Educational Design Research in Mozambique; Starting Mathematics from Authentic Resources

Abstract

This article describes a research on learner-centred instruction in Mozambique, Africa. A starting point was the use of real-life resources, such as traditional art craft objects and authentic newspaper clippings. The study used a method which is termed 'design research'. This method aligns theory with practice and is geared towards improving educational practice. In two sub-studies, on geometry and on statistics, learner-centred instruction was facilitated through the use of worksheets with open- ended questions tailored for group work. The designs were tested in cyclic interventions and formatively evaluated through observation reports, interviews and assessment of learners' work. A decentralised, student-centred learning ecology proved to be feasible in overcrowded classrooms, typical in African education.

Full text

EDUCATIONAL DESIGN RESEARCH IN MOZAMBIQUE:
STARTING MATHEMATICS FROM AUTHENTIC RESOURCES
Pauline Vos1, Tiago G. Devesse2, and Assane Rassul2
1University of Groningen, Netherlands
2University Eduardo Mondlane, Mozambique
This article describes a research on learner-centred instruction in Mozambique,
Africa. A starting point was the use of real-life resources, such as traditional art craft
objects and authentic newspaper clippings. The study used a method which is termed
‘design research’. This method aligns theory with practice and is geared towards
improving educational practice. In two sub-studies, on geometry and on statistics,
learner-centred instruction was facilitated through the use of worksheets with open-
ended questions tailored for group work. The designs were tested in cyclic
interventions and formatively evaluated through observation reports, interviews and
assessment of learners’ work. A decentralised, student-centred learning ecology
proved to be feasible in overcrowded classrooms, typical in African education.
INTRODUCTION
In Mozambique all sectors of education suffer from weaknesses, whether it be
primary, secondary or tertiary education. Generally, mathematics education can be
characterized by teacher-centred instruction, chorus-recitation, shortages of materials
and facilities, un(der)qualified teachers, overcrowded classes, and a curriculum with
much theory and few links to learners’ lives. As a result, there are cognitive,
instructional and affective problems. Regarding the cognitive problems in the
mathematics classes, in general, learners learn to memorize formulas and algorithms,
needed for the immediate solution of exercises. For many learners, there is neither
logical sequence, nor any clear relationship between concepts. Mathematics is taught
as a deductive discipline, starting from definitions. For example, the official grade 10
mathematics curriculum document prescribes the drill of formulas, stating that the
formulas for area and volume of a cone should be imprinted by frequent repetitions
(Ministério da Educação, 1995). As a result, learners know mathematical formulas
without understanding them, leading to short-term retention and low motivation.
Instructional problems in Mozambican mathematics classes are related to the low
number of qualified teachers and to lack of instructional materials. The majority of
the teachers in primary education completed fewer than twelve years of education.
More than 80% of the mathematics teachers in secondary education are unqualified.
The few qualified teachers only teach the highest grades (grade 11-12). Also at
tertiary level, most lecturers have a degree equal to the level of the courses they
teach. As for the lack of instructional materials, there are few books, teaching
manuals and other publications. For governmental primary and junior secondary
schools, the Ministry provides schools with officially mandated textbooks, but the

