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Mathematics in everyday life A study of beliefs and actions

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Mathematics in everyday life
A study of beliefs and actions
Reidar Mosvold
Department of Mathematics
University of Bergen
2005
Preface
This study is a Dr. philos (doctor philosophiae) study. The study was financed and supported by
Telemarksforsking Notodden (Telemark Educational Research) through Norges Forskningsråd (The
Research Council of Norway).
The study has not been done in isolation. There are several people that have helped me in different
ways. First and foremost, I am immensely grateful to Otto B. Bekken for his advice and support
during recent years. He has assisted me in taking my first steps into the research community in
mathematics education, and without him there would not be a thesis like this.
I will also express my deepest gratitude to all the teachers who let me observe their teaching for
several weeks, and for letting me learn more about their beliefs and teaching strategies while
connecting mathematics to everyday life situations. It has been rewarding to meet so many of you
experienced teachers. You are supposed to be anonymous here, so I am not allowed to display your
names, but I thank you very much for having been so nice, and for having collaborated with me in
such a great way.
I am grateful to Telemarksforsking Notodden, for giving me this scholarship, and for letting me
work where it suited me best. Special thanks go to Gard Brekke, who has been in charge of this
project. My work with this thesis has been a pleasant journey, and I am now looking forward to
spend some time working with the colleagues in Notodden.
I would also like to thank Marjorie Lorvik for reading my thesis and helping me improve the
English language.
In the first part of my study I was situated in Kristiansand, and had the privilege of participating in
one course in research methodology with Maria Luiza Cestari and another course with Barbara
Jaworski. I feel lucky to have been given the opportunity to discuss my own research project in the
initial phase with them. This was very helpful for me.
At the beginning of my project I spent an inspiring week in Bognor Regis, in the southern part of
England. I want to give special thanks to Afzal Ahmed and his colleagues for their hospitality
during that visit, and for giving me so many ideas for my project. I look back on the days in Bognor
Regis with great pleasure, as a perfect early inspiration for my work.
I also spent some wonderful days in the Netherlands together with Otto. B. Bekken and Maria Luiza
Cestari. There I got the opportunity to meet some of the most important Dutch researchers in the
field. Thanks go to Jan van Maanen for great hospitality in our visit to Groningen and Utrecht, and
to Barbara van Amerom for inviting us to the celebration dinner after we witnessed her dissertation
defence, and to Jan de Lange and his colleagues at the Freudenthal Institute for giving us some
interesting days learning more about their research and projects.
In the Spring of 2003 I was lucky to spend a month at UCLA and the Lesson Lab in California.
Thanks are due to Jim Stigler and Ron Gallimore for opening the doors at Lesson Lab to make the
study of videos from the TIMSS 1999 Video Study possible, and to Angel Chui and Rossella
Santagata for assisting with all practical issues.
I would also like to thank Ted Gamelin and his colleagues at the mathematics department at UCLA
for giving me some inspirational weeks there. Thanks are due to Carolina DeHart, for letting me
participate in her lessons for teacher students, and to the people of the LuciMath group for all the
information about this interesting project, and to Phil Curtis for letting me use his office while I was
in Los Angeles. I remember the days of work in that office where I had the best possible view of the
beautiful UCLA campus, with great pleasure.
i
I am immensely grateful for having been given the opportunity to meet and get to know some of the
most prominent researchers in the field on these visits, and on conferences that I have attended.
Last but not least, I also express my deepest gratitude to my parents, for always having encouraged
me and supported me in every possible way. I would also like to thank my wife, Kristine. Before I
started working with my PhD we did not even know each other. Now we are happily married. The
last years have therefore been a wonderful journey for me in many ways. Thank you for supporting
me in my work and thank you for being the wonderful person that you are! Thanks also to my
parents in-law for letting me use their house as an office for about a year.
Notodden, August 2004
Reidar Mosvold
Revised preface
During this process there have been many revisions to the original document, and it is difficult to
list them all. The most significant revisions, however, have been the changes and additions made to
chapters 6, 10 and 11. A chapter 1.6 has been added, and there have been additions and changes to
chapter 2 as well as chapters 7, 8 and 9. Here I would like to thank the committee at the University
of Bergen, who reviewed the thesis and gave me many constructive comments.
I would also like to thank my colleague Åse Streitlien for reading through my thesis and giving me
several useful comments and suggestions for my final revision.
Many things have happened since last August. The main event is of course that I have become a
father! So, I would like to dedicate this thesis to our beautiful daughter Julie and my wife Kristine. I
love you both!
Sandnes, August 2005
Reidar Mosvold
ii
Notes
In the beginning there are some general notes that should be made concerning some of the
conventions used in this thesis. On several pages in this thesis some small text boxes have been
placed among the text. The aim has been mainly to emphasise certain parts of the content, or to
highlight a quote from one of the teachers, and we believe this could assist in making the text easier
to read and navigate through.
Some places in the text, like in chapters 1.6 and 2.1, some text boxes have been included with
quotes from Wikipedia, the free encyclopedia on the internet (cf. http://en.wikipedia.org). These
quotes are not to be regarded as part of the theoretical background for the thesis, but they are rather
to be considered as examples of how some of the concepts discussed in this thesis have been
defined in more common circles (as opposed to the research literature in mathematics education).
The data material from the study of Norwegian teachers (cf. chapters 8 and 9) was originally in
Norwegian. The parts from the transcripts, field notes or questionnaire results that have been quoted
here are translated to English by the researcher. The entire data material will appear in a book that
can be purchased from Telemark Educational Research (see http://www.tfn.no). This book will be
in Norwegian, and it will contain summaries of the theory, methodology, findings and discussions,
so that it can serve as a complete (although slightly summarized) presentation of the study in
Norwegian as well as a presentation of the complete data material.
The thesis has been written using Open Office (http://openoffice.org), and all the illustrations, charts
and tables have been made with the different components of this office suite. Some of the
illustrations of textbook tasks as well as the problem from the illustrated science magazine (cf.
chapter 8.10.3) have been scanned and re-drawn in the drawing program in Open Office to get a
better appearance in the printed version of the thesis.
iii
Table of contents
1 Introduction.......................................................................................................................................1
1.1 Reasons for the study.................................................................................................................1
1.2 Aims of the study ......................................................................................................................2
1.3 Brief research overview.............................................................................................................4
1.4 Research questions.....................................................................................................................5
1.5 Hypothesis..................................................................................................................................6
1.6 Mathematics in everyday life.....................................................................................................6
1.7 Summary of the thesis..............................................................................................................11
2 Theory..............................................................................................................................................15
2.1 Teacher beliefs.........................................................................................................................15
2.2 Philosophical considerations....................................................................................................19
2.2.1 Discovery or invention?...................................................................................................21
2.3 Theories of learning.................................................................................................................23
2.4 Situated learning......................................................................................................................24
2.4.1 Development of concepts.................................................................................................25
2.4.2 Legitimate peripheral participation..................................................................................27
2.4.3 Two approaches to teaching.............................................................................................28
2.4.4 Apprenticeship ................................................................................................................30
2.5 Historical reform movements..................................................................................................31
2.5.1 Kerschensteiner’s ‘Arbeitsschule’....................................................................................32
2.6 Contemporary approaches........................................................................................................33
2.6.1 The US tradition...............................................................................................................33
2.6.1.1 The NCTM Standards..............................................................................................33
2.6.1.2 High/Scope schools..................................................................................................34
2.6.1.3 UCSMP – Everyday Mathematics Curriculum........................................................35
2.6.2 The British tradition.........................................................................................................37
2.6.2.1 The Cockroft report..................................................................................................37
2.6.2.2 LAMP – The Low Attainers in Mathematics Project...............................................38
2.6.2.3 RAMP - Raising Achievement in Mathematics Project...........................................40
2.6.3 The Dutch tradition..........................................................................................................43
2.6.3.1 Realistic Mathematics Education.............................................................................45
2.6.4 Germany: ‘mathe 2000’...................................................................................................47
2.6.5 The Japanese tradition......................................................................................................50
2.6.6 The Nordic tradition.........................................................................................................51
iii
2.6.6.1 Gudrun Malmer........................................................................................................51
2.6.6.2 Speech based learning..............................................................................................52
2.6.6.3 Everyday mathematics in Sweden............................................................................54
2.7 Everyday mathematics revisited..............................................................................................56
2.8 Transfer of knowledge?...........................................................................................................60
2.9 Towards a theoretical base.......................................................................................................63
3 Real-life Connections: international perspectives...........................................................................67
3.1 The TIMSS video studies.........................................................................................................67
3.2 Defining the concepts..............................................................................................................68
3.3 The Dutch lessons....................................................................................................................70
3.3.1 Real-life connections........................................................................................................70
3.3.2 Content and sources.........................................................................................................71
3.3.3 Methods of organisation...................................................................................................72
3.3.4 Comparative comments....................................................................................................72
3.4 The Japanese lessons...............................................................................................................74
3.4.1 Real-life connections........................................................................................................74
3.4.2 Content and sources.........................................................................................................75
3.4.3 Methods of organisation...................................................................................................76
3.4.4 Comparative comments....................................................................................................77
3.5 The Hong Kong lessons...........................................................................................................78
3.5.1 Real-life connections........................................................................................................78
3.5.2 Content and sources.........................................................................................................79
3.5.3 Methods of organisation...................................................................................................79
3.5.4 Comparative comments....................................................................................................81
3.6 Summarising............................................................................................................................81
4 Norwegian curriculum development...............................................................................................83
4.1 The national curriculum of 1922/1925....................................................................................83
4.2 The national curriculum of 1939..............................................................................................84
4.3 The national curriculum of 1974..............................................................................................85
4.4 The national curriculum of 1987..............................................................................................86
4.5 The national curriculum of L97 ..............................................................................................87
4.5.1 The preliminary work of L97...........................................................................................88
4.5.2 The concept of ‘mathematics in everyday life’................................................................89
4.6 Upper secondary frameworks..................................................................................................93
4.7 Evaluating L 97 and the connection with real life...................................................................94
4.8 Curriculum reform and classroom change...............................................................................98
iv
5 Textbooks........................................................................................................................................99
5.1 The books.................................................................................................................................99
5.2 Real-life connections in the books.........................................................................................100
5.2.1 Lower secondary textbooks ...........................................................................................100
5.2.2 Upper secondary textbooks............................................................................................102
5.3 Textbook problems................................................................................................................103
5.3.1 ‘Realistic’ problems in lower secondary school.............................................................103
5.3.1.1 Realistic contexts....................................................................................................103
5.3.1.2 Artificial contexts...................................................................................................106
5.3.1.3 Other problems with real-life connections.............................................................107
5.3.1.4 Comments...............................................................................................................109
5.3.2 ‘Realistic’ problems in upper secondary school.............................................................109
5.3.2.1 Realistic contexts....................................................................................................109
5.3.2.2 Artificial contexts...................................................................................................110
5.3.2.3 Comments...............................................................................................................113
5.4 Comparison of the textbooks.................................................................................................113
6 More on our research approach.....................................................................................................117
6.1 Research paradigm.................................................................................................................117
6.1.1 Ethnography...................................................................................................................119
6.1.2 Case study......................................................................................................................120
6.2 The different parts of the study..............................................................................................122
6.2.1 Classroom studies..........................................................................................................122
6.2.1.1 Planning meeting....................................................................................................123
6.2.1.2 Questionnaire..........................................................................................................124
6.2.1.3 Observations...........................................................................................................125
