3. Thinking about problems
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, make it hard
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.