ArticlePDF Available

Problem-centred teaching and modelling as bridges to the 21st century in primary school mathematics classrooms

Problem-centred teaching and modelling as bridges to the 21st century in primary school
mathematics classrooms
P. Biccard D.C.J. Wessels
Stellenbosch University Stellenbosch University
Moving mathematics classrooms away from traditional teaching is essential for preparing
students for the 21
century. Rote learning of decontextualised rules and procedures as
emphasized in traditional curricula and teaching approaches have proven to be unsuitable for the
development of higher order thinking. The ‘dream’ is to have skills (that employer seek for the
century) such as being able to make sense of complex systems or working within diverse
teams on projects [2: p. 316] fostered in mathematics classrooms, even at a primary school level.
In this paper it will be shown that the problem solving perspective that modelling emphasizes
includes competencies and skills that are essential in developing authentic mathematical thinking
and understanding. Results of a study on modelling competencies [1] will be presented to
highlight the growth of a problem solving mode of thinking. We will therefore explain that
modelling achieves important aims for mathematics education in 21
century. Modelling fosters
students’ abilities to
actualise existing (but not yet explicit) knowledge and intuitions; to make
inventions; to make sense and assign meanings; and to interact mathematically [10: p. 176],
thereby developing authentic mathematical thinking. The aim is to provide a perspective that
shows how modelling meets the challenge of changing mathematics classrooms.
Students need to learn mathematics with understanding since ‘things learned with understanding
can be used flexibly, adapted to new situations, and used to learn new things. Things learned
with understanding are the most useful things to know in a changing and unpredictable world’
[11: p.1]. Adaptability and flexibility in using mathematical knowledge is particularly important
when students solve contextual problems. Mathematical problem solving has many faces and
requires definition. Schroeder and Lester’s [3: p. 32, 33] three main descriptions of problem
solving are used for this paper. In a traditional sense, problem solving means solving ‘word’
problems as an extension of routine computational exercises. This can be seen as teaching for
problem solving – teaching of procedures takes place first and then problems specifically related
to the taught concepts are solved. In some progressive programs, students are taught about
problem solving and are taught to employ various methods or heuristics as options when faced
with a problem (e.g. drawing a table or graph etc). When students learn via or through problem
solving, problems are used to teach important mathematical concepts. When students interact
with modelling problems, they solve the problems in their own way with mental tools that they
already have available to them. The teacher facilitates by connecting different ideas that allow
students more sophisticated understandings through these connections. It is by solving problems
first and then building by connections between student ideas and representations that students
become adaptable and flexible and move toward a problem solving mode of thinking. Modelling
allows students to learn via problem solving and can be appreciated as a significant mathematics
teaching and learning opportunity.
The Problem-centred approach and modelling
Hiebert, Carpenter, Fennema, Fuson, Human, Murray, Olivier & Wearne [4] build a problem-
centred approach on Dewey’s principles of reflective inquiry. They work from the assumption
that understanding is the goal of mathematics education. Students solve problems at the outset of
a mathematics lesson and the process of solving, collaborating, negotiating and sharing leaves
‘behind important residue’ [4: p. 18]. As expressed by Human [5: p. 303] problems are used as
vehicles for developing mathematical knowledge and proficiency together with teacher-led social
interaction and classroom discourse.