6.2.1.4 Interviews...............................................................................................................128
6.2.1.5 Practical considerations and experiences...............................................................129
6.2.2 The TIMSS 1999 Video Study.......................................................................................130
6.2.3 Textbook analysis...........................................................................................................130
6.3 Triangulation..........................................................................................................................131
6.4 Selection of informants..........................................................................................................132
6.4.1 Teachers.........................................................................................................................132
6.4.2 Videos............................................................................................................................133
6.4.3 Textbooks.......................................................................................................................134
6.5 Analysis of data......................................................................................................................134
6.5.1 Classroom study.............................................................................................................135
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6.5.1.1 Questionnaire..........................................................................................................135
6.5.1.2 Observations – first phase of analysis....................................................................135
6.5.1.3 Observations – second phase of analysis................................................................139
6.5.1.4 Interviews...............................................................................................................140
6.5.2 Video study....................................................................................................................141
6.5.3 Textbooks.......................................................................................................................141
7 Questionnaire results.....................................................................................................................143
7.1 The questionnaire...................................................................................................................143
7.2 The Likert-scale questions.....................................................................................................143
7.2.1 Real-life connections......................................................................................................144
7.2.2 Projects and group work.................................................................................................145
7.2.3 Pupils formulate problems.............................................................................................146
7.2.4 Traditional ways of teaching..........................................................................................147
7.2.5 Re-invention...................................................................................................................147
7.2.6 Use of other sources.......................................................................................................148
7.2.7 Usefulness and understanding – two problematic issues...............................................149
7.3 The comment-on questions....................................................................................................150
7.3.1 Reconstruction................................................................................................................150
7.3.2 Connections with other subjects.....................................................................................151
7.3.3 Problem solving.............................................................................................................152
7.3.4 Content of tasks..............................................................................................................152
7.4 The list questions...................................................................................................................153
7.5 Comparison of teachers..........................................................................................................154
7.6 Categorisation........................................................................................................................157
7.7 Final comments......................................................................................................................158
8 Three teachers: Their beliefs and actions......................................................................................159
8.1 Curriculum expectations........................................................................................................159
8.2 Setting the scene....................................................................................................................160
8.3 Two phases............................................................................................................................160
8.4 Models of analysis.................................................................................................................160
8.5 Brief comparison....................................................................................................................161
8.6 Karin’s beliefs........................................................................................................................165
8.6.1 Practice theories.............................................................................................................165
8.6.2 Content and sources.......................................................................................................166
8.6.3 Activities and organisation.............................................................................................167
8.7 Ann’s beliefs..........................................................................................................................168
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8.7.1 Practice theories.............................................................................................................168
8.7.2 Content and sources.......................................................................................................169
8.7.3 Activities and organisation.............................................................................................169
8.8 Harry’s beliefs........................................................................................................................170
8.8.1 Practice theories.............................................................................................................171
8.8.2 Content and sources.......................................................................................................172
8.8.3 Activities and organisation.............................................................................................173
8.9 Into the classrooms................................................................................................................174
8.10 Harry’s teaching...................................................................................................................175
8.10.1 Fibonacci numbers.......................................................................................................175
8.10.2 Pythagoras’ theorem.....................................................................................................176
8.10.3 Science magazine.........................................................................................................178
8.10.4 Bicycle assignment.......................................................................................................179
8.11 Ann’s teaching.....................................................................................................................181
8.11.1 Construction of 60 degrees...........................................................................................181
8.11.2 Area of figures..............................................................................................................182
8.11.3 Size of an angle............................................................................................................183
8.11.4 Blackboard teaching.....................................................................................................184
8.12 Mathematics day..................................................................................................................185
8.13 Karin’s teaching...................................................................................................................187
8.13.1 Lazy mathematicians....................................................................................................187
8.13.2 Grandma’s buttons.......................................................................................................189
8.13.3 If I go shopping............................................................................................................190
8.13.4 Textbook teaches..........................................................................................................191
8.13.5 How many have you slept with?..................................................................................191
9 Five high-school teachers: Beliefs and actions..............................................................................193
9.1 Curriculum expectations........................................................................................................193
9.2 Questionnaire results..............................................................................................................194
9.3 Models of analysis.................................................................................................................198
9.4 Jane’s beliefs..........................................................................................................................198
9.4.1 Practice theories.............................................................................................................199
9.4.2 Content and sources.......................................................................................................199
9.4.3 Activities and organisation.............................................................................................200
9.5 George’s beliefs.....................................................................................................................201
9.5.1 Practice theories.............................................................................................................201
9.5.2 Content and sources.......................................................................................................202
vii
9.5.3 Activities and organisation.............................................................................................203
9.6 Owen’s beliefs.......................................................................................................................203
9.6.1 Practice theories.............................................................................................................204
9.6.2 Content and sources.......................................................................................................204
9.6.3 Activities and organisation.............................................................................................204
9.7 Ingrid’s beliefs.......................................................................................................................205
9.7.1 Practice theories.............................................................................................................205
9.7.2 Content and sources.......................................................................................................206
9.7.3 Activities and organisation.............................................................................................206
9.8 Thomas’ beliefs......................................................................................................................206
9.8.1 Practice theories.............................................................................................................206
9.8.2 Content and sources.......................................................................................................207
9.8.3 Activities and organisation.............................................................................................207
9.9 Into the classrooms................................................................................................................208
9.10 Jane’s teaching.....................................................................................................................209
9.10.1 Mathematics in the kitchen..........................................................................................209
9.10.2 Is anyone here aunt or uncle?.......................................................................................209
9.10.3 Techno sticks and angles..............................................................................................209
9.10.4 I am going to build a garage.........................................................................................210
9.10.5 Pythagoras....................................................................................................................210
9.11 George’s teaching................................................................................................................210
9.11.1 Trigonometry and Christmas cookies..........................................................................210
9.12 Owen’s teaching...................................................................................................................211
9.12.1 Areas............................................................................................................................211
9.13 The teaching of Thomas and Ingrid.....................................................................................212
9.13.1 Cooperative groups......................................................................................................212
10 Discussions and answers.............................................................................................................213
10.1 Activities and organisation..................................................................................................213
10.1.1 Cooperative learning....................................................................................................213
10.1.2 Re-invention.................................................................................................................215
10.1.3 Projects.........................................................................................................................218
10.1.4 Repetitions and hard work...........................................................................................220
10.2 Content and sources.............................................................................................................221
10.2.1 Textbooks.....................................................................................................................221
10.2.2 Curriculum...................................................................................................................224
10.2.3 Other sources................................................................................................................226
viii
10.3 Practice theories...................................................................................................................228
10.3.1 Teaching and learning..................................................................................................228
10.3.2 Vocational relevance....................................................................................................229
10.3.3 Connections with everyday life....................................................................................231
10.4 Answering the research questions........................................................................................235
10.4.1 Are the pupils encouraged to bring their experiences into class?................................235
10.4.2 Do the teachers use examples from the media?...........................................................235
10.4.3 Are the pupils involved in a process of reconstruction or re-invention?.....................236
10.4.4 What sources other than the textbook are used?..........................................................236
10.4.5 Do they use projects and more open tasks? .................................................................237
10.4.6 How do they structure the class, trying to achieve these goals?..................................237
10.4.7 Answering the main questions.....................................................................................237
11 Conclusions.................................................................................................................................241
11.1 Practice theories...................................................................................................................242
11.2 Contents and sources...........................................................................................................245
11.3 Activities and organisation..................................................................................................247
11.4 Implications of teacher beliefs.............................................................................................249
11.5 Curriculum - textbooks – teaching.......................................................................................252
11.6 Definition of concepts..........................................................................................................253
11.7 How problems can be made realistic...................................................................................254
11.8 Lessons learned....................................................................................................................256
11.9 The road ahead.....................................................................................................................257
12 Literature......................................................................................................................................261
13 Appendix 1: Everyday mathematics in L97.................................................................................273
14 Appendix 2: Questionnaire..........................................................................................................279
15 Appendix 3: Illustration index ....................................................................................................285
16 Appendix 4: Table index.............................................................................................................287
ix
Mathematics in everyday life
1 Introduction
1.1 Reasons for the study
As much as I would like for this study to have been initiated by my own brilliant ideas, claiming so
would be wrong. After having finished my Master of Science thesis, in which I discussed the use of
history in teaching according to the so-called genetic principle, I was already determined to go for a
doctorate. I only had vague ideas about what the focus of such a study could be until my supervisor
one day suggested ‘everyday mathematics’. Having thought about that for a while, many pieces of a
puzzle I hardly knew existed seemed to fit into a beautiful picture. I could only wish it was a picture
that originated in my own mind, but it is not.
