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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

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