function is a formula
(an equation).”
Solution of the equation
Result The result of the equation
is here stressed in the
comparison with the
previous category. Strong
conceptual character.
“„Function = solution
of the equation
(inequality).”
„Function – result… it is
possible to obtain it by
means of a graph or by a
calculation.”
Curve
Graph
Line
Set of points
Function is identifi ed
with a graph or a set
of points (which are
expressed graphically).
The explicit expression of
the relationship between
variables is missing here.
„Function is a graph
(straight line, curve).”
„Function is a line
passing the axis x a y.”
Graphical representation
of the dependence Function is represented
graphically; the
connection with an
equation or analytic
expression is here
obvious here.
Conceptual character of
the defi nition.
„Function = set of points
satisfying to some
equation, it is given by
means of a graph.”
„Function is a curve
made from the points by
the given equation.”
Action
Activity Function is understood as
some activity. The nature
of the concept is not
evident.
The procedural
defi nition.
„Function works,
regularly.”
„Function is a certain
action, which is going
on and we can see on the
graph how it is going
on.”
“„... if, then ...” The concept of function is
described but the nature
of the concept is not
obvious. The defi nition
can be tied in with both,
analytic formula and
graph.
“„Function is if we assign
one real number to one
number of the set X.”
„Function is if we make
the parallel line down
and the points are not
over them, but they can
be beside them.
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Alena Kopáčková
155
Epiphenomena,
properties Only some
epiphenomena of the
concept and properties of
some function are listed
here. The nature of the
concept is not obvious.
“„Function is a concept
that must have domain
and range.”
„Function has got the
values on the axes x and
y, it can have a maximum,
it can be one-to-one, …”
„It is possible to express
the function by the graph;
it can be odd, even,
bounded above or from
below.”
Separated models,
examples Here there are only
concrete examples
of function (by an
abbreviation, a formula,
a graph).
The concept of a function
is represented by a fi nite
set of concrete models.
„Function = sin, cos, tg,
cotg.”
„What is a function…y
= x2.”
„Function is a set of
points, it is created by
straight line, hyperbola or
parabola.”
Table 1. Classifi cation of the defi nitions of function given by
students
We compared our conclusions with the categorization of the
defi nitions described in Karsenty and Vinner (2002). They classifi ed
the responses of individuals by means of the classifi catory tool
that Dreyfus and Vinner prepared in 1989. Although their sample
of respondents was diff erent from ours and the researchers had
diff erent objectives (they studied mechanisms of evoking old school
knowledge) they described similar experiences to ours. The concepts
of function of Israeli adults were very close to those of our Czech
students (independent of the defi nition, which the respondents had
learned in school many years ago). The authors mentioned the same
diffi culties of classifi cation as we had and they too decided against
the a empt at sorting into correct/incorrect.
Defi nition and graphical representation of the function
The goal of this experiment was both to analyze the students’
“defi nitions” of function and characterize typical examples of
the concept and contra-concept, and to investigate how much the
students understood their own defi nitions . We wanted to recognize
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CONCEPT FORMATION OF THE CONCEPT OF FUNCTION
156
if the defi nitions were only pseudo-knowledge (knowledge without
understanding). We were interested to fi nd out if the students based
their ideas about function on their defi nition. The combination of
the fi rst and second tasks of the experiment helped us to answer
this. We frequently found the defi nition mastered but without
much comprehension (graphs o en in visible disharmony with the
student’s defi nition ).
Illustration of selected students’ work
Here we cite some student descriptions of a concept of function
and observe if there is a correspondence between the “defi nition”
of function given and its graphical representation. (The students are
not identifi ed by their real names. The fi rst graph of the pair always
stands for a function, whilst the second should be a “nonfunction”.)
Alex (15 years old) does not equate his second graph with his own
defi nition. His example of a function is a very “typical” representative
of a function, whilst he proposed a horizontal straight line as an
example of a “nonfunction”. We mentioned that the concept of
function was primarily associated with some change and that the
constant was not taken for a function. The same approach
was taken by Euler.
Figure 1. “A function is a graph
(straight line, curve).”
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Alena Kopáčková
157
Figure 2. “y = f(x) Mapping, which to x ∈ M, assigns a value f(x) ∈
M, where M is a set of numbers.”
Even a quite well formulated defi nition does not guarantee the
understanding of a concept. Be y (22 years, 3rd year student of
the Faculty of Education, future teacher of mathematics) defi ned
function as a mapping. She probably did not understand the concept
of mapping very well because she sketched a graph of a real sequence
as an example of a “nonfunction”.
17-year-old Cecil understands a function as some regularly working
object; however, she does not say what it is. Cecil probably takes
a function for an activity, or a process; she looks at the concept
procedurally. The appropriate representative of such a “regularly
working” function is for her a graph of a periodical function sin;
whilst the graph of
Figure 3. “Function works, regularly, it is
possible to solve. The function itself can
be solved by known rules.”
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CONCEPT FORMATION OF THE CONCEPT OF FUNCTION
158
a constant function represents for her something “irregular and not
working”. In the second graph the domain of the constant function
is visibly bounded, which should probably underline the non-
functioning and irregularity.
An imperfection and torso (probably also a discontinuity) were the
reasons why 19-year-old student Dan decided to sketch his graph of
“nonfunction”. He thought that the initial graph of a linear function
became the graph of a “nonfunction” by an omission of a part of a
straight line. He understands a function as a process.
Figure 4. “A function is a certain activity,
which is running in some form
and then we can visually see how it is running.”
Conclusion
We have studied the role of defi nition in concept formation by
students of diff erent ages. We saw that defi nition did not play a
large part in the ontogeny of the concept of function. The research
demonstrated that the majority of students who had met the general
defi nition of function in school at least once, had not been able to
create the universal model of function. False conceptions have
common features regardless of the age of the student (although the
number of false conceptions decreases with age). The comparison of
some student “mistakes” with the phylogeny of the concept leads us
to the idea that many of those “mistakes” are natural.
We see the following as reasons for misunderstandings of the concept
of function: historical parallels, the nature of the concept itself, vague
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Alena Kopáčková
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teaching in school of the defi nition, the way the topic “Functions”
is introduced to the school curriculum, and formalism of teaching
and learning mathematics. Each is worth studying. The results of the
research suggest it could be necessary and sensible to introduce the
general defi nition of function into the secondary school curriculum
(at age 15). If we want our students to build a be er understanding of
the concept of function we must expose them to a suffi cient number
of concrete examples. These should include empirical functions (i.e.
not to reduce the stage of separated models of representations),
solve not just standard and typical problems and use all the ways of
describing functions (graphs, tables, formulas, words). The functions
should permeate throughout the school curriculum and not only be
part of one isolated school subject.
References
Bos, H. J. M. (1975) Diff erentials, Higher-Order Diff erentials and the
Derivative in the Leibnizian Calculus, in Arch Hist Exact Sci, 14, pp.
1–90.
Bourbaki, N. (1994) Elements of the History of Mathematics, Berlin,
Heidelberg: Springer-Verlag.
Cantor, M. (1900, 1901) Vorlesungen über Geschichte der Mathematik
– Band 2, 3, B.G. Leipzig: Teubner.
Church, A. (1956) Introduction to Mathematical Logic, New Jersey:
Princeton University Press.
DeMarois, P. and Tall, D. (1999) Organizing Principle or Cognitive
Root? in Proceedings PME 23 (Vol. 2). Haifa, pp. 257–64.
Edwards, C.H. Jr. (1979) The Historical Development of the Calculus
Heidelberg, Berlin, New York: Springer-Verlag.
Even, R. (1990) Subject Ma er Knowledge for Teaching and the Case
of Functions, in: Educational Studies in Mathematics 6 (Vol. 21), pp.
521–44.
Even, R. (1993) Subject-ma er Knowledge and Pedagogical Content
Knowledge: Prospective Secondary Teachers and the Function
Concept, in Journal for Research in Mathematics Education 2 (Vol.
24). The National Council of Teachers of Mathematics, Reston, pp.
94–116.
Hejný, M. (1990) Teória vyučovania matematiky 2, Bratislava: SPN.
Hejný, M. and Kuřina, F. (2001) Dítě, škola a matematika.
Konstruktivistické přístupy k vyučování Praha: Portál.
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CONCEPT FORMATION OF THE CONCEPT OF FUNCTION
160
Juškevič, A. P. (1970, 1972) Istorij a matematiki 2, 3, Nauka: Moskva.
Karsenty, R. and Vinner, S. (2002) Functions, Many Years A er
School: What do Adults Remember? In Proceedings of the 26th
Annual Conference of the International Group for the Psychology of
Mathematics Education (Vol. 3), Norwich, pp.185–92.
Kopáčková, A. (2001a) Fylogeneze pojmu funkce, in Dějiny matematiky,
Vol. 16, Matematika v proměnách věků, Prometheus Praha, pp. 46–80.
Kopáčková, A. (2001b) Some Aspects in Relation to the Concept
of Function at School, in Proceedings of The 8th Czech – Polish
Mathematical School, UJEP, Ústí nad Labem, pp. 109–14.
Kopáčková, A. (2001c) Understanding of Functions, in Proceedings
of The First International Conference on Applied Mathematics and
Informatics at Universities, Gabčíkovo, SR, pp. 326–29.
Kopáčková, A. (2002) Nejen žákovské představy o funkcích, in: Pokroky
matematiky, fyziky a astronomie, 47/2, Praha, pp. 149–61.
Kopáčková, A. (2003a) How Not Only Czech Students Think About
Functions, in Mathematics Education (Reviewed Papers), UK/ Praha:
University of Warwick, pp. 47–52.
Kopáčková, A. (2003b) Thesis – Pojmotvorný proces konceptu funkce.
UK/Praha. (Unpublished.)
Schliemann, A.D., Caraher, D.H. and Brizuela, B.M. (2001) When Tables
Become Function Tables, Proceedings of the 25th Annual Conference of
the International Group for the Psychology of Mathematics Education
(Vol. 4), Utrecht 2001, pp. 145–52.
Smith, E.E. (1991) Concepts and Thought, in The Psychology of Human
Thought, Cambridge: Cambridge University Press.
Tall, D. and Gray, E. (1985) Symbols and the Bifurcation between
Procedural and Conceptual Thinking, Canadian Journal of Science,
Mathematics and Education 1, pp.80–104.
Tall, D. and Vinner, S. (1981) Concept Image and Concept Defi nition
in Mathematics with Particular Reference to Limits and Continuity,
Educational Studies in Mathematics, 12, pp. 151–69.
Youschkevitch, A.P. (1976) The Concept of Function up to the Middle of
the 19th Century, Arch Hist Exact Sci, 16, pp. 37–85.
Contact
Dr Alena Kopáčková, KMD
Fakulta pedagogická
Technical University of Liberec
Charles University of Prague
alena.kopackova@vslib.cz
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Lucia Ilucová
161
Abstract
Aims and partial results of a pilot study on the possible implementation
of polygonal tessellations in mathematics education are discussed in this
chapter. The fi rst part covers the historical background, background to the
topic of tessellation, and the aims and methodology of the research. This is
illustrated by interviews in which selected real life problems were solved,
and pictures of polygonal tessellations created, by pupils.
Key words: imagery, interior angles in polygons, polygonal shapes,
quadrilaterals, tessellation
Introduction
A tessellation of a plane is a family of sets – called tiles (or cells)
– that cover a plane without gaps or overlaps (Grünbaum and
Shephard, 1977). Tessellations are also described as tilings, pavings,
parque ings or mosaics .
Figure 1. Samples of Moorish tessellations
TESSELLATIONS BY POLYGONS IN
MATHEMATICS EDUCATION
Lucia Ilucová, Charles University of Prague*
162
TESSELLATIONS BY POLYGONS IN MATHEMATICS EDUCATION
162
The word tessellation is derived from the English verb tessellate,
which comes from the Greek word τέσσαρες (tessares) meaning
‘four’. The Greeks called the die τέσσερα because any side of it has
four edges. The Latin word for die is tessera and its diminutive is
tessella or tesserula. In Webster’s New Collegiate Dictionary (1989),
the word ‘tessera’ is used for a small tablet of wood, ivory, etc. used
as a token, ticket, or tablet in ancient Rome.
The art of tessellation has been known since the origin of the history
of civilization. When people began to build houses and fortifi cations,
they tried to fi ll space or planes. The fi rst houses, churches and castles
were built of broken stones which created random tessellations. Later,
arrangements of prismatic stones and bricks frequently took on a
more regular pa ern. In Gothic architecture, stained glass windows
(mosaics with colored glass pieces connected by lead strips) are
seen.
Diff erent sorts of tilings, complicated pa erning of ceilings or
walls with tiles, or paving of fl oors, are examples of tessellations
documenting human creativity.
Every known human society made use of tiling and pa erns in
some form. Portraits of people and scenes of nature o en occurred
on intricate mosaics from the Mediterranean region. On the other
hand, the Islamic religion strictly prohibits depictions of humans
and animals which might result in idol-worship, so that Moors and
Arabs created art utilising a number of primary forms: geometric,
arabesque, fl oral and calligraphic. Moorish architecture in Spain and
Islamic culture in the Middle East have shown graceful examples
of planar tessellations with rich ornaments. The Alhambra Palace in
Spanish Granada is one of the best-known and best-preserved relics
of the Islamic civilization on the Pyrennean peninsula.
Planar tessellations also occur in the modern art of the twentieth
century, namely in creations of the Dutch artists M.C. Escher (1898–
1972) and P. Mondrian (1872–1944). In fact, Escher was fascinated and
deeply inspired by Alhambra’s mosaics and later used some motifs
derived from them in his prints. He utilised mathematical imagination
as a tool for his graphics; he o en fi lled planes with fi gures of people
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Lucia Ilucová
163
and animals (Eight Heads, 1922, Day and Night, 1938) or connected
planes and space (The Reptiles, 1943). When speaking about his
work (Ranucci and Teeters, 1977) he said: ‘Although I am absolutely
without training or knowledge in the exact sciences, I o en seem to
have more in common with mathematicians than with my fellow
artists.’ Mondrian o en used nets and grids to depict reality. In this
way he tried to avoid diff erences between fi gure and background,
between mass and antimass (Composition with Net 8, Chessboard
with Dark Colors, 1919; Broadway Boogie Woogie, 1942–1943)
(Golding, 2000).
The fi rst serious mathematical study of tessellations, Harmonices
Mundi, was wri en in 1619 by J. Kepler. He described the geometric
properties of planar tessellations by regular polygons. However, his
contribution to astronomy was so monumental that his geometric
investigations were largely forgo en for almost 300 years.
In 1975, papers about planar tilings formed by pentagons were
published in Scientifi c American. It has been an interesting problem
for mathematicians because there has not been any rule for their
general construction. M. Rice, a San Diego housewife with no formal
education in mathematics but with enormous enthusiasm, started
to discover new and until then unknown types of pentagons tiling
the plane. Recently B. Grünbaum and G.C. Shephard elaborated the
detailed theory of planar tessellations and symmetries of pa erns
(Grünbaum and Shephard, 1987).
Some part of the mathematical theory concerning tessellations is
elementary, but it contains a rich supply of interesting and surprising
problems which can be solved by pupils of various ages.
Mathematical background
Here I will discuss the problem of some types of planar tessellations
formed by polygons.
Let us consider the three following mathematical statements which
are important for this research (the fi rst is part of the mathematics
curriculum and is given without proof here):
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TESSELLATIONS BY POLYGONS IN MATHEMATICS EDUCATION
164
Statement 1: The sum of the measures of all interior angles in every
quadrilateral is equal to 360°.
Statement 2: Every quadrilateral tessellates the plane by itself.
(The existence of tessellations built up of the arbitrary identical
quadrilateral tiles is an example of this statement.)
Proof:
Let α, β, γ, δ be the interior angles of a general quadrilateral. The sum
of their values is equal to 360˚. Every quadrilateral can be arranged in
a tessellation as shown in the following picture:
Four quadrilaterals meet at each vertex and the corresponding
interior angles form together a 360˚ angle, so that there are no gaps
and overlaps. (The same statement also holds for triangles, whereas
only some specifi c types of other polygons tessellate the plane by
themselves.)
Statement 3: Only three regular polygons tessellate the plane by
themselves: triangles,
squares and hexagons.
Proof (for equilateral triangle):
The value of all interior angles of equilateral triangles is equal to
60˚. Six triangles meet at each vertex and six corresponding interior
angles form the angle of 360˚ (= 6. 60˚). Thus there are no gaps and
overlaps.
Proposed research questions
When looking for possible research questions, I was able to draw
on my previous experience because I had already worked with two
pupil groups (14/15 years, 15/16 years) on problems concerning
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165
tessellations (in 2002 and 2003). Both groups were large and,
moreover, several other problems had to be covered in the lessons;
so that there was not suffi cient time for a more detailed examination.
However, this experience served as a basis for the present qualitative
experiment.
In the research I indicate cognitive and interactive phenomena which
can be divided into fi ve groups. They can be characterised as the
answers to the following questions; some of them have a wider scope
and do not concern only the area of tessellations.
Cognitive phenomena
1. Means of expressions and bilingualism
What verbal and nonverbal communication tools do children use?
How successful is the communication between children forming
pairs and between pairs and experimenter?
How do children manage to overcome bilingualism? (The interviews
were held in Czech – solvers – and Slovak – experimenter; these
languages are close enough but there are some diff erences, e.g. in
the terminology of quadrilaterals: square – čtverec/ štvorec, rectangle
– obdélník/ obdĺžnik, general – obecný/ všeobecný)
2. Imagery
The imagery could be defi ned for my research as ‘... the basic psychic
function, which is important for the psychic visualization of the
events that are not actual, in the constructive and reconstructive
meanings ... (Půlpán et al., 1992)
Which tools do children choose to solve proposed problems?
What strategy do they use – what do they do?
Do they improve their strategy by using their previous experience?
Do they improve their strategy if they have been unsuccessful?
How many diff erent arrangements of tiles and pa ernsdo they
manage to fi nd?
3. Mathematical content
Which mathematical concepts do children use? (The terminology of
quadrilaterals, the properties of quadrilaterals and angles ...).
Do they understand them?
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TESSELLATIONS BY POLYGONS IN MATHEMATICS EDUCATION
166
Do they fi nd out if and why every quadrilateral can be used as a
tile?
4. Proofs and justifying
Do children justify their answers?
Do they feel the need of a proof?
How do they prove their statements?
Interactive phenomena
5. Interpersonal relations
How do children work together?
What are their mutual relations as partners in looking for the solution
of the problem?
How do they react when they have not been successful and are told
that the problem does have a solution?
The relation between imagery and tessellations (2 above) seems to
be the most interesting in this research; and the question of whether
children fi nd out if and why every quadrilateral can be used as a tile
( 3 above) is also important .
Methodology
Interviews
Semi-structured clinical interviews with pairs of pupils carried out
separately were chosen for this research. Communicative children
were not interrupted, otherwise supplementary questions were
posed (why did you do that? can you explain it?). For subsequent
analysis, interviews should be recorded; a video camera and voice
recorder are essential.
The tools for solving the problems are set up on the table and none
of them is given preference (the children choose convenient tools
for their work by themselves). The tools can be divided into two
groups: the main tools (coloured, white and squared A4 format
paper, ‘the building set’, pencils) and the supplementary tools
(scissors, ruler, coloured pens); ‘the building set‘ is my working
name for the set of paper models of quadrilaterals. It consisted of
10–14 paper pieces for all selected types of quadrilateral (square,
rectangle, parallelogram, trapezoid, general convex – trapezium
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Lucia Ilucová
167
– and nonconvex quadrilateral) and was made in cardboard by the
experimenter before the interviews.
Quadrilaterals have been chosen as tiles because some of tessellations
formed by them are the most frequently occurring in real life (square
tiles, rectangular pa erns) and the mathematical background
concerning quadrilateral tessellations can be easily discovered by the
majority of children. Five types of quadrilaterals in ‘the building set’
are those which are the most frequently handled in school geometry,
only the sixth one (a general nonconvex quadrilateral) is uncommon.
(Operations with triangles and regular types of other polygons are
made up on similar principles.)
Manipulations with models of quadrilaterals proceed step by step
from the simplest shape – square – to general convex and nonconvex
shapes. Thus pupils can repeat the hierarchy of the quadrilaterals or
start with its understanding. (When solving problems children also
observed the properties of the quadrilaterals – length of the sides,
measure of the interior angles, parallelism of the opposite sides, and
so on.)
Figure 2. Quadrilaterals forming ‘the building set’
The problems presented are formulated as real life questions that can
be answered by YES or NO and should be substantiated, e.g. by a
picture or a mosaic formed by the paper models of quadrilaterals.
Problems:
1. When tiling the wall in the bathroom, can we use identical tiles of
a square shape?*
2. Can we use identical tiles of a rectangular shape to cover the
fl oor?*
3. Can we use identical tiles of a parallelogram shape?*
4. Can we use identical tiles of a trapezoid shape?*
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TESSELLATIONS BY POLYGONS IN MATHEMATICS EDUCATION
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5. Can we use identical tiles of this shape?* (a general convex
quadrilateral – trapezium)
6. Can we use identical tiles of this shape?* (a general non-convex
quadrilateral)
(The concepts ‘convex’ and ‘non-convex’ are unknown to some pupils
of these age levels, so that such quadrilaterals have to be indicated.)
*The fi rst four problems have several solutions, hence the children
are asked in all problems: Is the proposed tiling the unique possible
one? Can these tiles be arranged otherwise?
A er solving these particular problems, the pupils are asked the
following questions:
• Is it possible to use identical tiles of an arbitrary quadrilateral
shape?
• If YES, why?
• (Did you examine all possible quadrilateral shapes?)
• If NO, fi nd out which tiles of that quadrilateral shape we cannot
use to cover the fl oor.
The fi rst question concerned the square tiles in the bathroom,
the other ones concerned the parquets; why were the problems
formulated that way?
Questionnaire and homework
A er closing the interviews, the pupils are asked to fi ll in a
questionnaire and to do homework. The questions concern their
mathematical eff orts, dealings with their cooperative partner and
their relation to the subject of mathematics. The answers (especially
questions 3. and 4, see below) are subsequently analysed. The aim
of the homework is to draw tessellations formed by other polygons
(triangle, pentagon, hexagon etc). Observation during the interviews
could be slightly subjective, hence it is not regarded as the most
important source of information (but it can help to explain some
phenomena, e.g. observed nervousness of children could explain
their approach to the problem). The children’s answers are compared
mutually and with my (assumed) correct answers.
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Questionnaire questions
1. What is the mathematics grade on your school report? Are you
satisfi ed with your results in mathematics or do you want to
improve?
2. How do you get along with the boy/girl that you solved problems
with? Are you good friends?
3. Did you like the solved problems? Are they diff erent from those
ones solved in the mathematics lessons? What is the diff erence?
4. Which quadrilaterals seems to you to be the most interesting
and why?
Homework
1. Have you got a tiled bathroom or kitchen at home? Have you got
tiles at home? If yes, draw them.
2. Think about tiles of other shapes, e.g. triangular, pentagonal,
hexagonal, and so on, and draw them.
The resulting drawings of the tessellations made by the polygons
diff erent from the quadrilaterals will be analysed by the same method
as that of the case study.
mohli
(In my previous experience pupils created designs of tessellations for
wall, pavements or fl oor tilings. Most of them created tessellations
formed by very complicated shapes which could not be simply
justifi ed mathematically. I will therefore focus on the polygonal
tessellations formed by one type of polygon.)
Experimental se ing
Five pairs of children aged from 13 to 16 were interviewed.
Interviews lasted from 17 to 27 minutes and were held in the
classroom a er lessons. The names of the children, their ages and
simple characteristics of the pairs are given in the following table.
Children cooperated very well within pairs. The pairs M+P1 and K+T
decided voluntarily to take part in the experiment. (These children
were friends.) The other pairs were selected by their teacher to my
requirements (I wanted three pairs: girls, boys and mixed). The
teacher probably chose friends for pairs (but was not asked to do
so).
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170
Table 1. Children interviewed in the research
Names Age
and sex Date of
experiment Comments (experimenter’s
observation)
Michael (M)
Petr (P1) 13, M February
2004 easy to work with, very
communicative pair
Kristína (K)
Tereza (T) 13, F February
2004 easy to work with, but they were
a li le bit bored by this activity
and o en laughed
Lukáš (L)
Jan (J) 16,15,
MMay 2004 they were li le nervous (I
suppose because of not knowing
me )
Hana (H)
Veronika (V) 15, F May 2004 although H was the main verbal
communicator of the two, V was
not just a passive solver
Gábina (G)
Petr (P2) 14, F, M May 2004 easy to work with, both children
reacted quickly to my questions
Partial results of experiment
Illustrations from the interviews
From the transcriptions of the interviews, the following extracts were
chosen to illustrate the children’s thinking and solving processes
when answering two selected questions. The answers to the fi rst
question comprise the ‘mathematical’ base of the problems. The
second question concerns the imagery of pupils – the square tiles
and rectangular tiles are frequently encountered and I believed that
the shape of tiles could be more complicated.
The question ‘Is it possible to use identical tiles of an arbitrary
quadrilateral shape?’ was answered as follows:
• M: Yes, it could be, but it goes from diffi cult to even more
diffi cult.
• P1: Sure, it would be possible. But it would be very complicated.
• P1: So that if you have these angles, so you can put them
together.
(This sentence seems to be the closest one to the general mathematical
theorem.)
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• J: I don’t know know … (long pause) …if it were perhaps some
regular one perhaps parallelograms fi ing together... but
with… some arbitrary quadrilateral, then it wouldn’t go... it
wouldn’t fi t together.
• H: Yes, I think so. Except this one (she points at her unsuccessful
tessellation by nonconvex quadrilateral)... so that... no... (they
laugh).
• V: Just those ones which we managed.
• G: No.
The question ‘The fi rst question concerns the square tiles in the
bathroom, the other problems concern the parquet;, why do I
formulate problems that way?’ was answered as follows:
• K: Because... In fact... Because this one (the tessellation by
nonconvex quadrilateral) wouldn’t work for the wall... it looks
rather like tiles.
• T: Rather it would be very diffi cult to use.
• K: But it would be strange to look at.
• T: ... Maybe it would be diffi cult to handle.
• K: It’s also complicated and these (tessellations by squares) are
so simple.
• L: That is ... that walls are usually tiled by square tiles. So it
isn’t so diffi cult to rearrange, to fi nd that combination and the
square has all its sides of the same length.
• H: There are usually these tiles in the bathroom... the square
ones. It would look very strange if there were these ones (she
points at the nonconvex quadrilateral’s tessellation).
• V: It’s the simplest way, isn’t it?
• G: Because these tiles are mainly squares or of rectangular
shape
• P2: I haven’t seen tiles like these as yet (he points at the
nonconvex quadrilateral tessellation).
• G: Nor me.
• G: They seems to me (she thinks about it)
• P2: ...so kitscher.
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TESSELLATIONS BY POLYGONS IN MATHEMATICS EDUCATION
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Notes on the course of the interviews
The following notes concern the set of proposed research questions
for cognitive phenomena.
My interest in mathematics was known to children so that they
were rather surprised by my fi rst question. (Perhaps they expected
‘something more mathematical’.) Also further problems sometimes
evoked laughter. But except for K+T, the children wrote or told me
that they were interested in the problems.
When I prepared the problems, my idea about tessellation was
already too mathematical. It meant that in my thinking the planar
tessellation was an infi nite planar covering without gaps or overlaps.
But the children understood the tessellations as covering a bounded
planar area.
As far as expression and bilingualism were concerned, all pairs were
communicative. Even when some of them were li le nervous at the
beginning of the interview, working with them was easy. Sometimes
children communicated between themselves in a low voice or
by eye contact; they also used hand gestures as a complementary
communicative tool. There was no problem with bilingualism.
The most interesting area of research is the imagery of children in
the work with tessellations. The answers to the required problems
had to be substantiated by using ‘the building set’ (the paper models)
or by drawing the corresponding tilings. Only one pair (L+J) solved
the fi rst three problems by drawing the tessellations (drawing the
tessellation by trapezoids was too diffi cult for them), the other pairs
used the paper models from the beginning of the interview; only
the pair M+P1 also utilised scissors for some additional problems
emerging during the interview. All pairs found at least two tessellating
arrangements for squares, rectangles and parallelograms (M+P1 was
the most successful pair). Pairs M+P1 and K+T took advantage of their
previous solution (rectangle: ‘it is like the square, but there are two
joined squares now’, rhomboid: ‘it is similar to rectangles’; trapezoid:
‘when we put them like that, it forms a parallelogram there’), the other
pairs tiled randomly (in my opinion). Handling the general convex
and nonconvex quadrilaterals was the most diffi cult for the children.
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173
Three pairs were not successful with one of them; a er being told
that the problem could be solved, they tried to fi nd the solution and,
failing again, they at least admi ed ‘…maybe it could, but we would
need more time to think about the arrangement …’. The pair G+P2
was successful with a general nonconvex quadrilateral but not with a
general convex one (trapezium); they found the principle ‘the shorter
sides together, the longer sides together’ but they were not able to
use it for the general convex type again.
In the area of mathematical content, three pairs correctly used the
terminology of the quadrilaterals (they used their names fl uently in
their slang language), the two youngest pairs (K+T and M+P1) made
some mistakes. But only these pairs (K+T and M+P1) intuitively
found out why every quadrilateral tessellates the plane by itself:
notice the underlined sentence from interview (M+P1) ‘So that if you
have these angles, so you can put them together.’ And (K+T) ‘If we
put the shorter sides together and the longer ones, too... and we get
the circle.’
The proofs were absent in these interviews, there was no need of them
because children thought they examined all possible quadrilateral
shapes (‘and the other quadrilaterals look alike as well as those in
“the building set”’).
Questionnaire
The completed questionnaires gave me basic information about the
children (e.g. whether children forming pairs were really friends).
Children were interested in the posed problems, they answered the
questions Did you like the solved problems? Are they diff erent
from those ones solved in the mathematics lessons? What is the
diff erence?:
• I liked the problems, mainly because statements have to be
defended and to be proved.
• We don’t do problems like this, only sometimes we combine and
glue the paper pieces.
• Yes, they were interesting, perhaps we are usually do more
counting in mathematics lessons.
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TESSELLATIONS BY POLYGONS IN MATHEMATICS EDUCATION
174
• The problems were diff erent from usual mathematical lessons.
We spend more time counting than playing with pa erns in the
lessons.
Seven children answered the question Which quadrilateral seems to
you to be the most interesting and why? Saying that just a general
nonconvex quadrilateral is interesting (or ‘strange’) because e.g. ‘it
isn’t as regular as the square’, ‘ I would have to fi nd a combination
for its arrangement’ or ‘it has got a nonconvex angle and hence it
is something absolutely diff erent from a square for example’.
(Manipulating the nonconvex quadrilaterals was a very interesting
process. It is not a ‘common’ quadrilateral occurring in school
geometry. Some people have to count the number of its sides to
persuade themselves that this polygon is really a quadrilateral.)
Tessellations created by pupils
Further research on this question will try to answer the following
questions (they enlarge the set of questions concerning children’s
imagery):
What polygonal shapes and their arrangements do they use?
Why do they use these polygons?
Were they inspired by typical tessellation encountered in real life
(e.g. common parque ings, honeycombs)?
What tool do they use to draw them?
Interviews were held with fi ve pairs of children, but the drawings
of the tessellations were given in by only three pairs (the teacher of
pairs M+P1 and K+T insisted upon doing homework, the teacher
of the other three pairs let the homework be optional). Two girls
(K+T) worked together, the other children prepared their drawings
separately. This part of the homework was based on the handling of
the polygons diff erently from the quadrilaterals; in spite of this, P1
drew the tessellation by the nonconvex quadrilaterals. The simplest
polygon, triangle, occurred only in one case (H). Children drew the
tessellations with black pens and black pencils. The tessellations of
the K+T pair were drawn on squared paper, the tessellations of M
and P1 on lined paper and H’s ones on plain paper. (The drawings
of tessellations below were the most interesting because of the
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Lucia Ilucová
175
polygons used. Some of them are commonly used for pavements,
e.g. Michael’s ones.)