In my MS thesis I indicated a theory of genesis that not only concerned incorporating the history of
mathematical ideas, methods and concepts, but was more a way of defining the learning of
mathematics as a process of genesis, or development. This process could be historically grounded,
in what we might call historical genesis (or a historical genetic method), but we could also use
concepts like logical genesis, psychological genesis, contextual genesis or situated genesis of
mathematical concepts and ideas to describe the idea. The genetic principle is not a new idea, and it
is believed by many to originate in the work of Francis Bacon (1561-1626), or even earlier. Bacon’s
‘natural method’ implied a teaching practice that starts with situations from everyday life:
When Bacon’s method is to be applied in teaching, everyday problems, the so-called specific cases,
should be the outset, only later should mathematics be made abstract and theoretical. Complete
theorems should not be the starting point; instead such theorems should be worked out along the way
(Bekken & Mosvold, 2003b, p. 86).
Reviewing my own work, I realised that genesis principles (often called a ‘genetic approach’) could
be applied as a framework for theories of learning with connections to real life also. When I
discovered this, my entire work suddenly appeared to fall into place like the pieces of a marvellous
puzzle. Since I cannot regard the image of this puzzle as my work only, I will from now on use the
pronoun ‘we’ instead of ‘I’.
A genesis perspective could be fruitful when studying almost any issue in mathematics education. In
this study we were particularly interested in ways of connecting mathematics with real or everyday
life. We wanted to focus on the development of these ideas in history and within the individual.
Starting with an interest in connecting mathematics with real life, or what we could now place
within a paradigm of contextual genesis, we also decided to focus on teachers and their teaching
(particularly on experienced teachers). The idea of studying experienced teachers could be linked
with a famous statement that occurred in one of Niels Henrik Abel’s notebooks, and this could also
serve as an introduction to our study:
It appears to me that if one wants to make progress in mathematics one should study the masters and
not the pupils (Bekken & Mosvold, 2003b, p. 3).
This statement was initially made in a different connection than this, but we believe that it is also
important to study ‘master teachers’ if one wants to make progress in teaching. This is why we in
our study chose to focus on experienced teachers particularly. Behind that choice was an underlying
assumption that many teachers have years of experience in teaching mathematics, and many of these
teachers have some wonderful teaching ideas. Unfortunately the experience and knowledge of a
1
1 Introduction
teacher all too often dies with the teacher, and his ideas do not benefit others. We believe that there
should be more studies of master teachers in order to collect some of their successful ideas and
methods. These ideas should be incorporated in a common body of knowledge about the teaching of
mathematics.
1.2 Aims of the study
The focus of interest in this study is both connected with content and methods of work. The content
is closely connected with ideas of our national curriculum (which will be further discussed in
chapter 4). We wish to make a critical evaluation of the content of the curriculum, when it comes to
the issues of interest in this study, and we wish to make comparisons with the national development
in other countries.
There have been national curricula in Norway since 1890, and before that there were local
frameworks ever since the first school law was passed around 1739. Laws about schools have been
passed, and specific plans have been made in order to make sure these laws were followed in the
schools. The ideas about schools and teaching have changed over the years. We have studied a few
aspects of our present curriculum, and this will serve as a basis for our research questions and plans.
Norway implemented a new national curriculum for the grades 1-10 in 1997. The general
introductory part also concerned upper secondary education (in Norway called ‘videregående
skole’). This curriculum has been called L97 for short. Because it is still relatively new, we have not
educated a single child throughout elementary school according to L97. Its effects can therefore
hardly be fully measured yet, and the pupils who start their upper secondary education have all gone
through almost half of their elementary school years with the old curriculum. Long-term effects of
the principles and ideas of L97 can therefore hardly be measured at this time. Only a small number
of the teachers in the Norwegian elementary school today have gone through a teacher education
that followed this new curriculum, and all of them have their experience from schools and teachers
that followed older curricula. However, in spite of all this one should expect the teaching in
elementary and upper secondary school to follow the lines of L97 now (at least to some extent).
L97 was inspired by the Cockroft report (Cockroft, 1982), the
NCTM standards (NCTM, 1989) and recent research in
mathematics education. The aims and guidelines for our
contemporary national curriculum appear as well considered, and
the curriculum itself has an impressive appearance. In our
classroom studies we wanted to find out how the principles of L97 have been implemented in the
classrooms. A hypothesis suggests that most teachers teach the way they have been taught
themselves. Experience shows that there is quite a long way from a well-formed set of principles to
actual changes in classrooms. Another issue is that every curriculum is subject to the teacher’s
interpretation. Because of this we do not expect everything to be as the curriculum intends. But we
do believe that many teachers have good ideas about teaching and learning, and it is some of these
good ideas that we have aimed to discover. Together with the teachers we have then reflected upon
how things can be done better.
The teaching of mathematics in Norwegian schools is, or at least should be, directed by the national
curriculum. In any study of certain aspects of school and teaching, L97 is therefore a natural place to
start. We will look at a few important phrases here:
2
We want to find out how the
principles of L97 are
implemented in classrooms.
Mathematics in everyday life
The syllabus seeks to create close links between school mathematics and mathematics in the outside world.
Day-to-day experience, play and experiment help to build up its concepts and terminology (RMERC, 1999, p.
165)
Everyday life situations should thereby form a basis for the teaching of mathematics. ‘Mathematics
in everyday life’ was added as a new topic throughout all ten years of compulsory education.
Learners construct their own mathematical concepts. In that connection it is important to emphasise
discussion and reflection. The starting point should be a meaningful situation, and tasks and problems
should be realistic in order to motivate pupils (RMERC, 1999, p. 167).
These two points: the active construction of knowledge by the pupils and the connection with
school mathematics and everyday life, has been the main focus of this study. L97 presents this as
follows:
The mathematics teaching must at all levels provide pupils with opportunities to:
carry out practical work and gain concrete experience;
investigate and explore connections, discover patterns and solve problems;
talk about mathematics, write about their work, and formulate results and solutions;
exercise skills, knowledge and procedures;
reason, give reasons, and draw conclusions;
work co-operatively on assignments and problems (RMERC, 1999, pp. 167-168).
The first area of the syllabus, mathematics in everyday life, establishes the subject in a social and
cultural context and is especially oriented towards users. The further areas of the syllabus are based on
main areas of mathematics (RMERC, 1999, p. 168).
Main stages Main areas
Lower
secondary
stage
Mathematics
in everyday
life
Numbers
and
algebra
Geometry Handling
data Intermediate
graphs and
functions
Intermediate
stage Mathematics
in everyday
life
Numbers Geometry Handling
data
Primary
stage Mathematics
in everyday
life
Numbers Space
and shape
Table 1 Main areas in L97
As we can see from the table above, ‘mathematics in everyday’ life has become a main area of
mathematics in Norwegian schools, and this should imply an increased emphasis on real-life
connections.
Although a connection with everyday life has been mentioned in previous curricula also, there has
been a shift of focus. The idea that the pupils should learn to use mathematics in practical situations
from everyday life has been present earlier, but in L97 the situations from real life were supposed to
be the starting point rather than the goal. Instead of mathematics being a training field for real life
the situations from real life are supposed to be starting points. When the pupils are working with
these problems they should reach a better understanding of the mathematical theories. This is an
3
1 Introduction
important shift of focus, and in our study we wanted to investigate
how teachers have understood and implemented these ideas in their
teaching.
The ideas of the curriculum on these points were examined in this
study. The curriculum content was also examined, and we aimed at
finding out how the textbooks meet the curricular demands, as well as how the teachers think and
act. We have observed how these ideas were carried out in actual classrooms and then tried to
gather some thoughts and ideas on how it can be done better.
Connections with real life are not new in curricula, and they are not specific for the Norwegian
tradition only. New Zealand researcher Andrew J.C. Begg states:
In mathematics education the three most common aims of our programs are summed up as:
Personal – to help students solve the everyday problems of adult life;
Vocational – to give a foundation upon which a range of specialised skills can be built;
Humanistic – to show mathematics as part of our cultural heritage (Begg, 1984, p. 40).
Our project has built on research from other countries, and we wish to contribute to this research. In
research on mathematics education, mathematics is often viewed as a social construct which is
established through practices of discourse (Lerman, 2000). This is opposed to a view of
mathematics as a collection of truths that are supposed to be presented to the pupils in appropriate
portions.
1.3 Brief research overview
The work consisted of a theoretical study of international research, a study of videos from the
TIMSS 1999 Video Study of seven countries, a study of textbooks, a study of curriculum papers,
and a classroom study of Norwegian teachers, their beliefs and actions concerning these issues.
In the theoretical study we investigated research done in this area, to uncover some of the ideas of
researchers in the past and the present. We focused on research before and after the Cockroft Report
in Britain, NCTM (National Council of Teachers of Mathematics) and the development in
curriculum Standards in the US, research from the Freudenthal Institute in the Netherlands, the
theories of the American reform pedagogy, the theories of situated learning and the Nordic research.
Through examining all these theories and research projects, we have tried to form a theoretical
framework for our own study.
The contemporary national curriculum, L97, was of course the most important to us, but we have
also studied previous curricula in Norway, from the first one in 1739 up till the present. We have
tried to find out if the thoughts mentioned above are new ones, or if they have been part of the
educational system in earlier years. This analysis served as a background for our studies. The
curriculum presents one set of ideas on how to connect mathematics with real life, and the textbooks
might represent different interpretations of these ideas. Teachers often use the textbooks as their
primary source rather than the curriculum, and we have therefore studied how the textbooks deal
with the issue.
The main part of our study was a qualitative research study, containing interviews with teachers, a
questionnaire survey, and observations of classroom practice. This was supported by investigations
of textbooks and curriculum papers, analysis of videos from the TIMSS 1999 Video Study, and a
review of theory. The qualitative data were intended to help us discover connections between the
4
Situations from real life
are supposed to be a
starting point.
Mathematics in everyday life
teachers’ educational background and their beliefs about the subject, teaching and learning on the
one hand, and about classroom practice and methods of work on the other hand.
1.4 Research questions
A main part of any research project is to define a research problem, and to form some reasonable
research questions. This was an important process in the beginning of this study, and it became
natural to have strong connections with the curriculum. The national curriculum is, or should be, the
working document of Norwegian teachers. We have been especially interested in how they think
about and carry out ideas concerning the connection with everyday life.
