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

Authors:
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
pbiccard@yahoo.com dirkwessels@sun.ac.za
Abstract
Moving mathematics classrooms away from traditional teaching is essential for preparing
students for the 21
st
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
21
st
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
st
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.
Introduction
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
st
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
st
century.
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.
Results
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.
Conclusion
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.
References
[1] Biccard, P. 2010. An investigation into the development of mathematical modelling competencies in
Grade 7 learners. Unpublished MEd dissertation. Stellenbosch University.
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for the Future in Mathematics Education. Mahwah:New Jersey. Lawrence Erlbaum Associates.
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[3] Schroeder, T.L. & Lester, F.K. 1989. Developing understanding in mathematics via problem
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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.
Article
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.