The problems are opportunities for students explore
mathematics and come up with reasonable methods for solutions [11: p. 8]. The role of the
teacher and the student changes which means that the classroom takes on a different culture. It is
in the problem-centred classroom culture where the real benefits to student learning lie. Students
have regular opportunity to discuss, evaluate, explain, and justify their interpretations and
solutions [6: p. 6]. This can only be achieved if the teacher allows this discussion to take place
without presenting or demonstrating set procedures to solve problems. It is this change in focus
in the classroom away from ‘teacher thoughts’ to ‘student thoughts’ [13: p. 5] that epitomizes
student-centred methods such as the problem-centred approach and modelling. The classroom
culture now takes on what Brousseau [7: p. 30] termed an ‘adidactical situation’. The teacher, in
an adidactical situation, does not attempt to tell the students all. Brousseau also explains that the
‘devolution’ [7: p. 230] of a problem is fundamental in adidactical situations. This happens when
the teacher provokes adaptation in the students by the choice of problems put to them. The
problems must also be such that the student accepts them and wants to solve them. The teacher
refrains from suggesting the knowledge, methods or procedures he/she is expecting or wanting to
see. The teacher seeks to transfer part or all of the responsibility of solving the problem on the
student. It is this adaptation to the adidactical situation that allows students to learn
meaningfully. It is this adaptation to a problem-centred approach that allows a growth of
mathematical understanding, adaptability and flexibility. This in turn promotes a problem
solving mode of thinking that is so necessary in the 21
century workplace.
The problem-centred approach in mathematics education allows us to understand modelling and
its place in effective mathematics teaching and learning. Modelling goes beyond problem solving
in that the important questions of when and why problems are solved as well as whose thoughts
ideas and constructs are used when solving problems. In defining what a problem solving
capacity or mode of thinking entails, the constructs of [10] are used. When students solve
problems, they should be provided with opportunities to: actualise existing (but not yet explicit)
knowledge and intuitions; make inventions; make sense and assign meanings, and interact
mathematically [10: p. 176]. These four constructs encompass what it means to solve problem
with understanding and flexibility. It is often difficult for teachers to elicit existing knowledge
from students since there is an array of different understandings and levels of thinking in a single
classroom. Modelling allows students to verbalise their current ways of thinking and improve on
these ways of thinking. In making inventions, students are able to use their own ways of thinking
in constructing a response to the problem. Modelling tasks allow students to produce meaningful
solutions that keep the context of the problem in sight. While students work collaboratively on
modelling tasks they do make sense and assign meaning since they have to communicate their
thinking and ideas while interacting mathematically with each other in order to make progress.
These four constructs underline what it means to develop a problem solving mode of thinking
since they encompass student understanding, adaptability and flexibility in solving problems. It
also underlines what students need to learn meaningful mathematics in the 21
Mathematical modelling goes beyond problem solving since students ‘create a system of
relationships’ [12: p. 110] from the given situation that can be generalised and reused. Although
students are solving problems when modelling, a modelling approach means that students must
display a wider and deeper understanding of the problem. Modelling goes beyond problem
solving because students structure and control the problem – not only solve it. The aim of this
paper is to show that modelling tasks allow students and teachers access to significant problem
solving that bridges student understanding and student problem solving abilities. The
development of a problem solving mode of thinking that results from student involvement with
modelling tasks is presented in this paper. Problem solving and modelling problems specifically
hold a reciprocal interdisciplinary relationship with other knowledge fields. Modelling problems
for mathematics classrooms are applicable to and can be sourced from fields outside
mathematics such as engineering, architecture, commerce and medicine to name a few.
The study
The main study [1] investigated the development of modelling competencies in grade 7 students
working in groups. Partial results will be presented in this paper. Twelve grade 7 students were
selected to work in three groups of four students in each group. The results of only one of the
groups working on the first (of three) task are presented in this paper. This group comprised
students whose mathematics results in a traditional setting the previous year were considered
“weak”. The groups solved three model-eliciting tasks over a period of 12 weeks in weekly
sessions of about one hour. The results from this group’s discussions around the task – Big Foot
is presented.
Task 1: Big Foot taken from [9: p. 123].
Example of footprint (size 24) given to students. Groups had to find the height/size of this
person and also provide a ‘toolkit’ on how to find anyone’s height/size from their footprint.