It was of particular interest for us to identify the views of the teachers, when connections with
everyday life were concerned, and to see how these views and ideas affected their teaching. A
reasonable set of questions might be:
To these questions we have added a few sub-questions that could assist when attempting to answer
the two main questions and to learn more about the strategies and methods they use to connect with
everyday life:
Being aware of the fact that it is hard to answer these questions when it comes to all aspects of the
mathematics curriculum, it is probably wise to focus on one or two areas of interest. The strategies
for implementing these ideas in the teaching of algebra might differ from the strategies used when
teaching probability, for instance. We chose to focus on the activities and issues of organisation
rather than the particular mathematical topics being taught by the teachers at the time of our
classroom observations.
The two main research questions might be revised slightly: How can teachers organise their
teaching in order to promote activities where the pupils are actively involved in the construction of
mathematical knowledge, and how can these activities be connected with real life? The sub-
questions could easily be adopted for these questions also. From the sub-questions, we already see
that pupil activity is naturally incorporated into these ideas. It is therefore fair to say that activity is a
central concept, although it is an indirect and underlying concept more than a direct one.
5
1) What are the teachers’ beliefs about connecting school
mathematics and everyday life?
2) What ideas are carried out in their teaching practice?
Are the pupils encouraged to bring their experiences into
class?
Are the pupils involved in a process of reconstruction or
re-invention?
What sources other than the textbook do teachers use?
Do the teachers use examples from the media?
Do they encourage projects and open tasks?
How do they structure the class, in trying to achieve
these goals?
1 Introduction
Important questions that are connected with the questions above, at least on a meta-level, are:
How do we cope with the transformation of knowledge from specific, real-life
situations to the general?
How does the knowledge transform from specific to general?
How does the knowledge transform in/apply to other context situations?
These are more general questions that we might not be able to answer, at least not in this study, but
they will follow us throughout the work.
1.5 Hypothesis
Based on intuition and the initial research questions, we can present a hypothesis that in many
senses is straightforward, and that has obvious limitations, but that anyhow is a hypothesis that can
be a starting point for the analysis of our research.
The population of teachers can be divided into three groups when it comes to their attitudes and
beliefs about real-life connections. Teachers have multiple sets of beliefs and ideas and therefore
cannot easily be placed within a simplified category. We present the hypothesis that teachers of
mathematics have any of these attitudes towards real-life connections:
Positive
Negotiating (in-between)
Negative
We believe that the teachers in our study can also be placed within one of these groups or
categories. Placing teachers in such categories, no matter how interesting that might be, will only be
of limited value. We will not narrow down our study to such a description and categorisation.
Instead we have tried to gather information about the actions of teachers in each of these categories
when it comes to real-life connections, and we have also tried to discover some of the thinking that
lies behind their choices. A main goal for our study is therefore to generate new theory, so that we
can replace this initial model with a more appropriate one. Such knowledge can teach us valuable
lessons about connecting mathematics with real life, at least this is what we believe.
Our interest was therefore not only to analyse what the teachers thought about these matters and
place them within these three categories, but to use this as a point of departure in order to generate
new theory. We not only wanted to study what beliefs they had, but also to study what they actually
did to achieve a connection with everyday life, or what instructional practices they chose. It was our
intention to study the teaching strategies a teacher might choose to fulfil the aims of the curriculum
when it comes to connecting mathematics with everyday life; the content and materials they used
and the methods of organising the class.
1.6 Mathematics in everyday life
This thesis is based on the Norwegian curriculum (L97) because this was the current curriculum at
the time of our study. The national curriculum is the main working document for Norwegian
teachers, and the connection with mathematics and everyday life has been the key focus here.
6
Mathematics in everyday life
Naturally our definitions of concepts will be based on L97, but unfortunately the curriculum neither
gives a thorough definition, nor a discussion of the concepts in relation to other similar concepts.
Several concepts and terms are used when discussing this and similar issues in international
research. We are going to address the following:
(mathematics in) everyday life
real-life (connections)
realistic (mathematics education)
(mathematics in) daily life
everyday mathematics
In Norwegian we have a term called “hverdagsmatematikk”, which could be directly translated into
“everyday mathematics”. When teachers discuss the curriculum and its presentation of mathematics
in everyday life, they often comment on this term, “everyday mathematics”. The problem is that
“hverdagsmatematikk” is often understood to be limited only to what pupils encounter in their
everyday lives, and some teachers claim that this would result in a limited content in the
mathematics curriculum. The Norwegian curriculum does not use the term “everyday mathematics”,
and the area called “mathematics in everyday life” has a different meaning. For this reason, and to
avoid being connected with the curriculum called Everyday Mathematics, we have chosen not to use
the term “everyday mathematics” as our main term. International research literature has, however,
focused on everyday mathematics a lot, and we will therefore use this term when referring to the
literature (see especially chapters 2.6.6.3 and 2.7).
The adjective “everyday” has three definitions (Collins Concise Dictionary & Thesaurus):
1) commonplace or usual
2) happening every day
3) suitable for or used on ordinary days
“Daily”, on the other hand, is defined as:
1) occurring every day or every weekday
2) of or relating to a single day or to one day at a time: her home help comes in on a daily
basis; exercise has become part of our daily lives
“Daily” can also be used as an adverb, meaning every day.
Daily life and everyday life both might identify something that occurs every day, something regular.
Everyday life could also be interpreted as something that is commonplace, usual or well-known (to
the pupils), and not necessarily something that occurs every day. Everyday life could also identify
something that is suitable for, or used on, ordinary days, and herein is a connection to the complex
and somewhat dangerous term of usefulness. We suggest that daily life could therefore be a more
limited term than everyday life. In this thesis, we mainly use the term everyday life. Another
important, and related, term, is “real life/world”.
The word “real” has several meanings:
1) existing or occurring in the physical world
2) actual: the real agenda
7
1 Introduction
3) important or serious: the real challenge
4) rightly so called: a real friend
5) genuine: the council has no real authority
6) (of food or drink) made in a traditional way to ensure the best flavour
7) Maths involving or containing real numbers alone
8) relating to immovable property such as land or buildings: real estate
9) Econ. (of prices or incomes) considered in terms of purchasing power rather than
nominal currency value
10) the real thing the genuine article, not a substitute or imitation
From these definitions, we are more interested in the
“real” in real life and real world, as in definition 1 above.
We could say that real life and real world simply refer to
the physical world. Real-life connections would thereby
imply linking mathematical issues with something that
exists or occurs in the physical world. Real-life
connections do thereby not necessarily refer to something
that is commonplace or well-known to the pupils, but
rather something that occurs in the physical world. If we,
on the other hand, choose to define real-life connections as
referring to something that occurs in the pupils physical
world (and would therefore be commonplace to them),
then real-life connections and mathematics in everyday life have the same meaning. To be more in
consistence with the definitions from the TIMSS 1999 Video Study as well as the ideas of the
Norwegian curriculum, L97, we have chosen to distinguish between the terms real world and real
life. When we use the term “real world” we simply refer to the physical world if nothing else is
explained. Real life, however, in this thesis refers to the physical world outside the classroom.
As we will see further discussed in chapter 4, mathematics in everyday life (as it is presented in the
Norwegian curriculum L97) is an area that establishes the subject in a social and cultural context
and is especially oriented towards users (the pupils). L97 implies that mathematics in everyday life
is not just referring to issues that are well-known or commonplace to pupils, but also to other issues
that exist or occur in the physical world.
This thesis is not limited to a study of Norwegian teachers, but also has an international approach,
through the study of videos from the TIMSS 1999 Video Study. In the TIMSS video study the
concept “real-life connections” was used. This was defined as a problem (or non-problem) situation
that is connected to a situation in real life. Real life referred to something the pupils might encounter
outside the classroom (cf. chapter 3.2). If a distinction between the world outside the classroom and
the classroom world is the intention, then one might argue that the outside world and the physical
world, as discussed above, are not necessarily the same. We have chosen to define the term “real
world” as referring to the physical world in general, whereas “real life” refers to the (physical)
world outside the classroom. We should be aware that there could be a difference in meanings, as
far as the term “real life” is concerned. Others might define it as identical to our definition of real
world, and might not make a distinction between the two. The phrase “outside the classroom” is
used in the definition from the TIMSS 1999 Video Study, and the Norwegian curriculum also
makes a distinction between the school world and the outside world. This implies that our notion of
the pupils’ real life mainly refers to their life outside of school, or what we call the “outside world”.
8
REAL LIFE
“The phrase real life is generally used to
mean life outside of an environment that is
generally seen as contrived or fantastical,
such as a movie or MMORPG.
It is also sometimes used synonymously with
real world to mean one’s existence after he or
she is done with schooling and is no longer
supported by parents.”
http://en.wikipedia.org/wiki/Real_life
Mathematics in everyday life
We do not thereby wish to claim that what happens in school or inside the classroom is not part of
the pupils everyday life, but for the sake of clarity we have chosen such a definition in this thesis.
When we occasionally use the term “outside world”, it is in reference to the curriculum’s clear
distinction between school mathematics and the outside world.
“Realism”, as in realistic, is also an important word in this discussion. It is defined in dictionaries
as:
1) awareness or acceptance of things as they are, as opposed to the abstract or ideal
2) a style in art or literature that attempts to show the world as it really is
3) the theory that physical objects continue to exist whether they are perceived or not
Realistic therefore also refers to the physical world, like the word real does. The word realistic is
used in the Norwegian curriculum, but when used in mathematics education, it is often in
connection with the Dutch tradition called Realistic Mathematics Education (RME). We should be
aware that the Dutch meaning of the word realistic has a distinct meaning that would sometimes
differ from other definitions of the term. In Dutch the verb “zich realisieren” means “to imagine”, so
the term realistic in RME refers more to an intention of offering the pupils problems that they can
imagine, which are meaningful to them, than it refers to realness or authenticity. The connection
with the real world is also important in RME, but problem contexts are not restricted to situations
from real world (cf. van den Heuvel-Panhuizen, 2003, pp. 9-10). In this thesis, the word realistic is
mostly referring to authenticity, but it is also often used in the respect that mathematical problems
should be realistic in order to be meaningful for the pupils (cf. RMERC 1999, p. 167).
Wistedt (1990; 1992 and 1993), in her studies of “vardagsmatematik” (which could be translated
into everyday mathematics), made a definition of everyday mathematics where she distinguished
between:
1) mathematics that we attain in our daily lives, and
2) mathematics that we need in our daily lives.
