Section 3

Ways to solve number problems

Introduction

Learning Outcomes

By the end of this section, you will have:

used strategies to explore pupils’ ways of solving mathematical problems;
distinguished between deep and superficial features of mathematical problems;
used techniques to develop thinking about thinking in your pupils.

Problem solving is an interesting way to develop your pupils’ mathematical thinking. Pupils have to work out what calculations need to be done before they can find the answer. This means sorting the information given to establish what it is they need to find out and how to do it. This will help them make explicit their mathematical thinking, and understand and recognise the deep features of a mathematical problem. You might find it useful to think of why problem solving is important. Some reasons are given in Resource 1: Why problem solving is important.

Page 1

‘Thinking about thinking’, or meta-cognition, is a powerful means for helping pupils understand and recognise the ‘deep’ features of particular kinds of problems, and how to solve such problems. The first step towards such thinking is to give pupils the opportunity to talk about the problems they are trying to solve and how they are trying to solve them. When pupils are explaining their thinking, it is important to listen and not dismiss any ideas. There are many different ways of solving mathematical problems (see Resource 1). You may be surprised at how many other ways pupils find, other than the way you may have expected them to use.

Case Study 1: Listening to pupils’ voices in mathematics

Nomonde in South Africa reminded her pupils that, when they go home from school, there isn’t only one way to get home: there are many possible ways. Some are shorter, some longer, some safer, some more interesting. She told them it was the same with mathematics problems – there is often more than one way to get to the right answer, and looking at the different ways might be interesting. Nomonde put the following questions on the board.

Sipho has 24 stones. He gives 9 stones to a friend. How many stones does he have left?
Thembeka eats 7 sweets every day. She has 42 sweets. For how many days does she have sweets?
The teacher buys 25 packets of crayons. There are 12 crayons in each pack. How many crayons does she have?

Next she asked the pupils to answer the questions using any method they liked. She gave her pupils ten minutes to answer the questions. She checked their answers and then asked one or two to explain how they worked out each question. Nomonde listed these methods to find the answers and made a note of which methods were most popular. She reminded her pupils about the different routes to school.

Activity 1: Helping pupils think

Try this activity yourself first, preferably with two or more colleagues. Then try it with your pupils. Ask your pupils to try to answer Nomonde’s three questions by working individually. Split the class into groups of four or five and ask them to take turns to explain carefully to each other how they worked out their answers. Next, ask the groups to make a list of the strategies used, then ask these questions:

Did you all have the same answer?
Did you all work it out in the same way?

How many different ways can your group find to work out a correct answer for each question? List these on the board. Explain how important it is to your pupils to try different ways to solve problems to help their mathematical thinking.

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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

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?

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Problem solving can be adapted so that every pupil can contribute. For example, all pupils can discuss what makes a problem easy or difficult to solve. It can be the variations in the superficial features – for example, using large numbers, decimals or fractions rather than small integers – that often make a problem harder to solve. Sometimes, setting a question in a ‘context’ can make it easier, but sometimes this can distract pupils from the deep features of the problem, so they may not easily see how they are meant to solve it. When pupils begin to see the deep features of a problem, they can also begin to ‘see through’ the superficial features, so they recognise the underlying task. Pupils can then confidently tackle any task with the same deep features. See Resource 2 for important factors for you to consider when setting and solving problems with your class.

Case Study 3: Make it easy

Agnes was working with her pupils on the topic of division. She wrote three division problems on the board:

Kofi has 12 oranges, and 3 children. If he shares the oranges equally, how many should each child get?
Divide 117 by 3.
Amma has 20 Gp for travelling to work. She spends 3 Gp each day on a taxi. One day, she doesn’t have enough money for the taxi. How many days has she travelled to work? On the day her money runs out, how much extra does she need for the taxi that day? You might like to use pretend paper coins based on real coins to help with this activity – see Resource 3: Coins of Ghana).

She asked pupils in groups of four to try to answer these problems together. After ten minutes, Agnes asked her pupils which problems were easier or harder to answer. Together they made two lists on the board – ‘things that make the problems hard’ and ‘things that make the problems easier’. Agnes asked the groups to find how many different ways they could solve the problems they had been given. She said she would reward the group that found the most ways by displaying a ‘maths champions’ certificate, with their names on it, on the classroom wall.

Key Activity: Pupils writing their own tasks

Make a list on the board of ‘things that make the problems hard’ and ‘things that make the problems easier’.
Ask your pupils, in groups, to write three questions of their own. They should make one question easy, one harder and one very hard.
After ten minutes, ask the groups to swap the problems they have written with another group and to solve the questions they have been given by the other group.
Ask the groups to report back. Were the ‘very hard’ questions really much harder than the ‘easy’ questions? What made the questions hard or easy? Revisit your lists on the board – is there anything pupils want to change or add now about making problems hard or easy?
Ask them to make up problems for homework related to their local community e.g. about the number of trees, the cost of a taxi.
Next day, share these in class and ask pupils to solve them.