KS1 and KS2 Maths – setting rich activities

Does the way you set tasks in maths lead to rich activity, or are children just scratching the surface, asks Mike Askew?

I’ve brought in a friend to meet you today. Her name is Polly and…she’s a parrot. The thing is, Polly is a mathematical genius. She can do any calculation you give her. Any calculation at all. Who would like to give Polly a calculation to do?”

So begins one of my favourite maths lessons with five to seven-year-olds. (I’ve not had enough of the bottle to try with older kids.) OK, Polly is a glove puppet, and my ventriloquism skills are worse than amateurish, but the children are always happy to suspend disbelief.

• “Ten and ten?” asks a child.
• “Five,” says Polly.
• There are murmurs from the class.
• “Three times four.?”
• “Five.”
• The murmurs get louder.
• “A hundred plus a hundred?”
• “Five.”

Murmurs give way to shrieks of laughter.

Having milked this for as long as possible, I ‘fess up.

“OK, it’s not true. Polly isn’t a mathematical genius. In fact she only knows one mathematical word – five. Now here’s what I’d like you to do. Go off, work with a partner if you like, and make up a maths test for Polly that will make her look like a genius.”

Yes, it’s the old ‘the answer is five, make up some questions’ task, but given this – albeit imaginary and playful – purpose, the children make up much richer calculations through their desire to make the dumb bird look smart.

Task and activity Tasks are central to mathematics lessons. But I want to distinguish between task – what teachers set for children to do – and activity; the subsequent mathematics in which children engage to carry out the task.

If children are going to engage in rich mathematics, there needs to be a certain ‘gap’ between the task a teacher sets and the children’s subsequent activity. In the case of ‘Polly’, the task draws children into richer mathematical activity than if they were simply asked to come up with a number of questions to which the answer is five.

As another example, think about two different ways that children might be introduced to place value.

Place value 1

The teacher puts up a selection of two-digit numbers on the board and gets children to show each number using base ten blocks, e.g. 25 is modelled with two ten sticks and five ones. She points out to the children how the two in 25 corresponds to the number of tens and the five to the number of ones. In the second part of the lesson, the children complete a worksheet that has a table of two-digit numbers and columns to record the number of tens and number of ones.

Place value 2

The teacher sets up the scenario of selling fruit at a market. Bowls of fruit are put out that each contain 10 pieces of fruit. A number of two-digit numbers on the board represent how many pieces of fruit are in larger boxes, e.g. 25 apples. Working in pairs, the children figure out how many full bowls of 10 pieces of fruit can be made from the larger boxes, and how many pieces will be left over. In the second part of the lesson, the teacher puts children’s solutions up on the whiteboard, and holds a class discussion: what do pupils notice about how many items of fruit there are in total, the number of bowls that can be made from these, and the number of leftovers?

I suggest that the second place value lesson is based around a more ‘rich’ mathematical experience than the first. There is a problemsolving aspect to the lesson and, based on their solutions to the problems, children are encouraged to reason through the number of groups of 10 that can be made from two-digit numbers. Through that reasoning, they can make the connection between organising quantities into groups of tens (and ones) and the place value notation that we use. It is important that children develop this strong understanding of the relationships between quantities and notation.

In the first lesson, although base ten blocks are used initially to model the relationship between quantity and notation, children quickly move to an activity which is focused primarily on manipulating symbols and is less likely to lead to deep understanding of place value. Filling in a worksheet on place values is still useful in helping children become fluent in writing down the number of tens or ones in a numeral, but we must not confuse being fluent at writing ‘35 is three tens and five ones’ as an indication of understanding what that means. For example, after the second lesson, children may be able to figure out there are 12 tens in 125, whereas as the learning from the first lesson is likely to lead them to say there are only two tens in 125.

Rich tasks or rich mathematical activity?

Some tasks have the potential to promote more ‘rich’ mathematical activity than others. By rich, I mean engaging learners in real mathematical activity, particularly reasoning and problem-solving. Some writers prefer to talk about ‘rich tasks’, but this invites an interpretation of rich tasks as having to be elaborate or complicated.

Tasks themselves do not make a lesson rich in mathematics, but how they are carried out makes a big difference. Consider the following two similar tasks:

• Using each of the digits 1, 2, 3, 4, and any operations, make some different numbers.
• Using each of the digits 1, 2, 3, 4 and any operations, can you make all the numbers from 1 to 20?

At first glance, the second version may appear more ‘rich’ as it contains the specific challenge of making a complete set of numbers. The first version is vague; children may think they’ve finished when they’ve done one or two, and there’s no clear challenge involved.

But imagine the first version gets played out along the following lines (and I’m grateful to colleague, Mike Ollerton, for introducing this version to me). After the children have had a minute or two, the teacher stops them and collects on the board some of the different numbers they have made. The class spends a few minutes talking about any numbers that more

than one child has managed to make using the digits, and whether there is more than one way to make any of them. For example, nine could have been found from 21 – (3 x 4) or from (3 x 4) - 1 - 2. The teacher suggests that, later on, children might choose a number and investigate how many different ways they can think of to make it.

Looking at the numbers that have been made, the teacher points out that some of these numbers are consecutive: 9, 10, 11. Someone has found 13, so if 12 can be found, then a string of five consecutive numbers would have been made. She challenges the children, individually or in pairs, to create the longest string of consecutive numbers, still As a staff, choose a mathematical topic and find – from a textbook or the Internet – two different tasks each intended to teach an aspect of that topic (you could prepare a selection of these or ask colleagues to seek some out). Working together in small groups, colleagues imagine being the learner having to complete each activity. Questions to discuss include: making each number from the digits 1, 2, 3 and 4.

What are the advantages of this second way of introducing the challenge? First, the activity has a low threshold – all children can begin the activity. By allowing children to work on the activity for only a few minutes, the teacher has the opportunity, by sharing the initial answers, to subtly advance the children’s thinking. For example, some may have not thought to put the digits together to make two-digit numbers. In drawing the children’s attention to strings of consecutive numbers, there is more chance they will become engaged and curious about producing these, in a way that simply being asked to find all the numbers from 1 to 20 is less likely to do.

Introduced in this way, the activity also has a high ceiling - children can choose where to start with their string of consecutive numbers and some may be provoked to continue over several days to extend their string. If a child gets stuck on one string of consecutive numbers, then they are free to work on a different string of numbers. And everyone succeeds, whether your individual string is five numbers long or 25. On the other hand, if you get stuck on finding the numbers from one to 20 then there’s nowhere else to go. The ‘richness’ of tasks is lies not in the tasks themselves, but in when and how they are worked on. We don’t need loads more problems for children to work through, there are more than enough out there. We do need to think how to work with tasks and children to promote rich activity.