The Norwegian curriculum certainly intends a teaching where the pupils learn a mathematics that
they can use in their daily lives, but it also aims at drawing upon the knowledge that pupils have
attained from real life (outside of school). When L97 also implies that the teacher should start with a
situation or problem from real or everyday life and let the pupils take part in the reconstruction of
some mathematical concepts through a struggle with this problem, it is not limited to either of these
points. When the phrase “mathematics in everyday life” is used in this thesis, it almost exclusively
refers to the topic in the Norwegian curriculum with this same name. The term “everyday life”,
when used alone, is considered similar to the term “real life”, as discussed above, and we have often
chosen to use the phrase ‘real-life connections’ rather than ‘connections with mathematics and
everyday life’ or similar. This choice is mainly a matter of convenience. Our interpretation of
mathematics in everyday life (as a concept rather than a curriculum topic) is derived from the
descriptions given in L97. In short, mathematics in everyday life refers to a connection with
something that occurs in the real or physical world. It also refers to something that is known to the
pupils. In this thesis we are more concerned with how teachers can and do make a connection with
mathematics and everyday life, and thereby how they address this specific area of the curriculum.
While everyday mathematics, at least according to the definition of Wistedt, has a main focus on
mathematics, the concept of mathematics in everyday life has a main focus on the connection with
real or everyday life.
9
1 Introduction
We should also note that some people make a distinction between everyday problems and more
traditional word problems (as found in mathematics textbooks), in that everyday problems are open-
ended, include multiple methods and often imply using other sources (cf. Moschkovich & Brenner,
2002). If we generalise from this definition, we might say that everyday mathematics itself is more
open-ended.
The last term - everyday mathematics - is also the name of an alternative curriculum in the US,
which we discuss in chapter 2.6.1.3. Everyday Mathematics (the curriculum) and “everyday
mathematics” (the phrase) are not necessarily the same. The Everyday Mathematics curriculum has
a focus on what mathematics is needed by most people, and how teachers can teach “useful”
mathematics. We have deliberately avoided the term useful in this thesis, because this would raise
another discussion that we do not want to get stuck in. (What is useful for young people, and who
decides what is useful, etc.) We do, however, take usefulness into the account when discussing the
motivational aspect concerning transfer of learning in chapter 2.8.
Wistedt’s definition, as presented above, is interesting, and it includes the concept of usefulness.
Because the Norwegian phrase that could be translated into “everyday mathematics” is often used in
different (and confusing) ways, we have chosen to omit the term in this thesis. Everyday
mathematics, as defined by Wistedt, implies a mathematics that is attained in everyday life. L97
aims at incorporating the knowledge that pupils bring with them, knowledge they have attained in
everyday life, but we have chosen refer to this as connecting mathematics with real or everyday life
instead of using the term everyday mathematics. Another interpretation of everyday mathematics,
again according to Wistedt, is mathematics that is needed in everyday life. L97, as well as most
other curriculum papers we have examined, presents intentions of mathematics as being useful in
everyday life, but the discussion of usefulness is beyond the scope of this thesis.
To conclude, our attempt at clarifying the different terms can be described in the following way:
“mathematics in everyday life” refers to the curriculum area with this same name, and
to the connection with mathematics and everyday life
“real life” refers to the physical world outside the classroom
10
Illustration 1 Many concepts are involved in the discussion
Real life Everyday life
Daily life
Realistic Everyday mathematics Outside world
School mathematics
Real world
Mathematics in everyday life
“real world” refers to the physical world (as such)
“everyday life” mainly refers to the same as real life, and we thereby do not distinguish
between ‘real-life connections’ and ‘connections between mathematics and everyday
life’ or similar
“daily life” refers to something that occurs on a more regular basis, but is mainly
omitted in this thesis
“everyday mathematics” both refers to a curriculum, but also to a distinction between
mathematics that is attained in everyday life and mathematics that is needed in everyday
life.
1.7 Summary of the thesis
The main theme of this thesis is mathematics in everyday life. This topic was incorporated into the
present curriculum for compulsory education in Norway (grades 1-10), L97, and it was presented as
one of the main areas. We have studied how practising teachers make connections with everyday
life in their teaching, and their thoughts and ideas on the subject. Our study was a case study of
teachers’ beliefs and actions, and it included analysis of curriculum papers, textbooks, and videos
from the TIMSS 1999 Video Study as well as an analysis of questionnaires, interviews and
classroom observations of eight Norwegian teachers.
Eight teachers have been studied from four different schools. The teachers have been given new
names in our study, and the schools have been called school 1, school 2, school 3 and school 4.
Schools 1 and 2 were upper secondary schools. We studied one teacher in school 1 (Jane) and four
teachers in school 2 (George, Owen, Thomas and Ingrid). Schools 3 and 4, which were visited last,
were both lower secondary schools. We studied two teachers in school 3 (Ann and Karin) and one
teacher in school 4 (Harry). All were experienced teachers.
We used ethnographic methods in our case study, where the focus of interest was the teachers’
beliefs and practices. All mathematics teachers at the four schools were asked to answer a
questionnaire about real-life connections. 20 teachers responded (77% of all the mathematics
teachers). The eight teachers were interviewed and their teaching practices observed for about 4
weeks. These three methods of data collection were chosen so as to obtain the most complete
records of the teachers’ beliefs and actions in the time available.
In chapter 2 the theoretical foundations of the study are presented and discussed. Here,
constructivism, social constructivism, social learning theories, situated learning and transfer of
learning are important concepts. The thesis also aims at being connected with international research.
An important aspect of the thesis is therefore a study of videos from the TIMSS 1999 Video Study
(cf. Hiebert et al., 2003). This part of our study was conducted in May 2003 while the author was in
residence at UCLA and at Lesson Lab as a member of the TIMSS 1999 Video Study of
Mathematics in seven countries. Videos from Japan, Hong Kong and the Netherlands were studied
to investigate how teachers in these countries connected with real life in their teaching. This study
of videos is presented in chapter 3 and it aims to give our own study an international perspective.
The Norwegian national curriculum, L97 (RMERC, 1999), implies a strong connection of
mathematics and everyday life. This is supposed to be applied in all 10 years of compulsory
education, and it is also emphasised (although not as strongly) in the plans for upper secondary
education. Chapter 4 is a presentation and discussion of the curriculum ideas concerning
mathematics in everyday life. We also present how these ideas were present in previous curricula in
Norway.
11
1 Introduction
The curriculum is (supposed to be) the working document for teachers, but research shows that
textbooks are the main documents or sources of material for the teachers (cf. Alseth et al., 2003).
Chapter 5 is a study of the textbooks that were used by the eight teachers in this study. We have
focused on how these textbooks deal with real-life connections, and especially in the chapters on
geometry (lower secondary school) and trigonometry (upper secondary school), since these were the
topics most of the teachers were presenting at the time of the classroom observations.
Chapter 6 gives a further presentation and discussion of the methods and methodological
considerations of our study. The different phases of the study are discussed, and the practical
considerations and experiences also. A coding scheme from the TIMSS 1999 Video Study was
adopted and further adapted to our study, and, in a second phase of analysis, a list of categories and
themes were generated and used in the analysis and discussion of findings.
The findings of our study constitute an important part of this thesis, and chapters 7-9 give a
presentation of these. The questionnaire results are presented in chapter 7, with the main focus on
the Likert scale questions. They represent some main ideas from the curriculum, and the teachers’
replies to these questions give strong indications of their beliefs about real-life connections. 35% of
the teachers replied that they, often or very often, emphasise real-life connections in their teaching
of mathematics, and so there was a positive tendency. The classroom observations and the
interviews were meant to uncover if these professed beliefs corresponded with the teaching
practices of the teachers.
Chapter 8 is a presentation of the findings from the study of three teachers in lower secondary
school (Ann, Karin and Harry). They were quite different teachers, although all three were
experienced and considered to be successful teachers. Harry was positive towards real-life
connections, and he had many ideas that he carried out in his lessons. Ann was also positive towards
the idea of connecting with everyday life, but she experienced practical difficulties in her everyday
teaching, which made it difficult for her to carry it out. Karin was opposed to the idea of connecting
mathematics with everyday life and she considered herself to be a traditional teacher. Her main idea
was that mathematics was to exercise the pupils’ brains, and the textbook was a main source for this
purpose, although she did not feel completely dependent on it.
In chapter 9 we present the findings from the pilot study of five teachers from upper secondary
school (Jane, George, Owen, Thomas and Ingrid). They teach pupils who have just finished lower
secondary school. They follow a different curriculum, but the connections with everyday life are
also represented in this. Jane taught mathematics at a vocational school, and she focused a lot on
connecting with everyday or vocational life. Her approach was different from Harry’s, but she also
had many ideas that she carried out in her teaching. George was positive towards real-life
connections, but he had questions about the very concept of everyday life. He believed that school
mathematics was a part of everyday life for the pupils, and their everyday life could also be that they
wanted to qualify for studies at technical universities etc. Owen seemed to be positive towards real-
life connections in the questionnaire, but he turned out to be negative. He was a traditional teacher,
and he almost exclusively followed the textbook. Thomas and Ingrid were teaching a class together,
and this class was organised in cooperative groups. Neither Thomas nor Ingrid had a significant
focus on real-life connections.
We have observed teachers with significantly different beliefs and practices. Some were opposed to
a connection with everyday life, some were not. Our study has given several examples of how real-
life connections can be implemented in classrooms, and it has provided important elements in the
discussion of how mathematics should and could be taught. Chapter 10 presents a more thorough
discussion of the findings as well as answers to the research questions, while chapter 11 presents the
conclusions of the present study and the implications for teaching. This chapter also presents a
discussion of the connection between curriculum intentions and the implementation of these
12
Mathematics in everyday life
intentions in the textbooks and finally in actual teaching practice. A discussion of how problems can
be made realistic is also presented, as well as comments about the lessons learned (according to
research methods etc.) and the road ahead, with suggestions for how to change teachers’ beliefs and
teaching practice.
There are many approaches to teaching. Our study has aimed at giving concrete examples of how
teaching can be organised in order to connect mathematics with everyday life and thereby follow the
suggestions of L97.