Figure 2. Hana’s tessellations
Figure 3. Kristína and Tereza’s tessellations
Figure 4. Michael’s tessellations
Figure 5. Petr’s (P1) tessellations
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TESSELLATIONS BY POLYGONS IN MATHEMATICS EDUCATION
176
Conclusions
The topic examined is very close to real life (architecture, art, nature)
and children had interesting creative ideas in the interviews. They
started to use their imagery when choosing strategies and selecting
tools and also in creating the drawings of tessellations. But discovering
easy mathematical principles in kitchen tiling or room parque ing
seemed to be more diffi cult for children than I had expected. The
cause may possibly be that children are not accustomed to look for
mathematics in real life nor to apply it as a tool to solving real life
problems.
The mathematics of bathroom tilings or fl oor parque ings is
somewhat unexpected even for adults. A didactic program was held
for teachers of mathematics in February 2004 at our faculty: Two Days
with Didactics of Mathematics. I contributed a workshop entitled
Parque ings, tilings, mosaics and geometry. Some participants
of this workshop solved the same problems in a similar way to
the children – namely at random. A erwards, they were asked to
answer the question ‘Is it possible to use identical tiles of an arbitrary
quadrilateral shape?’ and they used the words common to the
children taught by them.
• It can be done, because we tested all ‘sorts’ of quadrilaterals.
• It can, because some pupils were successful in the class (I wasn’t)
and the teacher praised them.
• Yes, it can, because I’ve been successful until now.
(I had already heard similar answers from the pupils in my previous
experience.)
In my research I focus especially on the role of tessellations for
developing the imagery of pupils. However, there are many areas of
geometry in which the understanding of the tessellations could be
stimulating:
• terminology and properties of polygons,
• measure of the interior angles of polygons and the sum of them,
• transformations (in the case of the Escher-type tessellations),
• symmetries.
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Lucia Ilucová
177
It seems that the problem of tessellations is troublesome in spite of
their broad practical applications. In general, this may perhaps be
because of their complete absence from our mathematical curricula.
References
Golding, J. (2000) Paths to The Absolute, Princeton: Princeton
University Press.
Grunbaum, B. and Shephard, G.C. (1987) Tilings and Pa erns, New
York: W. H. Freeman and Company.
Grunbaum, B. and Shephard, G.C. (1977) Tilings By Regular Polygons,
Mathematics Magazine, 5, pp. 227–45.
Ranucci, E. R. and Teeters, J.L. (1977) Creating Escher-Type Drawings,
Palo Alto: Creative Publications.
Pŭlpán, Z., Kuřina, F. and Kebza, V. (1992) O Představivosti A Její Roli
V Matematice [On imagery and its role in mathematic (in Czech.),
Praha: Academia.
Webster’s New World Dictionary (1989), New York: Prentice Hall,
General Reference.
Contact
Lucia Ilucová
Charles University, Prague
lucia_i@post.sk
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TESSELLATIONS BY POLYGONS IN MATHEMATICS EDUCATION
178
179
Ilmārs Kangro, Kaspars Politers
179
Abstract
The technique of determining learning styles according to Honey and
Mumford and studying the interrelation of learning styles using the
SPSS program, is considered. Theoretical and practical ways of acquiring
knowledge are considered in studies of mathematics with the application of
computer technologies. The research has been carried out at The University
of Latvia and Rezekne Higher Education Institution.
Key words: Honey and Mumford learning style questionnaire,
learning styles (activist, refl ector, theoretic, pragmatic), chi-square-
based measures, mathematics as a tool and an object, computer
technologies in mathematical studies.
Introduction
The strategy of economic and social development of the European
Union adopted in Lisbon (March 2000) applies to society as a
whole. Therefore university students should also be involved in the
research. This also applies to Latvia where higher education and
research guidelines must be implemented. The main problem of
the pedagogical process is to optimise the relationship between the
knowledge given to students and the knowledge obtained by them.
From the three main roles of consumer, observer and participant, the
student must choose the last.
DETERMINING, STUDYING AND
USING LEARNING STYLES – ONE
ASPECT OF THE TECHNIQUE OF
THE HIGHER SCHOOL
Ilmārs Kangro, Rēzeknes Augstskola, Rezekne
Higher Education Institution.
Kaspars Politers, Liepājas Pedagoģij as Akadēmij a,
University of Latvia
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DETERMINING, STUDYING AND USING LEARNING STYLES ...
180
The crucial issue, therefore, in the research of a cognitive activity
(i.e., learning styles) is that of the coincidence of students’ learning
styles with methods of training in order to increase their educational
achievements.
Research into cognitive styles is connected with the choice of
strategies regarding the process management and organisation of
education.
A ainment of results in studies is possible in diff erent ways,
depending on the style of the cognitive action – by emphasizing
theoretical (information, analysis and generalization) or practical
(implementation, observation, experiment and improvement)
approaches (Figure 1).
Figure 1. Ways of acquiring new knowledge
There is a serious problem related to the acquisition of theoretical
knowledge and its practical implementation in studies of
mathematics. Mathematics usually has a double nature – it is both a
means towards the studies of other sciences as well as the object of
study in the acquisition of mathematical knowledge itself. However,
the present generation responds more to the former than the la er
(Hoyles, Newman and Noss, 2001).
The question arises: is there a way in which, whilst emphasizing the
priority of the fi rst characteristic of mathematics, it is still possible to
181
Ilmārs Kangro, Kaspars Politers
181
preserve the clarity and content of its second aspect without losing
the a raction of mathematics and its usefulness among students?
A possible solution is to study the interconnectedness of the
double nature of mathematics and modern technologies (e.g., the
mathematical systems Mathematica, Maple, Derive, etc.) which
are expressed in the following mutual relations (Blomhoj, M., 2002,
Carreira, S. 2001, Dubinsky and Tall, 1991, Kent and Noss, 2000, Noss,
2001, Tall, 1991, Аладьев and Богдявияюсь, 2001): the person and
the double nature of mathematics; the person and new information
technologies; new information technologies and mathematics.
Determining diff erent kinds of students’ cognitive spheres (e.g.
Learning Styles) is one of the appropriate ways of solving the problem.
Cognitive styles are the psychic features that are subordinated to
people’s individual peculiarities in developing mental experience.
Their main functions are the establishment of objective mental
representations of the surrounding reality and the control of the
aff ective states. They allow for the treatment of cognitive styles
as metacognitive abilities, which form the basis of regulating
intellectual activity. Therefore individual styles infl uence not only
the productivity of the individual intellect but also the peculiarity of
the personality and his/her social behaviour (Холодная, 2002).
As to the theoretical and practical aspects, it was possible to distinguish
the most eff ective learning styles using Honey and Mumford’s
method (1995). These learning styles, which have been devised on
the basis of the D. Kolb model (Kolb, 1984), raise awareness of the
following polar measurements of intellectual opportunities in pupils
/ students: 1) Concrete / abstract types of thinking; 2) Processing the
information at the following levels: a) action, b) supervision.
In fi nding the right approach to promoting a student’s interests, it is
important to know him/her, to discover his/her needs and wishes,
learning style, interests and the preferred method of his/her cognitive
activity.
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DETERMINING, STUDYING AND USING LEARNING STYLES ...
182
Discussion and results
The methodological foundation of the study is based on using the
methods of determining the learning styles.
To ascertain learners’ a itudes, we prepared a test on “The Most
Appropriate Learning Style”, consisting of an 80-statement
questionnaire taken from the Learning Style Questionnaire (Honey,
and Mumford, 1995), Appendix). The respondents’ answers to
the statements enabled us to determine the level (very high, high,
moderate, low, very low) of their learning style (activist, refl ector,
theorist, pragmatist). The highest level of the four learning styles
is considered to be dominant and no more than two should be
chosen.
Experimental and data analysis
The experiment was conducted with 68 masters degree students
from the University of Latvia studying marketing at the faculty of
Economics and Management, and 69 fi rst-year students from Rezekne
Higher Education Institution studying at the faculty of Engineering
and Economics.
Figure 2 shows the distribution of all the respondents (137) from
the University of Latvia and Rezekne Higher Education Institution
according to their learning styles and their levels.
Figure 2. Distribution of respondents according to learning styles
and their levels
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Ilmārs Kangro, Kaspars Politers
183
Figure 2 indicates that the active style prevails among very high-level
learning styles; the refl ector style prevails among high-level learning
styles; the activist style prevails among moderate-level learning
styles; and the theorist and pragmatist styles prevail among low and
very low-level learning styles. Among the fi ve learning style levels,
moderate l is the most prevalent. The activist learning style prevails
at this level, but the other three moderate-level styles are represented
approximately to the same degree.
Analyzing the answers of 68 respondents at the Faculty of Economics
and Management who were studying for the master’s degree in
marketing, we obtained the following indicators about learning
styles. Thirty percent of the respondents are inclined towards a single
learning style whilst 50% of the respondents incline towards two
styles. In most cases, one of these two learning styles is the theorist.
This is proof of a strong desire to acquire theoretical knowledge.
The relationship of the learning styles on respondents’ ages was
also analyzed. The ages ranged from 20 to 27 years. The theorist
and pragmatist learning styles are characteristic of all respondents
irrespective of their age. Respondents between the ages of 20 to 23
prefer the theoretical style. This can be explained by their a itude
towards the importance of objectively -based knowledge and their
rejection of subjectivism and intuition. It is interesting to note that
respondents between the ages of 23 and 25 display a sharp increase
in preference for the refl ective style. We may conclude that at this age
self-evaluation and the desire to understand oneself become very
active; there is some reluctance to make decisions and to support
untraditional ideas. This is the age when short work experience is
obtained.
We observed that as work experience increases, so does the
number of supporters of refl ector and theorist learning styles. The
explanation could be that work experience develops one’s capacity
for self-evaluation and objectivity. The individual understands his
or her intellectual potential, the theoretical basis for processes and
situations, the need to analyse cause and eff ect relationships. These
are factors which shape an individual’s a itude and approach to
learning.
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DETERMINING, STUDYING AND USING LEARNING STYLES ...
184
28% of the fi rst year students of Rezekne Higher Education Institution
have one prevalent learning style but 62% display two prevalent
learning styles. These learning styles are the activist and refl ective
but when found in pairs the dominating styles are: activist/refl ective
and activist/theoretical learning styles.
However, the fi ndings do not provide information about the
relationship among diff erent learning styles. As a considerable
number of respondents are not inclined towards more than two
learning styles it is important to investigate the mutual relationships
of the learning styles. To conduct this research the Chi-Square-based
measures were used on the computer programme SPSS, release 9.0
(Norušis, 1993). Table 1, column 3 (Value) shows the value of the
measure of association – the contingency coeffi cient – the relationship
among diff erent pairs of learning styles.
The underlined pairs of the learning styles should be considered
statistically important (Table 1, Nos 1, 2, 4 and 6), approx. Sig. <0,05
(Norušis, 1993). It should be added that the contingency coeffi cient
value must not be directly correlated with the correlation coeffi cient
value of the parametric statistics. For example, if a correlation
coeffi cient value within the range of 0,3–0,4 indicates a weak
correlation of the analyzed features, then a contingency coeffi cient
value within this range indicates a medium strong correlation
(Krastiņš, 1998).
Table 1 reveals that the combination of particular learning styles
with the theorist style (which was typical of the University of Latvia
students studying for the master’s degree) is also the same for most
of the younger (fi rst year students) respondents. This is clear because
three out of four pairs of learning styles (Table 1, Nos 2, 4 and 6)
include the theoretical learning style.
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Ilmārs Kangro, Kaspars Politers
185
Table 1. The measure of association – value of contingency coeffi cient
calculated among the learning styles
During the study the following characteristic features of the learning
styles were observed.
Activists are extroverts by nature; they are active and ready to tackle
the tasks. They o en begin working although they have not listened
to the end of the teacher’s instruction. This increases the probability
of making mistakes and demands special patience and readiness to
help from the teacher in correcting the mistake and fi nishing the task.
Mistakes are a normal phenomenon of studying, teachers should help
students to correct them and understand the reasons themselves.
Theorists do not usually hurry to start solving the problem – they
prefer to be clear about the theoretical substantiation, the structure of
the problem, its connections and formulas. They want to see things
and the phenomena in a context, to be able to understand the task
and perform it conscientiously. Before beginning the work they
usually ask the teacher questions so as to gain a full understanding of
a theoretical substantiation and the logic of a problem. This demands
responsiveness from the teacher because students’ questions are a
normal part of any study situation and questions should not be
considered either good or bad.
Pragmatists are openly focused on practical activities and tasks
connected with them give them satisfaction; they usually like
practical problems. They usually do not like theoretical subjects
and frequently feel bored in lessons. They are highly motivated
to study those problems in which they can see the practical value
and understand where and how that knowledge could be put into
practice. These students willingly choose the inductive way of
solving the problem – from the concrete to the general, therefore
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DETERMINING, STUDYING AND USING LEARNING STYLES ...
186
teachers should fi nd ways of demonstrating the practical purposes of
the problems studied.
Refl ectors, similarly, usually are in no hurry to start doing the
educational problem – they search in their previous experience for
the basis for its performance. These students are more o en introverts
by nature and are comfortable in lessons if they have enough time to
draw on their existing knowledge and skills, which are necessary
to do the new problem. They expect help from the teacher to do the
task and they usually worry whether there will be enough time for
the new task.
The prevalent learning styles and their levels allow the teacher to
vary his/her activities in working with students who have diff erent
specialities and diff erent backgrounds (e.g. their preliminary
knowledge, the profession they choose and their motivation
(Kangro, 1999, 2001, 2004), personal qualities and skills (Garleja and
Kangro, 2002), particular features of the relationship between their
professional competence and social behaviour (Garleja and Kangro,
2003), etc.). This information is especially useful for developing and
implementing those courses which need an individualized and
diff erentiated approach, e.g. mathematics.
Learning styles practice gives effi cient information to the teacher
about his/her students; teachers become be er acquainted with their
students, the importance of informal relations increases. It facilitates
the cooperation between teachers and students. As a result students
become more convinced of their strengths, positive motivation and
satisfaction with the course in general develops (Burke, 2000).
Learning styles in the context of applying modern information
technologies
In the context of the theoretical and practical importance of
knowledge acquisition, the application of computer technologies in
studying mathematics must be referred to.
The simplicity and convenience of computing procedures and the
vivid and expressive visualization of images allows the application
of computer systems to a wide range of diff erent theoretical and
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Ilmārs Kangro, Kaspars Politers
187
practical problems, as well as to the performance of scientifi c and
academic tasks.
Solving problems with the help of a PC seriously eases the routine
of performance, makes tiresome actions more tolerable and saves
school hours. This all gives space for students’ individual learning
and so helps in implementing the learning styles approach (Kalis
and Kangro, 2003b, 2004a). It is possible to combine the
inductive and deductive presentation of the problem as well as to
demonstrate it with the help of cognitive visualization. It ensures that
the representatives of diff erent learning styles can choose the most
eff ective way for themselves of solving the given problem. Students
in course evaluations point out particular sections of the mathematics
course as useful: for example, the elements of fi nance mathematics
and linear programming, ordinary and partial diff erential equations,
as they demonstrate the mathematical modelling process and reveal
be er how mathematics could be used (Kangro, 2004).
A conclusion should be drawn that the application of computers
during training has many advantages. For example, it is possible to
modernize the mathematics course with the help of vivid illustrative
material and connect it with diff erent sciences and fi elds of practice
which have been treated separately due to their complexity. Thus it
is possible to make interdisciplinary connections (Kalis and Kangro,
2003a, 2004a, Noss, 1999). It facilitates the co-operation between
teacher and students in doing independent tasks, because it can
take into account students’ learning styles and develop not only
students’ traditional mathematical competence but also their social
and communicative skills (Garleja and Kangro, 2002).
The symbolic language of modern computer systems, for example,
Maple, Mathematica a/o is quite ‘user-friendly’, as in many cases there
is no essential distinction between the mathematical concept being
studied and the defi nition of the operator of the computer system for
representing the concept. It is possible to acquire simultaneously the
necessary mathematical concept (for example, plane, surface, level
curves) and the corresponding two- or three-dimensional graphical
operator to construct the research concept (Kalis and Kangro,
2003a, Kangro, 2000). The opportunities are especially a ractive for
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188
representatives of the active and pragmatic learning styles because
the illustration of the researched concept and a method (operator)
for its reception (image) is evident together. It is easier to bring
together the visual and structurally understandable concept with
the necessary context and to indicate the practical usefulness of the
concept (Norušis, 1993).
For practical realization of algorithms using computers it is not
necessary to know any special language of programming. It is enough
to acquire the elementary standard operators of mathematical systems
(Maple, Mathematica, a/o), for example, in solving the diff erential
equations, input and output of data, actions with fi les and data array,
formation of cycles etc. (Kalis and Kangro, 2003a, 2004a).
These possibilities usually add interest to the research problem for the
representatives of the theoretical and refl ective learning styles. They
are especially keen to devise the plan to complete a task, choosing
the necessary operators, testing their operability and comparing the
practical data obtained with the theoretically forecast data.
Using the PC it is possible to receive a visual image of the research
problem (to build up the link Concept Defi nition -> Concept Meaning-
> Concept Image) (Carreira, 2001, Kalis and Kangro, 2003a, 2004b).
Representatives of all learning styles usually appreciate the
possibilities of using mathematical systems. These allow for
presentation of knowledge both in the conceptual structure mode
(mainly in visual form) which includes concrete-practical level
(formation of knowledge and skills operating with real pa erns of the
objects and processes investigated, teaching models); visual-spatial
level (application of visual models – pictures, drawings, graphs,
tables, etc.); abstract-symbolic level (usage of the instruments of the
scientifi c language, for example, numbers, signs, etc. and its diff erent
combinations, for example, expressions, formulas, equations, etc.)
and the procedural structure mode (mainly in the form of (Kalis and
Kangro, 2004b, Mayer, 1989, Skemp, 1987).
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Ilmārs Kangro, Kaspars Politers
189
Conclusion
1. The intellectual abilities of students are diff erent. They depend on
the talent, interests and level of knowledge, life experience, age, and
the professional competence of an individual.
2. The technique for discovering one’s learning style and a suitable
research programme makes it possible to process a large amount
of data and obtain reliable information about the distribution of
respondents’ learning styles. It is then easy to use the SPSS computer
program to discover the relationship among the learning styles.
3. The results make it possible for each student to understand his or
her learning style and it is especially important for fi rst year students
because it allows them to organize their studies be er.
4. The distribution of learning styles allows teachers to organize the
learning process be er, to take into account the individual features
of students’ learning styles and to develop an individual approach to
the learning process.
5. The discovery and description of students’ individual styles
are necessary in order to create conditions during training for the
formation of a personal cognitive style. Thus, students with diff erent
learning styles should have an opportunity to choose, within a
uniform educational space, a method of training congruent with their
own learning style. At the same time, an educational environment
should be created that stimulates each student to enlarge his or her
repertoire of behaviour styles.
6. The application of ICT ensures the individualization of the teaching/
learning process within the curriculum. Consequently, teaching/
learning and didactic materials can be designed in order to help
each student to choose his/her own teaching/learning method and
improve his/her teaching/learning process based on the individual
learning style.
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Burke, K. (2000) A Paradigm Shi : Learning-Styles Implementation
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Practical Approaches to Using Learning Styles in Higher Education,
Westport: Bergin and Garvey, pp. 85–94.
Carreira, S. (2001) There’s a Model, There’s a Metaphor: Metaphorical
Thinking in Students’ Understanding of a Mathematical Model,
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Dubinsky, E., and Tall, D. (1991) Advanced Mathematical Thinking
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Dordrecht: Kluwer Academic Publishers, pp. 231–43.
Garleja, R., Kangro, I. (2002) The Analytical Evaluation of the
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Garleja, R., Kangro, I (2003) Interaction of the Professional Competence
and Social Behaviour, in L. Frolova (ed.) Management, vol. 660, Riga:
University of Latvia, Zinātne (in Latvian) pp. 25–43.
Garleja, R., Kangro, I (2004) Complex Management of Mathematical
Knowledge and Skills, in Proc. of the Int. Conference: Teaching
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Appendix
Questionnaire ‘‘Using Your Learning Styles’’
Instruction
The aim of this questionnaire is to fi nd out your most characteristic
learning style in order to help you evaluate its appropriateness for
your cognitive potential. There is no limit on the time to fi ll in the
questionnaire; it could be 10 to15 minutes or more. The precision of
the results depends on your honesty in completing it. If you disagree
with a question more than you agree with it, write “NO” opposite
the number of that particular question. If you agree more than you
disagree, then write “YES”.
1. I have strong beliefs about what is right and wrong, good and
bad.
2. I o en act without considering the possible consequences.
3. I tend to solve problems using a step-by-step approach.
4. I believe that formal procedures and policies restrict people.
5. I have a reputation for saying what I think, simply and directly.
6. I o en fi nd that actions based on feelings are as sound as those
based on careful thought and analysis.
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Ilmārs Kangro, Kaspars Politers
193
7. I like the sort of work where I have time for thorough preparation
and implementation.
8. I regularly question people about their basic assumptions.
9. What ma ers most is whether something works in practice.
10. I actively seek out new experiences.
11. When I hear about a new idea or approach I immediately start
working out how to apply it in practice.
12. I am keen on self-discipline such as watching my diet, taking
regular exercise, sticking to a fi xed routine, etc.
13. I take pride in doing a thorough job.
14. I get on best with logical, analytical people and less well with
spontaneous, ‘irrational’ people.
15. I take care over the interpretation of data available to me and
avoid jumping to conclusions.
16. I like to reach a decision carefully a er weighing up many
alternatives.
17. I’m a racted more to novel, unusual ideas than to practical ones.
18. I don’t like disorganised things and prefer to fi t things into a
coherent pa ern.
19. I accept and stick to laid down procedures and policies as long as
I regard them as an effi cient way of ge ing the job done.
20. I like to relate my actions to a general principle.
21. In discussions I like to get straight to the point.
22. I tend to have distant, rather formal relationships with people at
work.
23. I thrive on the challenge of tackling something new and
diff erent.
24. I enjoy fun-loving, spontaneous people.
25. I pay meticulous a ention to detail before coming to a
conclusion.
26. I fi nd it diffi cult to produce ideas on impulse.
27. I believe in coming to the point immediately.
28. I am careful not to jump to conclusions too quickly.
29. I prefer to have as many sources of information as possible – the
more data to think over the be er.
30. Flippant people who don’t take things seriously enough usually
irritate me.
31. I listen to other people’s points of view before pu ing my own
forward.
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194
32. I tend to be open about how I’m feeling.
33. In discussion I enjoy watching the manoeuvrings of the other
participants.
34. I prefer to respond to events on a spontaneous, fl exible basis
rather than plan things out in advance.
35. I tend to be a racted to techniques such as network analysis, fl ow
charts, branching programmes, contingency planning, etc.
36. It worries me if I have to rush out a piece of work to meet a tight
deadline.
37. I tend to judge peoples’ ideas on their practical merits.
38. Quiet, thoughtful people tend to make me feel uneasy.
39. People who want to rush things o en irritate me.
40. It is more important to enjoy the present moment than to think
about the past or future.
41. I think that decisions based on a thorough analysis of all the
information are sounder than those based on intuition.
42. I tend to be a perfectionist.
43. In discussions I usually produce lots of spontaneous ideas.
44. In meetings I put forward practical, realistic ideas.
45. More o en than not, rules are there to be broken.
46. I prefer to stand back from a situation and consider all the
perspectives.
47. I can o en see inconsistencies and weaknesses in other people’s
arguments.
48. On balance I talk more than I listen.
49. I can o en see be er, more practical ways to get things done.
50. I think wri en reports should be short and to the point.
51. I believe that rational, logical thinking should win the day.
52. I tend to discuss specifi c things with people rather than engaging
in social discussion.
53. I like people who approach things realistically rather than
theoretically.
54. In discussions I get impatient with irrelevancies and digressions.
55. If I have a report to write I tend to produce lots of dra s before
se ling on the fi nal version.
56. I am keen to try things out to see if they work in practice.
57. I am keen to reach answers via a logical approach.
58. I enjoy being the one that talks a lot.
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Ilmārs Kangro, Kaspars Politers
195
59. In discussions I o en fi nd I am the realist, keeping people to the
point and avoiding wild speculations.
60. I like to ponder many alternatives before making up my mind.
61. In discussions with people I o en fi nd I am the most dispassionate
and objective.
62. In discussions I’m more likely to adopt a ‘low profi le’ than to take
the lead and do most of the talking.
63. I like to be able to relate current actions to a longer-term bigger
picture.
64. When things go wrong I am happy to shrug it off and ‘put it
down to experience’.
65. I tend to reject wild, spontaneous ideas as being impractical.
66. It’s best to think carefully before taking actions.
67. On balance I do the listening rather than the talking.
68. I tend to be tough on people who fi nd it diffi cult to adopt a logical
approach.
69. Most times I believe the end justifi es the means.
70. I don’t mind hurting people’s feelings so long as the job gets
done.
71. I fi nd the formality of having specifi c objectives and plans
stifl ing.
72. I’m usually one of the people who puts life into a party.
73. I do whatever is expedient to get the job done.
74. I quickly get bored with methodical, detailed work.
75. I am keen on exploring the basic assumptions, principles and
theories underpinning things and events.
76. I’m always interested to fi nd out what people think.
77. I like meetings to be run on methodical lines, sticking to laid
down agenda, etc.
78. I steer clear of subjective or ambiguous topics.
79. I enjoy the drama and excitement of a crisis situation.
80. People o en fi nd me insensitive to their feelings.
The learning styles are characterized by the quality of psyche. The
following learning styles are indicated:
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196
Activist
Strengths: 1) Flexible and open minded; 2) Happy to have a go; 3)
Happy to be exposed to new situations; 4) Optimistic about anything
new therefore unlikely to resist change.
Weaknesses: 1) Tendency to immediately obvious action without
thinking; 2) O en takes unnecessary risks; 3) Tendency to do
too much themselves and hog the limelight; 4) Rush into action
without suffi cient preparation; 5) Get bored by implementation /
consolidation.
Refl ector
Strengths: 1) Careful; 2) Thorough and methodical; 3) Thoughtful; 4)
Good at listening to others and assimilating information; 5) Rarely
jump to conclusions.
Weaknesses: 1) Tendency to hold back from direct participation; 2)
Slow to make up their minds and reach a decision; 3) Tendency to be
too cautious and not take enough risks; 4) Not assertive – they aren’t
particularly forthcoming and have no ‘small talk’.
Theorist
Strengths: 1) Logical ‘vertical’ thinkers; 2) Rational and objective; 3)
Good at asking probing questions; 4) Disciplined approach.
Weaknesses: 1) Restricted in lateral thinking; 2) Low tolerance for
uncertainty, disorder and ambiguity; 3) Intolerant of anything
subjective or intuitive; 4) Full of ‘should, ought and must’.
Pragmatist
Strengths: 1) Keen to test things out in practice; 2) Practical, down
to earth, realistic; 3) Businesslike – gets straight to the point; 4)
Technique oriented.
Weaknesses: 1) Tendency to reject anything without an obvious
application; 2) Not very interested in theory or basic principles; 3)
Tendency to seize on the fi rst expedient solution to a problem; 4)
Impatient with waffl ing; 5) On balance, task oriented not people
oriented.
If the respondent has given the answer “yes” to a particular statement,
then this statement obtains the value of number 1 but if it is “no”,
then the number is 0. Using Table 1, we can fi nd the number of points
for each learning style (e.g. the number of points for respondent A-1
is shown in Table 2).
197
Ilmārs Kangro, Kaspars Politers
197
Each learning style is divided into 5 levels (Table 3). For example,
according to Table 3, respondent A-1 (Table 2) corresponds to the
moderate level of the active learning style (7 points), the high level of
the refl ective learning style (17 points), the high level of the theoretical
style (14 points) and the low level of the pragmatic style (11 points).
Appendix Table 1. Conformity of learning style to numbers of
statements
Appendix Table 2. Points gathered by respondents concerning all
the learning styles
Appendix Table 3. Evaluation of the a itudes of respondents
Contact
Ilmārs Kangro
Rēzeknes Augstskola
Rēzekne, Latvij a
kangro@ru.lv
Kaspars Politers
Liepājas Pedagoģij as Akadēmij a,
Liepāja, Latvia
k.politers@inbox
198
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198
199
Sigmund Ongstad
199
Abstract
In April 2004 24 teachers of Pedagogy, Mother Tongue and Mathematics
Education in general teacher education were interviewed as part of the
project Wholeness in Teacher Education? The interviews focused mostly
on the relationship between general didaktik and disciplinary didaktik
(Norwegian: ‘fagdidaktikk’; Swedish, Danish and German: ‘fagdidaktik’).
In Norway there is, at least on the rhetorical level, a strive towards Bildung,
‘wholeness’, multidisciplinary education, coherence and a will to focus on
a single educational path for years 1–13; in short a will towards general
teacher education. On the other hand, there are more local, personal,
and disciplinary tendencies to focus on the particular and on the school
subject. The interviews revealed a variety of positionings between these two
extremes.
This paper presents some of the positionings of three Mathematics teachers in
general teacher education in Norway, and clusters their view of fag/didaktik
into examples of diff erent ‘profi les’. In spite of the fact that general didaktik
(taught in pedagogy) has lost much of its infl uence in Norwegian teacher
education in the last few decades, the didaktik of particular disciplines has
not yet fi lled this gap. Nevertheless there is apparently a new tendency,
perhaps in all Scandinavian countries, partly within, partly between, the
disciplines, to move from narrower to more open perceptions of the didaktik
of the discipline(s). However this ‘openness’ generates new dilemmas and
TEACHER OF MATHEMATICS
OR TEACHER EDUCATOR
POSITIONINGS AND
PROBLEMATISATIONS
Sigmund Ongstad, Oslo University College, Faculty
of Education
200
TEACHER OF MATHEMATICS OR TEACHER EDUCATOR? ...
200
confl ict; a development which surfaced in the interviews. This paper focuses
in particular on the possible consequences for the discipline of mathematics
education in general teacher education.
Methodologically the approach is explorative and oriented towards the
validity of didactic concepts rather than driven by overall hypotheses. An
important aim is to problematise critically the crucial role of the teacher
educators’ positionings as an element in achieving more coherence in teacher
education. Thus their qualitative reasoning is seen as more relevant than a
quantifi cation of ‘how many would mean what’.
Keywords: general didactics, subject didactics, mathematics teacher
education
Introduction
This article stems from Wholeness in General Teacher Education?
(Ongstad, 2001, a project which develops theories of triadic
communication in order to study im-/balances between aesthetics,
epistemology and ethics in diff erent subjects and their didaktik.
The critical focus on wholeness is concerned with ‘compatibilities’
between major elements in teacher education. The project analyses
textbooks and wri en curricula in mathematics education and
mother tongue education for student teachers. In addition 24 teacher
educators, in mathematics education (8), MTE (8) and pedagogy
(8),have been interviewed about their views on didactic aspects in
general teacher education.
