/ 7 March 2011

Investigating the issue

In my previous column I made the point that starting the year with an investigation would reveal the problem-solving strategies, the attitudes and even the mathematical knowledge and skills of the children in your class. In many ways, learning mathematics is one big investigation.

In this column we will conduct the investigation in detail.

What is the greatest product that can be made from the numbers that add up to a given number?

At the heart of being able to investigate is to understand what you are being asked to do. To some children, the instruction provided above may seem quite difficult. Do the children know the mathematical meaning of the word product? I recommend that you start out by posing the problem and asking the children if they can figure out what is being asked and then asking them to explain their understanding to the class.

You could tighten your request by saying: “Can you try to work out what this means by thinking about the number eight?”

By asking children first to try to figure things out for themselves, you are telling them that you have faith in their abilities. This confidence motivates children to try. You are also signalling that you are not going to tell them things that they can work out for themselves. Children quickly learn to sit back and wait for the teacher to “do it”.

The way you phrase the instruction to the task/investigation should be age-appropriate. The way I have phrased this task is appropriate for intermediate phase children who have been exposed to the word “product”. In the case of younger children you might pose the task by means of an actual example.

Having given children a chance to make sense of the problem, it is time to discuss what they have worked out. You should mediate a discussion that reveals the following meaning for the number eight:

8=5+3 and 5×3=15

As much as an investigation can introduce important mathematics, an investigation can also be about developing investigating and problem-solving techniques and strategies.

The solution itself does not have to matter. It is the process that is valuable. You will find that, the more you use investigations in class, the more you stimulate the curiosity and interest of the children.

In the case of eight, having clarified what is meant by the instruction, you can now ask questions such as: “Is there a larger product?”

Chances are good that somebody in the class will have realised that:

Now, be careful. Do not rush away from eight too fast. There is much to be gained from looking at the solutions that did not yield large products. For example:

This reveals quite clearly that using one does not give a very large product.

As the teacher, you should continually ask questions such as: “Have you noticed any patterns?” and “Can you see a rule?”

Now it is time to ask: “What about the numbers nine and 10?”

I have deliberately asked about two numbers. As children find a solution to one they will have noticed some patterns and trends and will want to find out if they apply to other numbers as well.

Given enough time — and this is a critical comment — many children will have found that the largest product they can make for nine is 24 and for 10 is 32.

Why is “given enough time” so important? In a study of a large number of Japanese and American classrooms, researchers found that American teachers typically pose a question and wait a very short time before they answer the question themselves. By contrast, Japanese teachers gave the children ample time to work on the problem and then expected them to explain what they found.

Having established that the largest product for 9 is 24 and for 10 is 32, it is again important to ask if the children noticed any “patterns” or “rules”. Have they noticed anything that is making it easier to find the largest product? This discussion helps those who are still struggling to learn from their classmates. It also forces those doing the explaining to reflect on what they have done and to refine their understanding and approach.

The kinds of observations we might expect to hear are children explaining that using one is a bad idea and using larger numbers also does not seem to work as in 7 + 3 = 10 and 7 × 3 = 21. The optimal solution seems to be one in which a lot of twos, threes and fours are used. The possibility exists that some children will explain that using 2 + 2 or 4 is exactly the same, since 2 × 2 = 4.

After this discussion, you would now ask the class to apply their observations to 11 and 12, seeing if they can establish the largest product a little faster this time. In the case of 11, they will find that: 2 × 2 × 2 × 2 × 3 = 48 and in the case of 12 that: 2 × 2 × 2 × 2 × 2 ×2 = 64.

When do you stop an investigation? In the case of grades 4 and 5 we may have reached a point where we have done enough. The children will have investigated, identified patterns and — in case you did not notice — done a lot of arithmetic! In the case of older children, you may now extend the investigation by asking: “Is it possible to predict what the largest product will be?” The point is that the children have by now developed a strategy that they can use to find the largest product for any number, but using this strategy every time can become tedious. As mathematicians, we want to see if we can see a pattern in the pattern and start predicting what will happen.

For now I am going to leave it there. I hope you will continue the investigation. Should you get stuck and want a hint or want to tell me what you and your class found, feel free to email me at: [email protected].

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