2. Identifying different types of problem
With any mathematical task or problem you set your pupils, there are ‘deep’ features – features that define the nature of the task, and strategies that might help solve it.
Almost all mathematics problems have these deep features, overlaid with a particular set of superficial features. As a teacher, you have to help your pupils understand that once they have recognised the superficial features, changing them does not have any effect on how we solve the problem. The strategies for solving a problem remain the same. (See Resource 2: Ways to help pupils solve problems.)
Case Study 2: The essence of the problem
Amma wrote this problem on the board:
In one family, there are two children: Charles is 8 and Osei is 4. What is the mean age of the children?
Some pupils immediately wanted to answer the question, but Amma told them that before they worked out the answer, she wanted them to look very closely at the question – at what kind of a question it was. Was there anything there she could change that would not alter the sum?
Some pupils realised that they could change the children’s names without changing the sum. Amma congratulated them.
She drew a simple sum on the board (1+1=2) and then said, ‘If I change the numbers here,’ (writing 2+5=7) ‘it is not the same sum, but it is still the same kind of sum. On our question about the mean, what could we change, but still have the same kind of sum?’
Some pupils suggested they could change the ages of the pupils as well as the names.
Then Amma asked, ‘Would it be a different kind of sum if we talked about cows instead?’
They kept talking in this way, until they realised that they could change the thing being considered, the number and the property of these things being counted, all without changing the kind of sum being done.
The pupils then began writing and answering as many different examples of this kind of sum as they could imagine.
Activity 2: What can change, what must stay the same?
Try this activity yourself first.
- Write the following question on your chalkboard: Mr Ogunlade is building a cement block wall along one side of his land to keep the goats out. He makes the wall 10 blocks high and 20 blocks long. How many blocks will he need in total?
- Ask your class to solve the problem.
- Check their answer.
- Next, ask your pupils in groups of four or five to discuss together the answer and what can be changed about the problem, yet still leave it essentially the same so it can be solved in the same way.
- Ask the groups to make up another example, essentially the same, so that the basic task is not changed.
- Swap their problem with another group and work out the answer.
- Do they have to solve this new problem in the same way?