A simple hypothesis is that there exists a political and common
sense-based stereotyping of curricular goals: Math trains brains,
Mother tongue (education) tongues and Arts hearts, even if teachers
resist such banalities. To avoid simplifi cations, another project
hypothesis is that the pedagogical triangles between teacher, subject
and learner (Künzli, 1998; Westbury, 1998) should be seen as mutual
communicational triads (Habermas, 1984, Bakhtin, 1986, Halliday,
1994). This allows for further diff erentiation of both research objects
and methodological approaches. The project analyses, by using an
approach called discursive positioning(s), how ‘teacher’, ‘subject’ and
‘learner’ relate in a systemic way to ‘expressivity’, ‘referentiality’ and
‘addressivity’ in educational u erances or aesthetics, epistemology
201
Sigmund Ongstad
201
and ethics in textbooks, classrooms and curricula. Diff erent balances
of these domains will shi across borders, cultures, times and
political ideologies, and create educational trilemmas rather than
just dilemmas. A paradigmatic shi to the triadic understanding of
communication may therefore, among other approaches, contribute
to a more profound understanding of the potential for coherence
between interdisciplinary elements (Ongstad, 2004).
In addition to studies of curricula and text books it has been of
interest to hear the voices of teacher educators. However, given the
unclear situation in the fi eld at large a quantitative project would be
of less interest. It seemed more appropriate to do a qualitative study,
focusing on diff erent specifi c positionings to didaktik in general and
to subject didaktik in particular.
A continuation to the interviews was a follow-up project at Oslo
University College that was concerned with the multi-disciplinarity
of the main subjects in general teacher education (Ongstad, 2003).
Both the project and the follow-up study revealed a strong will
to create more coherence, coordination, and cooperation at the
college, clearly supported by faculty leaders. However there were
nevertheless ‘centrifugal’ forces of diff erent kinds pointing in other
directions.
Furthermore the textbook situation and the rapid shi s of wri en
curricula for general teacher education (there had been three major
reforms over ten years) had contributed to a more unbalanced
situation. Interviews would help in clarifying how teacher educators
related to this fl uid situation. There were already strong indications
from the culture of general teacher education that disciplinary teachers
at the college (and probably elsewhere) tended to see professionality
as more restricted as disciplinarity (Kvalbein, 1998). In some cases
one could even claim that professionality and disciplinarity were
seen as counterparts.
The interviews and the foci for this study
Four faculties of education were contacted. I used lists of employees
accessible on their websites and contacted people in fulltime,
tenured, positions, in alphabetical order, with one eye on gender. All
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TEACHER OF MATHEMATICS OR TEACHER EDUCATOR? ...
202
24, six at each college, two in each of the three subjects, were sent
the interview guide. The interviews were recorded in their offi ces,
lasting approximately 90 minutes, and notes were taken. For the
interviews in this article no changes were made.
The interview focused mostly on the relationship between general
didaktik and disciplinary didaktik. I sometimes use the Norwegian
word ‘fag’ for discipline or subject, and didaktik for didactics
or education, because these concepts are not necessarily easy
to translate into English. In Norway there is, on the one hand, a
curricular impetus towards Bildung, ‘wholeness’, multidisciplinary
education and coherence. And also a desire to focus on a common
educational approach for all the years from 1–13 for all schools in
Norway (Braathe and Ongstad, 2001). In short there is a will towards
general teacher education. On the other hand, however, there are
more personal, local, regional, and disciplinary characteristics which
focus also on the particular and on the school subject. Hence even
the wider project focuses the im-/balances of these tendencies in
curricula and textbooks, and among teacher educators.
One hypothesis for this study is that shi s in curricular infl uence
and importance of disciplines and school subjects are related to
the communicative ‘nature’ of the discipline. . Simplifi ed, that is to
say that major changes in society, culture and economy over time,
and o en in paradigmatic ‘waves’, will aff ect the balance between
aesthetics, epistemology and the ethics of (and between) disciplines.
Thus for instance mathematics may increase its infl uence when
there are strong political demands for epistemological ‘needs’ and
competitiveness in science and technology. Similarly MTE may
be seen as important when there is a stress on advanced literacy,
democracy and communicational skills in business and in public
life.
A further tendency is for the disciplines/subjects to take up the
challenge if they are serious about contributing to general Bildung.
Increased competition between kinds of professional knowledge
may then lead to increased didaktisation of established and new
disciplines (Hertzberg, 1999; Ongstad, 1999). In other words,
continuous change in society forces institutionalized fi elds of
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Sigmund Ongstad
203
knowledge (such as fag) to develop a meta awareness (didaktik) to
position the discipline appropriately. Hence Peter Elbow has claimed
that the question What is English? is actually the answer, suggesting
that self-awareness should be part of any discipline (Elbow, 1991).
The imprecisions of these generalisations, however, is not
coincidental. The need for sharper sub-hypotheses derives from the
fact that the tendencies and trends now seem more unclear than ever.
This situation can be seen as a crossroads between modernity and
postmodernity. Hence a function of the interviews is to search for
how teacher educators resonate and prioritise in these unsafe waters,
or in other words, how they position themselves and their disciplines
under the infl uence of increased didaktisation.
I have chosen three interviews for this article, partly randomly,
however with one eye on their diff erences, and made a profi le of each.
Regarding ‘fi ndings’ the interviews revealed a variety of positionings.
In spite the fact that general didaktik (taught in pedagogy) has
lost much of its infl uence and teaching time in Norwegian teacher
education over the last few decades, the didaktik of disciplines
has not yet fi lled this gap. Nevertheless there apparently is a new
tendency, perhaps in all Scandinavian countries, partly within the
disciplines, partly between them, to move from more narrow to more
open perceptions of the fag-didaktik (Sjøberg, 2001; Schüllerqvist,
2001; Schnack, 2004). However this ‘openness’ in turn generates new
dilemmas and confl ict lines, a development which surfaced even in
the interviews, not least in the questions like: – what is ‘mathematics?’
and what is ‘Norwegian?’ Therefore the focus will be on mathematic
education as part of general teacher education.
General comments on the profi les
In a loose sense, generally and somewhat subjectively, the clustered
aspects in the right hand ‘column’ represent a more traditional
orientation. The arrows hint that the aspect is not a category, and
that the person’s view just tends more towards that direction. The
question mark is used when I, as interpreter, lacked clear signals
in the interview to place the person between the two positions. If
someone has many question marks, this can mean at least three
things. Firstly that the aspects in general were not suitable to position
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TEACHER OF MATHEMATICS OR TEACHER EDUCATOR? ...
204
this person’s didactic profi le. Secondly that the interview did not
give the interpreter good enough information. Thirdly it might even
mean that the person deliberately avoided being placed in categories
or between given aspects.
Further it should be noted that a radicalization towards more
didaktisation is probably a fairly new tendency. Therefore it would
be quite a surprise if there were many x’s on the le hand side
when summarized. It should be noted that person A has been asked
whether she thinks she has been ‘correctly’ positioned, not B and C.
A agreed on all positionings. It should be added though that it is not
a necessary norm for the validity of these profi les that the interpreter
and the interviewee always should agree. People may not be aware
of their relative positions in a randomly chosen group. By contrast
interpreters may be blinded by their perceptions of the /concepts.
Person A’s orientation towards...
Def. of
fagdidaktik open/wide X narrow
Fagdidaktik
as theroy X practice
thinking X doing
explicit X implicit
integrated X separate
meta X concrete
process X product
Fagdidaktik
close to pedagogy X disciline
“wholeness” X disciplinarity
student/pupil ? subject/discipline
multi-disciplinarity X “one-disciplinarity”
Fagdidaktik
focused early in the study ? later
Mathemat-
ics more as fagdidaktik X discipline
general Bildung X disciplinary content
Teacher edu-
cation less school-dependent X more school-dependent
205
Sigmund Ongstad
205
student oriented X disciplinary
A view of
one selfe radical/diff erent X mainstream
didactically moving X didactially stable
4 5(2?) 7
Comments
If we state that the signifi cance of a given profi le is related to the
number of x-es on the right or the le hand side, then A can be said
to be positioned as (relatively) traditional, at least compared to B
and C (whose profi les are given below). It can be suggested that two
question marks probably means that this person is not too diffi cult
to position or is generally willing to take standpoints in didactic
questions. Firstly what becomes clear is that A in general defends
the direction mathematics has taken toward mathematic education
(or didaktik of mathematics), but that in practice there is a lower
threshold of mathematical knowledge that can not be ignored. (A er
the interview the Ministry of Research and Education decided that
grade 3 in mathematics from upper secondary school is the lowest
limit for ge ing into general teacher education in Norway.)
She (subject A) also holds that subject knowledge is a necessary
premise for the ability to consider didactic questions. However
when it comes to developing a coherent mathematical education into
a tool for general Bildung she has stronger doubts. At least two of
her comments will help position A more generally in this respect: A
subject didaktician is something diff erent from a teacher educator
and Then I’m tempted to say – are we going to have general teacher
education or are we going to have subject teacher education? Why
do we cling to general teacher education? The fi rst statement reveals
that A sees a split between the two concepts; the second, which is
partly a rhetorical question, may tells us in addition that A wants
to downplay the degree of general education in favour of a more
disciplinary one.
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TEACHER OF MATHEMATICS OR TEACHER EDUCATOR? ...
206
Person B’s orientation towards...
Def. of
fagdidaktik open/wide X narrow
Fagdidaktik
as theroy X practice
thinking X doing
explicit X implicit
integrated X separate
meta X concrete
process X product
Fagdidaktik
close to pedagogy X disciline
“wholeness” X disciplinarity
student/pupil ? subject/discipline
multi-disciplinarity X “one-disciplinarity”
Fagdidaktik
focused early in the study X later
Mathemat-
ics more as fagdidaktik X discipline
general Bildung X disciplinary content
Teacher edu-
cation less school-dependent X more school-dependent
student oriented X disciplinary
A view of
one selfe radical/diff erent X mainstream
didactically moving X didactially stable
5 10(1?)
Comments
B not only ends up in the middle, he even claims that he has
deliberately taken up this position and gives an explicit reason. He
has spent a long life in school, working with pupils, and with more
than just one school subject. He took his Masters degree relatively late
in his career and by background he is ‘enculturated’ to mathematics
and to pedagogy (educational research). His perception of fag-
didaktik is nevertheless relatively radical, partly explained perhaps
by his more recent Masters thesis. He seems to have met fresh ideas
207
Sigmund Ongstad
207
about didaktik in his study. He has wri en a booklet on mathematic
didaktic for his students.
B’s most signifi cant position is the statement that the importance of
mathematics is rather overestimated. He is quite radical here, since
most mathematicians believe that mathematics should be compulsory
from year 1 to year 13. (It should be added that his masters thesis
was about the ‘need’ for and use of mathematics in practical work
and business life.) While placing B in the middle position we should
not forget that in his own department he sees himself and one other
as didactic radicals, warning that he may not be representative of
other mathematicians at this college.
Person C’s orientation towards...
Def. of
fagdidaktik open/wide X narrow
Fagdidaktik
as theroy X practice
thinking X doing
explicit X implicit
integrated X separate
meta X concrete
process X product
Fagdidaktik
close to pedagogy X disciline
“wholeness” X disciplinarity
student/pupil X subject/discipline
multi-disciplinarity X “one-disciplinarity”
Fagdidaktik
focused early in the study X later
Mathemat-
ics more as fagdidaktik X discipline
general Bildung ? disciplinary content
Teacher edu-
cation less school-dependent X more school-dependent
student oriented ? disciplinary
208
TEACHER OF MATHEMATICS OR TEACHER EDUCATOR? ...
208
A view of
one selfe radical/diff erent X mainstream
didactically moving X didactially stable
9 7(2?) 0
Comments
C is by far the most ‘radical’ of the three quoted here. In another article
I have presented the profi le of three MTE teachers. Two of those had
a Ph.D. thesis. It is tempting to believe that doctoral theses fuel the
process of didaktisation at least on a personal level, perhaps even
in departments and in general, since those working to, or already
having achieved, a thesis o en work at the cu ing edge of a discipline.
However there might be other reasons, ones that are inherent in
mathematics education. The whole fi eld of Mathematics Education/
the didaktik of mathematics has developed in that direction for quite
a while (Biehler et al., 1994; Niss, 2001; Sierpinska and Kilpatrick,
1998). There may be a cumulative eff ect in this particular teacher
education where several disciplines have moved towards a more
didactic direction. C, being responsible for the relationship both to
the leaders and to other departments in connection with a particular
project which aims for greater coherence, might be more willing to
integrate mathematics into general teacher education.
However there are other reasons for a move in this direction. C feels
that he has been developing his thinking over the last years and that
philosophy has played an important role in his now more generalised
view on learning.
Comments on comments
We can not generalise without increasing the number of interviews
signifi cantly. Even if we added the fi ve other interviews we might
not get a valid general picture. What these three cases can tell us
though is that most of the suggested aspects are evident and do have
an impact on the perceptions of teacher educators in mathematics
education. It should be emphasised again that the most radical of the
interviewees has a Ph.D. and that the other two gained their Masters
degrees late in their careers. Those degrees were signifi cantly not in
mathematics but in mathematics education.
209
Sigmund Ongstad
209
It is probable however that this sample is too small. There are still
several possible alternative profi les, for instance a strongly disciplinary
one. By contrast, in this study eight mathematics teachers have been
asked what they think their own position is in their departments and
relative to teachers in Norway in general. There really does seem to
be a general agreement that mathematics education in Norway is
signifi cantly didakticised.
This could lead us to the conclusion that mathematics education is
more ‘radical’ than the other fag/disciplines in teacher education.
However such a view is problematic if ‘radical’ is taken to be an
overall description of the profi le. When it comes to Bildung, to general
education, to general enculturation to a complex society, mathematics
education does not seem to be particularly eager or willing to work in
that direction (Braathe and Ongstad, 2001; Ongstad, 2003 and 2004).
Kvalbein’s research is relevant here. She has found that there is (still) a
strong general reluctance among (disciplinary) teachers as indicated
in her data, to avoid ‘professional orientation’. Even if she is not just
focusing on fagdidaktik, both Kvalbein’s and my own study indicate
that the ‘empty space’ that occurred a er the reduction of pedagogy
in general teacher education in the early 1990s has still not been fi lled.
Or, put diff erently, one of the reasons may be that even signifi cantly
disciplinary teachers are hesitant to address the dilemma of how and
to what extent fag/discipline should be balanced with other aspects
in general teacher education. It is tempting to see these profi les as
symptoms of a dri towards more coherence, and less antagonism,
between disciplines and pedagogy. Such a conclusion could be risky,
however, since there it has is no quantitative basis (yet). I would still
hold that didaktisation is at work among these three interviewees.
Problematisation
One way of making the familiarity of the confl ict between
pedagogy and the other teacher educational disciplines strange is
to see institutionalized knowledge not as products but as processes.
Hence pedagogy, as well as mathematics and other fags, can be
seen in a narrative or developmental perspective. When nation
building is no longer of prime political interest, but has given long
lasting benefi ts to a discipline and its members, a dilemma occurs
210
TEACHER OF MATHEMATICS OR TEACHER EDUCATOR? ...
210
when the political signals are more economical. The function of
mathematics as a discipline is now foregrounded, which educators
in the Norwegian fi eld of mathematics fi nd positive a er a long life
in the shadows. (Mathematics was not compulsory in Norwegian
general teacher education before 1992!) Many in the profession are
less enthusiastic about the emphasis put on economical arguments
and on competitiveness in international tests. Besides certain deep
ideologies within mathematics and mathematics education do not
work for an increased cooperation with other disciplines (Braathe
and Ongstad, 2001).
Finally, how likely is it that A, B and C in the future will see themselves
just as much teacher educators as teachers of mathematics and its
didaktik? Before I try to answer that, it should be pointed out that
when pedagogues or general educationalists ask such a question, an
implicit ideological premise is o en that pedagogy teachers are more
‘real’ teacher educators than disciplinary teachers. The idea is related
to the view that pedagogy is closer to the profession than the content
disciplines. This is probably more the case for primary teacher
education than for lower and upper secondary. By and large I think
the profi les warn against the belief that teacher educators could be
easily moved towards greater cooperation. On the other hand none
of these three seems directly in opposition to the idea.
References
Bakhtin, M. (1986) Speech Genres and Other Late Essays. Austin:
University of Texas Press.
Biehler, R. et al. (eds) (1994) Didactics of Mathematics as a Scientifi c
Discipline. Amsterdam: Kluwer Academic Publishers.
Braathe, H.J. and Ongstad, S. (2001) Egalitarianism meets ideologies
of mathematical education – instances from Norwegian curricula
and classrooms. ZDM, Vol. 33 (5).
Elbow, P. (1990) What is English? NY and Urbana: MLA/NCTE.
Habermas, J. (1986) Kommunikativt handlande. Texter om språk
rationalitet och samhälle. Göteborg: Daidalos.
Halliday, M.A.K. (1994) An Introduction to Functional Grammar.
Second Edition. London: Arnold.
211
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211
Hertzberg, F. (1999) Å didaktisere et fag – hva er det? In Nyström, C.
and Ohlsson, M. (eds) Svenska i utveckling. FUMS Report, no. 196)
Uppsala: Uppsala University.
Hudson, B., Buchberger, F., Kansanen, P. and Seel, H. (1999a) (eds)
Didaktik/Fachdidaktik as Science(s) of the Teaching Profession?
TNTEE Publications, Vol. 2 (1).
Hudson, B..(1999b) Mathematic Didaktik (Teaching-Learning
Mathematics), in Hudson, B., Buchberger, F., Kansanen, P. and
Seel, H. (eds) Didaktik/Fachdidaktik as Science(s) of the Teaching
Profession? TNTEE Publications, Vol. 2 (1).
Hudson, B., Ongstad, S. and Pepin, B. (2002) Didaktik der Mathematik:
teaching-learning mathematics. Linz: Linz University/EMDID.
h p://ive.pa-linz.ac.at/emdid/latest/default.html
Kvalbein, I.A. (1998) Lærerutdanningskultur og kunnskapsutvikling.
Ph.D. thesis, Oslo University, PFI/UiO.
Künzli, R. (1998) The Common Frame and the Places of Didaktik, in
Gundem, B. and Hopmann, S. (eds) Didaktik and/or Curriculum. An
International Dialogue. New York: Peter Lang.
Niss, M. (2001) Den matematikdidaktiska forskningens karaktär
och status. I. B. Grevholm, (red.) Matematikdidaktik – et nordisk
perspektiv. Lund: Studentli eratur.
Ongstad, S. (1999) Sources of ‘didaktization’. On Defi ning Disciplines
and Their ‘(fag) didaktik’ Across Borders, Illustrated with Examples
from Mother Tongue Education (MTE), in Hudson, B. Buchberger, F.,
Kansanen, P. and Seel, H. (eds) Didaktik/Fachdidaktik as Science(s)
of the Teaching Profession? TNTEE Publications, Vol. 2 (1).
Ongstad, S. (2001) Positioning Aesthetics, Epistemology and Ethics
in Didaktik of Subjects. Paper at ECER 2001. Lille. September.
Ongstad, S. (ed.) (2003) Koordinert lærerutdanningsdidaktikk? Ideer
og erfaringer. Hio-rapport. nr. 14.
Ongstad, S. (2004) Språk, kommunikasjon og didaktikk. Bergen:
Fagbokforlaget/LNU.
Ongstad, S., van de Ven, Piet-Hein and Buchberger, I. (2004) Mother
tongue education didaktik. Linz: Trauner Verlag.
Schnack, K. (ed.) (2004) Didaktik på kryds og tværs. Copenhagen:
DPU.
Schüllerqvist, B. (2001) Ämnesdidaktik som lärarutbildningens mi fält,
in Schüllerqvist, B. and R. Nilsson (eds) (2001) Lärarutbildningens
212
TEACHER OF MATHEMATICS OR TEACHER EDUCATOR? ...
212
ämnesdidaktik. Gävle: Högskolan i Gävle. Gävle: HS-institutionens
series no. 5.
Sierpinska, A. and J. Kilpatrick (eds) (1998) Mathematics Education
as a Research Domain: A Search for Identity. (Book I/II). Amsterdam:
Kluwer Academic Publishers.
Sjøberg, S. (2001) (ed.) Fagdebatikk. Fagdidaktisk innføring i sentrale
skolefag. Oslo: Universitetsforlaget.
Westbury, I. (1998) Didaktik and Curriculum Studies, in Gundem, B.
and S. Hopmann (eds) Didaktik and/or Curriculum. An International
Dialogue. NY: Peter Lang.
Internet references to the project(s)
h p://www.math.uncc.edu/~sae/dg3/dg3.html
h p://www.math.uncc.edu/~sae/dg3/ongstad.pdf
h p://www.emis.de/journals/ZDM/zdm015i1.pdf
h p://www.ped.gu.se/konferenser/amnesdidaktik/ongstad.pdf
h p://www.aare.edu.au/03pap/ong03276.pdf
h p://ive.pa-linz.ac.at/emdid/latest/doc/pdf/MTEcover.pdf
h p://tntee.umu.se/publications/v2n1/pdf/ch13.pdf
h p://www.aare.edu.au/03pap/ong03274.pdf
www.ingenta.com/isis/browsing/TOC/ingenta;jsessionid=2orkat81a
i6gs.crescent?issue=pubinfobike://sage/j275/2002/00000019/00000003
h p://www.luna.itu.no/Fokusomraader/dlo/1060157614.2/view
http://program.forskningsradet.no/kupp/uploaded/nedlasting/
kunnskapsstatus.pdf (page 31 –51)
h p://www.palgrave.com/products/Catalogue.aspx?is=140391690X
h p://home.hio.no/~sigmund/english/eng.htm
h p://www.ilo.uva.nl/development/iaimte/News/The%20Scientifi c
%20Commission%20for%20Mother%20Tongue%20Education1.htm
Contact
Prof. Dr. Sigmund Ongstad
Oslo University College, Faculty of Education
Sigmund.Ongstad@lu.hio.no
213
Hans Jørgen Braathe
213
Abstract
This paper explores Habermas’s theory of the pragmatics of communication
to fi nd support for both theory and method in the analysis of student teachers’
texts. First the project is seen in the context of teaching and learning as being
acts of communication and presents the methodology for analysing student
teachers’ u erances in the context of teacher education. Secondly it gives
an account of Habermas’s reference to Bühler’s complex language signs to
give reason for his triadic validity claim and further discusses Habermas’s
reference to Meads’ theory of subjectivity to explain Habermas’s concept of
‘Individuation through Socialisation’ as part of his theory of identity and
lifeworld. Thirdly the paper gives an account of Habermas’s relationship to
the critic of his rational pragmatics. Finally we will see how the strengths
and weaknesses in Habermas’s theory can help refl ection on methods of
analysing and especially into the problem of validating.
Keywords: Habermas, pragmatics of communication, student
teachers, validating
Introduction
Habermas’s theory of formal pragmatics serves as a theoretical
underpinning for his theory of communicative action and it contributes
to an ongoing philosophical discussion of problems concerning truth,
rationality, action and meaning. The unit of analysis in the theory of
VALIDATING AS POSITIONING(S)
— A DISCUSSION OF HABERMAS’S
FORMAL PRAGMATICS
Hans Jørgen Braathe, Oslo University College,
Faculty of Education
214
VALIDATING AS POSITIONING(S)? — A DISCUSSION OF HABERMAS’S ...
214
communicative action is the use of linguistic expressions or u erances
which areoriented towards reaching understanding. (Habermas,
1998). Every u erance is u ered, according to Habermas, with a
validity claim in a dialogic process with the aim of achieving mutual
understanding. ‘Such an agreement can be achieved by both parties
only if they accept the u erance involved as correct. Agreement
with regard to something is measured in terms of the intersubjective
recognition of the validity of an u erance that can in principle be
criticized.’ (Habermas, 1998, 227) This validity claim was initially
divided into four diff erent classes: intelligibility, truth, normative
rightness, and sincerity (Habermas 2001, 85). However they are later
reduced to three as intelligibility is presupposed as necessary for a
dialogue to continue.
Every speech act as a whole can always be criticized as invalid from
three perspectives; as untrue in view of a statement made (or of the
existential presuppositions of the propositional content), as untruthful
in view of the expressed intention of the speaker, and as not right in
the view of the existing normative context (or the legitimacy of the
presupposed norms themselves). (Habermas 1998, pp. 76–77).
I will relate these three claims to the problem of validating student
teachers’ u erances communicated in teacher education.
Learning as communication
My project in Mathematics Education, ‘Analysis of positionings of
student teachers’ identities in their mathematics educational texts’,
takes a communicational view of teaching and learning as its theoretical
framework for analysing student teachers’ texts (Rommetveit, 1972,
Habermas, 1984, Ongstad, 2004a). The texts analysed are those
produced by the students during their compulsory mathematics
course in their teacher education. The project discusses writing in a
professional se ing and will link theoretically and methodologically
to the analysis and interpretation of texts in a wide sense. ‘Text’ in
this connection will also include mathematical text, that is, seeing
mathematics as a semiotic sign system. The texts/u erances must be
given meaning with reference to the context/genre they are expressed
in. As part of these analyses it will be necessary to see the diff erent
and shi ing contexts and genres, including the ideologies, that
215
Hans Jørgen Braathe
215
surround the u erances (Ongstad, 1997). Ideology is broadly defi ned
as the unspoken premises for communication (Braathe and Ongstad,
2001). It is something we think from, not on (Riceour, 1981). Genres
are therefore carriers of ideology (Bakhtin, 1986). Semiotic simply
means that I want to study more than only language, as I see all
cultural expressions and u erances as meaningful signs. Genres can
be described as kinds of communication. We communicate through
u erances. U erances are any suffi ciently closed use of sign that
makes sense. We communicate by u ering and by giving u erances
meaning. All u erances are u ered and interpreted related to
expectations of genres, i.e. contexts that helps us to understand the
u erance. Genres are, as already mentioned, ideological, i.e. they give
tacit premises for the participants’ positioning in the communication
(Ongstad, 2004b)
Positioning as a triadic discursive concept
My methodology of analysing texts uses this concept of ‘positioning’
as a point of departure. Positioning as it is developed and used by
Ongstad (1997, 2004b) takes its initial inspiration from Bakhtin’s
essay ‘The problem of speech genres’ (Baktin, 1986, 60–102) where
he identifi es Bakhtin’s communicative elements necessary for an
u erance to communicate in a dialogic relation. These elements are:
— delimiting (from former u erances)
— positioning (the u erance as such by...)
expressing (by which expressivity becomes a constitutive
aspect)
referring a semantic content (by which referentiality becomes
a constitutive aspect)
addressing (by which addressivity becomes a constitutive
aspect)
— fi nalising (the u erance as a whole by...)
fi nalising forms (by which form as aspect contributes to
wholeness)
semantic exhausting (by which content as aspect contributes
to wholeness)
ending speech will (by which intention as aspect contributes
to wholeness)
(Ongstad, 2004b)
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VALIDATING AS POSITIONING(S)? — A DISCUSSION OF HABERMAS’S ...
216
Ongstad (2004b) holds that the genres are also to be seen as triadic
in the same sense as the u erance, that they simultaneously give
potential for expressing, referring and addressing. The three aspects
are seen as parallel, inseparable, reciprocal, simultaneous processes.
All learning presupposes some kind of communication – either it takes
place as interaction between individuals, or between individuals and
groups, or it takes place through interaction in environments that
combine theoretical studying with practical training. One is u ering
in teaching and learning situations and the u erance is given
meaning. U erance will both be the theoretical concept and also
the object of analysis. The students’ u erances in praxis, in tutorial
situations, tasks, lectures, discussions, books, project assignments,
questions or answers are all u erances that belong to the teaching/
learning process. The analyses of the students’ texts can be viewed as
u erances within teacher education. Regarding every u erance as an
inseparable triadic unit is what inspires the theoretical methodology
I apply, hence I analyse the texts from 1) referential, 2) expressive and
3) addressive aspects in the texts. This ‘triangulation’ can give insight
into how the student (respectively) positions him/her self in relation
fi rst to mathematics and the teaching and learning of mathematics,
second, to his/her own emotions and experiences and third to the
classroom, teachers and others. Thus a specifi c focus of this project
will be to apply this ‘triangulation’ and to develop methods for
analysing its infl uence on the development of students’ identities in
their capacity as teachers of mathematics, through their writing of
mathematics educational texts.
Identity is used as a concept that fi guratively combines [2] the intimate
or personal world, with [1] the collective space of cultural forms and
[3] social relations. (Holland et al., 1998, p. 5. [My numbering])
I see identities as social products, lived in and through communicative
activity. Therefore they must be conceptualised as they develop in
social communicative practice. In this way I see learning and the
development of identity as two aspects of the same process. Identities
can then be used as a concept situated in social practices (Braathe and
Ongstad, 2001, Ongstad and Braathe, 2001. It is presupposed and
expected that the students will develop an identity as teachers, and
217
Hans Jørgen Braathe
217
a view of students, school subjects and teaching/learning that will
give them suffi cient confi dence and competence to join Norwegian
primary and lower secondary schools. Thinkers like Bakhtin,
Goff man, Foucault and Bourdieu all, in their own way, support the
argument that individuals develop social identities through the
(communicative) genres – wri en and oral – that society off ers them.
Empirical studies show this to be the case when students try out
norms of writing and diff erent genres they meet in universities and
colleges (Freedman, 1992, Ivanic, 1994, Ongstad, 1997, Morgan, 1998,
Evans, 2000).
The problem of validation: three dimensional texts
Starting from semiotic genre theory I shall validate my interpretations
in terms of a communicative learning process. I will compare these
validations with Habermas’s validity claims stated in his theory of
formal pragmatics.
Validation of communication implies evaluation of u erances. Taking
as a point of departure a triadic concept of discursive positioning
allows for a validation related to the three main dimensions of the
u erance. The validity of 1) the referential dimension, can be seen to
mean whether the phenomena referred to are true or false — in other
words to epistemology and the degree of ‘objectivity’. The validity of
2) the expressive dimension, can be understood as meaning that what
is expressed is in accordance with the u erer’s inner emotions and
intentions — in other words the (estimated) quality of subjectivity
and sincerity. Finally the validity of 3) the addressive dimension, can
be interpreted to mean whether the u erance is pragmatically sound,
whether it is a right or wrong (action), for more than the u erer, in
other words the quality of normativity (or intersubjectivity).
Validity has also been an explicit issue for Bakhtin, and he has
connected the issue of validity to the u erance as a semantic whole:
The whole u erance is no longer a unit of language (and not a
unit of the ‘speech fl ow’ or the ‘speech chain), but a unit of speech
communication that has not mere formal defi nition, but contextual
meaning (that is, integrated meaning that relates to values – to truth,
beauty, and so forth – and requires a responsive understanding, one
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that includes evaluation). The responsive understanding of a speech
whole is always dialogic by nature. (Bakhtin, 1986, p. 125)
The validity is also seen by Bakhtin as connected to the dialogic
situation and related to the possible response to the u erance:
But the u erance is related not only to preceding, but also to
subsequent, links in the chain of speech communion. When a speaker
is creating an u erance, of course, these links do not exist. From
the very beginning, the u erance is constructed while taking into
account possible responsive reactions, for whose sake, in essence,
it is actually created. As we know, the role of the others for whom
the u erance is constructed is extremely important. […] From the
very beginning, the speaker expects a response from them, an active
responsive understanding. The entire u erance is constructed, as it
were, in anticipation of encountering this response. (Bakhtin, 1986,
p. 94)
Bakhtin understands the u erances as dialogic links in never-ending
chains of communication and at the same time as triadic. Hence the
Bakhtinian dialogism could be seen as dynamics between the three
positioning aspects and their endless sub-variations on the one hand,
and dynamics between the triadic u erance and the triadic genre on
the other (Ongstad, 2004b). The main point is that none of these three
extremes can occur as categories; u erances are placed in between,
that is within the triadic communicational space, and so are validities
(Ongstad, 2004b).
In my comments and descriptions of student teachers’ texts I will
take into consideration the discursive contexts or genres within
which they are u ered. In the students’ texts it might be possible to
trace their own experience of learning mathematics at school, their
self image as mathematics learners, the theories they met at teacher
training college (both in theory lessons and in wri en material), and
the practice teaching they met during their teacher education.