Supporting material: rulers, tape measures, calculators
Table 1: Task Instruction for Big Foot
Students solved and presented their solution as a group with minimal teacher/researcher
intervention. These students had not been exposed to a problem-centred approach nor had they
solved modelling tasks before. Each group presented their solution to the other groups and
students were encouraged to question each other’s models. The contact sessions were audio
recorded and transcribed. Transcriptions were coded for each competency for the main study and
coded again for the results presented in this paper. The competencies identified for the main
study were: understanding, simplifying, mathematising, working mathematically, interpreting,
validating, presenting, using informal knowledge, planning and monitoring, a sense of direction,
student beliefs and arguing. How students develop and refine a problem solving mode of
thinking is highlighted in this paper. The four constructs [10: p. 176] were used to code the data
from the transcriptions and to structure the discussion in the next section. This assisted in
establishing to what extent students working in groups were engaging a problem-centred
paradigm when solving model-eliciting problems.
The results presented are from the group’s solutions processes for Task 1: Big Foot. This was
their very first modelling task so it exhibits the impact modelling tasks have on students thinking.
Furthermore it highlights the mathematical learning opportunities that are implicit in a modelling
task. In the transcripts R stands for researcher.
Actualising existing knowledge and intuitions
This group had an intuitive idea that there was a universal foot to height ratio although this took
place in the second session. The first session was taken up by a seeming avoidance of
mathematising the task. Once they had decided to take action on their own intuitions they were
successful in producing a model for this task.
M: Ok wait, why don’t I take my foot and divide it by my height, times a 100
R: why do you times it by 100?
M: Because that is how you find your percentage so we can find out ...I am saying that when we
do his (Big Foot) then we must get the same...
The students introduced the idea of a percentage on their own accord, but it later transpired that
they used the ‘times by 100’ to remove the decimal number that resulted from their division.
This group also had an intuitive idea that Big Foot had to be very tall and they were able to use
this to interpret and validate their progress which allowed them to make progress in their solution
process. They had taken a number of measurements including across their hips which they called
their ‘width’.
M: 58 (a group member’s height) divided by 2 is 28. Similar to his width - they measured 27 as
this person’s ‘width’.)
N: So he is 30 inches!
M: No he can’t be, that’s too short.
If student do bring their own ideas and constructs forth and they act on these ideas it is clear to
see how ‘making inventions’ is possible. This would not be possible if students are offered
methods or procedures by the teacher.
Making inventions
This group ‘invented’ their model to assist them in resolving Big Foot.
M: I divided my feet to my height and I timesed it by 100 and I got 15.
N: Yes...
M: So now I have to try get 16 times by what to get 15 again because it is a human. Then that
will be right.
On their presentation sheet (see Fig 1) they had written:
The solution is to take his foot size and divide it by an estimated number, multiply that by 100.
The result should be 15-20.
Although they did not see a connection here between multiplying and dividing (surprising for
this year of their schooling), they invented a way around this of ‘estimating’ the multiplicand so
that the result would be 16. Once this group had found the foot length to height ratio of all their
group members, they had four different (although very close) ratios. They then realized they
needed more data and tried more people. After trying three more people they found that 16
seemed to be a common ratio. Although they never used the term ‘mode’, this is a construct that
they ‘invented’ by understanding that they needed this from their set of data. When questioned:
M: R is 16 and N is 16. Then we must use 16.
R: why did you decide that N and I have the right measurements?
M: Because you are the most.
Making sense and assigning meaning
After calculating that 15 was one of the group member’s foot/height ratio, they continued to
work through the rest of the group, other people in the room as well as continuing this at home
and with other students at school. They were clearly in control of this ‘method’ or model
although it was not an elegant approach it was meaningful and they were able to assign meaning
to other areas of the model.
M: Divided by (known height) and times 100 and let hope it equals something nearby 15 and 20.
N: 62 divided by 11, ag no sorry the other way; 11 divided by 62 times 100 is
M: Yes I told you. 17. So I equal 15 and you 17.
M: OK it (the quotient) might be 15, 16 or 17. So he (Big Foot) might be: 98, 97 or 96.
When looking at their presentation sheet- they understood that if the ratio was 15, then their
estimated height was too short, or it the ratio was 19, then their estimated height was too tall.
They were able to assign meaning to a fairly complicated model which is surprising since they
achieved lower mathematics results than average in a traditional setting.
