/ 14 October 2010

Why are we doing this?

In my columns so far, I have said that I believe so-called “maths anxiety” is not so much an anxiety or fear of the subject itself but rather a reaction to the sense of powerlessness children experience when they think about “doing mathematics”.

I have illustrated ways in which we can help children develop a strong sense of numbers — crucial if they are to feel confident when doing mathematics. I have also shown how learning mathematics by solving problems can help children to make more sense of what they are doing and feel more confident about the subject.

One of the most common questions children ask is: “Why are we doing this?” And the answer they get is often something to do with the topic being in the syllabus and/or the examination. Though this is true, this reason alone is a poor one for learning mathematics.

As I work with teachers across South Africa, my challenge to them is that they should have some sense of why the topic we are studying is important.

In the foundation phase we need to learn about the basic operations of mathematics because they are needed to solve the problems we face in day-to-day life. As I have illustrated in earlier columns, realistic and meaningful problems can provoke children into quite naturally performing actions that we describe (summarise) using the words addition, subtraction, multiplication and division.

The role of the teacher is, firstly, to set the problems that provoke children to behave in certain ways, and then to help them to develop increasingly sophisticated ways of doing what they do quite naturally.

In my previous column, I illustrated how particular sharing problems naturally give rise to parts of a whole and how we need conventions to name these parts of a whole — resulting in what we know to be fractions. Having “created” fractions, we then need to be able to perform operations with fractions. We need to add, subtract, multiply and divide with fractions. The need for these operations arises naturally from meaningful and realistic problems. The following problems are suitable for the end of the foundation phase and/or early intermediate phase.

I am convinced that any child who understands the meaning of the notation 1½, 1¼, and so on, can solve each of these problems. If the teacher has introduced the problems as interesting and important, creating the necessary excitement concerning them, then the children in the class are unlikely to ask the question: “Why are we doing this?”

Quickly try to solve the problems yourself. The first two can be solved by what we would call repeated addition or multiplication, whereas the third is actually a division problem. Did you notice something — you can solve these problems without knowing that they are repeated addition, multiplication and/or division problems. The problems provoke a natural reaction in you because it is possible to visualise what is happening.

Some readers may be wondering about the addition of unlike fractions, the role of equivalent fractions and the rules that we associate with the multiplication (multiply the tops and multiply the bottoms) and the division (turn upside-down and multiply) of fractions. Of course, we need to get to that and we will, but not in this column.

The point is not whether or not we can get there (because we can), the point is that the teacher needs to think about the problems she chooses very carefully.

Problems should not be chosen to keep children busy. Problems need to be chosen with the deliberate purpose of provoking a certain reaction from the child and that reaction is the mathematics that we are trying to develop.

Some readers may be arguing that the examples I use in these columns are all focused on the early years. There are two reasons for this: firstly, I am trying to keep the columns accessible to more readers and, secondly, the purpose of the problems is more transparent.

I now want to make a cautionary remark. You will have noticed how all the problems I have used are so-called real-life problems. This is because real-life problems are the basis of all the basic mathematics that we develop in the early years.

The caution I want to raise is that I have no expectation of our using real-life problems throughout the school years. There is no real-life analogy for negative numbers and there is certainly no real-life problem that will provoke an action in children that reveals that a negative times a negative is positive.

This is quite simply because negative numbers are an abstract mathematical construct — we invented negative numbers so that we could solve equations such as 5 + x = 3.

Operations with negative numbers are, however, necessary — if we can add, subtract, multiply and divide with positive numbers then we certainly want to be able to do so with negative numbers as well.

So how do we use problems to develop these operations? The answer is simply this: the question of what it means to add, subtract, multiply and divide with negative numbers is the problem. In other words, we say to children: “Now that we have these numbers we call negative numbers, we need to know how to operate with them. What do you think it means to add, subtract, multiply and divide with negative numbers?”

And the teacher needs to have a plan about how to help children answer the question. In my next column I will show how we do so.

The underlying message of today’s column is this: to enable children to see the purpose of what they are doing in class, teachers need to ask the question “why are we doing this?” long before they start the lesson and they need to structure their lesson so that the purpose is clear and the purpose almost always has to do with solving some or other problem.

In the early years, the problems are more close to real life in nature; in the later years, the problems become more abstract and mathematical in nature.

Aarnout Brombacher is a private maths consultant. For more information go to www.brombacher.co.za