Hence in this project when I look for the referential aspects of the
u erance, related to the mathematics or the teaching and learning of
mathematics, I look for truth or falsity. When I look for the expressive
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aspects of the u erance, I look at form and what this form expresses
regarding feelings and experiences. When I look for the addressive
aspects of the u erance I look for normativity, in the sense of usefulness
related to the role of a mathematics teacher in primary school. Here
we fi nd much in common with Habermas’s validity claim in his
theory of the pragmatics of communication. In the rest of this chapter
I will examine how Habermas’s theory supports the methodology
of the ‘analysis of positioning’. Methodologically I face a challenge,
because as I focus on all three aspects simultaneously, looking for
one aspect alone means ignoring the two others. Habermas points
to this challenge by holding that: ‘Admi edly, only one of the three
validity claims can be thematically emphasized in any explicit speech
act.’ (Habermas, 1998, 77). Habermas mostly focuses on these three
aspects but normally all u erances will be in-between.
Habermas and validity claims
Habermas’s fi rst concern is the development of a critical theory of
society, and his a empt to work out a theory of communicative
rationality is of primary importance in this context. At the centre of
both Habermas’s theory of communicative action and his account
of communicative rationality, is the thesis that u erances – as the
smallest unit of communication – raise various kinds of validity
claims. In earlier works Habermas used the term ‘ideal speech
situation’ to refer to the hypothetical situation in which the strong
idealisation implicit in everyday communication action would be
satisfi ed (Habermas, 1984), however he now dissociates himself from
this formulation (Habermas, 1998, 367–68). Forms of argumentation
that come suffi ciently close to satisfying these strong idealisations
are called ‘discourse’ by Habermas. A ‘discourse’ is a refl ective
dialogue on the universal validity claims of a proposition whose
participants seek to achieve, through argumentation, a rational
consensus (Bordum, 2001). The universal validity of a contested
claim is possible in principle only if the argumentation can be carried
on long enough. However contested claims can be overthrown by
be er arguments. In this way Habermas argues for fallibility as he
suggests that the premises for arguments can change. In this way
he avoids foundationalism (Bordum. 2001, 204). Habermas separates
the question of validity from the question of existence by using the
words ‘Geltung’ and ‘Fakticität’ (Bordum, 2001, 34).
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Habermas’s ‘discourse’ has, like Baktin’s, a dialogic view of
communication, that is, that all u erances are parts of a dialogue. In
this dialogue the participants set validity claims in the sense that it is
expected that the u erer is able to give reasons for the claims, and is
expecting a ‘yes or ‘no’ to his/her claims. Every u erance, according
to Habermas, is someone u ering something to someone. In the
context of my project the texts I am analysing can be seen as those of
the student teachers u ering something about mathematics, and/or
didaktics of mathematics, to me as their teacher. That is to say that
their u erances must be seen as responses in a dialogic relation.
In teacher education the educators are se ing the premises for
the dialogue and there is accordingly an asymmetrical situation.
Nevertheless the expressive and normative aspects of the u erance
can be seen here as more symmetrical than the referential aspect. This
is linked to the ideology of teacher education within which students
are encouraged to, and expected to, use critical and autonomous
thinking. When it comes to the development of their identity as
mathematics teachers this is of great importance. Therefore their
validation is of great importance in my analysis of positioning.
Given Habermas’s concern with mechanisms of societal integration,
he focuses on the coordinating power of language. His formal
pragmatic examination of everyday language takes as a starting
point the structure of linguistic expressions rather than the speaker’s
intentions. He argues that the communication-theoretic approach
expounded by the German psychologist Karl Bühler in the 1930s
suggests a fruitful line of inquiry for investigations into language
as social coordination.1 He also maintains that Bühler’s organon
model of language, although it needs to be modifi ed in certain
ways, provides a basis for a more adequate account of meaning and
understanding than any of the contemporary analytic approaches to
meaning as they stand. (Habermas, 1992, 57ff )
Bühler puts forward a schema of language functions that places the
linguistic expression in relation to the speaker, the world, and the
hearer. He starts from a model of a linguistic sign used by a speaker
with the aim of coming to an understanding with a hearer about objects
and states of aff airs. The linguistic sign functions simultaneously
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as symbol (by virtue of its being correlated with objects and states
of aff airs), as symptom (by virtue of its dependence on the sender
whose subjectivity it expresses), and as signal (by virtue of its appeal
to the hearer, whose internal or external behaviour it steers) (Bühler,
1934).
Habermas a empts to release Bühler’s schema from its origins in a
particular psychology of language and, by expanding the linguistic
approach, to develop it to give a broader interpretation of each of
the three functions mentioned by Bühler. At the same time, he tries
to retain what he regards as Bühler’s basic insight: that language
is a medium that fulfi ls three mutually irreducible but internally
connected functions. Bühler draws a ention to the fact that
linguistic expressions that are used communicatively (as opposed to
strategically) function (a) to represent states of aff airs (or something
in the world that confronts the speaker), (b) to give expression to the
speaker’s intentions or experiences, and (c) to enter into a relationship
with a hearer. Habermas, following Bühler, claims that these three
functions cannot be reduced to a single one. Thus, the three aspects
involved in u ering a linguistic expression are the following: I (the
speaker)/come to an understanding with a hearer/about something
in the world. Habermas identifi es these as the three structural
components of u erances: the propositional, the expressive, and the
illocutionary.
Both the main aspects of an u erance: the content/reference, the
form/structure, and the use/act (to use Ongstad’s notional set, 2004b),
as well as the subjective respective evaluation of those aspects, point
to validity as a problem. Depending on where one positions the
research, and accordingly the object, one faces diff erent validity
expectations. Regarding the dilemma of choice Habermas holds:
The validity-theoretical interpretation of Bühler’s functional scheme
off ers itself as a way out of the diffi culties of speech-act theory because
it does justice to all the three aspects of a speaker coming to an
understanding with another person about something. It incorporates
within itself the truth contained in the use-theory of meaning and
at the same time overcomes the types of one-sidedness specifi c to
intentionalistic and formal semantics. (Habermas,1992, 73).
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and
A validity-theoretic interpretation of Bühler’s functional scheme
further leads to the assumption that with a speech act ‘MP,’ S takes
up relations simultaneously to something in the objective world, to
something in the subjective world and to something in the social
world. (Habermas, 1992, 76).
I share with Ongstad the opinion that in the cultural domain the
tension between these three aspects o en makes a one-dimensional
validity inadequate. If, for instance, we relate this to the fi eld of
didaktics, it is clear that didaktics can never isolate itself to a purely
scientifi c regime of validities (Ongstad, 2004a). The subjectivity of
the person (both the learner and the teacher) has to be balanced
not only with the traditional validity of ‘objectivity’, but also with
the ‘pragmatic’ validity of acting (normativity/intersubjectivity)
(Ongstad, 2004b).
Habermas and individuation
Habermas has been criticised for the universalism of his pragmatics. In
the eyes of some of his critics; this universalism indicates insensitivity
to the claims of the individual over and against the universal, and
thus insensitivity to related themes of otherness and diff erence
(Habermas, 1992, xi; Løvlie, 2001). In ‘Individuation through
Socialisation: On George Herbert Mead’s Theory of Subjectivity’
Habermas (1992, 149–204) discusses the individual, otherness, and
diff erence. He uses the model of intersubjective communication that
Mead used to explicate the structure of the individual. Mead retains
the ‘mirror-model’ of self-consciousness familiar from German
Idealism in which the subject only comes upon itself via the mediation
of its object. Now however this ‘object’ is understood not from the
third-person perspective of an observer, but from the second-person
perspective of a participant in a linguistic communication – the other
is an alter ego. Mead used the term ‘me’ to give expression to this
structure of the self as a second person to another second person
(Habermas, 1992, xvi). The self is intersubjectively constituted, and
the relation to a community is what makes the practical relation to
self possible. Habermas distinguishes between conventional and
postconventional forms of communicative actions. Conventional
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forms are bound to local communities and are preconditions for
postconventional forms, that are connected to what he calls universal
communities consisting of all possible alter egos. Conventional forms
of communicative actions in local communities do not necessarily
require rational reasons from the u erer for validating the u erance.
These, however, will be required in postconventional forms. Regarded
as part of a development of identity in a didaktik context this could
be seen as introducing new elements as postconventional forms of
communicative actions. These new elements require ‘yes’ or ‘no’ as
validations in the communicative community. Identity formation is,
again in Habermas’s view, connected to these two forms.
Among the universal and unavoidable presuppositions of action
oriented to reaching understanding, is the presupposition that the
speaker qua actor lays claim to recognition both as an autonomous
will and as an individual being. And indeed the self, which is able
to assure itself of itself through the recognition of this identity by
others, shows up in language as the meaning of the performatively
employed personal pronomen in the fi rst person. […] The universal
pragmatic presuppositions of communicative action constitute
semantic resources from which historical societies create and
articulate, each in its own way, representation of mind and soul,
concepts of the person and of action, consciousness of morality, and
so on. Wherever relations are more or less formalized, […], legal
norms relieve one of responsibility of a moral kind; at the same time,
anonymous and stereotypical behaviour pa erns leave li le room
for individual characterizations. However, the reciprocally raised
claims to recognition for one’s own identity are not completely
neutralized, even in rigorously formalised relationships, as long as
recourse to legal norms is possible; the two moments are preserved
in the concept of the legal person as the bearer of subjective rights.
(Habermas, 1992, p. 191)
This account of individuation connects to Habermas’s universal
pragmatics, since both require a universal community of discourse.
According to Habermas this relationship between the supposition
of a universal community and the individual is complementary
(Habermas, 1992, xviii).
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The teacher-education community, in which the communication
I am analysing takes place, containing the student teachers’ texts,
can, in Habermas’s terms, be seen as a postconventional community.
The individuation of the student teachers can be seen as situated in
this context. Teacher education, as does all teaching, has ambitions
to refl ect universal validity claims by introducing new elements all
the time, and thereby off ering student teachers a postconventional
community.
The encyclopaedic Bildung’s ideal, with a focus on epistemological
subject knowledge, o en excludes ethical and aesthetic discussions.
According to Habermas, to argue for schooling in a dialogue on
curriculum, one has to argue for all three aspects of validities
simultaneously (Schou, 2003). In the context of teacher education,
including mathematics education, the dialogue between students
and teacher education discourse must include all three validity
claims: school subjects and teaching and learning. This dialogue
presupposes a communicational rationality, and hence students’
ability to use rational reasoning to reach consensus.
Habermas suggests that the concept of communicative action
must be understood against the background of lifeworld. He uses
lifeworld to refer to the stock of implicit assumptions, intuitive
know how, and socially established practices, that functions as a
background to all understanding. The lifeworld as a background to
everyday processes of communication also functions as a resource.
It provides a reservoir of intuitively certain interpretations upon
which participants can draw. The three domains of the lifeworld
identifi ed by Habermas are those of: cultural reproduction, social
integration, and socialisation. These correspond respectively to
the three functional aspects of communicative action: reaching
consensus, coordinating action, and socialisation (the formation of
personal identities). The gap between communicative action and
the lifeworld widens as conventional modes of communication are
increasingly replaced by postconventional modes – to the extent that
actions oriented toward understanding increasingly rely on ‘Yes’
and ‘No’ responses to validity claims. These cannot be traced back to
prevailing normative consensus but are refl ective of critical and open
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processes of interpretation. This in turn leads to the pluralisation of
(overlapping) lifeworlds (Habermas, 1998, 215–56).
The lifeworlds are both a horizon and a resource for communicative
actions. In postconventional modes where lifeworlds overlap, like in a
teaching/learning situation, new elements have to enter the discourse
and raise validity claims to be contested in the ongoing dialogue. In
the analysis below I will take into account possible lifeworlds that
give horizons and resources for the student teachers’ u erances.
Habermas divides lifeworld into three components: inner nature
(subjectivity), outer nature (objectivity) and society (normativity).
All three will be of equal importance in the validation of the analysis
(Habermas, 1998, 247).
Habermas’s universalism
Habermas’s validity concept is underpinned by a concept of
rational consensus. Consensus is also based on rationality shared
by a community, and universal consensus has as an ideal that a
postconventional community can ultimately achieve validity/‘truth’
through consensus. ‘A proposition is true if it withstands all a empts
to refute it under the demanding conditions of rational discourse.’
(Habermas, 1998, 367). Accordingly this understanding of truth is not
based on empirical facts but is an element in the discourse (Løvlie,
1992). This concept of ‘truth’ has been criticised by Rorty (1999)
who criticises the idea of the dominant-free dialogue that converges
towards a consensus that can be measured by one standard of
validity.
However I have not been able to fi nd explicit criticism of the triadic
thinking of u erances always having three simultaneous aspects:
referential, expressive and addressive. Cooke (1994, 94) however
comments ‘that the idea that the speaker raises three claims may be
misleading and that it seems more accurate to say that every speech
act makes reference to three aspects of validity, or that every speech
act relies on three kinds of presuppositions.’ Cooke also suggests a
multi-dimensional, not just three-dimensional, concept of reason.
Habermas answers these critics by stating that:
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In order to cover the entire spectrum of possible validity claims, it
makes sense to start by posing the heuristic question: in what sense
can speech acts be negated as a whole? In answering this question we
hit upon precisely three sorts of validity claims: truth claims in regard
to facts that we assert with reference to objects in the objective world;
claims of the truthfulness (Wharha igkeit) of u erances to make
manifest subjective experiences (Erlebnisse) to which the speaker has
privileged access; and fi nally, claims of the rightness of norms and
commands that are recognised in an intersubjectively shared social
world. (Habermas, 1998, p. 317).
Refl ections
In using Habermas’s concepts to describe my project; I am analysing
u erances in the dialogic discourse in a postconventional community.
The focus for analysis is understanding how the development of
knowledge-understanding-identity operates for student teachers of
mathematics. The u erances from the students raise validity claims
that I have to relate to. In my analysis I will have to assume that the
students will be able to give rational reasons for their claims, and
therefore try to include these in the analysis. My analysis may in
principle be a ‘yes’ or ‘no’ to their claims, and I will again have to give
rational reasons for these. The dialogue/‘discourse’ requires validity
claims to be raised simultaneously on epistemological, aesthetic and
ethical grounds.
Habermas is seeing his concepts of communicational rationality
relative to understanding its dependencies on the concept of
lifeworld:
[…] this entire rationality complex, on which a society’s capacities for
interpretation and learning in all its dimensions depends, obviously
does not, as it were, stand on its own two feet but rather needs a
lifeworld background whose substance is articulated in the medium
of language: a lifeworld background that forms more or less suitable
contexts, and provides resources, for a empts to reach understanding
and to solve problems. (Habermas, 1998, p. 336).
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At the same time the three domains of the lifeworld are seen as
inseparable in giving the background for communicative practices,
and the lifeworld is constituted by these practices.
[T]he lifeworld does not constitute an organization to which
individuals belong as members, nor an association in which
individuals join together, nor a collective comprised of individual
participants. Rather, the everyday communicative practices in which
the lifeworld is centred are nourished by the means of an interplay of
cultural reproduction, social integration, and socialization that is in
turn rooted in these practices. (Habermas, 1998, p. 251).
Habermas’s concept of lifeworld, together with his concept of system,
constitute his communicational macro-concepts. Lifeworld, with its
three domains, has many similarities to the triadic understanding of
genre explained at the beginning of this chapter. I am using u erance,
ideology, genre and positioning as core concepts in my analysis.
I am interpreting Habermas’s use of ‘speech act’ to be the same as
‘u erance’ in the genre theory of Bakhtin. Speech act is strongly
related to Speech Act theory, and here referential and addressive
aspects are dominant; Speech Act theory is more directed to a
functional view of communication than to a triadic one. Habermas,
however, critiques Speech Act theory and develops the speech act in
a triadic sense — according to Bühler’s language functions in terms
of the corresponding validity claims (Habermas, 1992, 73). I follow
Ongstad (1997, 2004b, 2005), regarding ideology. The diff erence
touches upon a more general debate within modern philosophy, for
instance between Habermas and Ricoeur, where Habermas holds
the more traditional view that ideologies can be directly pointed
to and described, while Ricoeur argues that ideologies should be
seen more as contextual dynamics than as textual and object-like,
and hence more diffi cult to grasp in the process of u ering as a
phenomenon (Habermas, 1984, Ricoeur, 1981). Ricoeur has claimed
that an ideology should be seen not as something we think on, but
from (Ricoeur, 1981). In a text an ideology is in the co(n)-text, giving
value and deeper sense and meaning to the u erance. Such co(n)-
texts can be established by genres (Bakhtin, 1986, Voloshinov, 1973
and Medvedev, 1985) or discourses (Foucault, 1972 and Gee, 1999)
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or any other communicational macro-concept. Habermas’s macro-
concept(s) are lifeworld and system.
Examples
Below are examples of analyses of student teachers’ texts to
illustrate how Habermasian concepts, together with a Baktinian
understanding, can give insight into the development of students’
identity as mathematics teachers. The methodological focus here
is to illustrate how validating of the analysis of positioning can be
supported by use of this understanding.
Students’ and teachers’ beliefs in mathematics education have been
studied and given theoretical and practical focus (Leder, Pehkonen
and Törner, 2002), and also seen as underlying the identities of
teachers of mathematics (Lerman, 2001, 2002). These beliefs have
been identifi ed and diff erentiated between beliefs about mathematics
education, the self and the social context (Eynde, De Corte and
Vershaff el, 2002).
Habermas’s ideas have been used as a theoretical foundation for
approaches to the change in teachers’ and student teachers’ beliefs
and identities, and therefore to a change of practice in seeing teachers
in these changing processes as ‘critical practitioners’ (Atkinson, 2004).
Such approaches, in diff erent ways, seem to place their trust in the
ability of language and rationality to eff ect a be er understanding of
self, teaching and institutional contexts of teaching and learning.
Atkinson (2004) however holds that ‘[i]n […] the struggle to learn
to teach we are concerned with signifi cant psychic and social
processes in the formation of subjectivity […] which I believe cannot
be adequately conceptualised through the idea of transcendent and
rational subjectivity, presupposed by refl ective, refl exive and critical
discourses’. (p. 383).
Beliefs are mostly seen as psychological constructs in mathematics
education literature; they are private and can be observed only
through the student teacher’s actions. This can be done by answering
questionnaires, responding to interviews or u ering in educational
contexts. Student teachers are exposed to diff erent educational
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ideologies, and these will infl uence their development of identities
as teachers. In the Norwegian mathematics classroom there are
diff erent ideologies simultaneously represented by diff erent actors
(Braathe and Ongstad, 2001). Essentially these are ideological
confl icts within which the student teachers are struggling to form
their teacher identities. Identities and subjectivity are seen as
dialogically situated in and formed by genres/lifeworlds, and so can
have many expressions dependent on the context. In my analysis I
will point to the genres, and hence the underlying ideologies, that
can be identifi ed.
The following examples are from an ethnic Norwegian student, here
called Kari. All examples are translated from Norwegian by me. There
are one and a half years between the two examples during which
she has been active in learning about mathematics and mathematics
education.
Task 1
A er fi nishing the fi rst semester the students were given an
individual mathematical task to solve within a week. The task was
divided into two parts. The fi rst was to read and comprehend an
introductory text about number sequences and fi gure numbers, in
which there were some tasks to solve. The second was about using
fi gure numbers to create artistic decorations. The whole task, and the
fi nal text, was on pure and applied mathematics. Below is an extract
from Kari’s work.
The task illustrated below was to fi nd and describe the pa ern of the
number sequences given and explicitly to fi nd the 5th, 6th and 10th
number in the sequences.
h) 4, 8, 16, 32,….
i) 1, 4, 9, 16,….
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The fi gure above shows Kari’s explanations to these tasks.
The wri en text in h) is, in English: ‘The number increases multiplied
by two. The next number then becomes the double of the previous
number’.
The text in i) is: ‘The number increases with the number plus the next
prime odd number. The next number then increases parallel with the
prime odd numbers. That means the previous number plus the next
odd number.’
The expressive aspects of the u erance are related to form and what
this form can symptomatically express. One could interpret how she
uses the dots and arrows at the beginning either as a (rough) dra
to help her own thinking, or as a communicative u erance where
she explains how the next number in the sequence is constructed.
In both cases she uses an informal, nearly oral, genre. It is the same
with the wri en texts that are also in an informal genre, although we
can identify it as a ‘rule giving’ genre; it is wri en in an impersonal
voice, in the present tense and in general terms (it is about ‘the next
number’). In the last part she is se ing up a table for the next numbers
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in the sequence, this can be identifi ed as a technical genre, like those
in her mathematics books. This mix of genres could be described as
voices both from her school experience and also from the teaching of
sequences at teacher training college. The introductory text, and the
textbooks she reads, use a more formal mathematical style.
My reading of the form of this u erance, related to the subjective/
emotional aspect of the development of Kari’s identity as a
mathematics teacher, connects to the postconventional mode of
the teacher education of which she now is part and consequently
the widening gap she experiences in her lifeworld(s), which has
been dominated until now by her experience of learning school
mathematics as a student.
We see that she has got the answers correct, even the square numbers
in i), but this wri en explanation is not easy to evaluate. It is not
obvious that we get the next square numbers by adding the next
odd number to the previous square number. She has not tried to
explain this. It is therefore uncertain whether she has understood the
mathematics behind it; or if she has just seen that it holds good for
some examples and concludes that it holds for all cases; or if she just
remembers it as a fact from either the textbook or from classes; or if she
remember it as a geometrical pa ern from the demonstration in the
class.2 When it comes to the referential aspects of texts/u erances it
must be allowed that they may contain many possibilities, depending
on the researcher’s focus and also on the possible intentions of the
u erer. From the perspective of Habermas one would expect that
she would be able to give reasons for this statement, and therefore it
must be an open judgement which references she would argue for.
The addressive aspects of the u erance are related to normativity,
used here in the sense of usefulness related to the role of the
mathematics teacher in primary school. Usefulness here in a
Habermasian sense includes ethical values concerning teaching and
learning, and including beliefs on mathematics, children, teachers
and others. Kari’s explanatory text can be identifi ed as a ‘rule giving’
genre within mathematics, and as such part of the repertoire of the
teacher in training. According to Habermas I must suppose that in
her u erance she is pu ing forward a normative claim, and that
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she should be able to give rational reasons for this. The normative
claim could be read as part of an instrumental view on teaching and
learning mathematics, which could be seen as an element of her
lifeworld as part of her belief in school mathematics teaching.
My interpretation of Kari’s u erance is that it is dominated by
the expressive aspect, as its form symptomatically expresses her
insecurity with the formal mathematical genre, and at the same
time her incomplete explanation of the mathematics. All her wri en
texts are descriptive processes, which again connects to an oral
genre. The analysis of positioning, then, will evaluate the u erance
as dominant expressive, but the two other aspects are present and
as important when it comes to understanding her appropriation of
the mathematical genre/register. Her u erance as act is giving an
answer to a task set, and therefore her reference point is answering as
correctly as possible within the ‘task’ genre in (school) mathematics.
Task 2
The second example is in the same mathematical subgenre as the
fi rst example, but done a year and a half year later than the fi rst.
It is presented here to illustrate Kari’s possible development in
positioning as a teacher of mathematics.
The fi nal exam in the compulsory mathematics course for student
teachers, at Oslo University College, currently consists of two separate
parts: a one week ‘home’ exam where a group of four students give
a co-wri en presentation on a given task, and an individual oral
examination. As a condition for passing the fi nal exam the students
have to do an individual six hour mathematics test. This test contains
both didactical and mathematical questions. In this test there was a
task on number sequences:
Two number sequences are given:
1) 2,7,12,17,….
2) 1,3,9,27,….
a) Find the next two numbers in the sequence.
b) Find the recursive and the explicit formula for the two sequences.
c) Explain why the formulas are correct.
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Hans Jørgen Braathe
233
Kari’s texts on a) and b) are similar to her texts from the individual
task on number sequences referred to earlier. However to give the
recursive and explicit formulas for the sequences she gives the explicit
formula initially in its general form. This could be regarded as a sign
that she has had some experience with the recursive formula, but less
with the explicit one and therefore has to look it up in her textbook
and then translate it into the sequence at hand. This whole text is
without any explanations; these are all implicit. Here she is drawing
both on her experience as a student of mathematics in class, and also
from theory which she has been reading. However I see the overall
form much as part of her experience of the school mathematics genre.
She is still not into the genre of textbook mathematics in or able to
explain her thinking in that genre.
Below is her answer to c):
To explain the explicit formula for the arithmetic sequence she
fi rst explains the diff erent parts of the formula by translating the
mathematical symbols into everyday language.
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234
To explain the (n-1) part she writes: ‘The chosen number –1, to fi nd
the 10th number one has to divide nine times. 10 - 1 then.’ She explains
the general case by an example. To explain the formula she writes:
‘We therefore add the fi rst number in the sequence with the number
of times we shall make a jump ahead multiplied by the diff erence.’
In this u erance she is nearly tactile in the way she explains jumping
ahead along the number sequence.
This u erance, as in the previous example, uses a mix of genres.
However one genre seems dominant, the ‘Explaining’ or ‘Introduction’
genre as she uses ‘Here you take…’, addressing a individual ‘you’,
and also an inclusive ‘we’ in ‘We therefore add…’, and by both
explaining the general by an example and the tactile metaphor she
uses in explaining the explicit formula. This is a genre which is used
frequently in the mathematics texts she is meeting in the study, and
explaining by examples is used a lot both in educational texts and also
in teaching sessions at the college and in the practice schools. One
could see this as a sign of her appropriating the voices of didactical
genres. Her explanation of the formulas for the geometric sequences
is similar. Her u erance this time must also be seen as an act giving
an answer to a task set to her, but here I read her giving as correct
as possible an answer to this within the ‘task’ genre in mathematics
education, and as such it is positioned with an addressive dominance.
The normative claim identifi able in her didactical voice can again
be read as part of an instrumental view on teaching and learning
mathematics. This could be seen as an element of her lifeworld in
her belief in mathematics as a subject where you have to learn the
rules, and where you have true or false answers. Teacher education
in mathematics is arguing for a more relativistic and context sensitive
view of mathematics and mathematics education (Cooney, 2001,
Lerman, 2001, 2002).
As Kari is appropriating the mathematical genre/register, to
communicate subject ma er, she hopefully is developing her identity
as a teacher of mathematics. A teacher of mathematics must have
a language to communicate mathematics to the learners. To learn
theory is to learn a language in which to be able to communicate
about the subject ma er. From the examples above it can be seen
that she still has an oral, elementary school mathematics way of
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Hans Jørgen Braathe
235
presenting her mathematics. She is describing processes and is not
focusing on the mathematical objects as such.
Conclusion
The analysis of positioning, as it is inspired by Bakhtin (1986) and
developed by Ongstad (1997, 2004b), has been viewed here with the
concepts within Habermas’s theory of formal pragmatics. Habermas’s
validity claims have been used as ‘tools’ for validation by focusing on
the three aspects separately in the fi rst example, but trying to see
them from the dominant aspect in the second. The three aspects will
always simultaneously be present, but the dominance of the three
will vary dynamically. The dominance of the u erance will also be
related to the triadic dominance of the genre that the u erance is
u ered within. This last point is not taken care of by Habermas’s use
of lifeworld, by focusing each aspect of the analysis of positionings as
an approach will risk being a contradiction in terms since the whole
idea is to try to move on from the one-dimensional kind of analysis.
Footnotes:
1 Recent research has also shown that the Bakhtin circle has been
infl uenced by Bühler’s organon model in their development of the
triadic thinking on the dialogic theory of u erance (Brandist, 2004).
2 It can be demonstrated that her rule is correct: Look at: n2 = ((n-1)
+ 1)2 = (n-1)2 + [2(n-1) + 1], or in her words: ‘The number increases
with the number plus the next odd number. The next number then
increases parallel with the odd numbers. That means the previous
number plus the next odd number.’
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Contact
Hans Jørgen Braathe
Oslo University College
Faculty of Education
HansJorgen.Braathe@lu.hio.no
239
Kirsti Kislenko
239
Abstract
This chapter introduces a study on students’ beliefs and a itudes towards
mathematics teaching and learning in Norway and Estonia. It presents
some previous studies related to the main study and gives an overview of
the planned methodology to be used in this research. As the main study
in Estonia took place in the spring of 2006 many aspects are still under
consideration and will be fi nally decided as the study develops.
Key words: Aff ective domain, belief system, comparative design,
KIM-study, Likert-scale questionnaire, mathematics education,
methodological triangulation, students’ beliefs
Introduction
The area of my study is students’ beliefs about mathematics teaching
and learning. A itudes play an important role in mathematics
education (McLeod, 1989). During the last twenty years research
about beliefs and a itudes has grown a great deal and research in
this area has been carried out in many diff erent countries (Lester Jr.,
2002). However, there is still a lack of investigation into students’
views of mathematics in Estonia. Hopefully, the outcomes from my
research will help to enlighten the situation in mathematics classes
and the suggestions will lead to new ideas on how to improve
teaching and learning in Estonian school mathematics.
STUDENT’S BELIEFS AND
ATTITUDES TOWARDS
MATHEMATICS TEACHING AND
LEARNING – AN INTRODUCTION
TO THE RESEARCH
Kirsti Kislenko, Agder University College, Norway
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STUDENT’S BELIEFS AND ATTITUDES TOWARDS MATHEMATICS ...
240
The learning outcomes of students are strongly related to their beliefs
and a itudes about mathematics (Furinghe i and Pehkonen, 2000;
Leder, Pehkonen and Törner, 2002; Pehkonen, 2003; Schoenfeld,
1992; Thompson, 1992). Thus, to improve teaching and learning of
mathematics it is important to be aware of the infl uence of the beliefs
and a itudes that are held amongst the students.
The general aim of the research is to fi nd out what kind of beliefs and
a itudes towards mathematics Estonian and Norwegian students
hold, why these kinds of beliefs are held and whether a relationship
can be found between students’ beliefs and a itudes and their
mathematical performance.
Some research questions
Based on the aim of the study some research questions can be
formulated:
• What kind of beliefs and a itudes towards mathematics teaching
and learning do students from one urban area in Estonia and
students from one urban area in Norway hold?
• Why do students from one urban area in Estonia hold these
kinds of beliefs and a itudes towards mathematics teaching and
learning?
• Are there any diff erences and similarities between the students’
beliefs and a itudes from one urban area in Estonia and one
urban area in Norway? If there are, then what kind of diff erences
and similarities exist and what may be the reasons for these?
• Are the students’ performances in mathematics related to their
beliefs about mathematics?
As can be seen from the research questions two countries are included
in the study – Estonia and Norway. The Estonian case is based on the
students from one urban area in Estonia and the Norwegian case
is based on the students from one urban area in Norway. Students
from grade 7, 9 and the fi rst year in upper secondary school will be
investigated.
241
Kirsti Kislenko
241
Brief theoretical overview
In the area of aff ect (which covers concepts like belief, a itude,
emotion, value, motivation, feeling, mood, conception, anxiety, etc)
a single coherent well-defi ned theoretical framework is not found
(Goldin, 2004; Vinner, 2004). Some researchers focus on the role of
aff ect in mathematical thinking generally, and on problem solving
particularly (e.g. McLeod, 1989). Another view in research is to
concentrate on the role of emotions and a itudes in the learning
process and emphasise the social context (e.g. Op ’t Eynde, De Corte
and Verschaff el, 2001). Because of this distinction in research areas and
the diff erences in the epistemological perspectives of the researchers
‘there is a considerable diversity in the theoretical frameworks used in
the conceptualisation of aff ect in mathematics education’ (Hannula,
2004, p. 107). Therefore, the aim is not to present one basic theoretical
framework for the study in these following sections but to introduce
diff erent concepts and notions related to the domain of aff ect.