Fig 1: Group presentation sheet
Interacting mathematically
The following excerpt from the transcripts for this group shows how one group member explains
a fairly inelegant yet complicated model for Big Foot to the other members.
M: Look, I take your foot (length) right; the foot is 12 (inches), then I divide it by any estimated
number, like I will take, a number will come in my head and I will divide it (by the foot length)
and then multiply by 100. Probably (the result will be) over 20 or below 15. If it’s below 15, it
means the person is taller, if it’s over 20 it means the person is a bit shorter. Then you estimate a
bit lower until you get 15, 16 or 17.
The confluence of the problem-centred environment and modelling tasks present mathematics
education with a ‘developmental space’ for the learning of essential, meaningful mathematics [1:
p. 37].
The data presented in [6] suggested that a problem-centered instructional approach in which the
teacher and students engage in discourse that has mathematical meaning as its theme is feasible
in the public school classroom [6: p. 25]. The results of [8] suggest that a problem-centered
approach together with a change in teacher beliefs is a viable for reforming mathematics
classrooms. Furthermore, a modelling approach assists in developing student competencies in
problem solving, modelling and mathematics. Modelling tasks present an arena for teaching and
learning that assists teachers in understanding a problem-centred approach and to simultaneously
apply these principles in teaching. Modelling tasks can be used successfully by teachers and
students unfamiliar to problem solving or a problem-centred approach.
[1] Biccard, P. 2010. An investigation into the development of mathematical modelling competencies in
Grade 7 learners. Unpublished MEd dissertation. Stellenbosch University.
[2] Lesh, R., Yoon, C. & Zawojewski, J. 2007. John Dewey revisited - making mathematics practical
versus making practice mathematical. In Lesh, R., Hamilton, E. & Kaput, J.J. (eds). Foundations
for the Future in Mathematics Education. Mahwah:New Jersey. Lawrence Erlbaum Associates.
[3] Schroeder, T.L. & Lester, F.K. 1989. Developing understanding in mathematics via problem
solving. In Trafton, P.R. & Schulte, A.P. (eds). New directions in elementary school
mathematics. NCTM, Reston, VA.
[4] Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., Wearne, D.
1996. Problem solving as a basis for reform in curriculum and instruction: the case of
mathematics. In Educational Researcher. 25(4): 12-21.
[5] Human, P. 2009. Leer deur probleemoplossing in wiskundeonderwys (Learning via problem solving
in mathematics education). In Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie.
28(4): 303-318.
[6] Cobb, P., Yackel, E., Nicholls,J., Wheatley, G., Trigatti, B. & Perlwitz, M. 1991. Assessment of a
problem-centred second-grade mathematics project. In Journal for Research in Mathematics
Education. 22(1): 3-29.
[7] Brousseau, G. 1997. Theory of Didactical Situations in Mathematics. Didatique des Mathematiques,
1970-1990. Netherlands: Kluwer Academic Publishers.
[8] Wood, T. & Sellers, P. 1997. Deepening the analysis: longitudinal assessment of a problem-centered
mathematics program. In Journal for Research in Mathematics Education. 28(2): 163-186.
[9] Lesh, R., Hoover, M. & Kelly, A. 1992. Equity, Assessment, and Thinking Mathematically: Principles
for the Design of Model-Eliciting Activities. In I. Wirszup & R. Streit, (Eds.), Developments in
School Mathematics Around the World. Vol 3. Proceedings of the Third UCSMP International
Conference on Mathematics Education October 30-November 1, 1991. 104-129. NCTM: Reston.
[10] Murray, H., Olivier, A. & Human, P. 1998. Learning through problem solving. In A. Olivier & K.
Newstead (Eds.) Proceedings of the Twenty-second International Conference for the Psychology
of Mathematics Education.1:169-185. Stellenbosch, South Africa.
[11] Hiebert, J., Carpenter, T.P., Fennema, E., Fusion, K.C., Wearne, D., Murray, H., Olivier, A. &
Human, P. 2000. Making Sense: Teaching and Learning Mathematics with Understanding.
