/ 18 August 2010

In my previous column, I made the point that, although we study mathematics to solve problems, it is not always obvious that we can also use problems to teach mathematics. I illustrated how well-chosen problems can provoke a child to perform mathematical procedures without necessarily knowing the mathematical term associated with the procedure.

This is a natural response to a given stimulus — the mathematical terms are merely man-made labels.

In this column, I want to use a series of division problems to illustrate how the teacher’s conscious act of making aspects of the problem more challenging over time can provoke children to use increasingly sophisticated ways of working — leading to what we know as the standard calculation algorithms.

Children will, of their own initiative, work in increasingly sophisticated ways provided they are provoked to do so. The illustrations accompanying this article show that what we think of as the standard long-division algorithm is no more than a condensed (sophisticated) way of doing what children do quite naturally.

The challenge for teachers is to manage children through the natural developmental stages so that they reach a point at which they are ready to use, in a meaningful way, the algorithms we associate with column arithmetic (addition, subtraction and multiplication) and long division.

Drawing one has been made by Sam, a young child (grade R or one), who is trying to solve this problem: Three friends want to share nine sweets equally between them. How must they do it?

Sam has drawn the three friends and the nine sweets and has allocated three sweets to each child. The drawing clearly shows all the elements of the story. The lines joining the sweets and the children illustrate the act of giving (sharing) the sweets.

In the next two drawings we observe a young boy, Ashwell, solving this problem: Four friends want to share 17 amagwinya (vetkoek) equally between them.

How must they do it? (There is quite deliberately an extra amagwinya in this problem — a remainder if you will).
In the first picture, we see that Ashwell is following exactly the same strategy that Sam used in the earlier example. Notice, however, that the lines linking the amagwinya and the friends are about to become quite jumbled and difficult to manage.
Ashwell has also noticed this and, in his second picture, we see how he has rubbed out his first attempt and adopted another approach that assigns each amagwinya to a child on a one-by-one basis until all the amagwinya have been used up.
At this stage, Ashwell has given the extra amagwinya to the first child.

What I want you to notice from Ashwell’s problem is that because the teacher has increased the values in the problem — in particular, the amount to be shared out — Sam’s method is no longer appropriate and Ashwell has developed a more sophisticated and appropriate approach to solving the problem.

Let us fast forward a year or two to a grade three class. Susan has solved the problem: Three friends want to share R81 equally between them. How must they do it?

Notice how, in this problem, the number of things to be divided among the three friends is now so large that to draw all of the R81 would take too long. Instead, Susan has become more efficient and is using numbers. Unlike Sam and Ashwell, she no longer draws the children, either — she simply writes a letter for the names of each of the three friends.

What this solution continues to have in common with Sam and Ashwell’s solutions is that we can still see the R81 that has been shared out and we can also see quite clearly that each child gets the same amount — R27.

Notice how, by increasing the number of things to be shared, the teacher has provoked the child into moving to a method that is more sophisticated than that of Ashwell yet still makes just as much sense.

Children have an enormous capacity for solving problems and developing methods that are appropriate to the situation.
What happens when both the number of objects to be shared (divided) among some children and the number of children have been increased to, say: 338 ÷ 13?

Because the divisor (the number of children) has become quite large, the child solving the problem might choose to draw only one child and not all 13 and the working might be shown alongside.

The child solving the problem has kept track of the number of objects to be shared and has stopped sharing when there is nothing more to share.

This solution is not that far from the thinking that underpins the long-division algorithm.

By means of a few simple illustrations of children’s work, I have tried to show you how a teacher can provoke children to develop increasingly sophisticated ways of solving a problem (in this case a division problem).

I have also shown how the child’s working continues to be related to the problem — throughout the solutions we can see all the elements of the problem.

The increasingly sophisticated approaches used by these children are typical of children as they develop. I like to say that they illustrate the developmental trajectory that most children will, in all likelihood, follow as they become increasingly confident and as the problems become more demanding; provoking the child to find a more efficient solution method.

The challenge for teachers is to be aware of the trajectory that children follow and to know how to develop problems that “pull” the children along the trajectory.

To make that last comment in a different way, teachers have a critical role to play to ensure that children move along the developmental trajectory — a path that shows both increasing sophistication in terms of thinking and increasing compression (more squashed into less space) in terms of writing.

Teachers must, however, have both a sense of the trajectory and the skill of asking the right questions that will encourage (provoke) the child to using increasingly sophisticated methods.

In closing, what I have tried to illustrate is that children can, over time and with appropriate management by their teachers, reach a point at which they perform standard algorithms with understanding and appreciation for what they are doing.

There is a great deal of well documented evidence to show that the alternative method of teaching these algorithms prematurely and without the child’s understanding causes so much damage to children’s experience of mathematics that few survive.

In my next column, I will illustrate how problems can be used to introduce the fraction concept.

Aarnout Brombacher is a private mathematics consultant