Improving your numeracy

Teaching through problem-solving: an introduction
In this module we introduce teaching through problem-solving for South African teachers.

We begin our discussion with an activity so that we can build upon your own current experience and understanding.

Activity 1

Working with fractions is often a challenge for both children and adults. We will therefore explore the teaching of fractions as a practical introduction to this unit''. ''How would YOU teach children how to ‘add’ fractions? Think about it first, then jot down some notes on the lesson steps and activities you would follow.''

Now we will visit two classrooms and see how other teachers teach this concept. Read through the two case studies and then answer the questions that follow. Case study 1: Mr Ntombela

Case study 2: Ms Khumalo Ms Khumalo says:

In the last two lessons we talked about fractions. Can anybody remember what a fraction is? Can you give me some examples? In this lesson I want to see if we can use what we already know to solve a problem. We will work in pairs. Listen carefully to the problem: Mpho and Thabo were each given a bar of chocolate in their lunch box. On the way to school they decided to eat some of their chocolate. Now Mpho has only a third of her chocolate bar left and Thabo has only a sixth of his chocolate bar left. How much chocolate do they have left between them? Try drawing diagrams to show how much chocolate they have left. She provides several similar problems on task cards. She then asks one of the learners to explain how they solved the problems and invites other learners to ask questions and to propose alternatives. Only once she is convinced that the learners understand the concept does she get them to think about the more formal ‘quick’ way of doing it. The learners attempt two similar problems (fifths and tenths, quarters and eighths) and one more difficult example (thirds and quarters) for homework and for discussion in class the following day.

Activity 2 ''1)	Which of these two approaches is most like the way that you teach? 2)	Which of these two approaches do you prefer and why? 3)	Which of these two approaches allows for meaningful construction of ideas? Explain your answer.''

The first approach, that of proceeding directly to formal mathematics and the use of ‘rules’, has the advantage that it is quick and easy for the teacher and some students will be able to ‘read between the lines’ and answer similar questions correctly.

A disadvantage of the first approach is that if learners misunderstand or misapply the rule, they will usually not realise they have made a mistake and may not be able to think their way through to a correct solution. This approach will often lead to errors like the following: $$1/3+2/6 = 3/9$$ The second approach has the disadvantage that it is initially more time consuming, both in terms of planning outside the classroom and the time taken to complete and discuss tasks inside the classroom. In addition, the fact that the teacher has not specified any rules to follow means that she must be able to cope with a variety of divergent, and sometimes erroneous, thinking. An advantage of this approach is that learners get to talk about and explore the issues in a meaningful way. With support and guidance from their peers and the teacher, they should be able to reason their way through the underpinning principles and not only avoid the kind of error outlined above but also be better able to think their way through more irregular examples. They should also need less drilling. The second approach focuses on problem-solving as a way of developing understanding of the concepts involved.

What then is problem solving? At the outset it is necessary to draw a distinction between problem solving and the doing of routine exercises. Nicholson (1992) explains: •	 In problem-solving one finds the solution to a particular situation by a means which was not immediately obvious. •	A problem-solving task is one that that engages the learners in thinking about and developing the important mathematics they need to learn. This can be contrasted with the traditional or stereotypical approach to teaching in which teachers explain a rule, provide an example and then drill the learners on similar examples.

Problem-solving has been described by many authors and researchers (e.g. Nicholson: 1992) as the essence of mathematics, and yet many learners spend most of their time on routine exercises. It must be stressed that whether something is a problem or not is dependent on the level of sophistication of the problem solver. A learner in grade 8 may be required to solve a problem in which the method and solution are not obvious, and yet the same problem given to an older child may be quite routine.

Hiebert et al (1997) have brought the problem solving approach for the teaching and learning of mathematics with understanding to the fore when they state:

We believe that if we want students to understand mathematics, it is more helpful to think of understanding as something that results from solving problems, rather than something we can teach directly.

Problem solving has been espoused as a goal in mathematics education since late 1970, with focused attention arising from NCTM’s ‘An Agenda for Action’ (Campbell & Boamberger: 1990).

However, problem solving should be more than a slogan offered for its appeal and widespread acceptance – it should be a cornerstone of mathematics curriculum and instruction, fostering the development of mathematical knowledge and a chance to apply and connect previously constructed mathematical understanding.

This perception of problem solving is presented in the Revised National Curriculum Statement for R-9 (schools) (Department of Education – Mathematics: 2001,16-18). Problem solving should be a primary goal of all mathematics instruction and an integral part of all mathematical activity. Learners should use problem-solving approaches to investigate and understand mathematical content.

A significant proportion of human progress can be attributed to the unique ability of people to solve problems. Not only is problem-solving a critical activity in human progress and even in survival itself, it is also an extremely interesting activity. Many pastimes such as games, puzzles and contests are in fact enjoyable tests of problem-solving abilities (Bel: 1982). The Cockroft Report (Nicholson: 1992) states that ‘The ability is to solve problems is at the heart of mathematics.’