13
Mathematics in everyday life
2 Theory
Our study is closely connected with two themes from the Norwegian national curriculum (L97), and
since these issues provide the basis for our research questions, we will briefly repeat them here:
The syllabus seeks to create close links between school mathematics and mathematics in the outside
world. Day-to-day experience, play and experiment help to build up its concepts and terminology
(RMERC, 1999, p. 165)
And the second:
Learners construct their own mathematical concepts. In that connection it is important to emphasise
discussion and reflection. The starting point should be a meaningful situation, and tasks and problems
should be realistic in order to motivate pupils (RMERC, 1999, p. 167).
Traditional school education may remove people from real life (cf. Fasheh, 1991), and L97 aims at
changing this. Mathematics in school is therefore supposed to be connected with the outside world,
and the pupils should construct their own mathematical concepts. We believe that these ideas are
not separated, but closely connected, at least in the teaching situation. It is also indicated in the last
quote that the starting point should be a meaningful situation. This will often be a situation from
everyday life, a realistic situation or what could be called an experientially real situation. The
Norwegian syllabus therefore connects these issues.
This theoretical part has two main perspectives: teacher beliefs and learning theories. Our study has
a focus on teacher beliefs, and it has a focus on the teachers’ beliefs about something particular.
This ‘something particular’ is the connection with mathematics and everyday life. We therefore
present and discuss learning theories and approaches that are somewhat connected with this. As a
bridge between the two main points of focus is a more philosophical discussion of the different
‘worlds’ involved.
Our aim is to investigate teachers’ beliefs and actions concerning these issues, and in this theoretical
part we will start by discussing teacher beliefs. Educational research has addressed the issue of
beliefs for several decades (cf. Furinghetti & Pehkonen, 2002).
2.1 Teacher beliefs
Beliefs and knowledge about mathematics and the teaching of mathematics are arguably important,
and in our study we aim mainly to uncover some of the teachers’ beliefs about certain aspects of the
teaching of mathematics. Research has shown that teachers, at least at the beginning of their careers,
shape their beliefs to a considerable extent from the experiences of those who taught them (cf.
Andrews & Hatch, 2000; Feiman-Nemser & Buchmann, 1986; Calderhead & Robson, 1991; Harel,
1994).
There are many different variations of the concepts ‘belief’ and ‘belief systems’ in the literature (cf.
Furinghetti & Pehkonen, 2002; McLeod & McLeod, 2002), but in many studies the differences
between beliefs and knowledge are emphasised.
Scheffler (1965) presented a definition, where he said that X knows Q if and only if:
15
2 Theory
i. X believes Q
ii. X has the right to be sure of Q
iii. Q
The third criterion, about the very existence of Q, is a tricky one. The very essence of
constructivism is that we can never know reality as such, but we rather construct models that are
trustworthy . Following a constructivist perspective, criteria ii and iii can be restated as follows
(Wilson & Cooney, 2002, p. 130):
iiR (revised). X has reasonable evidence to support Q.
One might say that beliefs are the filters through which experiences are interpreted (Pajares, 1992),
or that beliefs are dispositions to act in certain ways, as proposed by Scheffler:
A belief is a cluster of dispositions to do various things under various associated circumstances. The
things done include responses and actions of many sorts and are not restricted to verbal affirmations.
None of these dispositions is strictly necessary, or sufficient, for the belief in question; what is
required is that a sufficient number of these clustered dispositions be present. Thus verbal dispositions,
in particular, occupy no privileged position vis-á-vis belief (Scheffler, 1965, p. 85).
This definition provides difficulties for modern research, since,
according to Scheffler, a variety of evidence has to be present in
order to determine one’s beliefs. What then when a teacher
claims to have a problem solving view on mathematics, but in the
classroom he only emphasises procedural knowledge? The
researcher would then probably claim that there exists an
inconsistency between the teachers’ belief and his or her practice.
We might also say that each individual possesses a certain system
of beliefs, and the individual continuously tries to maintain the
equilibrium of their belief systems (Andrews & Hatch, 2000).
According to Op’t Eynde et al. (1999), beliefs are,
epistemologically speaking, first and foremost individual
constructs, while knowledge is a social construct. We might
therefore say that beliefs are people’s subjective knowledge, and
they include affective factors. It should be taken into
consideration that people are not always conscious of their
beliefs. Individuals may also hide their beliefs when they do not
seem to fit someone’s expectations. We therefore want to make a
distinction between deep beliefs and surface beliefs. These could again be viewed as extremes in a
wide spectrum of beliefs (Furinghetti & Pehkonen, 2002).
Another definition was given by Goldin (2002), who claimed that beliefs are:
(...) internal representations to which the holder attributes truth, validity, or applicability, usually stable
and highly cognitive, may be highly structured (p. 61).
Goldin later specified his definition of beliefs to be:
16
BELIEF
“Belief is assent to a proposition.
Belief in the psychological sense, is a
representational mental state that
takes the form of a propositional
attitude. In the religious sense,
‘belief’ refers to a part of a wider
spiritual or moral foundation,
generally called faith.
Belief is considered propositional in
that it is an assertion, claim or
expectation about reality that is
presumed to be either true or false
(even if this cannot be practically
determined, such as a belief in the
existence of a particular deity).
http://en.wikipedia.org/wiki/Belief
Mathematics in everyday life
(...) multiply-encoded cognitive/affective configurations, usually including (but not limited to)
prepositional encoding, to which the holder attributes some kind of (Goldin, 2002, p. 64;
original italics).
Another attempt of defining beliefs, which supports Goldin’s definitions, is to simply define beliefs
as purely cognitive statements to which the holder attributes truth or applicability (Hannula et al.,
2004). Hannula thereby wished to exclude the emotional aspect from beliefs, and he claimed instead
that each belief may be associated with an emotion (Hannula, 2004, p. 50):
If this distinction between a belief and the associated emotion were made, it would clarify much of the
confusion around the concept “belief”. For example, two students may share a cognitive belief that
problem solving is not always straightforward, but this belief might be associated with enjoyment for
one and with anxiety for the other.
A consensus on one single definition of the term ‘belief’ is probably neither possible nor desirable,
but we should be aware of the several types of definitions, as they might be useful in order to
understand the different aspects of beliefs (cf. McLeod & McLeod, 2002).
The view on teacher beliefs has changed during the years. In the 1970s there was a shift from a
process-product paradigm, where the emphasis was on the teacher’s behaviour, towards a focus on the
teacher’s thinking and decision-making processes. This led to an interest in the belief systems and
conceptions that were underlying the teacher’s thoughts and decisions (Thompson, 1992, p. 129).
Research on teacher beliefs has shown that there is a link between the teachers’ beliefs about
mathematics and their teaching practices (Wilson & Cooney, 2002). Studies like Thompson (1992)
suggest that a teacher’s beliefs about the nature of mathematics influence the future teaching
practices of the teacher (cf. Szydlik, Szydlik & Benson, 2003, p. 253). If a teacher regards
mathematics as a collection of rules that are supposed to be memorised and applied, this would
influence his teaching, and as a result he will teach in a prescriptive manner (Thompson, 1984).
On the other hand, a teacher who holds a problem solving view of mathematics is more likely to
employ activities that allow students to construct mathematical ideas for themselves (Szydlik, Szydlik
& Benson, 2003, p. 254).
Recent curriculum reforms indicate such a view of mathematics more than the earlier ones. When
faced with curriculum reforms, practising teachers often have to meet the challenges of these new
reforms by themselves. Their teaching practice is a result of decisions they make based on
interpretations of the curriculum rhetoric and experiences and beliefs they carry into the classroom
(Sztajn, 2003, pp. 53-54).
Change in teaching on a national basis would not only have to do with a change of curriculum and
textbooks, but it would also be connected with a change or modification of teachers’ beliefs about
mathematics, about teaching and learning mathematics, etc. Experiences with innovative curriculum
materials might challenge the teachers’ beliefs directly. Most teachers rely upon one or a few
textbooks to guide their classroom instruction, and they need guidance in order to change their
teaching practice (Lloyd, 2002, p. 157).
Ernest (1988, p.1) distinguished between three elements that influence the teaching of mathematics:
1) The teacher’s mental contents or schemas, particularly the system of beliefs concerning
mathematics and its teaching and learning;
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2 Theory
2) The social context of the teaching situation, particularly the constraints and opportunities it
provides; and
3) The teacher’s level of thought processes and reflection.
Such a model can be further developed into a model of distinct views on how mathematics should
be taught, like that of Kuhs and Ball (1986, p. 2):
Learner-focused: mathematics teaching that focuses on the learner’s personal construction of
mathematical knowledge;
Content-focused with an emphasis on conceptual understanding: mathematics teaching that is
driven by the content itself but emphasizes conceptual understanding;
Content-focused with an emphasis on performance: mathematics teaching that emphasizes
student performance and mastery of mathematical rules and procedures; and
Classroom-focused: mathematics teaching based on knowledge about effective classrooms.
Thompson (1992) continues the work of Ernest (1988) when she explains how research indicates
that a teacher’s approaches to mathematics teaching have strong connections with his or her systems
of beliefs. It should therefore be of great importance to identify the teacher’s view of mathematics as
a subject. Several models have been elaborated to describe these different possible views. Ernest
(1988, p. 10) made a distinction between (1) the problem-solving view, (2) the Platonist view, and
(3) the instrumentalist view. Others, like Lerman (1983), have made distinctions between an
absolutist and a fallibilist view on mathematics. Skemp (1978), who based his work on Mellin-
Olsen’s, made a distinction between ‘instrumental’ mathematics andrelational’ mathematics
(Thompson, 1992, p. 133). In the Californian ‘Math wars’, we could distinguish between three
similar extremes: the concepts people, the skills people, and the real life applications people
(Wilson, 2003, p. 149).
Research on teacher beliefs could be carried out using questionnaires, observations, interviews, etc.,
but one should be cautious:
Inconsistencies between professed beliefs and instructional practice, such as those reported by
McGalliard (1983), alert us to an important methodological consideration. Any serious attempt to
characterize a teacher’s conception of the discipline he or she teaches should not be limited to an
analysis of the teacher’s professed views. It should also include an examination of the instructional
setting, the practices characteristic of that teacher, and the relationship between the teacher’s professed
views and actual practice (Thompson, 1992, p. 134).