number of copies is insufficient to satisfy the needs of all learners (Mira, 2000). The
shortage of books at senior levels forces teachers to use foreign books and make
learners copy the content (either from the blackboard or by reading out aloud). With
the shortages being structural, teachers have become used to the situation. Even in
cases of sufficient facilities, teachers still stick to the routine of orally transmitting
definitions and theorems through chorus-recitation.
Mathematics education in Mozambique, like in many other countries, also faces
problems in the affective domain. For many learners, the language of teaching
(Portuguese) differs from their mother tongue. Also, the content of mathematics
education does not link with learners’ context. Mozambican learners perceive
mathematics as being of little use for understanding the world around them (Januário
et al., 2002). They see mathematics as strange, as coming from another world and
being imposed upon them. Moreover, many learners live with uncertainty of their
abilities and with fear of failure, especially in mathematics.
In this situation, a research in mathematics education was started, aiming at
integrating authentic experiences from learners’ context with mathematical concepts.
The objectives of the study were manifold. First, we wanted to show, that modern
Mozambican society is powerful and rich in resources, to such an extent that it can
provide mathematics education with many applications. Second, we wanted to create
a learning environment, in which the teacher-centeredness was reduced. For this
objective we planned to design prototype materials in such a way that whole-class
lecturing could be largely avoided, and discussion among learners enhanced. If the
materials were well-designed they could set an example for how dynamic classrooms
can be organized. Finally, with lesson materials that complied with the mandated
textbooks, we hoped to demonstrate that innovative instructional approaches can be
embedded within the frame of the central curriculum demands.
DESIGN RESEARCH
The Mozambican complexity of teaching and learning asked for a research approach,
which faces the conditions of learning. Thus, the study was conceived as a design
research. Design research in mathematics education goes back to the writing by Hans
Freudenthal (1991), who explained mathematics as a human activity and who insisted
on design research as the core of mathematics education.
Burkhardt and Schoenfeld (2003) also advocate design research. They state that
traditional educational research does generally not lead to improved practice, due to
lack of credible models. However, an engineering approach to design educational
processes leads to refined ideas and materials, which are robust across a wide range
of contexts. Also Wittmann (1995) suggested that the tension between research and
practice can be eased through the design of substantial learning environments,
exemplified by arithmetic puzzles that lead to lively classrooms at primary level.
Design research is a methodology that goes beyond the teaching experiment, in
which one attempts to establish an existence proof, which shows that ‘something’ can

be done in class (Lesh & Kelly, 2000). Cobb et al. (2003) explain how in design
research educational researchers engineer improvement by bringing about new forms
of learning in order to study them. They introduced the idea that design research
serves to study a learning ecology, to emphasize that learning takes place in an
environment consisting of interacting systems and not as a set of activities or as a list
of factors. A learning ecology contains aspects such as the teacher and his/her
instruction, tasks and problems, modes of dialogue, ‘norms’ for participation in
discussions, tools and aids and the way teachers conduct whole-class communication.
Using a design approach enabled us to include socio-cultural and situated analyses
into the research, going beyond cognitive and psychological aspects.
The current research was inspired by studies carried out in the United States and the
Netherlands (Breiteig, Huntley & Kaiser, 1993; Gravemeijer, 1994), in which
mathematical modelling activities were tailored as levers for the construction of
conceptual understanding. It was guided by the following research question: to what
extent can authentic resources be a starting point for assisting learners in the effective
formation of concepts. The research comprised two sub-studies. The first focused on
geometry at grade 10 level. It started from locally produced art craft objects, such as
drums, baskets, huts and fish traps (Figure 1). This study was carried out by the
second author (Devesse, 2004).
Figure 1: Example of a river fish trap (Niassa Province, Mozambique)
The second sub-study focused on Statistics. It started from newspaper clippings,
which were cut from the local daily paper Notícias and other Mozambican
publications, with themes such as suicide, domestic violence, maize prices and
employment. The target group consisted of university students in the Social Sciences.
This study was carried out by the third author (Rassul, 2004). The first author was
both supervisor and research assistant in the two sub-studies.
METHODOLOGY
The research was organized in four phases: Analysis, Development, Testing, and
Evaluation. In the Analysis phase, contexts of learning mathematics in Mozambique
and anticipated problems were analysed. This phase also contained an orientation on
the creation of worksheets, a phenomenon generally out of sight in Mozambican
education, despite the general availability of photocopiers and printers. In the planned
design, most instruction would be conveyed in written form. For inspiration, the
second and the third author made a visit to The Netherlands to study the use of
photocopies and textbooks as part of instructional methods. When learners work with
paper-based materials, classroom communication is no longer centred on the teacher.
The contrast with Mozambique was evident.