The defi nition of belief
The notions belief and a itude cannot easily be defi ned. Leder and
Forgasz claim that:
In everyday language, the term ‘belief’ is o en used loosely and
synonymously with terms such as a itude, disposition, opinion,
perception, philosophy, and value. Because these various concepts
are not directly observable and have to be inferred, and because of
their overlapping nature, it is not easy to produce a precise defi nition
of beliefs. (2002, p. 96)
Some of the defi nitions that can be found in literature relate the term
belief to the notions of motivation and conception. For example,
Kloosterman (2002) sees the direct connection between the belief
and eff ort. Hart (1989) and Thompson (1992) point out that beliefs
can be understood as a subclass of conceptions and on the other
hand the conceptions are explained as a subset of beliefs (Pehkonen,
1994). Probably the easiest to understand and thus the broadest, is
the defi nition given by Rokeach (1972). He says that ‘a belief is any
simple proposition, conscious or unconscious, inferred from what
a person says or does, capable of being preceded by the phrase “I
believe that…”’ (p. 113).
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242
I would rather reject McLeod’s (1989) idea to see a person’s aff ective
domain as an aggregate of beliefs, a itudes and emotions and
agree with Pehkonen (2003) who claims that a person’s beliefs can
be located somewhere between two domains. He notes: ‘beliefs are
situated in the “twilight zone” between the cognitive and aff ective
domain, since they have a component in both domains’ (Pehkonen,
2003, p. 1).
According to the dictionary, belief is something that one believes in
strongly, connected with faith, something that does not change easily.
A itude can be the mentality, the view of some phenomenon and
emotion is unsteady blind impulse which can vary to a great extent.
‘Beliefs and a itudes are generally thought to be relatively stable and
resistant to change, but emotional responses to mathematics change
rapidly’ (McLeod, 1989, p. 246). In this paper the defi nition of the
belief is adopted from Pehkonen (2003) who understands ‘beliefs as
an individual rather stable subjective knowledge, which also includes
his/her feelings, of a certain object or concern to which tenable
grounds may not always be found in objective consideration’ (p. 2).
The defi nition of the a itude given by Triandis (1971) that ‘a itudes
involve what people think about, feel about, and how they would
like to behave toward an a itude object’ (p. 14, emphasis original)
seems to be an appropriate one for this context.
Belief system
In the questionnaire used in my study (see the discussion below)
there are statements that are grouped into classes. This assortment
is based on the idea of belief system. Talking about mathematics, an
individual’s mathematics-related belief system is called his/her view
of mathematics (Pehkonen, 2003; Schoenfeld, 1985). The human
person’s belief system is dynamic, changeable and when individuals
evaluate and assess their experiences and beliefs, then they are
restructuring their system continuously (Thompson, 1992). Based
on observations, Green (1971) has pointed out three dimensions of
belief system where he emphasises the relations between the beliefs
in the system.
First, the structure of the belief system is quasi-logical. Some beliefs
are primary and some derivative. For example, a student believes that
243
Kirsti Kislenko
243
mathematics is useful for his/her life (this is a primary belief). And
thus she/he thinks that it is important (1) to work hard in mathematics
lessons, and (2) to work with problem solving, and to try to relate the
exercises to everyday life (these are derivative beliefs).
Secondly, Green (1971) talks about central and peripheral beliefs
in the system. Central beliefs are more important and held most
strongly, whereas peripheral ones can be changed more easily. From
my point of view it is from experiences, practice and affi rmation
that one’s own beliefs become more central. For instance, a new
teacher at school might hold more peripheral beliefs, which are more
changeable and negotiable. Experienced teachers’ beliefs are mainly
more central and thus more deep-rooted.
Thirdly, Green (1971) uses the word ‘cluster’. It means that beliefs
occur in clusters, they are not independent of each other. The
clusters are in weak relationships or are not connected at all and this
phenomenon can explain the contradictions in pupils’ and teachers’
beliefs (Pehkonen, 2003).
Previous studies
KIM study
In 1995 the KIM1 project collected national data on students’
understanding of key concepts in the national mathematics curriculum
in Norway. Students’ performances related to one particular area
of mathematics, namely Measurements and Units, were linked to
their a itudes. 105 grade 6 classes and 90 grade 9 classes took part.
Amongst those students approximately 900 were selected according
to their birthdays. The study is based on the data from 891 grade 6
students and 893 students from grade 9. The same schools were asked
to participate in the a itude study later in that school year which
made it possible to compare the mathematics test to questions about
their thoughts about mathematics and the teaching and learning
of mathematics. The conclusions of this project were as follows:
those pupils who stated a positive interest towards mathematics,
on average, performed be er on the mathematics test than their
fellow students. There was a strongly signifi cant connection between
the performance in the test and the self-confi dence in both grades
(Streitlien, Wiik and Brekke, 2001). Therefore, it seems that positive
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244
a itudes towards mathematics lead in general to greater motivation
to learn, and vice versa, high performance in mathematics, combined
with the experience that one achieved well in the subject, leads to
positive a itudes toward mathematics.
Pilot study
I carried out a pilot study about students’ beliefs in mathematics in
Norway in the autumn of 2004. As a pilot project involves a much
smaller sample because of the issues related to time and cost (Gorard,
2001) I therefore chose, largely because of familiarity with the teacher
and the convenient location of the school, one ordinary Norwegian
class from one urban area in Norway in grade 9 and gave the
students the questionnaire. There were 25 pupils who participated
in this study. I used with permission the aforementioned KIM
questionnaire (Streitlien, Wiik and Brekke, 2001) in order to be able
to compare the results. Since my study was a pilot, the instrument
was the main focus. I interpreted the results from the pilot study as
revealing that 9th grade students in this class show more positive
beliefs about mathematics than students from the 9th grade about
ten years ago. Further investigation is required to reveal whether this
is just a coincidence or if it is a sign of a change over time in a itudes
and beliefs. My aim is to do this as the study progresses (for more see
Kislenko, Grevholm and Breiteig, 2005).
Methods and methodology
Methodology
In my study, methodological triangulation is used. Methodological
triangulation – that is to use diff erent methods on the same object
of the study – is recommended because dependence on only one
method can bias or distort the researcher’s illustration of the case she/
he is studying (Cohen, Manion and Morrison, 2000). Alba Thompson
(1992) and Gilah Leder and Helen Forgasz (2002) mention that several
researchers apply this methodology to their work for investigating
beliefs and a itudes (e.g. Frykholm, 1999; Raymond, 1997; Williams,
Burden and Lanvers, 2002). My research methodology is highly
similar to the methodology of the study that Erkki Pehkonen and
Günter Törner carried out in 1994 for investigating German teachers’
beliefs. They emphasise the triangulation in their data-gathering
process and their instruments were interviews, questionnaires and
245
Kirsti Kislenko
245
self-estimations (Pehkonen and Törner, 2004). I agree with their
justifi cation where they note that:
The reason for this decision [to use methodological triangulation]
was the fact that the research methods seem to act as a fi lter through
which a researcher experiences his/her surroundings selectively. The
simultaneous use of several data-gathering methods adds to the
researcher’s possibilities to grasp complex reality. (Pehkonen and
Törner, 2004, p. 26)
Therefore, two types of research strategy are used in my study
– quantitative (questionnaire) and qualitative (interview, lesson
observation). The questionnaire is used to investigate students’ beliefs
and a itudes towards mathematics teaching and learning, thus it is
related to the fi rst, the third and the fourth research question. Lesson
observations and interviews are planned to enlighten the perspectives
of students and teachers. This aims for a general picture of the
mathematics classroom and possible relations between a itudes
and the practical situation, and to have ideas on how teaching and
learning can be improved based on the information that will be
gathered. It is related to the second and the third research question.
Research participants
The study in Norway was carried out in the spring of 2005 as one
part of the LCM-project within its KUL2 program. My interest there
was students’ beliefs and a itudes towards mathematics teaching
and learning. To keep as many similarities with the KIM project
from 1995 as possible (for the comparison analysis) the students in
grades 7, 9 and the fi rst year in upper secondary school from one
urban area of the country participated in the research. These were
the same students who participated in the study carried out by Irene
Skoland Andreassen (2005) in which mathematical performance of
the students was explored. The number of schools which took part,
and which were related with the KUL program, was 6 and around
370 students completed the questionnaire.
As the research in Estonia took place only in the spring of 2006 the
following description of the data collection process is preliminary
and still under consideration. The same number of schools (around
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STUDENT’S BELIEFS AND ATTITUDES TOWARDS MATHEMATICS ...
246
10) will take part. These schools will be selected from the schools
that collaborate and participate in the school-practice program with
Tallinn University (my home university) and which are located in
urban areas. Some part of the data in the study will be collected
through the questionnaire and it will be given to all participants.
Three classes (7th grade, 9th grade and fi rst year in upper secondary
school) will be picked out from the participating schools and in these
three classes lesson observations and interviews (with students and
teachers) will be carried out.
Data sources
Questionnaire
A questionnaire is a common instrument to study beliefs and a itudes
(Leder and Forgasz, 2002) and several researchers have used it as a
main tool (e.g. Graumann, 1996; Pehkonen, 1994; Pehkonen, 1996;
Pehkonen and Lepmann, 1994; Perry, Howard and Tracey, 1999;
Tinklin, 2003; Vacc and Bright, 1999; Williams, Burden and Lanvers,
2002). In my study one of the instruments for investigating students’
beliefs is the questionnaire elaborated in 1995 for the aforementioned
project called KIM. The questionnaire uses Likert-scale type
responses.
In the pilot study the original questionnaire from the year 1995 was
used but in the main study in Norway there were some changes.
Based on the research questions and the pilot study classes, students’
beliefs about the environment in school and diff erences between
boys and girls were eliminated. Being a participant in the KUL-
LCM project in Norway, the questionnaire was extended by adding
a section on students’ beliefs about ICT. This extension gave 2 extra
classes of questions within 14 extra items. Thus, fi nally there were 126
questions divided into 13 groups in the questionnaire. These were:
mathematics as a subject; learning mathematics; an individual’s own
mathematical abilities; an individual’s own experiences (security)
during mathematics lessons; teaching of mathematics; learning a
new topic in mathematics; environment in class; teaching tools in
mathematics lessons; using a computer in an individual’s spare
time; using ICT in mathematics (computer); an individual’s own
evaluation of the importance of mathematics; evaluation for teaching
mathematics; mathematics and the future.
247
Kirsti Kislenko
247
The questionnaire was web-based. This meant that the students
fi lled in the questionnaire that was made available on an Internet
web-page. All the students were issued with the codes that they used
to log into the questionnaire page.
The questionnaire used in the Norwegian study will be translated into
Estonian and will be used in the study in Estonia. As one of the aims
of this study is to make a comparative analysis then the translation
must be carried out with a high degree of caution. Gorard’s (2001)
suggestion will be followed during the study where he recommends
that:
… if you are working in one language and translating your instrument
into another language before completion (a common process for
overseas students), then use the techniques of back translation as
well. In this, the translated version is translated back into the original
language by a third person as a check on the preservation of the
original meaning. (p. 91)
As a check on this method a pilot study in one class in Estonia will
have been carried out during the autumn of 2005.
Lesson observation
The aim of the lesson observations is to build a general picture of the
situation in the observed mathematics classrooms in Estonia. I am
pursuing the research with the idea that some diff erences between
the students’ a itudes in diff erent countries will emerge because of
the economic, social and cultural diff erences that are infl uential on
the teaching and learning of mathematics. But to be aware of these
economic, social and cultural diff erences, lesson observations are
necessary. Some lesson observations are also planned in Norway
in relation to the same issue. These will not be as frequent and
consistent as in Estonia but they are necessary for highlighting the
classroom culture. The importance will emerge when the comparison
is discussed. My aim is not to go deeply into issues in the classroom;
so much as to give a generalized view of behaviours, relations and
events that are happening in the classroom.
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248
Observations in Estonia will last about two months. At present the
decision is that not all lessons in all classes will be examined. The
proposed number of the lessons observed in a week will be about
10.
Interviews
Interviews are not planned in the study in Norway. These will be
carried out only in the Estonian research. Two classes of people will
be interviewed – teachers and students. Interviews with the teachers
will be individual. These will be carried out at the same time as the
lesson observations and also in advance. Interviews with teachers are
planned to gain a general picture of the classroom culture and their
thoughts about mathematics and being a mathematics teacher. Most
of the interviews with the students will be group interviews. The idea
so far is that students will be grouped according to their answers a er
they have fi lled in the a itude questionnaire. The groups will include
the students who hold more or less similar a itudes in mathematics
(for example groups like ‘positive’, ‘rather neutral’, and ‘negative’).
Personal interviews with the students are not planned in the study at
present but are still under consideration.
Interviews with the teachers will start before the lesson observations
begin in order to have so-called ‘starting point ideas’ from teachers.
Further interviews will take place every two weeks. The frequency
of the group interviews with the students is still open and will be
decided later.
Mathematical tests
In relation to the research question ‘are the students’ performances
in mathematics related to their beliefs about mathematics?’ the main
source of the data in Norway is from Irene Skoland Andreassen
who carried out mathematical tests amongst the same students who
participated in my a itude study in Norway (Andreassen, 2005). This
enables me to analyse the connection between students’ beliefs and
mathematical performances. I plan to contribute with one student
in Estonia who is willing to help me to carry out mathematical tests
in Estonia before the a itude study, in the autumn of 2005. This
possibility is still under consideration.
249
Kirsti Kislenko
249
Empirical data
The data collection in Norway has already fi nished as the web-
questionnaire was closed on 15 June 2005. Some primary tentative
results from this study will be presented at the NORMA05
conference in Trondheim in Norway and published in the conference
proceedings.
Acknowledgement
The LCM project is supported by The Research Council of Norway
within its KUL programme, Kunnskap, utdanning og læring
(Knowledge, Education and Learning, Norges Forskningsråd, project
no. 157949/S20).
Footnotes
1 KIM – Kvalitet I Matematikkundervisningen, translated as Quality
in Mathematics Teaching.
2 KUL – Kunnskap, utdanning og læring, translated as Knowledge,
education and learning.
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Contact
Kirsti Kislenko
Agder University College
Faculty of Mathematics and Science
Norway
kirsti.kislenko@hia.no
253
Dimitris Chassapis
253
Abstract
Each scientifi c discipline constructs its own objects of inquiry: selecting,
defi ning and describing real world phenomena and thus conceptually re-
constituting them as scientifi c objects. This constructive process makes up
both the objects of inquiry and the scientifi c discipline itself. This process
is primarily carried out by the scientifi c research activity but is critically
infl uenced not only by the researchers but also by diff erent kinds of
knowledge-users, among whom policy makers and teachers play distinct
roles. This chapter, building on relevant research reviews, aims to provide
an exploratory look at mathematics education research from a standpoint
of spo ing the aspects, which are selected and emphasized as essential
characteristics of the phenomena studied. It is claimed that contextual aspects
as represented by social, political and cultural factors have not been given
suffi cient a ention by mainstream mathematics education-related research
in contrast to factors related to either psychological or instrumental aspects
of mathematics thinking, learning and teaching, resulting in negative
impacts on mathematics education knowledge.
Keywords: Cultural factors, political factors, social factors, scientifi c
objects
THE OBJECTS OF MATHEMATICS
EDUCATION RESEARCH:
SPOTTING, AND COMMENTING
ON, CHARACTERISTICS OF THE
MAINSTREAM
Dimitris Chassapis, Aristotle University of
Thessaloniki
254
THE OBJECTS OF MATHEMATICS EDUCATION RESEARCH: SPOTTING, ...
254
Introduction
A few preliminary remarks, briefl y formulated, are necessary to
clarify the viewpoint adopted in the arguments expressed here.
Each scientifi c discipline has its own objects of inquiry, constructed
by the discipline and constituting fundamental components of the
discipline itself. Each scientifi c discipline constructs its own objects
of inquiry selecting, defi ning and describing real world phenomena
and thus conceptually re-constituting them as scientifi c objects.
This constructive process makes up both the objects of inquiry
and the scientifi c discipline itself, since it presupposes using, and
contributing to the development of, a conceptual system as well as a
research methodology appertaining to the particular discipline. This
process is primarily carried out by the scientifi c research activity
but is critically infl uenced not only by the researchers as the leading
actors but also by diff erent kinds of discipline knowledge-users,
among whom policy makers and teachers play decisive roles. In this
account, the scientifi c research activity, however, as it is defi ned by
the mainstream view of all involved agents:
• identifi es, describes and grades the objects considered as
important for inquiry phenomena;
• defi nes the relevant general issues and the specifi c questions
that are posed ;
• opts for the methods and instruments proper for their
investigation and
• makes and assigns meaning to the results of these investigations,
providing the framework for their evaluation and their
utilisation, both in the scientifi c and the social sphere.
This chapter examines the objects of mathematics education, aiming to
raise questions concerning the orientation of mathematics education
research in relation to the construction of its scientifi c objects. This
presupposes that the pa erns of production, accumulation and use
of mathematics education-related knowledge is today a major issue
of concern.
An indispensable comment is in order. It is obvious that any a empt
to deal with such an issue exhaustively is doomed to failure. Therefore
255
Dimitris Chassapis
255
focal points have to be selected, choices made, and constraints
encountered, all of which refl ect the personal limitations, tastes and
biases of the author. Although it is intended to approach the topic
as a cartographer rather than as a critic, the thoughts expressed
necessarily involve not only objective data but also statements of a
fairly personal nature.
The foundation of mathematics education as a research fi eld, and
accordingly as a scientifi c discipline, may be considered as an output
of the great a empts to reform mathematics education that took place
during the early 1960s. In 1968 Educational Studies in Mathematics
was inaugurated and the next year the Journal for Research in
Mathematics Education, the two earliest, formative and leading
journals of the discipline. Ever since, mathematics education has
become established as an academic discipline on the international
scene, as confi rmed by the existence of the growth in university
departments, research grants and projects, academic programmes
and degrees, national and international organisations and bodies,
many journals and publication series, national and international
conferences, and so forth, all devoted to mathematics education.
Mathematics education as a scientifi c discipline is nowadays in
a mature stage of development and its fundamental features can
now be considered and discussed. In my opinion, mathematics
education as a scientifi c discipline is to be considered in a non-
restrictive view as the study of phenomena and processes actually
or potentially involved in the learning and doing of mathematics of
any kind, and the study of how this learning and doing can be, and
is, infl uenced and fostered among others by teaching, by the use of
media, by diff erent representations, or by the social organisation of
mathematical activity.
Thus mathematics education is clearly a certain area of human
activity whose content, object and goal is mathematics at diff erent
levels and in diff erent forms. This does not suggest, in any way, an
existence of mathematical knowledge outside and independent of
the respective activities. Therefore, mathematics education can be
considered as the study of the relationships between mathematics
and human beings taken in their widest variety.
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256
Keeping this broad defi nition in mind, three mutually dependent
conceptual fi elds may be considered constitutive of the mathematics
education object of study:
• a mathematical fi eld, being the content of learning and doing
mathematics,
• a psychological fi eld, since learning and doing mathematics
involves individual processes, and
• a sociological fi eld, since learning and doing mathematics are
social processes.
A balanced formation and development of mathematics education
as a discipline has to articulate fundamental aspects of both
psychological and sociological fi elds with mathematical concepts
and processes.
On the contrary, as will be claimed, mainstream mathematics
education-related research has ignored critical social, political and
economic aspects of its objects of inquiry. Kilpatrick (1992), for
instance, clearly states that the origins of mathematics education
lie jointly in mathematics and psychology – in mathematics for a
grounding in the content being studied, and in psychology for the
tradition of disciplined inquiry into the workings of the mathematical
mind. Thus, mainstream mathematics education research devaluing
the sociological aspect emphasises individualisation of learning and
doing mathematics; accordingly shaping its objects of inquiry and
consequently the related knowledge produced.
Such a claim is not new and many scholars have expressed similar
concerns. Tate (1997), for instance, argues that mathematics education
research tends to be narrowly focused, restricted to the disciplines of
mathematics and psychology. Reyes and Stanic (1988) and Secada
(1992) claimed that mathematics education researchers have virtually
ignored issues of poverty and social class. And Hudson (2001) agrees
in stressing that this phenomenon can be clearly seen in the fact that
the International Group for the Psychology of Mathematics Education
(PME) has become the major international forum for mathematics
education research.
257
Dimitris Chassapis
257
A review of selected surveys of mathematics education research
studies
There have been a number of surveys of mathematics education
research that have addressed some aspects of the aims of our venture.
The concerns of these surveys vary from being mainly pedagogical
to sociological and they also a empt to capture qualities pertaining
to the research activity characteristic of the fi eld. These surveys
contribute to sketching a rough picture of the fi eld, pointing to some
of its developments over time and are selectively summarised below
from the viewpoint of this chapter.
Kieran (1994), in one of the early review studies, off ers a retrospective
look at mathematics education research, presenting interviews with
two leading researchers looking back over a period of 25 years.
The interviews are followed by an analysis of articles published in
Journal for Research in Mathematics Education (JRME) in its fi rst 25
years (1969–1994) organised on the remarks made by the interviewed
researchers. She argues that in the time period examined there has
been a shi in mathematics education research towards integrating
learning with understanding and studying them together, as well
as an increasing orientation towards interactionist studies drawing
mainly on ideas inspired by Vygotskian approaches.
Niss (2000), in his plenary address to the 9th International
Congress on Mathematical Education, presented an account of
research on mathematics education ‘based on sample observations
obtained from probing into research journals, ICME (International
Congress on Mathematical Education) proceedings and other
research publications from the last third of the 20th century’. He
gave examples from these publications in his review of issues and
questions; objects and phenomena; research methods; results; and
emerging problems and challenges of the mathematics education
research fi eld. Whilst Niss did not claim to have been systematic,
he traced developments across the years from mathematics curricula
and ways of teaching it (research in the 1960s and 70s), to studies of
learning and its conditions at the level of individual students and of
mathematics classrooms as conditioned by a variety of factors such as
mathematics as a discipline; curricula; teaching; tasks and activities;
materials and resources, including textbooks and information
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THE OBJECTS OF MATHEMATICS EDUCATION RESEARCH: SPOTTING, ...
258
technology; assessment; students’ beliefs and a itudes; educational
environment, including classroom communication and discourse;
social relationships amongst students and between students and
teacher(s); teachers’ education, backgrounds, and beliefs; and so forth
(research in the 1980s). Beyond these he reviewed outside social,
cultural and linguistic infl uences on the individual and the classroom
(research in the 1990s). Overall, Niss concluded that ‘today, research
on the learning of mathematics is probably the predominant type of
research in the discipline of mathematics education’.
Adopting a diff erent conceptual and methodological viewpoint,
a study of Lubienski and Bowen (2000) provides a broad look at
mathematics education research published between 1982 and 1998.
They utilise the database of the Educational Resources Information
Center (ERIC) to count and categorise more than 3,000 articles
from 48 educational research journals and to identify the number
of published articles relating to gender, ethnicity, social class, and
disability. A ention has also been given to grade levels, mathematical
topics, and general educational topics in conjunction with each social
group considered. The main conclusion of Lubienski and Bowen is
that the majority of mathematics education research over the period
examined seemed to focus on student cognition and outcomes, with
less a ention to contextual or cultural issues. In addition, research
on gender was found to be more prevalent and integrated into
mainstream US mathematics education research, in comparison with
research on ethnicity, class, and disability. In particular, Lubienski
and Bowen found that student cognition was the most popular of
the categories considered, relating to 49% of the 3,011 articles (by
contrast only 18% of all ERIC items related to cognition) followed
by student achievement with 23% of the related articles. Teacher
actions (20% of the articles), curriculum (17%), technology (15%),
student characteristics (15%), and student aff ect (12%) were also
found to have received signifi cant a ention. Teacher education
(6%), student assessment (5%), educational environment (5%), and
students in classrooms (4%) were found to have received the least
a ention by mathematics education research. Relative to the entire
pool of 3,011 published articles, gender, ethnicity, social class, and
disability were found to have received less a ention by comparison
with curriculum, technology, and teacher education. Only 623 of the
259
Dimitris Chassapis
259
3,011 mathematics education articles were found to be related to
at least one of these categories. Gender received the most a ention
with 323 articles, and disability was second with 193 articles. There
were 112 articles pertaining to ethnicity and 52 discussing social
class. Although slightly more than half of the 3,011 identifi ed articles
appeared in mathematics education journals, the non-mathematics
education journals were found to contain more than two thirds of
the mathematics education articles considering social groups. Class,
ethnicity, and disability received relatively less a ention among
the 3,011 mathematics education articles than among the 510,241
items in the entire ERIC database, but the percentage of articles on
gender among the 3,011 articles was about double the percentage
of ERIC items relating to gender (see detailed data in Appendix 1).
Overall, the results paint a broad, rough picture showing a body of
research that gives considerable focus to cognition and achievement,
primarily in Grades K–12, with signifi cant a ention to integers and
problem solving. Authors conclude that mainstream mathematics
education research has tended to focus on ‘cognition without context
or culture’.
The study of Lubienski and Bowen expanded in time span and social
group categories was considered by Chassapis (2002). Adopting a
similar rationale of collection and analysis of data, this study provides
data and comments on mathematics education research articles
published in educational journals from various categories over thirty
years, from 1971 to 2000. The ERIC Database was also utilised as a
source and articles were identifi ed which reported or commented on
research relating to social groups defi ned along lines of social class or
socio-economic status; gender or sex; ethnicity or race; demographic
minority; and economic or educational disadvantage. Topics related
to teaching and learning of mathematics most frequently appearing as
focal points of mathematics education research were also identifi ed.
Based on the results of a simple quantitative analysis of the relevant
data series, it is argued that in mathematics education-related research
the issues of social groups are given insuffi cient a ention. However
they are relatively equivalent to the a ention given in educational
research as a whole throughout the time period considered. On the
other hand, the issues of mathematics achievement and learning
were identifi ed as the primary topics of interest over the thirty years
260
THE OBJECTS OF MATHEMATICS EDUCATION RESEARCH: SPOTTING, ...
260
of the study (see detailed data in Appendix 2). Chassapis concluded
that this prevailing approach in the fi eld of research in mathematics
education unavoidably results in less probing investigation into the
particular interrelationships of mathematics education to contextual,
social, political and cultural factors.
The study of Hanna and Sidoli (2002) explores the origins of the
international journal Educational Studies in Mathematics (ESM) in
1968 and traces its later development as it responded to changes in
mathematics education. First, in chronological order, the contributions
of its editors are examined in defi ning its spirit, policy and procedures,
as they directed its growth and its transformation into a leading
journal of research in mathematics education. Secondly, a statistical
profi le of ESM articles is presented by content area, educational
issue, level of schooling and research method. Furthermore a close
look at the special issues of ESM, each dedicated to a single topic, is
carried out analysing their refl ections of the changing concerns of the
mathematics education community.
The educational issue categories considered were: 1) aff ective issues,
2) cognitive issues, 3) epistemological issues, 4) didactical issues,
5) pedagogical issues, 6) reform and curricular issues, 7) social and
cultural issues, 8) historical analyses, 9) technology, 10) language, 11)
imagery and visualisation, 12) gender and ethnicity, 13) qualitative
methodology, 14) quantitative methodology, and 15) assessment. As
emphasised by the authors, the issue categories used were chosen
relatively recently by the editors of ESM to refl ect the issues dealt
with in papers submi ed to ESM and those thought by the editors
to be most relevant to current research in mathematics education.
Thus, these current educational issues are not necessarily those that
were high on the agenda of mathematics education researchers one
or two decades ago. Given this proviso, Hanna and Sidoli conclude
that cognitive issues, which had already a racted fairly signifi cant
a ention in the decade 1970–79, became dominant in the later
decades 1980–89 and 1990–99, where they accounted for over 29%
and over 32% respectively of all ESM papers. Like cognitive issues,
social issues showed a steady growth in relative weight over the
three decades. Concern for social issues in the teaching and learning
of mathematics, as refl ected in ESM, had been almost non-existent
261
Dimitris Chassapis
261
in the fi rst decade, but rose to just under 11% by 1980–89 and then
to almost 15% by 1990–99. Authors also underline that the issue of
gender and ethnicity, which accounted for just over 4% of the papers
in the fi rst decade of ESM, rose to over 11% in 1980–89, but then
dropped back to a li le under 6% in 1990–1999 (see detailed data in
Appendix 3).
Lerman and Tsatsaroni (2004) provide a survey of mathematics
education research based on a representative sample of the last 12
years of the papers published in the Proceedings of the International
Group for the Psychology of Mathematics Education (PME), and of
two leading journals: Educational Studies in Mathematics (ESM) and
Journal for Research in Mathematics Education (JRME). Authors
have developed a particular tool for recording and analysing the
texts of the mathematics education research community by drawing
broadly on Bernstein’s work. A distinction has been made between
an orientation towards the theoretical or towards the empirical,
according to which domain has been privileged in the reported
research. Lerman and Tsatsaroni found out that the predominant
theories throughout the period examined for all three types of text
are traditional psychological and mathematics ones, but there is an
expanding range of theories used from other fi elds. Mathematics
education texts were categorised as follows, using the type of
theory employed: psycho-social studies, sociology/sociology of
education/socio-cultural studies and historically orientated studies,
linguistics/social linguistics and semiotics, philosophy/philosophy of
mathematics, educational theory/educational research/neighbouring
fi elds of mathematics education and curriculum studies. An overall
conclusion drawn by Lerman and Tsatsaroni is that the predominant
fi elds from which researchers draw in all three journals are traditional
psychological and mathematical theories. However, over the period
papers drawing on traditional psychology and mathematics show
a decreasing trend in PME and ESM (from 73.1% to 60.5% for PME;
and from 63.4% to 51.6% in ESM), but have increased in the case
of JRME (from 54.8% to 57.9%). In correspondence, the published
papers drawing on sociological and socio-cultural theories are on the
increase (from 3.0% to 9.9% in PME, from 3.7 to 11.6% in ESM, and
from 1.6 to 7.9 in JRME) but they are all below 12%. Also, a noticeable
increase is recorded, over the time period investigated, in the use
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THE OBJECTS OF MATHEMATICS EDUCATION RESEARCH: SPOTTING, ...
262
of linguistics, social linguistics and semiotics in all three journals,
though the number of papers drawing on these are still very small
(see detailed data in Appendix 4).
Conclusion
The mathematics education research analyses reviewed supported
the claim put forward in this chapter that social, political or economic
issues related to learning and doing mathematics have received,
from the foundation of the discipline until recently, insuffi cient
a ention from mainstream research, in contrast to issues related
to psychological factors, which have been approached mostly in
individualistic terms. Excluded from this conclusion are gender or
sex issues related to mathematics education, although the type of
analysis employed in the relevant studies cannot reveal whether or
not these issues were associated with larger, societal structures and
groups.
In my view, the infl uence of a psychological perspective and
individualisation may be considered as the primary factors which
have shaped the objects and the phenomena studied, as well as the
prevailing conditions and terms of inquiry in mathematics education
research. The infl uence of absolutist philosophies of mathematics
in their various forms, which regard mathematical knowledge as
pure isolated knowledge, somehow above and beyond the social
sphere, and so being value-free and culture-free, has probably made
a considerable contribution to this situation. As has the powerful
impact of Piagetian psychology of cognitive development on the
fi eld of mathematics education, as in every educational fi eld. Piaget’s
model of cognitive development stage theory fi ts nicely and at the
same time supports an individualised, almost social-context free,
approach to learning and thinking. Also, Piaget’s preference for
a logical-mathematical description of the structures underlying
cognition made his theory extremely a ractive to mathematics
education researchers and teachers.
However, another factor should not be underrated which is related
to political issues involving funding of research projects, mainstream
journals’ publication policies and processes, incongruities between
the researchers’ ideologies and the predominant culture of academia.
263
Dimitris Chassapis
263
Secada (1995) argues, and in my experience quite rightly, that
mathematics education researchers who raise serious social concerns
tend to be pushed out of the limelight and held to higher standards
of proof than other researchers.