These inconsistencies might also be related to the significant discrepancy between knowledge and
belief. Research has shown that although the teachers’ knowledge of curriculum changes has
improved, the actual teaching has not changed much (Alseth et al., 2003). The reason for this might
be that it is possible for knowledge to change while beliefs do not, and what we call knowledge
could be connected with what Thompson (1992) called professed views. Research has also shown
that pre-existing beliefs about teaching, learning and subject matter can be resistant to change (cf.
Szydlik, Szydlik & Benson, 2003; Lerman, 1987; Brown, Cooney & Jones, 1990; Pajares, 1992;
Foss & Kleinsasser, 1996).
All these issues imply that educational change is a complex matter, and that we should be aware of
the possible differences between professed beliefs and the beliefs that are acted out in teaching. This
possible inconsistency between professed beliefs and instructional practice is a reason why we have
chosen a research design with several sources of data. We wanted to learn not only about the beliefs
of the teachers, but also about their teaching practices. If beliefs alone could give a complete image
of teaching, no researcher would need to study teaching practice. We wanted not only to study what
18
Mathematics in everyday life
the teachers said in the interviews or questionnaires (professed beliefs), but also to observe the
actual teaching practices of these teachers (instructional practice). We believe that such a knowledge
of the teaching practice and beliefs of other teachers is of importance to the development of one’s
own teaching.
All this taken into account, we study beliefs (and practice) of teachers because we believe, and
evidence has shown (Andrews & Hatch, 2000), that teachers’ beliefs about the nature of
mathematics do influence both what is taught and how it is taught. This is discussed by Wilson &
Cooney, 2002, p. 144:
However, regardless of whether one calls teacher thinking beliefs, knowledge, conceptions, cognitions,
views, or orientations, with all the subtlety these terms imply, or how they are assessed, e.g., by
questionnaires (or other written means), interviews , or observations, the evidence is clear that teacher
thinking influences what happens in the classrooms, what teachers communicate to students, and what
students ultimately learn.
In our study of teacher beliefs and their influence on teaching we wish to shed light on important
processes in the teaching of mathematics. Research has shown that teachers’ beliefs can change
when they are provided with opportunities to consider and challenge these beliefs (Wilson &
Cooney, 2002, p. 134).
Research has shown that the relationship between beliefs and practice is probably a dialectic rather
than a simple cause-and-effect relationship (cf. Thompson, 1992), and would therefore be
interesting for future studies to seek to elucidate the dialectic between teachers’ beliefs and practice,
rather than trying to determine whether and how changes in beliefs result in changes in practice.
Thompson also suggests that it is not useful to distinguish between teachers’ knowledge and beliefs.
It seems more helpful to focus on the teachers’ conceptions instead of simply teachers’ beliefs (cf.
Thompson, 1992, pp. 140-141). She also suggests that we must find ways to help teachers examine
their beliefs and practices, rather than only present ourselves as someone who possesses all the
answers.
We should not take lightly the task of helping teachers change their practices and conceptions.
Attempts to increase teachers’ knowledge by demonstrating and presenting information about
pedagogical techniques have not produced the desired results. (...) We should regard change as a long-
term process resulting from the teacher testing alternatives in the classroom, reflecting on their relative
merits vis-á-vis the teacher’s goals, and making a commitment to one or more alternatives (Thompson,
1992, p. 143).
Our study is not simply a study of teacher beliefs as such, but rather a study of teacher beliefs about
connecting mathematics with real or everyday life, and we aim at uncovering issues that might be
helpful for teachers in order to change teaching practice. Before we present and discuss theories and
research related to this particular issue, we have to make a more philosophical discussion.
2.2 Philosophical considerations
When discussing the connection with mathematics and everyday life, the outside world, the physical
world (or whatever we like to call it), there is a more basic discussion that we should have in mind.
This discussion, which is important in order to understand the entire issue that we discuss, is about
the very nature of what we might call ‘the mathematical world’ and ‘the physical world’. If we do
not include such a discussion, everything we say about the connections between mathematics and
everyday life maybe will make little sense.
19
2 Theory
We have already seen in the introductory discussion of concepts that our study deals with
conceptions of reality and what is ‘real’ to different people. In order to understand these issues
further, we might present a theory of three different ‘worlds’:
The world that we know most directly is the world of our conscious perceptions, yet it is the world that
we know least about in any kind of precise scientific terms. (...) There are two other worlds that we are
also cognisant of - less directly than the world of our perceptions - but which we now know quite a lot
about. One of these worlds is the world we call the physical world. (...) There is also one other world,
though many find difficulty in accepting its actual existence: it is the Platonic world of mathematical
forms (Penrose, 1994, p. 412).
The physical world and the mathematical world are most interesting to this discussion. Instead of
making a new definition of these worlds, we refer to Smith, who has a problem-solving approach to
this as opposed to Penrose’s more Platonic approach:
The physical world is our familiar world of objects and events, directly accessible to our eyes, ears,
and other senses. We all have a language for finding our way around the physical world, and for
making statements about it. This everyday language is often called natural, not because other kinds of
language are unnatural, but because it is the language we all grow up speaking, provided we have the
opportunity to hear it spoken by family and friends during our childhood.
I use the word “world” metaphorically to talk about mathematics because it is a completely different
domain of experience from the physical world. (...) Mathematics can be considered a world because it
has a landscape that can be explored, where discoveries can be made and useful resources extracted. It
can arouse all kinds of familiar emotions. But it is not part of the familiar physical world, and it
requires different kinds of maps, different concepts, and a different language. The world of
mathematics doesn’t arise from the physical world (I argue) - except to the extent that it has its roots in
the human brain, and it can’t be made part of the physical world. The two worlds are always at arm’s
length from each other, no matter how hard we try to bring them together or take for granted their
interrelatedness.
The language used to talk about the world of mathematics is not the same as the language we use for
talking about the physical world. But problems arise because the language of mathematics often looks
and sounds the same as natural language (Smith, 2000, p. 1).
This understanding of ‘the physical world’ has close relations to our definition of ‘real world’ (see
chapter 1.6). Penrose also takes up the discussion about the meaning of these different worlds:
What right do we have to say that the Platonic world is actually a ‘world’, that can ‘exist’ in the same
kind of sense in which the other two worlds exist? It may well seem to the reader to be just a rag-bag
of abstract concepts that mathematicians have come up with from time to time. Yet its existence rests
on the profound, timeless, and universal nature of these concepts, and on the fact that their laws are
independent of those who discover them. This rag-bag - if indeed that is what it is - was not of our
creation. The natural numbers were there before there were human beings, or indeed any other creature
here on earth, and they will remain after all life has perished (Penrose, 1994, p. 413).
The relationship between these worlds is of importance to us here, and Penrose presents three
‘mysteries’ concerning the relationships between these worlds:
There is the mystery of why such precise and profoundly mathematical laws play such an important
role in the behaviour of the physical world. Somehow the very world of physical reality seems almost
mysteriously to emerge out of the Platonic world of mathematics. (...) Then there is the second mystery
of how it is that perceiving beings can arise from out of the physical world. How is it that subtly
20
Mathematics in everyday life
organized material objects can mysteriously conjure up mental entities from out of its material
substance? (...) Finally, there is the mystery of how it is that mentality is able seemingly to 'create'
mathematical concepts out of some kind of mental model. These apparently vague, unreliable, and
often inappropriate mental tools, with which our mental world seems to come equipped, appear
nevertheless mysteriously able (...) to conjure up abstract mathematical forms, and thereby enable our
minds to gain entry, by understanding, into the Platonic mathematical realm (Penrose, 1994, pp. 413-
414).
Where Penrose talks about ‘mysteries’, Smith talks about a ‘glass wall’ between the world of
mathematics and the physical world:
Finally, the glass wall is a barrier that separates the physical world and its natural language from the
world of mathematics. The barrier exists only in our mind - but it can be impenetrable nonetheless. We
encounter the wall whenever we try to understand mathematics through the physical world and its
language. We get behind the wall whenever we venture with understanding into the world of
mathematics (Smith, 2000, p. 2).
Smith claims that major problems can arise when mathematics is approached as if it were part of
natural language. This indicates that the connection with mathematics and everyday life is far from
trivial, and that it can actually be problematic.
He explains further that mathematics is not an ordinary language that can be studied by linguists,
and it does not translate directly into any natural language. If we call mathematics a language, we
use the word “language” metaphorically (Smith, 2000, p. 2). Music is a similar language to
mathematics, and:
Everyday language is of limited help in getting into the heart of music or mathematics, and can arouse
confusion and frustration (Smith, 2000, p. 2).
This means that only a small part of mathematics can be put into everyday language. This coincides
with what some of the teachers in the pilot said, that mathematics in everyday life is important, but
mathematics is so much more than that...
To define what mathematics is, is not an easy task. It might refer to what people do (mathematicians
but also most normal people) or what people know. Smith claims that many people do mathematical
activities without being aware that they do so - they do without knowing (like in the study of
Brazilian street children, cf. Nunes, Schliemann & Carraher, 1993), and many of us recite
mathematical knowledge that we never put to use - we know without doing (cf. Smith, 2000, pp. 7-
9).
2.2.1 Discovery or invention?
When discussing what mathematics is, we often encounter a discussion of whether mathematical
knowledge was discovered or invented. People like Penrose, with a more Platonic view, would
probably say that mathematics is discovered, whereas social constructivists and others would argue
that mathematical knowledge is a construction of humans or rather the construction of people in a
society. The understanding of what mathematics is and how mathematics came into being also has
an influence of the way we think about teaching and learning of mathematics.
21
2 Theory
One would think that language is something that is discovered by every child (or taught to every
child). Yet studies of the rapidly efficient manner in which language skill and knowledge develop in
children has led many researchers to assert that language is invented (or reinvented) by children rather
than discovered by them or revealed to them. And no less psychologist than Jean Piaget has asserted
that children have to invent or reinvent mathematics in order to learn it (Smith, 2000, p. 15).
When curricula and theories deal with understanding of mathematics, they often include issues of
relating mathematical knowledge to everyday life, the physical world or some other instances. There
are, however, issues that should be brought into discussion here:
When I use the phrase “understanding mathematics,I don’t mean relating mathematical knowledge
and procedures to the “real world”. A few practical calculations can be made without any
understanding of the underlying mathematics, just as a car can be driven without any understanding of
the underlying mechanics (Smith, 2000, p. 123).