Portsmith: Heinemann.
[12] Doerr, H.M. & English L.D. 2003. A modeling perspective on students’ mathematical reasoning
about data. In Journal for Research in Mathematics Education. 34(2): 110-126.
[13] Petersen, N.J. 2005. Measuring the gap between knowledge and teaching: the development of the
mathematics teaching profile. Paper presented at the Michigan Association of Teacher Educators
Conference. October 28-29, 2005. Saginaw: Michigan.
... Recently PISA has become a benchmark for education in Indonesia as the Ministry of Education and Culture of the Republic of Indonesia states that to face challenges in the 21st century, the results of the PISA, TIMSS, and INAP tests are indicators of the success of education in Indonesia [2]. The role that mathematics can play to prepare students to compete in the 21st century can achieve by learning that connects with daily life problems [3][4][5]. Based on this reason to support students meeting the needs that exist in the 21st century in learning mathematics, that is by learning mathematics using real-life contexts or daily life problems and consider how mathematical assessment on an international scale used. ...
Full-text available
This research is a descriptive study that aims to describe the role of the solution plan as a scaffolding that helps students to solve mathematical modeling tasks. The subjects in this study were 32 students of grade VIII of Junior High School number 3 Ngaglik. Data collecting using observations and documents. The data collected in this study is in the form of field notes and students’ worksheets. During the learning process, students are given a worksheet about modeling mathematics and equipped with a solution plan, and then the researcher makes observations. The study results found that the solution plan can be used as a scaffolding to support students solve mathematical modeling problems, where the solution plan consists of four stages: understanding tasks, searching mathematics, using mathematics, and explaining results. The solution plan facilitates connecting students’ imagination with the context of the problem and connecting prior knowledge needed to solve mathematical modeling problems through questions or statements that do not directly lead to answers.
... Research (Biccard & Wessels, 2011) indicates that problem-focused approaches give students the opportunity to explore mathematics for themselves and offer sensible solutions. All attention in the classroom is focused on the problem to be solved, as a result of which there is an active discussion both among the students and between the teacher and the students. ...
... Mathematical modeling, in which unstructured problems are used in its process [65], can also be used with the same purpose. In addition to the problem solving activities, as mathematical modelling activities enable students not only solve a problem but also configure the problem and check, help students more than problem solving [28,63]. That is, while even a simple problem solving experience has four steps of modelling process, mathematical modelling problems consist of multi-modelling processes, in short [74]. ...
Full-text available
The aim of the present study is to investigate pre-service secondary mathematics teachers’ cognitive-metacognitive behaviours during the mathematical problem-solving process considering class level. The study, in which the case study methodology was employed, was carried out with eight pre-service mathematics teachers, enrolled at a university in Erzincan, Turkey. We collected data via think aloud protocols in which metacognitive supports are provided and not provided and with unstructured metacognitive protocols related to the process. The metacognitive support provided to the pre-service teachers and this case caused an increase in the numbers of cognitive-metacognitive behaviours in class level in general. Any pattern was not encountered for the sequences of cognitive-metacognitive behaviours in the process of mathematical modelling during the stages in which metacognitive support was given and was not given but some little patterns were detected. It was noticed that there was no relationship with respect to the metacognitive support and without metacognitive support between and frequencies of cognitive-metacognitive behaviour mathematical modelling progress. The 1st and 2nd grade-pre-service teachers claimed that the metacognitive support contributed to them such factors as understanding and detailing, the 3rd and 4th graders expressed that it enabled them to follow, reach alternative ways of solution. The participants, during the stage in which the metacognitive support was provided, demonstrated more success during the process of mathematical modelling compared with the first stage.
