I have emphasised before that frequent practice is necessary to give children more confidence in their ability to do mathematics. Those who do more mathematics every day are more likely to be successful in the subject than those who don’t.
I also made the point that this has implications for the planning of the teaching of mathematics. In this column I want to explain the reason for practice in a more -theoretical way. Consider a young child who is starting to learn about numbers. The child will, over time, know numbers in different ways. The earliest way is as words in a rhyme.
The child can rote count the number. Rote counting and learning the rhyme is important because it helps the child to establish the vocabulary associated with numbers. However, being able to rote count does not mean that the child has a sense of the “muchness” of a number or even the meaning of the words.
Parents of young children who can rote count are often delighted. They think of their children as “knowing” their numbers. This is not true. These children have no more sense of numbers than their parents know the meaning of the chorus of Auld Lang Syne that they sing so cheerfully every New Year’s Eve.
When the child starts to associate the words of the rhyme with objects, he or she is starting to count in a different way. We refer to this as rational counting. When we ask the child how many objects there are in a given pile, she will touch them, associating the words of the rhyme with the objects in the pile in sequence: one, two, three and so on.
The child will answer the question by counting the objects one, two, three and so on up to, say, 12 and repeat 12: “There are 12 objects.” If we repeat the question: “How many objects are there?” the child will in all likelihood start counting them again. Let us now fast forward a little to the point where we start calculating with numbers. Say we ask the child to calculate 5 + 7.
The child will, in all likelihood, count out five counters and count out another seven counters and will then count how many counters there are altogether, pronouncing that there are 12. What I am trying to illustrate is that the numbers five and seven at this stage in the child’s development still take their meaning from the process of counting out physical objects.
Think now about your own way of calculating 5 + 7. As an adult you no longer have to make the five or make the seven out of physical objects. You are able to calculate with the five and the seven, and many other numbers, for that matter, without having to think about the physical objects that gave the initial meaning to the words.
We say that the five and seven have become objects in their own right. They are no longer the result of the process of counting. The five and the seven that you calculate are objects to you in the same way that the physical objects are there for the child to calculate. Similarly, we can trace the development of other families of numbers.
At first, negative numbers are the result of subtraction. Minus 5 is the result of the process of subtracting 7 from 2. Fractions, at first, are the result of measuring and division. And so we can carry on, not only with numbers but also with almost all mathematical concepts. They start out as the result of an action — a process — on more primitive objects and with time they take on the status of an object themselves.
What is important in this discussion is to realise that we cannot go through our mathematical lives constantly giving meaning to a number by counting out objects. The number has to take on what I have referred to as object status. We have to help children to move beyond the more primitive “process” meaning of a mathematical idea to the more sophisticated “object” meaning.
Children initially and for a long time have to make the five and the seven and then suddenly there comes a moment when they start working with five and seven as objects. This sudden transition occurs only after they have done a great deal of work with the numbers produced by counting out objects in a variety of contexts.
Negative numbers become more than the result of subtraction and objects in their own right when children have calculated with negative numbers a great deal. Fractions become numbers in their own right only after children have worked with fractions as the result of measuring and division a great deal.
Children need a lot of practice. It is the teacher’s job to provide practice opportunities for children. Children develop mathematically by working with mathematical processes. It is the teacher’s role to provide the practice situations and opportunities. To be clear, what I have tried to emphasise is that practice serves at least two purposes.
We need to practise to get better. However, in the case of mathematics we also need to practise so that the mathematical ideas can mature from primitive processes to more sophisticated -mathematical objects.
Aarnout Brombacher is a private maths consultant. For more information go to www.brombaher.co.za