Smith also discusses what it means to learn mathematics, and he claims that everyone can learn it.
He does not thereby mean that everyone can or should learn all of mathematics, or even to learn
everything in a particular curriculum:
The emphasis on use over understanding is explicit in “practical” curricula supposed to reflect the
“needs” of the majority of students in their everyday lives rather than serve a “tiny minority” who
might want to obtain advanced qualifications. The patronizing dichotomy between an essentially
nonmathematical mass and a small but elite minority is false and dangerous. The idea that the majority
would be best served by a bundle of skills rather than by a deeper mathematical understanding would
have the ultimate effect of closing off the world of mathematical understanding to most people, even
those who might want to enter the many professions that employ technological or statistical procedures
(Smith, 2000, p. 124).
He also refers to the constructivist stance (which we will return to in chapter 2.3):
The constructivist stance is that mathematical understanding is not something that can be explained to
children, nor is it a property of objects or other aspects of the physical world. Instead, children must
“reinvent” mathematics, in situations analogous to those in which relevant aspects of mathematics
were invented or discovered in the first place. They must construct mathematics for themselves, using
the same mental tools and attitudes they employ to construct understanding of the language they hear
around them (Smith, 2000, p. 128).
This does not mean that children should be left on their own, but it means that they can and must
invent mathematics for themselves, if provided with the opportunities for the relevant experiences
and reflections.
The connection between mathematics and everyday life, which is evidently more complex than one
might initially believe, has often been dealt with through the use of word problems. These word
problems are often mathematical problems wrapped up in an everyday language:
It is widely believed that mathematics can be made more meaningful, and mathematics instruction
more effective, if mathematical procedures and problems are wrapped in the form of everyday
language. (...) But there are doubts whether many “word problems” - embedding (or hiding)
mathematical applications in “stories” - do much to improve mathematical comprehension. Such
problems need to be carefully designed and used in ways that encourage children to develop relevant
computational techniques. Otherwise, children easily but unwittingly subvert teachers aims by
showing the same originality and inventiveness they demonstrate in purely mathematical situations
(Smith, 2000, p. 133).
22
Mathematics in everyday life
What often happens, is that pupils find shortcuts, they search for key words, etc., to solve word
problems.
Children may appear to gain mastery but in fact find practical shortcuts and signposts that eventually
constitute obstacles to future progress. They usually prefer their own invented procedures to formal
procedures that they don’t understand (Smith, 2000, p. 133).
These are issues one should have in mind when discussing textbook problems (cf. chapter 5) in
general and word problems in particular.
2.3 Theories of learning
A number of studies (cf. Dougherty, 1990; Grant, 1984; Marks, 1987; Thompson, 1984) have
shown that beliefs that teachers have about mathematics and its teaching influence their teaching
practice. Our study has a focus on the teachers’ beliefs and practices as far as the connection of
mathematics with everyday life is concerned, and there are several issues concerning learning
theories that are important in this aspect.
When discussing learning and different views of learning, it is important to have in mind which
theory of reality we are building upon. Our conception of the physical world also accounts for our
conception of learning. To put it simply, we can view reality in a subjective or an objective way.
The objective tradition presents the world as consisting mainly of things or objects, which we can
observe in their true nature. This process of observation is completely independent of the person
observing, and the theory belongs to what we might call absolutism or empiricist philosophy.
Behaviourism builds on such an objective view. Behaviourists, or learning theorists, were interested
in behaviour, in activities that could be observed objectively and measured in a reliable way. This
psychological tradition claims that learning is a process that takes place in the individual learner,
who, being exposed to an external stimulus, reacts (responds) to this stimulus. The idea of stimulus-
response is central to the behaviourist theory of learning (cf. Gardner, 2000, p. 63).
Thoughts on what directs human behaviour (DNA, environmental influence or the individual itself)
influence our choice of psychological tradition. Various theories of human behaviour have been
developed: psychoanalysis, cognitive psychology, constructivism, social psychology, etc.
Our view of learning has a strong influence on our teaching. When we discuss how the teaching of
mathematics is connected to the pupils’ reality, we have already accepted a basic idea that learning
is something that occurs in an interaction between the pupil and the world he or she lives in. We
have thus entered the paradigm of social constructivism and socio-cultural theories, but this does
not necessarily imply that we believe knowledge is only a social construct.
According to the constructivist paradigm, any kind of learning implies a construction of new
knowledge in the individual. In some sense this construction takes place within a social context, but
the processes of construction must also be rooted in the individual person for the notion of learning
by the individual to provide meaning. Although textbooks might have a seemingly simple
definition, constructivism is a wide concept. It might be defined as a view that emphasises the active
role of the learning in the process of building understanding (cf. Woolfolk, 2001, p. 329), but
constructivism actually includes several theories about how people construct meaning. Broadly
speaking, we can distinguish between two different poles. On the one hand, constructivism is a
philosophical discipline about bodies of knowledge, and on the other hand it is a set of views about
how individuals learn (Phillips, 2000, pp. 6-7).
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2 Theory
There seems to be widespread current agreement that learning takes place when the pupil actively
constructs his or her knowledge. The construction of knowledge is seldom a construction of
genuinely new knowledge. It is normally more of a reconstruction of knowledge that is already
known to the general public, but new to the individual. Whether this construction occurs in a social
environment or is solely an individual process can be disputed. We call the former a social
constructivist view, and the latter a radical constructivist view. A radical constructivist view, as
presented by Glasersfeld (1991) will often enter the philosophical realm, and this view builds
strongly on the works of Piaget. Other researchers emphasise the idea of mathematics being a social
construction, and we thus enter the area of social constructivism (cf. Ernest, 1994; 1998). To make a
definite distinction is hard. We believe that the surrounding environment and people are important
in the construction process, and a process of construction normally takes place in a social context.
An emphasis on the context will soon lead to a discussion about the transfer of learning between
contexts (cf. Kilpatrick, 1992).
The ideas of social constructivism can be divided in two. First, there is a tradition starting with a
radical constructivist position, or a Piagetian theory of mind, and then adding social aspects of
classroom interaction to it. Second, there is a theory of social constructivism that could be based on
a Vygotskian or social theory of mind (Ernest, 1994).
Even reading and learning from a book can in some sense be viewed as a social context, since it
includes a simulated discussion with the writer(s). We can also emphasise the individual as a
constructor of knowledge. A social consensus does not necessarily imply that an individual has
learned something. Piaget was a constructivist, and he focused on the individual’s learning. Many
would call him a radical constructivist. But although he was advocating the constructivist phases of
the individual, he was also aware of the social aspects, and that learning also occurred in a social
context. Psychological theories, like other scientific theories, have to focus more on some aspects
than others. This does not mean that the less emphasised issues are forgotten or even rejected. In
constructivism one might focus on the individual, or learning as a social process. Classroom
learning is in many ways a social process, but there also has to be an element of individual
construction in this social process.
A term like ‘holistic’ might also be used to describe learning, and this can be viewed in connection
with descriptions of multiple intelligences as presented in the popular sciences. Gardner (2000, etc.)
is arguably the most important contributor to the theories of multiple intelligences. His theories
discuss and describe the complexities of human intelligence, and a teacher has to be aware of this
complexity in order to meet the pupils on their individual level. More recent theories of pedagogy
present concepts as contextual or situated learning, learning in context, etc. A main idea here is that
learning takes place in a specific context, and a main problem is how we are going to transfer
knowledge to other contexts. In our research we discuss how teachers connect school mathematics
and everyday life. This implies a discussion of teacher beliefs and their connection with teacher
actions. When discussing the connection between school mathematics and everyday life, we also
implicitly discuss transfer of learning between different contexts.
2.4 Situated learning
The teaching of mathematics has been criticised for its formal and artificial appearance, where much
attention has been paid to the drilling of certain calculation methods, algebra and the mechanical use
of formulas.
An alternative to this formal appearance is to work with problems in a meaningful context (cf.
RMERC, 1999), and to connect mathematics with everyday life. The theories of context-based
24
Mathematics in everyday life
teaching are significant, and in mathematics education theories of context-based learning are often
referred to as situated learning. A key point for such theories is that the context of learning, being
organised in school, to a strong degree must be similar to the context in which the knowledge is
applied outside school.
Situated learning is based on the idea that all cognition in general, and learning in particular, is
situated. We can perceive learning as a function of activity, context and culture, an idea which is
often in contrast with the experience we have from school. In school, knowledge has often been
presented without context, as something abstract. Situated learning is thereby a general theory for
the acquisition of knowledge, a gradual process where the context is everyday life activities. We
find these ideas also in what has been called ‘legitimate peripheral participation’, which is a more
contemporary label for the ideas of situated learning. According to this theory, learning is compared
to an apprenticeship. The unschooled novice joins a community, moving his way from the
peripheral parts of the community towards the centre. Here, the community is an image of the
knowledge and its contexts (cf. Lave & Wenger, 1991).
Situated learning should include an authentic context, cooperation and social interaction. These are
some of the main principles. Social interaction may be understood as a critical component. The idea
is simply that thought and action are placed within a certain context, i.e. they are dependent on locus
and time. We will take a closer look at the concept of situated learning and its development when
presenting some of the most important research done in the field.
2.4.1 Development of concepts
The studies of the social anthropologist Jean Lave and her colleagues have been important in the
development of the theories of situated learning. We sometimes use ‘learning in context’ or other
labels to describe these ideas.
Lave aimed at connecting theories of cognitive philosophy with cognitive anthropology, the culture
being what connects these in the first place. Socialisation is a central concept describing the
relations between society and the individual (Lave, 1988, p. 7).
Functional theory represents an opposite extreme to the ideas of Lave and others about learning in
context.
(…) functional theory treats processes of socialisation (including learning in school) as passive, and
culture as a pool of information transmitted from one generation to the next, accurately, with
verisimilitude, a position that has created difficulties for cognitive psychology as well as anthropology
(Lave, 1988, p. 8).
Such a functional theory also includes theories of learning:
(…) children can be taught general cognitive skills (…) if these ”skills” are disembedded from the
routine contexts of their use. Extraction of knowledge from the particulars of experience, of activity
from its context, is the condition for making knowledge available for general application in all
situations (Lave, 1988, p. 8).
Traditional teaching, in the form of lectures, is a typical example of this,<