In this age of “high” technology and “cold” touch, the role of teachers is growing. Teachers develop and change students’ lives, paving the way for lifelong learning and career success. Remembering the knowledge and skills acquired at school, students continue to draw strength from the support and love provided by teachers. The influence of teachers is usually deep and lasting. Creative and purposeful teachers engage and influence students, families, other teachers, school leaders and local communities. Even when most students studied at home during the Covid-19 pandemic, the issue of the quality of education remained the responsibility of teachers. All teachers have the same intention to help their students achieve their goals and succeed. The work of teachers is responsible and full of great challenges, as their needs and conditions are dictated by the students, the school, the local government, the state and even emergencies. Surveys of students conducted in 2021 and interviews with Latvian teachers of mathematics in focus group interviews indicated critical problems and revealed a worrying picture in the acquisition of mathematics in general education schools during the last two years in connection with distance online learning. In focus group interviews, teachers indicated that, given the background of the Covid-19 pandemic, they needed significant support from both local governments and policy makers to maintain emotional balance affected by technological progress, and from academics and scientists to understand the conditions and modalities how to learn math smarter, faster and more efficiently in the future. This article seeks answers to the question: How can we support those who help shape the future? The article presents the results of student surveys and teacher focus group interviews. As a solution to the problem, a framework model developed in cooperation with math teachers is proposed, using design thinking approaches and techniques. The model will further help to create a support system for technology-enhanced accelerated learning of mathematics, as well as provide innovative character and promote strategic use for STEM industries.KeywordsAI4MathTeacher support systemAccelerated learning of mathematicsInterdisciplinary approachesTechnology-enhanced learning
Full-text available
Three forms of mathematics education at school level are distinguished: direct expository teaching with an emphasis on procedures, with the expectation that learners will at some later stage make logical and functional sense of what they have learnt and practised (the prevalent form), mathematically rigorous teaching in terms of fundamental mathematical concepts, as in the so-called “modern mathematics” programmes of the sixties, teaching and learning in the context of engaging with meaningful problems and focused both on learning to become good problem solvers (teaching for problem solving) andutilising problems as vehicles for the development of mathematical knowledge andproficiency by learners (problem-centred learning), in conjunction with substantialteacher-led social interaction and mathematical discourse in classrooms.Direct expository teaching of mathematical procedures dominated in school systems after World War II, and was augmented by the “modern mathematics” movement in the period 1960-1970. The latter was experienced as a major failure, and was soon abandoned. Persistent poor outcomes of direct expository procedural teaching of mathematics for the majority of learners, as are still being experienced in South Africa, triggered a world-wide movement promoting teaching mathematics for and via problem solving in the seventies and eighties of the previous century. This movement took the form of a variety of curriculum experiments in which problem solving was the dominant classroom activity, mainly in the USA, Netherlands, France and South Africa. While initially focusing on basic arithmetic (computation with whole numbers) and elementary calculus, the problem-solving movement started to address other mathematical topics (for example, elementary statistics, algebra, differential equations) around the turn of the century. The movement also spread rapidly to other countries, including Japan, Singapore and Australia. Parallel with the problem-solving movement, over the last twenty years, mathematics educators around the world started increasingly to appreciate the role of social interaction and mathematical discourse in classrooms, and to take into consideration the infl uence of the social, socio-mathematical and mathematical norms established in classrooms. This shift away from an emphasis on individualised instruction towards classroom practices characterised by rich and focused social interaction orchestrated by the teacher, became the second element, next to problem-solving, of what is now known as the “reform agenda”. Learning and teaching by means of problem-solving in a socially-interactive classroom, with a strong demand for conceptual understanding, is radically different from traditional expository teaching. However, contrary to commonly-held misunderstandings, it requires substantial teacher involvement. It also requires teachers to assume a much higher level of responsibility for the extent and quality of learning than that which teachers tended to assume traditionally. Over the last 10 years, teaching for and via problem solving has become entrenched in the national mathematics curriculum statements of many countries, and programs have been launched to induce and support teachers to implement it. Actual implementation of the “reform agenda” in classrooms is, however, still limited. The limited implementation is ascribed to a number of factors, including the failure of assessment practices to accommodate problem solving and higher levels of understanding that may be facilitated by teaching via problem solving, lack of clarity about what teaching for and via problem solving may actually mean in practice, and limited mathematical expertise of teachers. Some leading mathematics educators (for example, Schoenfeld, Stigler and Hiebert) believe that the reform agenda specifi es classroom practices that are fundamentally foreign to culturally embedded pedagogical traditions, and hence that adoption of the reform agenda will of necessity be slow and will require more substantial professional development and support programs than those currently provided to teachers in most countries.Notwithstanding the challenges posed by implementation, the movement towards infusing mathematics education with a pronounced emphasis on problem solving both as an outcome and as a vehicle for learning seems to be unabated. Substantial work on the development of more effective means for professional development and support of teachers is currently being done.