Mathematics education, by its very nature, is a highly
multidisciplinary fi eld and this is refl ected in the broad variety of
conceptual structures and research methods employed. Therefore,
mathematics education concepts, like those of any interdisciplinary
subject, are many-faceted and their understanding depends on the
relations which may be considered to exist between their diff erent
components. Thus any interdisciplinary concept must be conceived
of ‘relationally’. For instance, learning or aff ect are not to be
conceived of as purely psychological concepts, just as assessment or
social class are not to be considered as purely sociological concepts.
If approached relationally learning contains social class within
it as an internally related component and vice-versa. Concepts
concerning social structures are, of course, generalisations, as are
concepts concerning psychological states and processes. No one
can assume that their inclusion in a mathematics education research
venture will cast dazzling light on the particular, and indisputably
complex, interrelationships of various aspects of learning and doing
mathematics. The exclusion, however, of the contextual social,
political or cultural factors from learning and doing mathematics,
eliminates any possibility of investigation, giving a distorted and
restricted form to the objects of mathematics education research.
References
Chassapis, D. (2002) Social Groups in Mathematics Education
Research: An Investigation into Mathematics Education-Related
Research Articles Published from 1971 to 2000, in P. Valero and O.
Skovsome (eds) Proceedings of the Third International Mathematics
Education and Society Conference, 1. Centre for Research in Learning
Mathematics, Danish University of Education, pp.273–81.
Ernest, P.: (1991) The Philosophy of Mathematics Education. London:
The Falmer Press.
Hertzberg, S. and Rudner, L. (1999) The Quality of Researchers’
Searches of the ERIC Database, Education Policy Analysis Archives,
7 (25), h p://epaa.asu.edu/epaa/v7n25.html [retrieved June 2004].
264
THE OBJECTS OF MATHEMATICS EDUCATION RESEARCH: SPOTTING, ...
264
Hanna, G. and Sidoli, N. (2002) The Story of ESM. Educational
Studies in Mathematics, 50(2), pp. 123–56.
Houston, J. (1995). The Thesaurus of ERIC Descriptors (13th edn).
AZ: Oryx Press, Phoenix.
Hudson, B. (2001), Holding Complexity and Searching for Meaning
– Teaching as Refl ective Practice, Journal of Curriculum Studies, 34,
1, pp. 43–57.
Kieran, C. (1994) Doing and Seeing Things Diff erently: A 25-Year
Retrospective of Mathematics Education Research on Learning,
Journal for Research in Mathematics Education, 25(6), pp. 583–607.
Kilpatrick, J. (1992) A History of Research in Mathematics Education,
in D. Grouws (ed.), Handbook for Research on Mathematics Teaching
and Learning. New York: Macmillan, pp. 3–380
King, K.D. and Mcleod, D.B. (1999) Coming of Age in Academe,
Journal for Research in Mathematics Education, 30, pp. 227–34.
Lerman, S. and Tsatsaroni, A., (2004) Surveying the Field of
Mathematics Education Research, Discussion Group 10 at the Tenth
International Congress on Mathematical Education, Copenhagen,
h p://www.icme-organisers.dk/dg10/Lermanpaper.pdf [retrieved
June 2004].
Lerman, S., Xu, G., and Tsatsaroni, A. (2003) A Sociological Description
of Changes in the Intellectual Field of Mathematics Education
Research: Implications for the Identities of Academics. Proceedings
of the British Society for Research in Learning Mathematics, 23(2),
pp. 43–48.
Lubienski, S.T. and Bowen, A. (2000) Who’s Counting? A Survey of
Mathematics Education Research 1982–1998, Journal for Research in
Mathematics Education, 31, pp. 626–33.
Niss, M (1999) Aspects of the Nature and State of Research in
Mathematics Education, Educational Studies in Mathematics, 40, pp.
1–24.
Niss, M (2000) Key Issues and Trends in Research on Mathematical
Education. Plenary address to Ninth International Congress on
Mathematical Education, Makuhari, Japan.
Reyes, L.H., and Stanic, G.M.A. (1988) Race, Sex, Socioeconomic
Status, and Mathematics. Journal for Research in Mathematics
Education, 19, pp. 26–43.
Secada, W.G. (1992) Race, Ethnicity, Social Class, Language, and
Achievement in Mathematics, in D.A. Grouws (ed.), Handbook
265
Dimitris Chassapis
265
of Research on Mathematics Teaching and Learning. New York:
Macmillan, pp. 623–60.
Sierpinska, A. and Lerman, S. (1996) Epistemologies of Mathematics
and of Mathematics Education, in Bishop, A., Clements, K., Keitel,
C., Kilpatrick, J. and Laborde, C. (eds): 1996, International Handbook
of Mathematics Education, Vols 1–2. Dordrecht: Kluwer Academic
Publishers, Chapter 22, pp, 827–76.
Sierpinska, A. and Kilpatrick, J. (eds) (1998) Mathematics Education
as a Research Domain: a Search for Identity – An ICMI Study, Vols
1–2. Dordrecht: Kluwer Academic Publishers.
Steen, L. A. (1999). Theories that Gyre and Gimble in the Wabe,
Journal for Research in Mathematics Education, 30, pp. 235–41.
Tate, W. F. (1997) Race-ethnicity, SES, gender, and language profi ciency
trends in mathematics achievement: An update. Journal for Research
in Mathematics Education, 28, pp. 652–79.
Contact
Assoc. Prof. Dr. Dimitris Chassapis
Aristotle University of Thessaloniki, School of Education
Greece
dchassap@eled.auth.gr
Appendix 1
Lubienski and Bowen (2000)
266
THE OBJECTS OF MATHEMATICS EDUCATION RESEARCH: SPOTTING, ...
266
Appendix 2
Chassapis (2002)
Number and percentage of education and mathematics education
research articles relating to each social group by years of publication
in decades
1971 – 1980 1981 – 1990 1991 – 2000 Total 1971
– 2000
All
ERIC Math A l l
ERIC Math A l l
ERIC Math A l l
ERIC Math
Research
Articles 37940
100% 857
100% 179581
100% 6177
100% 192045
100% 6965
100% 409566
100% 13999
100%
Social Class
or SES 282
0.7% 1
0.1% 1283
0.7% 41
0.7% 1511
0.8% 15
0.2% 3076
0.8% 57
0.4%
Ethnicity or
Race 437
1.2 3
0.4 1747
1.0 23
0.4 3347
1.7 63
0.9 5531
1.3 89
0.6
Gender or
Sex 2015
5.3 50
5.8 8700
4.8 401
6.5 9010
4.7 411
5.9 19725
4..8 862
6.2
Minority
groups 419
1.1 6
0.7 2591
1.4 85
1.4 4180
2.2 106
1.5 7190
1.8 197
1.4
Disad-
vantaged
groups
423
1.1 6
0.7 1449
0.8 31
0.5 2068
1.1 35
0.5 3940
1.0 72
0.5
At least one
of the social
group cat-
egories
3576
9.4 66
7.7 15770
8.8 581
9.4 20116
10.5 630
9.0 39462
9.6 1277
9.1
Note. Percentages in this table are column percentages.
267
Dimitris Chassapis
267
Number and percentage of mathematics education research articles
relating to teaching and learning topics by years of publication in
decades
1971 – 1980 1981 – 1990 1991 – 2000 Total 1971
– 2000
All
ERIC Math A l l
ERIC Math A l l
ERIC Math A l l
ERIC Math
Research
Articles 37940
100% 857
100% 179581
100% 6177
100% 192045
100% 6965
100% 409566
100% 13999
100%
Achievement 1818
4.8 135
15.8 8422
4.7 1123
18.2 9142
4.8 1214
17.4 19382
4.7 2472
17.7
Learning 3691
9.7 138
16.1 17388
9.7 915
14.8 24954
13.0 1412
20.3 46033
11.2 2465
17.6
Teaching 3068
8.1 111
13.0 15856
8.8 915
14.8 17681
9.2 1197
17.2 36605
8.9 2223
15.9
Teachers 5163
13.6 107
12.5 25881
14.4 792
12.8 31986
16.7 1244
17.9 63030
15.4 2143
15.3
Curriculum 2581
6.8 136
15.9 13319
7.4 962
15.6 14340
7.5 862
12.4 30240
7.4 1960
14.0
Cognition 1943
5.1 73
8.5 10702
6.0 694
11.2 9819
5.1 852
12.2 22464
5.5 1619
11.6
At least one
of the above
topics
18264
48.1 700
81.7 91568
51.0 5401
87.4 107922
56.2 6781
97.4 217754
53.2 12882
92.0
Note. Percentages in this table are column percentages.
268
THE OBJECTS OF MATHEMATICS EDUCATION RESEARCH: SPOTTING, ...
268
Appendix 3
Hanna, G. and Sidoli, N. (2002)
269
Dimitris Chassapis
269
Appendix 4
Lerman and Tsatsaroni (2004)
Theory type
PME ESM JRME
90-95 96-01 90-95 96-01 90-95 96-01
No. % No % No % No % No % No %
Traditional
psychological
& mathematics
theories
49 73.1 49 60.5 52 63.4 49 51.6 34 54.8 44 57.9
Psycho-social,
including re-
emerging ones
811.9 89.9 89.8 19 20.0 46.5 10 13.2
Sociology, So-
ciology of Ed,
socio-cultural
studies & His-
torically orien-
tated studies
23.0 89.9 33.7 11 11.6 11.6 67.9
Linguistics, so-
cial linguistics
& semiotics
00.0 22.5 11.2 55.3 23.2 67.9
Neighbouring
fi elds of Maths
Ed, science ed
and curriculum
studies
11.5 00.0 00.0 00.0 11.6 00.0
Recent broader
theoretical cur-
rents, feminism,
post-structural-
ism and psy-
choanalysis
11.5 00.0 89.8 11.1 00.0 11.3
Philosophy/
philo of math-
ematics
00.0 33.7 00.0 33.2 11.6 11.3
Ed theory and
research 23.0 00.0 11.2 11.1 23.2 00.0
Other 0 0.0 00.0 11.2 11.1 23.2 00.0
No theory used 4 6.0 11 13.6 89.8 55.3 15 24.2 810.5
Total 67 81 82 95 6 2 76
270
THE OBJECTS OF MATHEMATICS EDUCATION RESEARCH: SPOTTING, ...
270
271
Ioannis Michalis
271
Abstract
This study discusses graph comprehension, which is defi ned as the abilitys
of graph readers to derive meaning from graphs. It is classifi ed into
three levels: reading the data (i.e. literal), reading between the data (i.e.
interpretation, interpolation) and reading beyond the data (i.e. prediction,
extrapolation). The preliminary results of a pilot research project on the graph
comprehension of primary school students (grades 4, 5 and 6) are presented.
Students were given a questionnaire, comprised of ten graphs each with six
questions, covering the three comprehension levels. The sample consisted of
280 primary school students from the urban area of Thessaloniki and nearby
cities.
Keywords: data analysis, graph comprehension, primary school,
statistics
Introduction
Over the past fi een years, documents on mathematics education
reform (NCTM, 2000; DFES, 2001; AEC, 1994; M.E., 1993) have
highlighted the importance of including statistics and data analysis
throughout the school mathematics curriculum.
It is suggested that the teaching of statistics and data analysis in
school mathematics must be based on the statistical investigation
GRAPH COMPREHENSION OF
PRIMARY SCHOOL STUDENTS
Ioannis Michalis, Aristotle University of
Thessaloniki
272
GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
272
process (Graham, 1987), which typically involves four stages: posing
a question, collecting the data, analyzing the data, and interpreting
the results; a fi h stage could be added, which is communicating the
results (Kader and Perry, 1994).
It is important to look at students’ understanding as related to
concepts linked to this process in order to establish a be er learning
environment and to design the appropriate instructional material.
Located at the core of that process is the understanding of the
graphical representations or visual displays of data. As Shaughnessy
et al. (1996) state:
the current meaning of data analysis emphasizes organizing,
describing, representing, and analyzing data, with a heavy reliance
on visual displays such as diagrams, graphs, charts and plots (p.
205)
From an early age we meet diff erent kinds of graphs, charts, thematic
maps, and cartograms in educational as well as everyday situations.
Graphs are commonly used to depict mathematical functions, display
data from social and natural sciences, and specify scientifi c theories
in textbooks and other print media in and out of the classroom (Shah
and Hoeff ner, 2002). Skills in the critical reading and interpretation
of data, presented in these visual forms, are a necessity in our highly
technological society and represent one of the major components
of quantitative literacy. Refl ecting that view, The National Research
Council (1990) suggests:
Most obvious, perhaps, is the need to understand data presented
in a variety of diff erent forms and displays…. Citizens who cannot
properly interpret quantitative data are, in this day and age,
functionally illiterate.
Therefore, we need to know much more about how students (and
teachers) think about, and comprehend graphical representations.
The graphical presentation of data has a considerable history, from
the work of William Playfair (1759–1823) to the contemporary
innovations of John Tukey (Wainer, 1990). Playfair is credited with
273
Ioannis Michalis
273
the invention of the best known graphs; while EDA (Exploratory
Data Analysis) is a relatively new area of statistics, where the data
are explored by graphing techniques (Tukey, 1977). This approach/
philosophy has given a surge to the use of graphical representations
because such graphical techniques are much used in analyzing data.
The focus is on meaningful investigation of data sets with multiple
representations and li le probability theory or inferential statistics.
Graph defi nition and forms/types
Graphs have the potential of showing more than the values in the
data table. They can provide an overview of the whole data set and
can highlight specifi c characteristics that are not visible from the
numbers. They display the data and tell the truth (although they may
sometimes a empt to persuade); encourage comparison of diff erent
pieces of data; pack large amounts of quantitative information into a
small area; reveal the data at several levels of detail; provide impact;
communicate with clarity, precision and effi ciency; serve a defi ned
purpose of discovery, understanding and presentation; and are more
closely integrated with statistical and verbal descriptions of the
data.
Kosslyn (1994) defi nes a graph as ‘a visual display showing one or
more relationships between numbers’, while Fry’s (1984) description
is: ‘a graph is information transmi ed by the position of point, line
or area on a two-dimensional surface’ (p. 5). James and James (1992)
in their Mathematics Dictionary defi ne a graph as: ‘a drawing which
shows the relation between certain sets of numbers …Used to convey
a be er idea of the meaning of the data than is evident directly from
the numbers’. (p. 189)
The characteristics and features of the most common graphs and plots
used widely in newspapers, magazines and reports, and mentioned
in all curriculum documents around the world – including the
traditional forms of graphs and the new plo ing techniques (Curcio,
2001) – are presented below.
The picture graph (picture chart, pictogram, pictograph or pictorial
graph) is used to display discrete or categorical data using ideographs
or symbols to depict quantities of objects or people. Rectangular
274
GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
274
axes defi ne the fi eld of display. In its simplest form the ideograph
is in one-to-one correspondence with the item it represents. In more
complex displays, a many-to-one correspondence between the items
represented and the ideograph requires the use of a legend. In this
type of display, sometimes the size and the type of ideographs
mislead the audience either on purpose or accidentally.
The bar graph (bar chart) is also used to display discrete or categorical
data. Within rectangular axes that must be labelled, the heights of
rectangular bars of uniform width are proportional to the quantities
they represent. The bar graph may be set up either horizontally
(usually when the labels of the categorical data are long) or vertically
(the most common form). Discrete stratifi ed data (from particular
groups) may be compared in double or multiple bar graphs.
The histogram contains grouped data arranged like a vertical bar
graph. Only one set of data may be represented, the class intervals
(groups) must be equal, and both axes must contain a numerical
scale.
A line graph is used to display continuous data (change over a period
of time, or time-series data). The axes that intersect at a common
point (usually zero) are labelled and they defi ne the fi eld for the
display. On each axis, the units of division must be consistent. The
graphed points are connected by line segments. Two or more sets of
continuous data may be graphed on the same set of axes to create a
double or multiple line graph allowing the comparison of the data
sets.
A circle graph (area graph, pie chart, pie diagram or pie graph) is
used when data are to be compared to a whole or to diff erent parts
of a whole. The fi eld of display of a circle graph is defi ned by the
circumference of a circle. Sectors of the circle created by line segments
from the centre correspond proportionally to fractional parts of the
unit being analyzed.
A line plot is created by representing numerical data in the form of
x’s on a number line (like a primitive bar graph). This visual display
is easy to construct. The shape and the spread of the data are revealed
275
Ioannis Michalis
275
while having access to each individual datum. The median and the
mode are easily identifi ed. The number of items plo ed usually does
not exceed twenty-fi ve.
A stem-and-leaf plot provides a display of data that is created by
separating the digits in the data based on their place value. In a
regular stem-and-leaf plot, two columns identifi ed by place value
are established to list each digit. The digit with the higher place
value is the stem, and the digit with the lower place value is the leaf.
This type of plot usually contains more than twenty-fi ve data entries
and up to two hundred and fi y. Similar to the line plot, it is easy to
construct, and the shape and the spread of data are revealed while
having access to individual datum. The median and the mode can be
easily identifi ed. Back-to-back stem-and-leaf plots can be constructed
to compare two sets of related data.
A box plot (box-and-whiskers plot) uses fi ve summary points: the
lower extreme, the lower quartile, the median, the upper quartile
and the upper extreme. The plot has one axis in the form of a number
line arranged either horizontally or vertically. A rectangle is used to
represent the middle 50% of the data. The median is represented
by a line segment that partitions the rectangle to show the spread
of the upper and lower quartiles, proportional to the size of the
rectangle. The extremes are represented by points, and they are
connected to each end of the rectangle and are identifi ed in terms of
the interquartile range. This plot is used when analyzing more than
100 pieces of data. Unlike the line plot and the stem-and-leaf plot,
individual data are not identifi able.
According to Kosslyn (1989, 1994) almost all graphs have similar
structural components (graphic constituents). The similarity is not in
appearance, but in function, in the role that these constituents play
in how information is represented in a display and respectively how
it is interpreted.
The framework of a graph (axes, scales, grids, reference markings)
gives information about the measurement being used and the data
being measured. The most common framework has an L shape
with the horizontal line depicting the category axis (the data being
276
GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
276
measured) and the vertical line the value axis (providing information
about the measurement being used). Picture graphs, line graphs, bar
graphs, histograms and line plots have that L-shaped framework. Box
plots use a variation of that shape, while stem plots and tables have
T-shaped frameworks. Others, like pie graphs, have a framework
based on polar coordinates (Fry 1984).
The specifi ers are visual dimensions which are used to represent
data values. These might be the bars on a bar graph, the lines on a
line graph etc.
Labels are another component of a graph that name the type of
measurement being made, or the data to which measurement applies,
including the title of the graph.
Lastly, the background of a graph includes any colouring, grid, and
pictures over which the graph may be imposed. The background
serves no essential role in communicating the information conveyed
by a chart or graph.
Graph Comprehension
One of the reasons why graphs are so pervasive is that they seem
to make quantitative information easy to understand. Research
in statistics education, however, shows that graphs are diffi cult to
interpret for most people:
The increasingly widespread use of graphs in advertising and the
news media for communication and persuasion seem to be based
on an assumption, widely contradicted by research evidence in
mathematics and science education, that graphs are transparent in
communicating their meaning. (Ainley, 2000, p. 365).
We need to defi ne the processes and the factors that infl uence
graph comprehension in order to understand why it is diffi cult for
people to interpret graphs. In that sense we have to focus on graph
comprehension as reading and interpreting graphs; as the graph
reader’s ability to derive meaning from graphs (Friel et al., 2001).
277
Ioannis Michalis
277
In general, comprehension of information in wri en or symbolic
form involves three kinds of behaviours (Jolliff e, 1991) that seem to
be related to graph comprehension, namely:
• Translation
• Interpretation
• Extrapolation/interpolation
To translate between graphs and tables, one could describe the contents
of a table of data in words or interpret a graph at a descriptive level,
commenting on the specifi c structure of the graph (Jolliff e, 1991). To
interpret graphs one can look for relationships among specifi ers in a
graph or between a specifi er and a labelled axis. To extrapolate, which
is considered to be extension of interpretation, requires stating not
only the essence of the communication but also identifying some of
the consequences; one could extrapolate by noting trends perceived
in data by specifying implications.
Following that model the NCTM principles and standards for school
mathematics suggest that students in early elementary school should
be able to use graphs to identify what quantities are the highest and
lowest and to make comparisons between two single data points.
Late elementary students should be able to aggregate data and
evaluate the whole picture presented by a graph. Middle school
students should begin to develop ideas of statistical inference and
high school students are expected to use graphs to make inferences
(NCTM 2000).
These standards provide, above all, a framework for identifying
a hierarchy of question types for the graphs used in real world
tasks. The simplest questions require identifi cation of specifi c data
points or comparison of two or more data points and involve only
the extraction of explicit information. For this type of question, the
desired information is explicitly represented in the graph and the
graph reader is required only to locate and read the specifi c data
point. A more diffi cult question, requiring aggregation of data, is an
integration question. The graph reader must read off multiple data
points and then integrate the information using some kind of mental
operation (i.e. determining trends). The most diffi cult questions are
278
GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
278
those that require the user to make inferences and predictions. The
desired information is not explicitly represented, so the graph reader
must use extrapolation in order to extract it.
Gal (1998) suggested two similar types of question in interpreting
information in tables and graphs: literal-reading questions that
involve reading the data, or reading between the data and opinion,
and questions that focus on reading beyond the data. The la er types
require eliciting and evaluating opinions (rather than facts) about
information presented in representations.
Three levels of graph comprehension have emerged according
to relevant research (Bertin, 1983; Curcio, 1987; McKnight, 1990;
Carswell, 1992; Wainer 1992):
• An elementary level focused on extracting data from a graph
(locating, translating)
• an intermediate level characterized by interpolating and fi nding
relationships in the data as shown on a graph (integrating,
interpreting), and
• an advanced level that requires extrapolating from the data and
analyzing the relationships implicit in a graph (generating,
predicting).
Curcio’s (1987) terminology refers to the above three levels as:
read the data, read between the data and read beyond the data.
Shaughnessy et al. (1996) described a fourth aspect as looking behind
the data, involving consideration of the context within which data
sets arise. Friel, Curcio and Bright (2001) in an extended review of
graph comprehension research have identifi ed some critical factors
infl uencing graph comprehension, including:
1. Purposes for using graphs; there are two kinds of uses for graphs,
for analysis and communication. Graphs help us summarize data:
describe measures of interest, discover relationships or test models
that are based on our beliefs about how the world works. This is an
aspect that appears to be related to the school curriculum. The second
use is communicating our observations to others. That is, pictures
that intend to convey information about numbers and relationships
279
Ioannis Michalis
279
among numbers. These graphs o en called ‘presentation graphics’
and are typically more comprehensive than analytical graphs.
2. Characteristics of tasks in graph perception: The process of visual
decoding – which dimension associated with graphs (line, area,
position etc.) should be employed to facilitate graph use. Cleveland
and McGill (1984) have identifi ed ten graphical perception tasks
that form the basic perceptual judgments that a person performs
to decode visually quantitative information encoded on graphs
(specifi ers). For example, fi nding positions on a common aligned
scale (bar graph) can be processed more easily and more accurately
than determining area (pie chart). The nature of judgment tasks
– for example, comparison judgments (between absolute lengths
of bars) and proportional judgments (comparing individual slices
with the whole in pie charts). These tasks require point reading, or
integration of information across data points such as computations,
making comparisons and identifying trends. For example, the eff ect
of contextual se ing – a graph’s visual characteristics (syntax) and
the graph’s context (semantics). Because data are usually from real
world contexts, a graph reader must be able to describe, organize,
represent, and analyze and interpret data taking into account the
contextual frame of the data.
3. Characteristics of the discipline: spread and variation – data
reduction and scaling. Data type and size of data set – as stated above
the use of specifi c graphs is determined by the type and size of the
data. Another factor is the graph complexity – simple graphs should
be used early in the school while more complex ones later during
middle school.
4. Characteristics of graph readers; familiarity with the context,
domain knowledge, general learner characteristics (general
intelligence).
Similar classes of factors aff ecting the interpretation of graphs
have been stated by other researchers (Shah and Hoeff ner, 2002)
in reviewing graph comprehension research: ‘The characteristics
of the visual display; the viewers’ knowledge of graphical schemas
280
GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
280
and conventions; the content of the graph and the viewer’s prior
knowledge and expectations about the content’
Pilot research
Based on that theoretical framework, we conducted research into
graph comprehension of primary school students (4th, 5th and 6th
grades) to serve as a pilot to my fi nal research.
We visited three schools – one in the city of Thessaloniki, the other in
an industrial area in the outskirts of Thessaloniki and the third in the
small town of Chalkidiki – and we asked the students to complete a
questionnaire (see Appendix). The sample consisted of 280 students
(100 from 4th grade, 95 from 5th grade and 85 from 6th grade, 132
boys and 148 girls) who were given the questionnaire and asked to
complete the tasks without any further instruction in one hour.
The questionnaire included 10 graphs (2 picture graphs, 2 bar graphs,
a line plot, a line graph, a pie chart, a stem-and-leaf plot, a box plot
and a histogram) covering topics that were real or familiar to children
such as the height of children, books read by children, monthly
temperature, number of le ers in names, newborn babies’ weight,
birthdays etc. Most of the graphs were followed by six questions (in a
multiple choice format) intended equally to refl ect on the three levels
of graph comprehension, as stated above. In some graphs there were
more than six questions and open-ended.
We will discuss here the preliminary results of 4 out of the 10 tasks
(i.e. bar graph, line plot, line graph and box plot). Relevant research
(Curcio, 1987; Pereira-Mendoza and Mellor, 1991; Bright and Friel,
1998;) indicates that students experience few diffi culties with ‘read
the data’ questions, but they make errors when they encounter
‘read between the data’ questions. Such errors may be related to
mathematics knowledge, reading/language errors, scale errors, or
errors in reading the axes. ‘Read beyond the data’ questions seem to
be even more challenging. Students must make inferences from the
representations in order to interpret the data, for example, to compare
and contrast data sets, to make a prediction about an unknown case,
to generalize to a population, or to identify a trend.
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281
The results of our pilot study seem to be more or less similar. In
general, there were few problems with the literal reading of graphs
as shown in the table below:
Table 1: Success rate of the read-the-data questions
Graph Success
rate Graph Success
rate Graph Success
rate Graph Success
rate
Bar 1 94% Line-plot 1 34% Line-
graph 1 70% Box plot 1 93%
Bar 2 77% Line-plot 2 91% Line-
graph 2 88% Box plot 2 65%
Line-
graph 3 91% Box plot 3 42%
The fi rst question was ‘what does this picture tell you’, which requires
translation of the labels and titles of the graphs and as seen in the
table had a very good success rate in bar graph, box plot and line
graph. The line plot had no clear title, a fact that seemed to be very
confusing for the students. The question here was open-ended so 34%
of the students answered correctly at a very descriptive level while
54% were incomplete in their answers, focusing on the labels (‘the
names of a class’). Similarly, in the line graph 1 question 26% of the
students gave incomplete answers mostly combining the labels and
disregarding the title although there was a defi nite one. The answers
to these two questions from one student are quite interesting. He
wrote for the fi rst one ‘it is a pyramid of X’ and for the other ‘it is
a mountain’. A graph is a sign, and a sign is defi ned as something
that stands for something else or for someone (a student in our case)
(Baker, 2004). The fi rst something is mainly an inscription on paper
and the second something is a mental construction, a mathematical or
statistical object. It is acknowledged that a sign is always interpreted
as referring to something else within a social context.
The second (and third in two graphs) question(s) required locating
some specifi c information explicitly shown in the graph. In the bar 2
question 20% of the students answered ‘100 millimeters’’, obviously
a reading/language error, while in box plot questions 2 and 3 25%
and 22% respectively chose the lower and upper quartile to answer.
The students (and teachers) had never seen a box plot before and
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GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
282
that seem to have infl uenced their answers because they hadn’t had
the domain knowledge (Freedman and Shah, 2002) or the graph
schema (Pinker, 1990). Students encountered more problems with
the interpretation questions.
Table 2: Success rate of the read-between-the-data questions
Graph Success
rate Graph Success
rate Graph Success
rate Graph Success
rate
Bar 3 98% Line-plot 3 81% Line-
graph 4 68% Box plot 4 56%
Bar 4 61% Line-plot 4 11% Line-
graph 5 82% Box plot 6 37%
Line-plot 6 54% Line-
graph 6 62%
Line-
graph 7 32%
The third question on the bar graph had the highest success rate
of all in the test, while in the fourth almost 11% of the students
again confused the measurement unit and the rest (28%) made
computational or scale errors. In the line plot 3 question 5% of
students answered 13 (the highest number in the x axis). The
question that seemed to be frustrating for almost all students was
the line plot 4. It had the lowest success rate and 30 diff erent answers
with 10% of them being 6, which comes from adding the Xs on 4, 5,
11 and 12 le ers. In the sixth question 13% of students answered,
none confusing the axes. On the other hand, students seemed to be
be er in the interpretive questions on the line graph, although this
could not be seen in the success rate mainly because most of the
answers were only partly correct: in the line graph 4 and 5 questions
21% didn’t include dec and 8% didn’t include sep in their answer
whereas in line graph 7 question, 48% mention either feb/dec or jul/
aug. In the line graph 7 question 30% of students were wrong; mostly
answering descriptively ‘the weather is very warm during summer
and cold during winter, or July is the warmest and January is the
coldest month’.
Students encountered major problems in the read beyond the data
questions where they needed to infer or predict from the data
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283
represented to the graphs. Bar graph questions 5 and 6 were in that
category and students had a 57% and 72% success rate respectively.
Even though the last one was more than obvious, 20 % of the students
answered that ‘Chara is of average (normal) height for her age’. To
the more predictable question of line plot (5), 54% of the students
were correct with fair explanations (‘most of the student’s names
have 8 le ers’), while some pointed out that the name of the new
student would have 3, 4, 5, 11, 12, or 13 le ers (22% all together),
because there were few names with so many le ers. A noticeable 6%
answered 28, confusing the language of the question.
The answers of students to the (open-ended) read beyond the data
questions of the line graph and box plot show us why we should
use that kind of question and avoid the multiple choice format.
In the line graph students were asked to predict the lowest and
highest temperature of a typical day in May bearing in mind that the
monthly average temperature was 19.6 degrees Celsius. Nineteen
percent of the students used the lowest and the highest temperature
of the graph, that is 5 and 26, while 10% answered 18 to 20. There
were a number of students (almost 26%) who answered, correctly
using temperatures scales from 9 to 16 for the lowest and 20 to 29
for the highest. An interesting fact was that only few numbers were
integers. The majority of students used decimal numbers like (18.6
to 21.5). One girl didn’t answer, explaining that a er all, she wasn’t
a meteorologist.
The responses of students to the same question in the box plot were
even more interesting because they had to explain the answer. There
were 24 diff erent answers in a scale of 500 grammes to 5500 grammes.
Many of the students (19%) correctly used the median (3100) to
predict the weight of a heavy newborn brother or sister. Fourteen
percent and 8% used the low and the upper quartile (2800 and 3700),
while 5% and 4% used the two extremes (1500 and 4500). The most
common explanation was ‘because I weighed that much too’ but
there were a lot of explanations that need to be further examined
(with an interview perhaps). ‘I think it will be born at about 1500
grammes because my mom doesn’t eat much’, or ‘4000 grammes
because we all are overweight in our family’, or ‘1500, because my
mom is very thin’.
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GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
284
Discussion
The elementary mathematics curriculum in Greece that refers to data
analysis seems to adopt a superfi cial approach. Specifi cally, under the
topic of collection of data (ΥΠΕΠΘ, 1997), the curriculum suggests:
‘with appropriate activities students are expected to be capable
of collecting, organizing, interpreting and presenting data and
interpreting graphical displays (grade 4); To interpret and construct
graphical displays, to describe the concept and to fi nd the Mean and
to make estimations (grades 5 and 6)’. In the new revised national
curriculum (Government Gaze e, 1376, 2001) some indicative and
fundamental concepts of the so called interdisciplinary approach
are added only by a brief reference: ‘variation, system, organization,
space-time, unit-whole, similarity-diff erence, probability’.