In the Analysis phase, a number of instructional principles were formulated:
• Mathematics as a human activity, starting from experiences (not from
definitions) and developed through worthwhile activities.
• The use of worksheets for group work for facilitating learner-centeredness.
The worksheets were to include writing space, giving the researchers written
evidence of learners’ performance.
• The use of open-ended questions for discussions between learners. Of
course, not all questions were open-ended, but we were eager to include a
large number of discussion questions into the worksheets. These questions
asked for higher order thinking skills.
• The use of abundant authentic pictorial illustrations. Photographs of art craft
objects were included for visualization into the geometry materials. In the
statistics study, the newspaper clippings were scanned and pasted in their
authentic shape into the worksheets. In this way, learners would immediately
see that the themes were authentic and not primarily created for educational
goals. The authenticity would show them that they were dealing with themes
from their own daily life, from their cultural heritage or related to their
professional future.
• For the geometry study, which was geared towards a lower level of learning,
we included an additional instructional principle: the integration of
manipulatives to enable learners to really hold the objects in their hands and
learn about mathematical properties through many senses.
• For the statistics study, we included the integration of computers
(spreadsheets) to enable the handling of authentic data.
The second phase was the Development phase. This started with gathering
inspiration. For the geometry study, typical art craft objects were found at craft
markets and at the Natural History Museum in Maputo (the capital city of
Mozambique). For the statistics study, a large number of newspaper articles were cut
out. Simultaneously, curricula were studied to analyse to which mathematical
concepts the resources could be related, and at which level these would best suit for a
series of lessons. It was decided that central curriculum concepts in the geometry
study would be: cylinders and cones, which are taught from grade 7 onwards. The
central curriculum concepts in the statistics study were: mean, binomial distribution,
confidence intervals, sample size, and graphic representations. In this phase, we
decided on the target groups. For the geometry study, we decided to contact two
different lower secondary schools in Maputo and ask whether we could organise
interventions at grade 10 level in collaboration with teachers there. Because we came
as an external research team, we decided to limit the lesson series to four hours of
contact time. For the statistics study, we decided to stay within our own university
because the third author is lecturer of Statistics at the Faculty for Social Sciences. He
could organise interventions in the second year Statistics course for students of

Political Sciences, Anthropology and Sociology, in collaborations with two tutors.
The contact time of this intervention could be sixteen hours.
The first prototype worksheets were validated in an expert appraisal with subject
specialists. This made us limit the rigor of the terminology (to keep the language
accessible) and include the required formula for circumference and area (to avoid
memorization exercises). This yielded the second version ready for testing.
The Testing phase comprised an iteration of cyclic interventions. For example, in the
geometry study, three different interventions were organised. Each subsequent
intervention had a larger scale. The formative evaluation of an intervention lead to
improvements of the worksheets used in the ensuing intervention. The first
intervention was a trial with five learners and served to gain confidence with the
approach. The second intervention was carried out in a half-size class (22-25 learners,
grade 10). A formative evaluation showed that there were obstacles: the practicability
was still insufficient (some learners had no experience with scissors, making some
activities cumbersome and some models imprecise), and the efficiency towards
learners’ understanding needed fine-tuning (learners had a lower level of
understanding of cones than anticipated). The first problem was resolved by deleting
scissor exercises and adding pre-cut shapes; the second problem was addressed by
adding more tasks on comparing different cones.
The final intervention in both the geometry and the statistics study took place in a
crowded classroom, typical of African educational contexts (n=55-60), with groups
of three to five learners. The observations were recorded in field notes and
photographs. Each intervention was concluded with semi-structured interviews with
randomly selected learners. The interviews were audio taped and transcribed. In the
Evaluation phase, the instructional materials, the observation reports and the
interviews were summatively evaluated in light of the research question.
RESULTS
We ended up with a rich database, of which we can only present a small selection in
this paper. The research question asked: to what extent can authentic resources be a
starting point for assisting learners in the effective formation of concepts? The
interventions showed us, that the authentic resources in themselves did not directly
ask for mathematical activities. However, these resources were useful as curtain
raisers in the instructional design. The resources ignited interest and created a link
between extra-institutional experiences and mathematical content.
For example, the university students had already studied many newspaper articles,
but this had not helped them develop underlying statistical concepts. Now, the
worksheets asked them to think beyond a newspaper phrase, for example: “20% of
all women have been victim of sexual harassment during childhood”. Students were
asked to interpret this phrase and compare probabilities on different samples. This
lead to the discussion on dependent probabilities and on a required randomness of
samples:

“If there are five sisters, and some men in the family are a problem, then you can have
(that) all were abused.” (Observations on group work, Worksheet 1, Statistics study)
An example of the discussion on sample size:
“You cannot just put any five women together, and say: one of them was harassed.
Maybe not one of them was harassed. Or maybe all were harassed. It is an average, so
maybe if you take all women in Maputo, then one in five is harassed. But you will not
know which ones.” (Observations on group work, Worksheet 1, Statistics study)
The newspaper articles were rich in statistical resources. They enabled us to design
questions that made the learners grapple with underlying statistical concepts. The
authenticity of the themes triggered students’ motivation by revealing how statistics
matters for their professional future:
Student B (from the Interviews, Statistics study): The first year Statistics course, it was
limited to doing calculations, using formulas and very little interpretation. But these
exercises (points at the worksheets) are more involving, because we are studying Social
Sciences, and not Engineering or Economics. These exercises are more important than
the classical ones, they give a better opening and more understanding.”
In the geometry study, we observed a teacher holding up a miniature fish trap in front
of the blackboard, surprising the (urban) learners with their (rural) cultural heritage:
Episode 3 (from School B, lesson 2, Geometry study):
19. Teacher: Do you know the name of this art craft? Don't you?
20. Students: [in chorus] Noo....o.
21. Teacher: What is the purpose of this traditional object? In certain areas of the
country this thing is used as a fish trap. Or as a trap to catch rats. This is
called... fish trap.
22. Students: [in chorus] Fish trap.
24. Teacher: Trap, fish trap, just like this one [he shows another model of a fish trap].
This is also a trap for fishing.
22. Students: [in chorus] … for fishing.
The only mathematical activity that the fish trap model asked for was classifying (it is
a cone), but not for further concept formation. However, the worksheets introduced
cut-outs making learners relate two-dimensional and three-dimensional shapes and
discover rules (e.g. a larger sector yields a lower cone). A principal discovery for
many learners was the differences between the height of a cone, and the slanting
height (along the lateral surface).
Figure 2: Two circle sectors leading to different cones

One of the exercises made learners discuss how the two cones constituting a
traditional fish trap are interrelated, concluded by a multiple-choice exercise, on
which sectors could together make a fish trap (Figure 3). With the models given in
their hands, learners discussed intensively, holding the cones top-down, folding and
opening the circle sectors again and again. Despite the intensity of discussions, the
exercise was only resolved correctly by 60% of the groups.
Figure 3: Multiple-choice exercise: choose two circle sectors that together can be
folded into a traditional fish trap (answer: B and E)
In the instructional design we had orchestrated separate components, such as the local
Mozambican resources together with the group work, the open-ended questions and
so forth. The interventions showed us that these components together changed the
classroom dynamics. Learners could discuss mathematical concepts in their own
words because they sat in groups. But the group work would not have functioned as
vividly if it were not for the open-ended questions. The open-ended questions
triggered interest because their topics were linked to extra-institutional experiences
and made available through familiar resources.
As a result, the learning ecology changed in many aspects. The groups of learners sat
together, working on the tasks from the worksheet. It was the worksheet that
instructed them, not the teacher. Thus, the teacher became an outsider of learners’
activities. The customary class activities, in which learners follow and copy what the
teacher demonstrates, were changed as the learners were assembling each others
contributions within the groups. Here, the mode of dialogue changed, because
learners had to explain to each other. The following excerpt shows that it was not
always easy to exchange ideas within the groups:
Episode 30 (from the group interviews, Geometry Study):
23. Student A: We saw, we had different ways, but with the same destination. So, one
with an opinion, another with an opinion,…..we continued to discuss, and
then in the end we saw that the destination was the same!
24. Researcher: (..) Is there more? Yes, please.
25. Student B: We spoke of the same thing but with different words. There was some
confusion when we wanted to say things…we said one thing…they used a
different word to say the same thing. So it was difficult the discussion.
CONCLUSIONS
In this report we have presented a design research in mathematics education in
Mozambique, in which shared principles were applied in two completely different

settings, respectively at junior secondary and at tertiary level. Our starting point
demonstrated that modern Mozambican society is rich in resources, to such an extent
that it provides mathematics education with many applications. However, the
resources lead to concept formation only in conjunction with a number of
instructional design principles (starting mathematics from the applications;
worksheets with open-ended tasks for group work; many illustrations). The design
conveyed a learning environment, in which the central role of the teacher reduced.
Whole-class lecturing could be largely avoided, and discussion among learners
enhanced. Pivotal in the design was the worksheet, which decentralised classroom
communication and facilitated group work. This effect did not automatically emerge
from the use of traditional art craft or newspaper clippings.
The interventions in this research were atypical and small-scale, yet successful in the
Mozambican context of (over-)crowded classrooms. The combination of authentic
resources as a starting point for concept formation and instructional design principles
proved robust. These research findings could strengthen the Mozambican policies
that advocate educational innovations towards more student-centeredness.
Nevertheless, the large-scale enactment of these policies still has a long way to go.
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