Full-text available
Ten second-grade classes participated in a year-long project in which instruction was generally compatible with a socioconstructivist theory of knowledge and recent recommendations of the National Council of Teachers of Mathematics. At the end of the school year, the 10 project classes were compared with 8 nonproject classes on a standardized achievement test and on instruments designed to assess students' computational proficiency and conceptual development in arithmetic, their personal goals in mathematics, and their beliefs about reasons for success in mathematics. The levels of computational performance were comparable, but there were qualitative differences in arithmetical algorithms used by students in the two groups. Project students had higher levels of conceptual understanding in mathematics; held stronger beliefs about the importance of understanding and collaborating; and attributed less importance to conforming to the solution methods of others, competitiveness, and task-extrinsic reasons for success. Responses to a questionnaire on pedagogical beliefs indicated that the project teachers' beliefs were more compatible with a socioconstructivist perspective than were those of their nonproject colleagues.
Full-text available
We argue that reform in curriculum and instruction should be based on allowing students to problematize the subject. Rather than mastering skills and applying them, students should be engaged in resolving problems. In mathematics, this principle fits under the umbrella of problem solving, but our interpretation is different from many problem-solving approaches. We first note that the history of problem solving in the curriculum has been infused with a distinction between acquiring knowledge and applying it. We then propose our alternative principle by building on John Dewey’s idea of “reflective inquiry,” argue that such an approach would facilitate students’ understanding, and compare our proposal with other views on the role of problem solving in the curriculum. We close by considering several common dichotomies that take on a different meaning from this perspective
Longitudinal analyses of the mathematical achievement and beliefs of 3 groups of elementary pupils are presented The groups consist of those students who had received 2 years of problem-centered mathematics instruction, those who had received 1 year, and those who had received textbook instruction. Comparisons are made for the groups using a standardized norm-referenced achievement test from first through fourth grade. Next, student comparisons are made using instruments developed to measure conceptual understanding of arithmetic and beliefs and motivation for learning mathematics. The results of the analyses indicate that after 2 years in problem-centered classes, students have significantly higher achievement on standardized achievement measures, better conceptual understanding, and more task-oriented beliefs for learning mathematics than do those in textbook instruction. In addition these differences remain after problem-centered students return to classes using textbook instruction. Comparisons of pupils in problem-centered classes for only 1 year reveal that after returning to textbook instruction, these students' mathematical achievement and beliefs are more similar to the textbook group. Also included are exploratory analyses of the pedagogical beliefs held by teachers before and after teaching in problem-centered classes, and those held by teachers in textbook classes. The results of the student and teacher analyses are interpreted in light of research on children's construction of nonstandard algorithms and the nature of classroom environments.