Mathematic textbooks (which are the same for all students throughout
the country) include 8 teaching units, to cover these optimistic
suggestions, during all six years of primary school. As a result,
primary school students in our country have very limited domain
knowledge and familiarity with graphs and visual representations of
data. Nevertheless, this primitive situation is helpful in determining
some basic processes and features that take place during graph
comprehension.
Most of the students in our research faced their work with joy and
interest. They didn’t even take their break in order to fi nish the task
and there were no complaints at all. They asked for some explanations
but we preferred not to help, urging them to look at the picture and
answer what they understood. The fourth graders were struggling
to fi nish the task and many of them didn’t make it, mainly because
they were delayed by computations and language understanding.
Interestingly, there were no major diff erences at grade levels even
though sixth graders were slightly be er in most of the questions. In
some cases there were sex diff erences (for example, in bar 5 question
– 62% to 53% – and in bar 6 question – 64% to 78%) something that
has to be further examined. In general, girls were be er at completing
the tasks, with be er success rates in most questions.
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285
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Contact
Ioannis Michalis
Aristotle University of Thessaloniki, School of Education
Greece
gmichal@eled.auth.gr
Appendix
Bar graph (Adapted from Curcio (1987))
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GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
288
View the picture carefully and try to answer the questions below
1. What does this picture tell you?
α. The weight of four children of a family
β. The grades of four children of a family
γ. The height of four children of a family
δ. The age of four children of a family
2. How tall is Maria
α. 75 centimeters
β. 100 millimeters
γ. 125 millimeters
δ. 100 centimeters
3. Who was the tallest
α. Maria
β. Anna
γ. Helen
δ. Chara
4. How much taller was Helen than Anna
α. 25 centimeters
β. 50 centimeters
γ. 75 millimeters
δ. 75 centimeters
5. If Maria grows 5 centimeters and Anna 10 centimeters who will be
taller, and by how much?
α. Maria by 20 centimeters
β. Anna by 20 centimeters
γ. Maria by 5 centimeters
δ. Anna by 5 centimeters
6. If Chara is 5 years old, which of the following is a correct
statement?
α. Chara is much too short for her age
β. Chara could never be that tall for her age
γ. Chara is of average height for her age
δ. Chara is thin for her age
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Line plot
View the picture carefully and try to answer the questions below
1. What does the picture tell you?
……………………………………………………………………………
……………………………………………………………………………
……………………………………………………………………………
……………………………………………………………………………
……………………………………
2. How many children have 9 le ers in their names?
……………………………………………………………………………
……………………………………………………………………………
…………………
3. How many children are in the class?
……………………………………………………………………………
……………………………………………………………………………
…………………
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GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
290
4. What is the sum of the le ers of the 4 smallest names?
……………………………………………………………………………
……………………………………………………………………………
…………………
5. If a new child came to the class, how many le ers do you believe
that his/her name would have? Explain why.
……………………………………………………………………………
……………………………………………………………………………
…………………
6. How many children have more than 8 le ers in their names?
……………………………………………………………………………
……………………………………………………………………………
Line graph
View the picture carefully and try to answer the questions below
1. What does the picture tell you?
……………………………………………………………………………
……………………………………………………………………………
……………………………………
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Ioannis Michalis
291
2. Which is the coldest month of the year
……………………………………………………………………………
………………....
3. Which is the warmest month of the year
......................................................................................
4. For which months is the average temperature below ten degrees?
.........................................................................................................................
.......................
5. For which months is the average temperature above twenty
degrees
.........................................................................................................................
.....................................................
6. What is the diff erence between the warmest and the coldest
month
.........................................................................................................................
.....................................................
7. Which months have almost the same average temperature
.........................................................................................................................
.....................................................
8. Could you predict the lowest and the highest temperature of
a typical day of May having in mind that the monthly average
temperature of May is 19,6?
Lowest: …………. Highest: ……………………….
Box plot
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GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
292
View the picture carefully and try to answer the questions below
1. What does the picture tell you?
……………………………………………………………………………
…………………
2. According to the picture how much is the lightest baby’s weight?
α. 3100 grammes
β. 3700 grammes
γ. 2800 grammes
δ. 1500 grammes
……………………………………………………………………………
3. According to the picture how much is the heaviest baby’s weight?
α. 3700 grammes
β. 3100 grammes
γ. 4500 grammes
δ. 5000 grammes
……………………………………………………………………………
4.According to the picture, which of the following is a correct
statement?
α. Half of the babies are born with a weight below 2800 grammes
β. Half of the babies are born with a weight above 3700 grammes
γ. Half of the babies are born with a weight between 2800 and 3700
grammes
δ. Most of the babies are born with a weight below 2800 grammes
5. If a new brother or sister of yours was born tomorrow how much
do you think that he or she would weigh keeping in mind the
information provided by the picture?
……………………………………………………………………………
………… Why?
6. What is the likelihood that a new born baby will weigh 2800 to
3700 grammes?
α. 1/2. β. 1/4
γ. 1/3 δ. 1/5
Graph components
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293
Chart components
294
GRAPH COMPREHENSION OF PRIMARY SCHOOL STUDENTS
294
295
Aristarchos Katsarkas
295
Abstract
The growing literature of the last twenty-fi ve years on rational numbers is
concerned with the diffi culties that students have in this area, and the way that
teaching and learning must be connected. The multi-conceptual character of
rational numbers, and the lack of their development in elementary schools,
give them a central position among the many sources of misconceptions
and defi ciencies. The four subconstructs of rational numbers that Kieren
introduced – part-whole and measure, ratio, quotient and operator – are
widely accepted. The lack of knowledge and misconceptions of teachers are
examined here, as they are an important factor in the teaching process, which
within with the Greek curriculum, gives li le a ention to the development
of these subconstructs, thus leading to defi ciencies in learning.
Keywords: In-service education, primary teaching, rational numbers,
teachers’ knowledge
Introduction
The defi nition of rational numbers as ordered pairs of integers
(a, b), with b ≠ 0 and the construction of Q as an infi nite quotient
fi eld of the integers is not useful in elementary schools, because the
introduction of the rational numbers should be ‘rich in connections
among symbols, models, pictures, and context’ (Cramer et al., 2002,
p. 112).
KNOWLEDGE AND CONCEPTS OF
RATIONAL NUMBERS HELD BY
ELEMENTARY SCHOOL TEACHERS
Aristarchos Katsarkas, Aristotle University of
Thessaloniki
296
KNOWLEDGE AND CONCEPTS OF RATIONAL NUMBERS HELD BY ...
296
The most common symbol that we use for the representation of a
rational number is the fraction. As Freudenthal (1983, p. 134) argues:
‘Fractions are the phenomenological source of the rational number
– a source that never dries up.’
Over the past 30 years many things have been wri en about rational
number concepts as being ‘a formidable learning task’ (Behr et
al., 1983, pp. 92–93) and a serious obstacle in the mathematical
development of children. ‘When fractions and rational numbers as
applied to real-world problems are looked from a pedagogical point
of view, they take on numerous ‘personalities’ (Behr et al., 1992, p.
296).
In one of the latest approaches, Behr et al. (1992, p. 298) stated: ‘It
would appear, then, that fi ve subconstructs of rational number –
part-whole, quotient, ratio number, operator, and measure – which
have to some extend stood the test of time, still suffi ce to clarify the
meaning of rational number’, which seems to agree with Kieren,
Vergnaud and Freudenthal. These subconstructs are:
1. Part-Whole
2. Measure
3. Ratio
4. Quotient, and
5. Operator
Taking into account the theory that Dienes proposed, we see
clearly why knowledge and conceptualization of all the previous
subconstructs are indispensable. The Perceptual Variability Principle
that Dienes (1967 in Post and Reys, 1979) asserted suggests that
conceptual learning is maximized when children are exposed to
a concept in a variety of physical contexts or ‘suits of clothes’. His
Mathematical Variability Principle, also, asserts that if a mathematical
concept is dependent upon a certain number of variables, the
systematic variation of these is a prerequisite for eff ective learning
of the concept.
297
Aristarchos Katsarkas
297
With these thoughts, and within the previous analysis, Behr et al.
(1992, p. 315) identifi ed fi ve areas where mathematics curricula of
elementary and middle schools are ‘defi cient’:
1. Composition, decomposition, and conversion of units.
2. Operations on numbers from the perspective of mathematics
of quantity.
3. Constraint models.
4. Qualitative reasoning
5. Variability principles.
A characteristic of the Greek educational system is that, as curriculum
is constructed from the central educational service (Pedagogical
Institute) and in the vast majority of cases is followed ‘blindly’ by
teachers, these defi ciencies pass on to the instruction. A recent study
(Chassapis and Katsarkas, 2005) evidences that.
Rational numbers and recursive knowledge
Kieren (1993) characterized personal rational number knowledge in
terms of four kinds of knowing: ethnomathematical (E), intuitive (I),
technical-symbolic (TS), and axiomatic-deductive (AD) (see Fig. A),
and he gives the following descriptions:
‘The fi rst kind of knowing (E) is the kind that children, or adults
for that ma er, possess because they have lived in a particular
environment’ (p. 66). Intuitive (I) knowing of fractional numbers
entails the use of thinking tools, imagery, and the informal use of
fraction language. […] Technical-symbolic (TS) knowing is knowing
that is the result simply of working with symbolic expressions
involving fractional or rational numbers. […] Finally, axiomatic-
deductive (AD) is knowledge derived through logically situating a
statement in an axiomatic structure.’
Kieren (1993) in collaboration with Pirie, proposed a more enhanced
model of understanding as a dynamic, nonlinear process that they
regard as part of a recursive theory of mathematical understanding
which is shown in Figure B. A feature of this model is that, ‘the outer
level does not just appear and then become linked. It is in some way
already coherent with what has gone before’ (p. 74). This model
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KNOWLEDGE AND CONCEPTS OF RATIONAL NUMBERS HELD BY ...
298
suggests that ‘someone can function with fractions at a symbolic
level in a way that shows no connection to or awareness of previous,
more intuitive ways of knowing. […] But such “blindness” […] can
cause diffi culty. In forming an image or in formalizing, one may
develop an outer level of functioning that may be “wrong” even
though it is wrapped around “correct” intuitive knowing’ and also
“In knowing with understanding learners fold back to the inner level
of knowing”‘ (pp. 74–75).
The context of the research
As Post et al. (1988) assert: ‘We know from pilot investigations (for
example, Lesh and Schultz, 1983; Post et al., 1985) that many of the
same misunderstandings and “naive conceptualizations” that we
have identifi ed in youngsters also are prevalent among teachers (p.
200). This gives us the fuel for our research.’
1. Translations among several modes of representation.
‘When physical materials are used in instruction, they should
provide a concrete representation, or embodiment, of a mathematical
principle. When diff erent, yet appropriate, concrete materials are used
to develop the same mathematical idea, a “multiple embodiment” is
provided’ (Post et al., 1979, p. 353). The use of a single embodiment
or a single mode in instruction causes misunderstandings and
defi ciencies.
2. Perceptual distractors.
‘The extent to which a child [and teacher] is able to resolve confl icts
between visual information and their logical-mathematical thinking
is viewed as one of several important indicators of how solid or
tenuous is the child‘s understanding of the rational-number concept
in question’ (Behr and Post, 1981, p. 8). For example, the necessary,
but not directly used, marks on a number line may act as perceptual
distractors (Behr and Post, 1981).
3. Order, equivalence and quantitative notion of rational number.
Behr et al. (1983) observed that children’s ability to acquire a
quantitative notion of rational number is crucial to the development
of other rational number concepts. ‘One measure of children’s
quantitative notion of rational number is their ability to perceive the
relative size of the rational numbers in a pair or a larger set’ (Behr et
al., 1984, p. 324).
299
Aristarchos Katsarkas
299
4. Partition behavior and unit formation.
The concept of partitioning or dividing a region into equal parts,
or of separating a set of discrete objects into equivalent subsets,
is fundamental to an understanding of rational numbers. The
concept of a unit underlies the concept of a fraction. Formation and
reformation of units is important in constructing many rational
number subconstructs. Partition behavior can give us an insight in
how units are formed.
5. Recursive understanding.
Commenting on the results of the second NAEP, Carpenter and his
colleagues pointed out that ‘Students appear to be learning many
mathematical skills at a rote manipulation level and do not understand
the concepts underlying the computation’ (Carpenter et al. 1980, p.
47 in Post et al., 1982). Teachers have to act, during instruction, at all
levels of Kieren’s model in order to assist childrens’ eff orts, which
can be done only if they have a fi rm grasp of the rational number
concept.
Research questions
Are teachers able to:
1. make translations among several modes of rational number
representation?
2. resolve confl icts between visual information and their logical-
mathematical thinking?
3. perceive the relative size of the rational numbers in a pair or a
larger set?
4. make formation and reformations of units through several partition
behaviors?
5. use all levels of recursive understanding during instruction?
Method of research
For our research we have adopted a version of the method used by
Post et al. (1988) in his assessment of intermediate teachers’ knowledge
of rational number concepts. Thus, our research has three parts.
Part I includes a pilot-research questionnaire with 28 tasks which
examines: general knowledge for the quotient fi eld, equivalence,
dominant subconstruct, fraction as an entity, fraction operations,
choice of unit, partitive and quotitive division, embodiments
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KNOWLEDGE AND CONCEPTS OF RATIONAL NUMBERS HELD BY ...
300
with perceptual distractors, proportionality, connections between
representations, recursive knowledge, and decimals.
Part II includes interviews with a number of teachers who took part
in Part I. Subjects are male and female teachers in equal proportions,
with a wide range of length of service. The questions are declarative
of the answers they have given to the questionnaire and extend to
other areas as well.
Part III includes a small questionnaire with four tasks. Each task asks
teachers to solve one problem or correct a false answer to a problem
and then to explain thoroughly how they are planning to teach it in
their class.
Results and interpretations
Part I was concluded in June 2005. Thirty questionnaires were
collected. The subjects were in-service teachers in Greek elementary
schools (K-6 to K-12) from Macedonia (Northern Greece).
A statistical analysis of the responses gives the following fi rst
results:
Years of service and sex of the subjects play no signifi cant role in
their responses. Almost half of the subjects (53%) cannot recognize
the term ‘rational number’, although all of them know the term
‘fraction’. Part-whole is the dominant interpretation of the fractions.
The explanation given for the meaning of three-eighths uses (100%)
this interpretation. 53% of the subjects uses an area (‘pizza-circle’ or
rectangle) for the unit representation and 33% don’t use a graphical
representation at all. None uses the number line. When asked to
recognize a fraction from an area with 10 parts, 4 of them shaded, all
recognize 4/10 (or equivalent) or 6/10. None recognizes 2/3 (ratio) or
21/2 (conceptualizing the shaded area as the unit of measure) or 12/3.
73% give a false response to Noelting’s ‘orange juice’ test (1979, in
Behr et al., 1983), interpreting a ratio as a part-whole situation.
Quite a large number probably don’t have a fi rm concept of the
rational numbers. 31% cannot overcome the perceptual distractors of
non-related lines in a ‘pizza-circle’, and defi ne it wrongly or they can’t
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defi ne the fraction of an indicated part of it. Success in estimating the
results of addition and subtraction of proper fractions and location
on a number line ranges from 37% (6/7+2/3) to 47% (11/12-8/9) and
fi nally to 50% (4/9+3/7).
Two-thirds of the subjects recognize the fraction as an entity, but
seem not to have a fi rm knowledge of it. 60% cannot compare
two fractions without using the formal procedure of making them
homonyms (which was not allowed in that specifi c task). 53% cannot
defi ne correctly what the denominator shows.
In the task of fi nding the unit region of a given square with area
9/4, 46% cannot solve the problem at all and 21% use rulers and
algebraic equations. Only 33% use partitioning methods successfully.
Partitioning is a problem to 37% of the subjects when they try to divide
11 pizzas between 9 children, although the majority use higher forms
of partitioning.
The vast majority of the subjects prefer to use formal solutions to the
given tasks. 57% use a formal solution to a division school problem
of fractions and only 17% a graphical solution. This percentage
increases (23%) when the problem is not customary. In spite of this,
24% cannot solve a customary K-11 division problem. In addition,
37% don’t know how to calculate the division of fractions, and only
13% can give an explanation for the procedure of this calculation.
17% of them seem to have whole number ideas about fractions and
almost 20% use additive procedures in equivalence and order tasks.
Conclusions and discussion
Results from Part I of our research indicate that in-service teachers
don’t have clear ideas about rational number concepts. This seems to
be in agreement with previous research (see Post et al., 1988).
Most of the teachers use a single mode of representation in their
instruction (‘pizza circle’) which leads to certain interpretations
that seem to dominate their own thinking too. Most of them solve
‘successfully’ word problems using formal procedures and operations.
This seems to be in accordance with the school curriculum and the
way that schoolbooks represent rational numbers, thus leading to
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the defi ciencies that we have mentioned above. Behr et al. (1992)
state that: ‘Based on our own and others’ research in the content
domain of multiplicative structures, we have come to the realization
that many of the limited, alternative (or mis-) conceptions that
children and some adults (teachers of middle grades, for example)
have about many multiplicative concepts result from defi ciencies in
the curricular experiences provided in school. […] In the absence of
counter-experience or “high level” mathematical education, these
limited conceptions remain into adult life’ (p. 300). An indication that
gives support to the above is that 43% of the subjects in our research
view 1/2 as their favourite fraction, which is, as Jack (a 12 year old
child in Kieren (1993)) described it: the fraction that ‘I have known
the longest. I have known it since before I went to school’ (p. 56).
The previous indications need to be clarifi ed and examined further
through interviews and more specifi ed questions. Part II and III of
our ongoing research will we hope provide some declarative remarks
to our fi rst results.
References
Behr, M., Harel, G., Post, T., and Lesh, R. (1992) Rational Number,
Ratio, and Proportion, in D. Grouws (ed.), Handbook of Research
on Mathematics Teaching and Learning (pp. 296–333). New York:
Macmillan.
Behr, M., Lesh, R., Post, T.R., and Silver, E.A. (1983) Rational Number
Concepts, in R. Lesh and M. Landau (eds), Acquisition of Mathematical
Concepts and Processes (pp. 91–126). New York: Academic.
Behr, M., and Post, T. (1981) The Eff ect of Visual Perceptual Distractors
on Children’s Logical-Mathematical Thinking in Rational Number
Situations, in T. Post and M. Roberts (eds), Proceedings of the Third
Annual Meeting of the North American Chapter of the International
Group for the Psychology of Mathematics Education (pp. 8–16).
Minneapolis: University of Minnesota. (also available online 25/6/2005
h p://education.umn.edu/rationalnumberproject/81_1.html)
Behr, M., Wachsmuth, I., Post, T., and Lesh, R. (1984, November)
Order and Equivalence of Rational Numbers: A Clinical Teaching
Experiment. Journal for Research in MathematicEducation, 15(5),
323–41. (also available online 25/6/2005 h p://education.umn.edu/
rationalnumberproject/84_2.html)
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Chassapis, D. and Katsarkas, A. (2005) Fractional Number References in
Real Situations: The Elementary Schoolbooks Case, in Chassapis, D. (ed.)
Social and Cultural Dimensions in Mathematics Education (Proceedings
of 4th Two-days of Conversation for Mathematics Instruction, 19 and
20 March 2005, Aristotle University of Thessaloniki, (pp. 171–78).
Thessaloniki: Copy City. (in Greek)
Cramer, K.A., Post, T.R. delMas, R.C. (2002) Initial Fraction Learning by
Fourth- and Fi h-Grade Students: A Comparison of the Eff ects of Using
Commercial Curricula With the Eff ects of Using the Rational Number
Project Curriculum. Journal for Research in Mathematics Education,
33(2) 111–44.
Freudenthal, H. (1983) A Didactical Phenomenology of Mathematics (pp
vii–209). Dordrecht, Netherlands: D. Reidel.
Kieren, T.E. (1993) Rational and Fractional Numbers: From Quotient
Fields to Recursive Understanding, in Carpenter, T P., Fennema, E.,
Romberg, T.A. (eds), Rational Numbers – An Integration of Research
(pp. 1–10). Hillsdale, NJ: Lawrence Erlbaum Associates.
Post, T., Behr, M., and Lesh, R. (1982, April) Interpretations of Rational
Number Concepts, in L. Silvey and J. Smart (eds), Mathematics for
Grades 5–9, 1982 NCTM Yearbook (pp. 59–72). Reston, Virginia:
NCTM. (also available online 25/6/2005 h p://education.umn.edu/
rationalnumberproject/82_1.html)
Post, T., and Reys, R.E. (1979) Abstraction Generalization and Design
of Mathematical Experiences for Children, in K. Fuson and W. Geeslin
(eds), Models for Mathematics Learning. (pp. 117–39). Columbus, OH:
ERIC/SMEAC. (also available online 25/6/2005 h p://education.umn.
edu/rationalnumberproject/79_2.html)
Post, T., Harel, G., Behr, M., and Lesh, R. (1988) Intermediate Teachers’
Knowledge of Rational Number Concepts, in Fennema, et al. (eds),
Papers from First Wisconsin Symposium for Research on Teaching and
Learning Mathematics (pp. 194–219). Madison, WI: Wisconsin Center
for Education Research. (also available online 25/6/2005 h p://education.
umn.edu/rationalnumberproject/88_11.html)
Contact
Aristarchos Katsarkas
School of Education
Aristotle University of Thessaloniki
Greece
aristus@otenet.gr
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Maria Vlachou
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Abstract
Considering language as discourse, thus as a non-neutral means of
communication, this chapter a empts to analyse the primary school
teachers’ discourse of assessment in mathematics. Dominant assessment
discourses in pedagogical literature are examined, and offi cial texts of
curricula and relevant circulars are analysed, focusing on the assessment
of primary school mathematics. Our analysis shows diff erences in positions
and assessment practices of teachers, an appeal to a variety of discourses
for their justifi cation, as well as contradictions and ambiguities within the
offi cial discourse, which give rise to various tensions.
Key words: critical discourse, curriculum, mathematics assessment,
teachers’ assessment practices
Introduction
Adopting the theoretical premises of Halliday (1985) as well as
Critical Discourse Analysis (C.D.A.) by Fairclough (1989; 1995) and
Kress (1989), this study aims to investigate the picture of assessment
constructed within primary school teachers’ discourses, as well
as the role assigned to the teacher. The picture of assessment thus
constructed is examined as regards the meaning a ributed to it,
the functions and purposes it serves and the forms and criteria
THE DISCOURSE OF PRIMARY
SCHOOL TEACHERS FOR
ASSESSMENT IN MATHEMATICS
Maria Vlachou, Aristotle University of Thessaloniki
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highlighted. The language features of texts that contribute to it are
also examined.
In order to comprehend the texts that teachers produce to assess
students in mathematics, it is necessary to examine the frame of
discourses within which those texts are produced (Fairclough, 1989).
In our study, due to the limitations of the proceedings, we only
consider the pedagogical literature related to assessment in education
and examine offi cial texts on assessment, focusing on mathematics at
primary school.
tThe ways in which teachers justify their positions together with
the practices they use, the ways they position themselves within the
offi cial discourse and any tensions they may be experiencing are also
considered.
Discussion of assessment in general
The concept of assessment, aims, forms and criteria
Reviewing the assessment literature we notice a convergence of
opinions on the following criteria. Assessment is the systematic
process (and not the accidental observation) of students which asks
for data to be collected, for the collected information to be interpreted,
and for the existence of concrete criteria and predetermined
objectives.
In contrast to the traditional discourse of performance assessment,
which is only interested in the products of learning and not in
the internal thought processes that students follow, more recent
approaches (NCTM, 1995, 2000) see performance in a broader sense,
including elements such as creativity, thought process, a itudes, thus
broadening the concept of assessment to make it a dynamic process
that supports learning in a complex manner, rather than simply as a
means of quantifying students‘ traits (Matsagouras, 2004).
The main criteria of school performance assessment that appear
in the literature are those of: the average classroom student, the
student’s position with respect to the classroom norm (Gauss curve),
the curriculum and self-improvement (see Guskey, 1994, Hiotakis,
1997).
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Various forms of assessment appear in the literature to which
diff erent functions are a ributed, expressing diff ering perceptions
of the concept of assessment. Prevailing amongst them is the
distinction between formative and summative assessment. Scriven
was the fi rst to make this distinction (Tyler, 1967). Researchers
recognize the diffi culty of formative and summative assessment
co-existing (Torrance, 1993), as they come from diff erent theories of
learning. Researchers also point to the priority that should be given
to formative assessment in the classroom (Black and Wiliam, 2003).
Many researchers suggest that assessment results should be
formulated in recording of comments and/or in a descriptive level,
rather than assigning marks (Black and Wiliam, 2001).
The traditional theories of learning and the view of learning as a
knowledge construction process within specifi c contexts (Resnick,
1989) tend to be questioned nowadays. This has resulted in a shi from
the assessment model that looks at the performance of individuals
in context-free situations towards the model that is related to the
assessment of knowledge and skills within authentic situations that
have meaning for the students (Maclellan, 2001).
Researchers recognize the diffi culty of applying new assessment
methods to contexts where the dominant traditional discourse of
assessment confl icts with the alternative methods, thus preventing
their use for the purposes of promoting learning (Broadfoot, 1998:
447) and creating tensions (Lyons, 1998).
The necessity of assessment
The functions of assessment are more generally placed in a complex
framework of discourses that oppose each other. These discourses
put forward arguments for or against the necessity of assessment.
We found the following six dominant discourses:
the socio-economic discourse in favour of assessment, the sociological
against, the pedagogical-psychological in favour, the pedagogical-
psychological against, the educational-political in favour and the
methodological discourse against the assessment.
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The socio-economic discourse in favour highlights the social
functions of school assessment. The sociological discourse against the
institution of assessment focuses on the examinations and marking,
arguing that these contribute to the social categorisation of students,
its argument being based on a wealth of sociological research.
The pedagogical-psychological discourse in favour is aligned to the
school’s pedagogical mission. It is in tension with the pedagogical-
psychological discourse against, which is mainly focused on
examinations and marking, on how these are employed and on
their eff ects on the student’s personality from the pedagogical and
psychological point of view.
The educational-political discourse in favour focuses on the
educational system, aiming to improve it.
The methodological discourse against focuses on the ineff ectiveness
of assessment means and methods and on the subjectivity of
assessor.
Methodology
We considered the texts of Presidential Decrees1 and circulars
currently in eff ect (P.E. 8/1995 mi 121/1995, F7/228/T1/1561/15–11–96)
and related to mathematics assessment in Greek primary schools and
mathematics textbook supplements for the teacher. The teachers’ view
of mathematics’ assessment was examined through the transcripts of
semi-structured interviews of six teachers.
A method based on Halliday’s Functional Grammar (Halliday, 1985)
and on the interpretative techniques of Fairclough’s Critical Discourse
Analysis (Fairclough 1992a, 1989, Kress, 1989, Hodge and Kress,
1993) was used to analyse the offi cial texts and the teachers’ texts.
Thus characteristics of the language of these texts were examined and
the functions – ‘ideational’, ‘interpersonal’ and ‘textual’ – (Halliday,
1973) that they carry for the speaker/author and the reader/listener
were interpreted.
The ‘ideational’ aspects of texts to be analysed relate to the picture
constructed for mathematics assessment. They were analysed
primarily through the examination of types of processes (Halliday,
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1985:101–31) that take place in the discourses under examination
and are related to the task of assessment, the type of logical subjects,
ie the human or inanimate actors of these processes (ibid: 32–37).
The presence of a human being in a text or the absence thereof, and
the use of inanimate abstract nouns as actors of the processes, were
examined via the use of passive voice and nominalisations that relate
to social and ideological aspects of language (Fairclough, 1989, 1992a,
Hodge and Kress, 1993).
The teacher’s relationship to assessment, with the person to whom
the teacher’s discourse is addressed, the roles constructed for the task
of assessment and the degree of the teacher’s autonomy concerns the
‘interpersonal’ function of language. It was analysed through the use
of personal pronouns, modes and the text’s modality (Halliday; 1985:
86, Fairclough, 1989:129, 1992a:159 on modality, and Kress, 1989, on
their interpretation).
The structure of offi cial texts as a whole and the type of text that is
related to the ‘textual‘ function were examined through the ‘themes’
(Halliday, 1985) that dominate the text.
Drawing on the theory of C.D.A. by Fairclough, ‘member resources’
were also examined which the producers of texts draw on, in order
to justify their positions and practices.
The offi cial discourse of mathematics assessment
Examining the offi cial discourse of assessment in Presidential
Decrees, relevant circulars and curricula currently in eff ect in Greece,
we notice that there is an agreement between offi cial discourse
and teachers’ discourse regarding the pedagogical dimension of
assessment and its objectives. Offi cial texts draw on the researchers’
discourse to justify their positions, yet at the same time considerable
tension is apparent when it comes to applying pedagogical principles
of assessment in practice, through the use of school textbooks. We
notice contradictions between the offi cial texts of Presidential Decrees
and the teachers’ supplement. While in offi cial texts of P.D. there is
talk of continuous process with diagnostic and feedback purposes,
in the teacher’s supplement assessment appears to be considered
as a process separate from the other educational processes, since
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assessment is planned in two phases: while the students work with
their textbook and at the end of each main unit. The primary means
of assessment is the book’s predetermined ‘assessment criterion‘ that
is, a test in student’s textbook at the end of each unit.
Today in the assessment system for the primary school in Greece,
the numerical assessment coexists with the descriptive assessment.
These forms of assessment contradict each other, since the
descriptive assessment clearly has pedagogical aims and is linked to
the individual norm-referenced assessment and individual-specifi c
teaching, whilst the numerical assessment grades and categorizes
students and follows the classroom norm promoting comparison
and competition. With regard to the descriptive assessment and
the criteria of assessment which the offi cial texts suggest, explicit
instructions to teachers on how to apply them in practice are not
provided.
In the teacher’s textbook there are contradictions. For example,
the teacher’s textbook emphasizes the process of problem solving
that student follows and the same time proposes assessment tasks
which require that students demonstrate their knowledge and skills.
Methods of comprehension and assessment of the process that the
student follows are not described.
Within the offi cial texts, we see two dominant discourses in confl ict
with each other. On one hand we fi nd the pedagogical discourse
which is encoded in words and phrases of pedagogical content,
and on the other hand the educational-political discourse which
expresses the government’s intentions for education and is encoded
in the syntax of offi cial texts.
The main characteristics of offi cial texts are similar to those of scientifi c
texts: tendency to impersonal expression that adds objectivity to the
statements in the text, absence of personal syntax achieved with the
use of passive syntax and nominalisations, absence of actors, use of
processes that primarily concern defi nitions and relationships, use
of simple present in the indicative that expresses the modality of
certainty, specialized technical/scientifi c vocabulary (Kress, 1989).
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In the teacher’s supplement, the teacher is constructed as the one
who needs information, protection from insecurity and haphazard
planning, and as the executor of the proposed instructions and
predetermined activities. The diversity and the individuality of
students are not taken into account, as the existence of a single
school textbook and the universally applicable curriculum show.
This contradicts the offi cial educational policy that highlights
its assignment of priority to the provision of equal educational
opportunities and the non-selective and non-competitive character
of assessment.
The teachers’ discourse of mathematics assessment
Analysis of the teachers’ discourse showed that all teachers agree
with the offi cial texts and the researchers’ discourse as regards
the concept and objectives of assessment that make assessment a
necessity. Research fi ndings by Philippou and Panaoura (2000), as
well as Brown (2004), agree with the teachers’ opinion about the
diagnostic and formative role of assessment.