ENGLISH ABSTRACT: Mathematical modelling is becoming a popular teaching and learning approach in mathematics education. There is however a need within the modelling domain to identify exactly what modelling competencies are and how these competencies develop. This study examines how mathematical modelling competencies develop in Grade 7 students working in groups. Modelling is placed in the field of mathematics teaching and learning as a significant means of learning mathematics. Modelling competencies are identified and characterised from existing literature and explored through empirical generation and collection of data. The study is qualitative in nature and uses a mixed approach of design research and some aspects of grounded theory. Students’ progress through a modelling program is documented while the modelling competencies of students stereotyped as weak and strong are also investigated. The findings firmly support earlier research that competencies do develop in students who are exposed to modelling. A comprehensive picture of the modelling situation is presented since this study merges competencies from other studies into a detailed analysis of the modelling situation - it presents an authentic modelling situation of students working in groups and furthers the discussion on modelling competencies. The analysis of the data suggests that the development of modelling competencies is complex and interrelated but that competencies do develop progressively in groups involved in modelling tasks. Recommendations for additional studies include studies of a longer duration and a full investigation into the link between modelling and language ability. AFRIKAANSE OPSOMMING: Wiskundige modellering is besig om ‘n populêre onderrig- en studiebenadering in wiskundeonderwys te word. Daar is egter ‘n behoefte om die modelleringsbevoegdhede te identifiseer in hierdie veld en om te weet hoe hierdie bevoegdhede ontwikkel. Hierdie studie ondersoek watter bevoegdhede in wiskundige modellering by Gr.7 studente wat in groepe saamwerk ontwikkel. Modellering is in die studieveld van wiskundeonderrig en -leer geplaas as 'n betekenisvolle leerwyse in wiskunde. Modelleringsbevoegdhede word vanuit bestaande literatuur en navorsing geïdentifiseer en beskryf deur empiriese generering en versameling van data. Die studie is kwalitatief van aard en gebruik ‘n gemengde benadering van ontwikkelingsondersoek en sekere aspekte van begronde teorie. Studente se vordering in die modelleringsprogram is gedokumenteer terwyl die modelleringsbevoegdhede van gestereotipeerde swak en sterk studente ook ondersoek is. Die resultate bevestig vroeëre navorsing dat bevoegdhede ontwikkel word deur studente wat blootgestel is aan modellering. ‘n Omvattende beeld van die modelleringsituasie is in hierdie studie aangebied waardeur modelleringsbevoegdhede, soos geïdentifiseer in ander studies, tot ‘n gedetailleerde analise van die modelleringsituasie saamgevoeg word. Dit verteenwoordig dus ‘n outentieke modelleringsituasie van studente wat in groepe saamwerk en bevorder so die gesprek oor modelleringsbevoegdhede. Die analise van die data suggereer dat die ontwikkeling van modelleringsbevoegdhede kompleks en geïntegreerd is, en dat bevoegdhede progressief ontwikkel in groepe wat betrokke is by modelleringstake. Aanbevelings vir addissionele studies sluit langer ondersoektydperke in en 'n dieper ondersoek na die verband tussen modellering en taalvaardigheid. Thesis (MEd (Curriculum Studies))--University of Stellenbosch, 2010.
A modeling approach to the teaching and learning of mathematics shifts the focus of the learning activity from finding a solution to a particular problem to creating a system of relationships that is generalizable and reusable. In this article, we discuss the nature of a sequence of tasks that can be used to elicit the development of such systems by middle school students. We report the results of our research with these tasks at two levels. First, we present a detailed analysis of the mathematical reasoning development of one small group of students across the sequence of tasks. Second, we provide a macrolevel analysis of the diversity of thinking patterns identified on two of the problem tasks where we incorporate data from multiple groups of students. Student reasoning about the relationships between and among quantities and their application in related situations is discussed. The results suggest that students were able to create generalizable and reusable systems or models for selecting, ranking, and weighting data. Furthermore, the extent of variations in the approaches that students took suggests that there are multiple paths for the development of ideas about ranking data for decision making.
Equity, Assessment, and Thinking Mathematically: Principles for the Design of Model-Eliciting Activities
  • R Lesh
  • M Hoover
  • A Kelly
Lesh, R., Hoover, M. & Kelly, A. 1992. Equity, Assessment, and Thinking Mathematically: Principles for the Design of Model-Eliciting Activities. In I. Wirszup & R. Streit, (Eds.), Developments in School Mathematics Around the World. Vol 3. Proceedings of the Third UCSMP International Conference on Mathematics Education October 30-November 1, 1991. 104-129. NCTM: Reston.