Teachers consider that a fundamental obligation is to consult
offi cial texts on assessment. The degree of freedom allowed by the
educational system to deviate from the curriculum is interpreted in
various ways by the teachers. They take the initiative of the usage
of assessment results according to the students’ needs, modifying
the material to be taught, the teaching objectives or the proposed
teaching time for each unit. They also diff erentiate from each other as
regards the use of assessment criteria. Earlier research also indicates
the diff erent interpretation of offi cial policy by teachers (Yung, 2002).
The analysis showed that teachers don’t take into account ‘assessment
criteria‘ mainly because students tend to prepare for them at home.
The teachers make up their own tests for student assessment.
Teachers seem to use the same forms and methods of assessment.
They do not apply alternative methods because they lack information
on how to use them and because of lack of time. Research suggests
that there is a diffi culty in applying alternative assessment methods
(Black and Wiliam, 1998, Kahn, 2000) and recognizes the tensions
this causes to teachers (Broadfoot, 1998, Lyons, 1998).
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312
The analysis also showed that teachers do not apply descriptive
assessment, arguing that they lack information on how to apply it
and, again, due to lack of time. They mark wri en work and tests
despite the fact that offi cial instructions forbid it. This practice agrees
with Mavromatis’s fi ndings (1997).
Most teachers appear to believe in the counterbalancing role of the
teacher for students who come from diffi cult family backgrounds
and this agrees with fi ndings by Zbainos and Hallam (2002).
All teachers agree with the offi cial discourse regarding the main
objectives of mathematics education and the importance that should
be given to the process of problem solving that students follow.
However, when assessing, the majority of teachers appears to focus
more on results and on the existence of a single right answer or right
path to a solution, rather than on the process and variety of student
answers, hinting at an absolutist view of mathematics (Ernest, 1996).
This inconsistency refl ects the contradiction that exists between the
value which the offi cial discourse places on the solving process and
the concurrent requirement that students demonstrate skills and
precision of results, as required by the practical discourse of the
textbook.
All teachers recognize the lack of the assessment’s objectivity due
to the subjective criteria of each teacher. Variations are mainly to be
found regarding (1) the individual characteristics of students and the
kind of criteria that infl uence the assessment, vis a vis the need to
make students aware of them, (2) their view of the role of marking
and (3) their view of the ideal way to express assessment results.
With regard to the assessment criteria put forward by offi cial texts,
most participants agree on taking into account the eff ort spent,
performance in wri en tests, daily orals, all round participation in
the classroom and the student’s degree of interest. Homework and
student behaviour at school (which are criteria defi ned by the offi cial
discourse) appear not to infl uence the assessment of most teachers.
An overview of related research verifi es the variety of assessment
criteria that teachers employ and the diff erent prioritisations
applied (Gipps et al., 1996, Philippou and Christou, 1997, Zbainos
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and Hallam, 2002, Pilcher, 1994). Variations in assessment practice
spo ed by the analysis is confi rmed by research that supports the
idiosyncratic nature of assessment, which is based on the teacher’s
beliefs, knowledge and experience (Webb, 2004, Cizek et al., 1995)
and shows that diff erent teacher views lead to diff erent assessment
practices, even when the curriculum and assessment criteria are
predefi ned to a large extent (Watson, 1999).
Within the teachers’ discourse the role constructed for the teacher is
that of the local enforcer who is obliged to follow the offi cial policy
on assessment, yet, at the same time, ought to take into account
his/her students’ needs. He/she takes the initiative in modifying,
to a degree, the proposed assessment practices, depending on the
students’ individual situation, yet taking care not to stray far from
the general frame of assessment that the offi cial discourse shapes.
The role of participants as teachers/educators and as enforcers of
the offi cial discourse’s instructions appears to create tension which
teachers address with the following strategies: they yield to the
external authority or are infl uenced, each to a greater or lesser extent,
by the offi cial policy on assessment practices and adapt them, more
or less, to the particularities of the class (on tension, see: Baker and O’
Neil, 1994, Morgan, 1998, Black, 2001, Brown, 2003).
By investigating the discourses that infl uence teachers’ practices
and the sources of their arguments, it became apparent that teachers
mainly draw on the offi cial discourse and the discourse of researchers
with respect to the concept, objectives and necessity of assessment.
As regards marking, they draw on the sociological and pedagogical-
psychological discourse (for or against marking). A small percentage
argues based on common sense and their personal experience, except
for one participant who mainly draws on her personal experience.
As for the main linguistic features used in the interview transcripts,
which are linked to ‘ideational‘ and ‘interpersonal‘ functions of
language, not only similarities but also essential diff erences were
observed. Specifi cally these were: the processes that dominate are
mostly ‘material‘ and ‘mental‘ processes (Halliday, 1985); the choice
of a certain type of process puts the speaker in the position of the
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person who considers that the activities to which they refer involve
processes of this type.
With regard to the type of the human-subject of the processes in use
(where stated explicitly) – excluding cases in which the student is the
human actor – we note the following four categories:
1. A specifi c person, the interviewee herself/himself talking in
the fi rst singular, implying his/her personal involvement in
the activity stated in the text or that the views expressed are
personal opinions for which he/she assumes responsibility.
We couldn’t claim that the use of the pronoun ‘I’ carries any
ideological signifi cance (Fairclough, 1989).
2. The general participant ‘you’ used to claim commonality of
experience between transmi er and receiver (ibid.: 180),
whereas in other cases in order to share responsibility for the
transmi er’s statements with the general participant ‘you’ or
to make the receiver feel that the practices and convictions
of the transmi er also constitute convictions of the general
participant ‘you’ (ibid.).
3. The teacher, whose use as actor implies the speaker’s distancing
from the task of assessment.
4. The teachers’ community which, in Greek, is revealed by the
ending of verbs in the fi rst plural, and whose use indicates
that the speaker speaks on behalf of the teachers’ community
to which he/she belongs, which guarantees the authority of
his/her statements or shows the diff usion of responsibility to
this community (see Kress, 1989).
There exists minimum usage of inanimate subjects and, where present,
they refer to the impersonal authority of the government. More
generally, within interview texts, there appears some inconsistency
in the use of personal pronouns, while the dominant type of human
actor varies between texts.
The usage of passive voice and of nominalisations (infrequent) also
varies. They are mostly used in the discourse of two participants
who mainly use as their actor the general participating ‘you’. As a
result, the relationship constructed between speaker and listener
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and between speaker and assessment varies substantially among
texts according to the degree of familiarity with the listener and the
distance or ’ownership’ of the task of assessment. It is worth noting
the case of a text in which the absence of the fi rst person singular and
the high usage of passive syntax and nominalisations contribute to
the creation of a formal and impersonal relationship between speaker
and listener and express his/her tendency to distance himself/herself
from his/her statements, as well as from the task of assessment.
A second text also shows the speaker’s tendency to distance his/
herself and evade responsibility for statements concerning personal
assessment practices; in this text, the speaker does not employ the
fi rst person singular, thus avoiding a personal involvement in the
task of assessment.
The use of the simple present (indicative) is dominant in all texts,
a aching the modality of certainty which, combined with the use of
certainty or uncertainty modals, varies among texts. In most texts,
modals of uncertainty are used almost as frequently as modals of
certainty, implying a lack of certainty for numerous statements by
interviewees.
Conclusions
The diff erences between teachers’ assessment practices shown by
analysis of their discourse highlight the complexity of the assessment
process and raise issues of social justice for the students (Watson,
1999).
Teachers draw on diff erent, sometimes contradictory, discourses for
the justifi cation of their positions and practices. This renders obvious
the need for teachers to strengthen their awareness of the way in
which particular discourses position them and their awareness of
assessment practices towards which these discourses direct them. So
doing they can realize which discourses, values, beliefs and practices
they identify with, resist those with which they do not agree and face
eff ectively the tensions that they experience.
Additionally, contradictions and ambiguities within the offi cial
discourse were highlighted as well as inconsistency between revised
objectives and suggested assessment practices. All these require that
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positions and practices should be submi ed to scrutiny and that
teachers’ training programmes on assessment and how to apply
alternative methods should be planned.
Footnotes:
1 Author’s note: A Presidential Decree is a legislative instrument
widely used by the Hellenic Republic, whereby the President of the
Hellenic Republic signs off a specifi c item of legislation, thus pu ing
it into eff ect.
References
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Greece
vlachoum@eled.auth.gr
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Abstract
Considerable development of various activities associated with, or considered
as constitutive of, mathematics education has taken place during the latest
thirty years. In spite of this the scientifi c object of mathematics education
remains, to a great extent, vague. In consequence, the constitutional
conceptual system and the fundamental research methodologies of
mathematics education also remain ambiguous, since the scientifi c object,
the conceptual system and the privileged research methodologies of a
discipline are simultaneously constructed and developed through reciprocal
determinations, wherein each one presupposes the existence of the other two.
As a result, the epistemic status of mathematics education as a discipline
remains in question, and its contribution to the solution of problems arising
in the real world of mathematics classrooms is doubted. Relevant questions
are posed in this chapter and possible answers are traced by highlighting
three interconnected aspects. First, the historical conditions under which
mathematics education has been founded both as a discipline and as a research
fi eld, second, the fundamental characteristics of the relationship between
mathematics education theory and practice which has been developed, and
third its essential interdisciplinary nature.
Keywords: History of mathematics education, interdisciplinarity,
theory and practice
MATHEMATICS EDUCATION
AS A SCIENTIFIC DISCIPLINE:
IMPLICIT ASSUMPTIONS AND OPEN
QUESTIONS
Dimitris Chassapis, Aristotle University of
Thessaloniki
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322
Introductory remarks
In considering mathematics education as a discipline, what is its
scientifi c object? More precisely, what object of study is ascribed
explicitly to mathematics education or is deduced implicitly by the
relevant activities, publications, conferences and research programs
of researchers and educators of the community?
At fi rst glance, the answer to this question seems obvious and easy:
the learning and teaching of mathematics. I claim, however, that
the answer is neither obvious nor easy. This is because this answer
and any similar answer assumes that the ‘phenomenon of learning
mathematics’ has quite distinct characteristics from the phenomenon
of learning any other subject ma er. On the same basis, the teaching
of mathematics is assumed to be a diff erent activity from the teaching
of any other subject ma er, being an application of that branch of
knowledge that has emerged from the study of learning mathematics.
Such assumptions, however, may be considered neither self-evident
nor as being categorically accepted.
In consequence, the scientifi c status of mathematics education may
be questioned, and, by analogy, the scientifi c status of any other
discipline which is included in the class of the so-called ‘educational
sciences’, since all these disciplines are founded on the same
assumptions about particular learning phenomena.
This chapter raises questions and a empts to outline answers,
although fragmented and incomplete, adopting an epistemological
perspective grounded on the philosophical contributions of Luis
Althousser as they have been further developed and exemplifi ed
in the epistemology of science by the Greek philosopher Aristidis
Baltas (1990).
The main points are summarised here. According to Baltas’ analyses
(Baltas, 1990), each scientifi c discipline has its own object of inquiry,
which is constructed by the discipline itself. At the same time this
object of inquiry, being its scientifi c object, constitutes a founding
component of the discipline. Each scientifi c discipline constructs its
own object, selecting, defi ning and describing real world phenomena,
re-constituting conceptually and incorporating into its scientifi c
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object. This constructive process makes up simultaneously the
scientifi c object of the discipline and the discipline itself. This process
presupposes a conceptual system, and it produces this conceptual
system by contributing to its development. The same holds for the
using and formation of a research methodology appertaining to the
particular discipline. The object of inquiry, the conceptual system
and the research methodology of a discipline are, therefore, being
constituted at the same time by a process of reciprocal determinations
and controls, wherein each of the constituents presupposes the
existence of the others. This construction process makes its own
scientifi c object relatively autonomous from the object of any other
discipline and builds up to the meaning of its fundamental concepts
in relative autonomy. These two autonomies of a discipline, i.e. the
autonomy of its subject ma er and the autonomy of the meanings of its
fundamental concepts, defi ne the boundaries of its interdisciplinary
relations.
The emergence of mathematics education as a scientifi c discipline
As indicated by Baltas (1983), the formation of each scientifi c discipline
constitutes: ‘an especially complex process, which involves and brings
into action many diff erent and dissimilar elements of the whole social
practice in particular ways. Changes in material production and in the
life of people, technical achievements and innovations, novel social
experiences, tensions and ruptures in practical ideologies, changes
caused to theoretical philosophical diversifi cations and confl icts are
accumulated, interwoven and create a point of critical concentration,
where the partial pressures on the known are condensed, mutually
reinforced and lead to its destruction. This process of destruction is
at the same time the process which creates the scientifi c object of a
discipline, constitutes its conceptual system and instals the research
methodologies appertaining to the discipline. Namely, this is the
process which constructs, in a unifi ed movement, all the constituent
elements of the discipline, a process which is known, a er Bachelard,
as an “epistemological break”’.
Mathematics education as a scientifi c practice and as a research
fi eld has emerged in the same era as, and to a great extent by way
of, the radical school mathematics reforms of the 1960s. These
reforms had been inspired and guided by the Organisation for
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324
European Economic Cooperation (OEEC) – later on Organization for
Economic Cooperation and Development (OECD) as a response of
the USA and its allies to the technological challenge of the Soviet
Union posed by the launching of Sputnik on October 1957. This
was a challenge directly related to the Cold War. During the 1960s
distinguished mathematicians, mathematics education experts and
educational decision-makers from the USA and many Western
European countries met in seminars and conferences and worked
out a plan for the radical reform of school mathematics objectives,
curricula, text-books, and in-service preparation of teachers, known
a erwards as the ‘New Mathematics’ movement. (Fehr, 1961a,
1961b, 1964). UNESCO played an important role in these reforms
with publications and seminars aiming at the dissemination of ‘new
school mathematics’ in other countries besides the USA and Europe
(UNESCO, 1966).
Two pioneer publications were established, the journal Educational
Studies in Mathematics was founded in 1968 and two years later the
Journal for Research in Mathematics Education. These two journals
have heavily infl uenced the formation of mathematics education
as a discipline. At the same time many primary and secondary
school mathematics projects, which were developed by various
foundations and institutes (Nuffi eld-UK, Alef-Germany, Analogue-
France, Wiskobas-Holland, School Mathematics Study Group-USA
and others) have had a considerable impact on the conditions of the
foundation of mathematics education as a discipline.
In this societal and political framework, and directly related to the
activities of planning and implementation of school mathematics
reforms, mathematics education has emerged as a distinguished
scientifi c practice and therefore as a discipline and a research fi eld.
In following years, mathematics education has been developed as
a scientifi c discipline and has been established in the academy. The
foundation of university departments, provision of undergraduate
and postgraduate courses in almost every university around
the world, research and development projects, foundation of
national and international associations, organization of numerous
conferences, publication of books and journals devoted exclusively
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Dimitris Chassapis
325
to mathematics education, etc., are all circumstantial evidence of this
development.
The scientifi c object of mathematics education has, therefore, been
constructed by adopting and applying a ‘scientifi c’ view to the
phenomenon of ‘teaching mathematics’ in formal education. That
is, altering the view of a phenomenon, under the pressure of socio-
political necessities, has produced the scientifi c object of mathematics
education. Until then, the prevailing view approached mathematical
thinking and as a consequence mathematics learning as an individual
aptitude or as an innate capability. Therefore, mathematics teaching
addressed particular individuals, considered to be more or less
mathematically gi ed, and its contribution to mathematics thinking
was considered as more supportive than crucial.
Summing up, mathematics education emerged in a context created by
a new view of fi rst the teaching and then the learning of mathematics.
However, the formation of mathematics education in these terms
implied from its very beginning an ambiguity in the relationships
between its theory and its applications, as well as in the relationships
that both of them establish with educational policy.
Finally, in my view, the aspects of application took priority and
prevailed over the aspects of theory in the construction of the scientifi c
object of mathematics education, a fact that has moulded accordingly
its conceptual system as well as its research methodologies.
The relationships between theory and applications of mathematics
education
An inherited ambiguity exists in the relationships between theory and
applications of mathematics education, as well as in the relationships
that both maintain with educational policy. In my view, this ambiguity
is a result of the interdisciplinary status of mathematics education
and the impact of the socio-political conditions under which it has
emerged as a scientifi c discipline.
The research questions in mathematics education and the inquiries
which set them off are, as a rule, arising from, and referring to,
mathematics classrooms. Their study is not primarily justifi ed by
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326
theoretical requirements but on the grounds of their usefulness
in promoting eff ective mathematics teaching, i.e. in particular
applications of mathematics education. These applications are mostly
recommended by occasional and specifi c needs of mathematics
education policy-making and not by the demands of mathematics
education theory development. Most of the research papers
published in journals or presented at conferences, particularly in
the USA, justify their aims with reference to objectives or needs
arising from educational policy documents or induced by school
mathematics curricula and evaluation frameworks. At the same
time, as reported by Lerman and Tsatsaroni (2004), more than 85%
of the articles published during 1990–2001 in Educational Studies in
Mathematics and in Journal for Research in Mathematics Education
construct, project or promote an explicit pedagogical model for
teaching mathematics.
Thus, applications of mathematics education generated by
educational policy-making gain a clear priority over issues arising
from its theoretical evolution. This predominance of applications over
theory has two clear implications. First, a reduction in the scientifi c
objectives of mathematics education (and as a result a one-sided
development of the conceptual system), and second, an establishment
of a particular state of aff airs concerning the legitimization of the
resulting knowledge which, in turn, informs and shapes its content.
To a great extent, the criterion for the legitimization of knowledge
produced by mathematics education is not considered to be its truth,
however defi ned, but rather its potential usefulness in mathematics
educational policy-making. So, mathematics education is constantly
called to account either for employing not scientifi c but educational
norms or for adopting criteria imposed by prevailing educational
policies with consequent distortions and insuffi ciencies in its
theoretical evolution.
The assignment of a primary role to theory in its relationship to the
applications of mathematics education, as well as the a ainment of
complete independence from relevant ideological infl uences, remain
an ideal. These are, amongst other requirements, the necessary
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Dimitris Chassapis
327
conditions for the formation of mathematics education as a scientifi c
discipline, as an interdisciplinary subject.
The interdisciplinarity of mathematics education
As mentioned earlier, mathematics education has been formed
and evolves on an interdisciplinary basis. The boundaries of its
interdisciplinarity are defi ned by two relative autonomies. The
autonomy of its scientifi c object against the scientifi c objects of
other disciplines and the autonomy of meanings of its fundamental
concepts against the meanings, which these concepts possibly have
in the empirical world or in the conceptual system of any other
discipline (Baltas, 1983).
Assuming that the phenomenon of learning mathematics, and as a
consequence the activity of teaching mathematics, have quite diff erent
characteristics from the phenomenon of learning and the activity of
teaching any other disciplinary knowledge, mathematics education
emerged as a scientifi c discipline adopting and transforming
concepts, at fi rst, from three well-established disciplines. Mathematics
defi nes the learning and teaching content, psychology describes the
phenomena of human learning at the individual level and sociology
approaches the phenomena of learning and the activities of teaching
as socially and culturally embedded processes.
However, although social functions are assigned to the teaching of
mathematics in formal schooling and its outcomes, almost nothing
social is a ributed to the phenomenon of learning mathematics
itself. The approach to learning mathematics is mainly delimited
at an individual level by the mainstream of mathematics education
(e.g. Chassapis, 2002; Kilpatrick, 1992; Lerman et al.; 2003, Lubienski
and Bowen, 2000; Reyes and Stanic, 1988; Secada, 1992). Social
approaches from various standpoints to the study of learning
mathematics, individually or collectively, are always a requirement,
considering li le about the occasional acknowledgements of social
discriminations that are produced or are justifi ed by processes of
learning of mathematics (Chassapis, 2004).
In addition, the autonomy of the mathematics education object
of inquiry against the objects of mathematics and the objects of
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328
psychology is always a requirement. It is alsonecessary, although
to a lesser extent, to consider the autonomy of the constitutional
conceptual system of mathematics education in relation to these
disciplines, as well as against the current teaching empiricism.
The approach to mathematics education as a particular application
of mathematics, has many supporters infl uenced by a philosophy of
Platonism in mathematics. From the standpoint of Platonism – and
because of its epistemological premises concerning mathematical
knowledge – an approach to the phenomenon of learning mathematics
in terms which would permit or even facilitate its autonomy as a
scientifi c object seems to be, almost, unfeasible. An approach to the
learning of mathematics which adopts a Platonism for mathematics,
privileges the mathematics curricula as an object of inquiry, closely
associated with issues concerning the status of mathematics as a
teaching subject in formal schooling. Moreover, the teaching of
mathematics is considered as an activity inherently determined
by the discipline of mathematics itself, which self-determines its
didactics.
Another perspective, leading to similar outcomes, includes
various dominant trends in psychology, such as behaviorism or
cognitivism, which claim for themselves the phenomenon of learning
mathematics as an object of scientifi c inquiry. They utilise conceptual
sets of mathematics, and more broadly fundamental aspects of
mathematical thinking, as a preferred fi eld for applying and testing
their theories for human learning or cognitive functioning. As a
result, they incorporate the phenomenon of learning mathematics
in their fi eld of applications, questioning indirectly the founding
assumption of mathematics education: that the phenomenon of
learning mathematics has quite distinct and particular characteristics
from the phenomenon of learning any other scientifi c subject ma er.
E. Thorndike’s book (1922) The Psychology of Arithmetic is an
early exemplar of such an approach, which adopts the standpoint
of behaviourism. Further books promote approaches which adopt
perspectives on the learning of mathematics inspired by cognitive
psychology (for example, Dehaene, 1997, The Number Sense: How
the Mind Creates Mathematics).
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329
As mentioned above, the ambiguity of disciplinary boundaries of
mathematics education, as well as the relativity of its scientifi c object,
which inevitably results from such an ambiguity, is an indispensable
characteristic of any interdisciplinary relationships, as analysed by
Baltas (1983). Furthermore, an indispensable characteristic of the
interdisciplinarity of mathematics education is the production of
new knowledge, which meets the requirements of its theoretical
evolution as a scientifi c discipline and, at the same time, the
production of new knowledge, which is clearly useful and directly
applicable to mathematics teaching in schools. These expectations
are contradictory in several ways, a fact that explains many of the
characteristics of the relationships between theory and applications
of mathematics education.
Summarising, and to great extent simplifying, Baltas’ analysis (Baltas
1983) the production of scientifi c knowledge, which is a requirement
and a result of the theory development in any discipline, mathematics
education included, is primarily emanated and promoted by ‘internal’
processes to mathematics education. Questions and problems which
arise are posed and defi ned, mostly, in the context and in the terms
of the conceptual system of mathematics education, while their
answers and solutions are subjected to the research methodologies of
the discipline and to the broadly accepted standards of the scientifi c
practice.
On the other hand, the production of clearly useful and directly
applicable knowledge aims at the solution of practical problems of
school mathematics education. A practical problem in mathematics
teaching and learning in schools arises, primarily, in a social context
by processes ‘external’ to mathematics education and, as with every
phenomenon of everyday life, is many-sided. So, the solution of a
problem in school mathematics may not be approached in terms
which are determined uniquely or exclusively by the discipline of
mathematics education or even in strictly scientifi c terms, since a
multitude of social factors interfere. Consequently, any solution of a
practical problem of school mathematics education is not necessarily
subject to formal scientifi c standards and actually constitutes a
multitude of options or a fi eld of potential solutions. Any one
solution fi nally selected is not absolutely and uniquely included in
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330
the domain of mathematics education, being an outcome of broader
social interests and relationships (Baltas, 1983).
Concluding comments
Despite the fact that activities related to mathematics education have
exhibited a remarkable development since the 1960s, the scientifi c
object of mathematics education considered as a scientifi c discipline
remains vague and in consequence its constitutive conceptual system
remains fuzzy. Both are constituted at the same time by a process
of reciprocal determinations, wherein each of the constituents
presupposes the existence of the others. As a result, the scientifi c
status of mathematics education is questioned and its essential
contribution to school mathematics teaching and learning is thrown
into doubt. Recognising this problem and its negative outcomes, the
International Commission on Mathematics Instruction (ICMI) – an
old organisation involved in mathematics education issues expressing
mainstream options – raised issues concerning the ‘identity’ of
‘mathematics education as a research domai’ in a study conference
held in 1994 at the University of Maryland, USA. The proceedings
of this conference (Sierpinska and Kilpatrick, 1998) simply record
and confi rm the problem posed in this paper. The scientifi c object,
fundamental concepts, theoretical frameworks, research practices
and fi elds of applications of mathematics education are defi ned in
diff erent terms – in most cases incompatible or/and opposing amongst
themselves – in the 33 papers of the internationally known experts in
mathematics education research, in the fi ve working groups’ reports.
This is also the case in the editors’ summary, characteristically
entitled Continuing the Search (Sierpinska and Kilpatrick, 1998, vol.
2, p. 527–48).
Adopting the analysis of interdisciplinarity developed by Baltas
(1983, p. 39–44), I suggest that we have to accept that mathematics
education constituted on an interdisciplinary basis, and on the
aforementioned assumptions concerning the phenomena of learning
and teaching mathematics, is inherently characterised by two features.
First: a fl uidity in the defi nition of its scientifi c object in connection
with fundamental aspects of these phenomena and, second: an
ambiguity in the demarcation of its scientifi c object against related
disciplines and in particular mathematics, psychology and sociology,
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331
which include in their objects of study the same or diff erent aspects
of learning and teaching mathematics.
In conclusion, both the autonomy of the scientifi c object of
mathematics education and the autonomy of the meanings of the
concepts which constitute its conceptual system are in question.
However, a question is posed and discussed taking into account
the interdisciplinarity of mathematics education which imposes
a ‘scientifi c indeterminacy’ (Baltas 1983) in its approaches to the
practical problems of learning and teaching mathematics. Therefore,
the scientifi c status of mathematics education will remain an object
of an on-going, interesting, discussion.
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Contact
Assoc. Prof. Dr. Dimitris Chassapis
School of Education
Aristotle University of Thessaloniki
Greece
dchassap@eled.auth.gr
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SCHRIFTENREIHE DER PÄDAGOGISCHEN AKADEMIE DES BUNDES
IN OBERÖSTERREICH
(Hrsg.: Josef Fragner) sind bisher erschienen:
BAND 1:
Erwin Gierlinger (Ed.)
Refl ective Teaching and Action Research in Language Teaching: An International Ap-
proach – Background reading materials for an international introductory course in
refl ective teaching
BAND 2:
Brigi e Leidlmayer (Hrsg.)
Frauen – Bildung – Politik; Der Kampf um Bildung eine Frauensache
Beiträge zum Symposium: 100 Jahre Frauen an den Universitäten
BAND 3:
Erwin Gierlinger (Ed.)
Sharing the European Language Classroom:
A project in international classroom action research
BAND 4:
F. Buchberger, S. Berghammer, E. Winklehner (Eds.)
Integrating ICT into Institutions of Teacher Education
BAND 5:
Brian Hudson, Peter Schürz, Rauni Räsänen (Eds.)
Crossing Boundaries:
Pedagogical and Didactical Issues in the Social and Natural Sciences
BAND 6:
Peter Ganglmair (Hrsg.)
Begabungs- und Begabtenförderung
Eine Herausforderung für die Schule / Symposiumsdokumentation
BAND 7:
Josef Fragner (Hrsg.)
Pädagogische Essays
BAND 8:
M. Aigner, U. Greiner, A. Habichler
SEIN – ERKENNEN – HANDELN Philosophische Kontexte zur Pädagogik der Ge-
genwart
BAND 9:
F. Buchberger, S. Berghammer (Eds.)
Active Learning in Teacher Education
334
Autor
334
BAND 10:
Ewald Feyerer & Wilfried Prammer (Hrsg.)
10 Jahre Integration in Oberösterreich
Ein Grund zum Feiern!? Beiträge zum 5. Praktikerforum
BAND 11:
Reinhard Kroll, Brigi e Leidlmayer (Hrsg.)
Arbeit mit aggressiven Kindern – Ein Projekt an der ÜHS, Päd. Akademie/Bund
BAND 12:
F. Buchberger & S. Berghammer (Eds.)
Education Policy Analysis in a Comparative Perspective
BAND 13:
Brigi e Leidlmayer
Schulen auf dem Weg zur Lernenden Organisation – ein notwendiger Quanten-
sprung
Folgerungen und Konsequenzen für die LehrerInnenaus- und -fortbildung
BAND 14:
Ewald Feyerer & Wilfried Prammer (Hrsg.)
Eine kindgerechte Schule für alle – Beiträge zum 7. Praktikerforum
BAND 15:
J. Fragner, U. Greiner, M. Vorauer (Hrsg.)
Menschenbilder – Zur Auslöschung der anthropologischen Diff erenz
BAND 16:
Ewald Feyerer & Wilfried Prammer (Hrsg.)
Qual-I-tät und Integration – Beiträge zum 8. Praktikerforum
BAND 17:
Erich Mayrhofer
Schule als gelebter Widerspruch
Empirische Untersuchungen zur Schule an der Basis
BAND 18:
F. Buchberger, K. Enser (Eds.)
@-learning in Higher Education
BAND 19:
Michael Pichler, Herbert Pichler
scales chords & harmonic structures – jazz theory
BAND 20:
Siegfried Kiefer
Global Education Week
BAND 21:
Sigmund Ongstad, Piet-Hein van de Ven, Irina Buchberger
Mother tongue didaktik
335
Titel
335
BAND 22:
Elisabeth Marischler, Johannes Pögl, Simone Venhoda
Der Erwerb von Lese- und (Recht-)schreibkompetenz
BAND 23:
Christian Heitzinger, Josef Schütz
Begabungen fördern – Persönlichkeiten stärken
BAND 24:
Brian Hudson, Siegfried Kiefer, Mart Laanpere, Joze Rugelj
eLearning in Higher Education
BAND 25:
Siegfried Kiefer, Kari Sallamaa
European Identities in Mother Tongue Education
BAND 26:
Siegfried Kiefer
ECMA
European Counsellor for Multicultural Education
BAND 27: „students edition nr. 1“
Doris Neuhofer
Von einfachen Strichen bis zum komple en Alphabet
BAND 28:
B. Hudson, D. Enser (Eds.)
Researching the Teaching and Learning of Mathematics
BAND 29:
Gaby Weiner (Ed.)
Social Inclusion and Exclusion, and Social Justice in Education
BAND 30:
Siegfried Kiefer, Thomas Peterseil (Eds.)
Analysis of Educational Policies in a Comparative Perspective
BAND 31:
Karin Busch, Ulrike Reinhart
Begabungsförderung in jahrgangsgemischten Lerngruppen, Teil 1
BAND 32:
Karin Busch, Ulrike Reinhart
Begabungsförderung in jahrgangsgemischten Lerngruppen, Teil 2
BAND 33:
Eva Prammer-Semmler, Ulrike Prexl-Krausz, Katharina Soukup-Altrichter
LehrerInnen erforschen ihre Praxis:
Beispiele aus dem Lehrgang „Pädagogik und Fachdidaktik für LehrerInnen“
(PFL Grundschule und Integration)
336
Autor
336
BAND 34:
D. Bluma, S. Kiefer (Eds.)
Active Learning in Higher Education
BAND 35:
S. Kiefer, T. Peterseil (Eds.)
Global Education Week
BAND 36:
Siegfried Kiefer, Johanna Michalak, Ali Sabanci, Klaus Winter (Eds.)
Analysis of Educational Policies in a Comparative Educational Perspective
BAND 37:
Siegfried Kiefer, Kari Sallamaa (Eds.)
European Languages - Open Minds
BAND 38:
Karin Busch, Ulrike Reinhart
Tagungsband zum vierten Symposium „Begabtenförderung konkret“ 8. März 2006
BAND 39:
Brian Hudson, Josef Fragner (Eds.)
Researching the Teaching and Learning of Mathematics II
Alle Bände sind erschienen im Universitätsverlag Rudolf Trauner, Linz
