How withholding answers can help children rise to the challenge

More challenging work might put a downer on your maths lessons, but resist the temptation to cheer everyone up by revealing the solutions.

Learning mathematics is not a single set of skills. You could, for example, teach children how to carry out arithmetical calculations, but you’d be hard pushed to describe this as a mathematical education. A more balanced view is provided by an American publication called Adding it up, which has influenced many education models – including our own. Four of the five strands it identifies are now encapsulated in England’s new National Curriculum for mathematics: fluency, conceptual understanding (incorporated in fluency), reasoning and problem solving. The fifth, however, ‘productive disposition’, represents something rather different, and is notable by its absence.

The the authors of Adding it up¹ define productive disposition as an ‘habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.’ (p. 116). But if such a disposition is ‘habitual’ – in other words, learners have to develop it – how do we achieve that? I think there are two parts to answering this question. First we have to offer pupils the chance to work on challenging tasks. Second, we then have to help them develop the perseverance to continue working on such tasks, even if progress is not immediately apparent.

For me, these go hand in hand – if pupils don’t get challenged with mathematical tasks to which the answers are not immediately obvious, they won’t ever develop the habits of perseverance. On the other hand, if a pupil does not have some sense of what it feels like to persevere and ultimately succeed, then she will not rise to the demands of a challenging task and, instead, give up or wait to be helped. The sweet spot we need to achieve is where pupils are engaged in what some call ‘productive struggle’.

Engaging pupils in productive struggle, however, seems easier said than done. A number of research studies show that even when teachers have the best of intentions, once a lesson gets under way they are quick to remove the challenge from tasks by showing pupils what to do. The question is, if mathematical challenge is an important part of learning, why do we feel compelled to remove it? The work of the Nobel Prize winning psychologist Daniel Kahneman sheds light on what might be happening here.

Fast and slow thinking

Kahneman presents a wealth of evidence suggesting that our minds operate on two systems: one which is fast and intuitive (System 1, in Kahneman’s terms) and a second that is slower and deliberate (System 2). The Bat and Ball problem below illustrates this difference. As you read through it, see if an answer rapidly pops into your head.

Together a bat and ball cost £1.10
The bat costs £1 more than the ball.
How much does the ball cost?

If the answer ‘10p’ popped into your head, that’s your fast, System 1 thinking at work, which Kahneman argues is clustered around intuition, creativity (and gullibility!). In fact, 10p is not the right answer – after all, if the ball is 10p, then the bat must be £1.10, and so together they would cost £1.20. Figuring out the answer requires a shift into System 2 thinking, which is analytic and requires increased effort.

Kahneman’s theory is that System 1 thinking is linked to being in a good mood, while System 2 thinking provokes a feeling of slight sadness. If his thesis is correct (and I find his evidence compelling) then this suggests why the ‘mood’ of mathematics classrooms can change when pupils are presented with a challenge.

I know from my own teaching that things can be going swimmingly, but if I introduce a challenge the energy feels as though it has been sucked out of the room. It’s not a good feeling and I have to resist the temptation to help learners in the expectation that this will restore a good mood. While my helping might do that, the cost is that I remove the responsibility for thinking from the learners.

Expecting learners to shift into System 2 thinking and the difficulties that accompany this may explain why lessons that start out as challenging can often turn out to be quite routine.

Getting the classroom culture right

If we are to encourage learners to stick with challenging tasks, then it is not enough simply to present a challenge: the classroom climate has to encourage learners to persevere. Factors that have been shown to make a difference include:

• Teacher enthusiasm for the task.
• Taking time at the beginning of the lesson to set up the task. This does not mean telling learners how to succeed, but perhaps doing a simpler task that tunes learners into what the task is about – and letting children know in advance that the main challenge will be tricky, so they shouldn’t give up too quickly!
• Making it clear there will be many different ways to approach the task – which is fine!
• Giving learners time to engage with the task.
• Encouraging them to work in pairs and support each other.
• Bringing the class together after a short while to share ideas and initial progress.
• Making sure there is time at the end for pupils to share strategies.
• Summing up at the end of the lesson what opportunities for learning the challenge presented.

Challenge children to make connections

Challenges where pupils are provided with incomplete statements of equivalence require them to go beyond simply carrying out a calculation. These provide learning opportunities for thinking about the connections between addition and subtraction, and multiplication and division.

One way of setting up such challenges is to introduce the scenario of calculations that have been printed on a printer that’s running out of ink. Can the pupils help figure out what the missing numbers might be? In all cases, encourage pupils to find at least two different solutions.

• 20 + [ ] = 40 + [ ]
• 50 - [ ] = 75 - [ ]
• 20 + [  ] = [  ] + 5
• 50 - [  ] = [  ] - 12
• 12 x [  ] = 24 x [  ]
• 32 x [  ] = [  ] x 4
• 12 ÷ [  ] = 24 ÷ [  ]
• 32 ÷ [  ] = [  ] ÷ 4

Encourage learners to work systematically

Rather than setting money problems where the items bought are listed and the total cost has to be found, provide situations where multiple buys of two items are made and the total cost is given. The challenge is to find out how many of each item has been bought. Such problems provide opportunities for learning about working systematically. For example:

Sam went shopping for some t-shirts and shorts. She found some t-shirts that were £4 each and some shorts for £9 each. Altogether, Sam spent £48. How many t-shirts and pairs of shorts did she buy?

Lawan bought a pick-and-mix bag of chews and jellies. The chews were £1.80 each and the jellies £1.40. Lawan spent exactly £20. How many chews and how many jellies did he buy?

Be flexible with fractions

When finding fractions of quantities, a typical problem would be to find, say, 1/4 of 12. Changing this around to find the total quantity, given some fractional information, adds to the challenge. It can help children learn to work flexibly with fractions, and how to link their understanding of fractions with division.

Ruksana bought a box of kiwi fruits. She gave her friend 6 kiwi fruits. This was a quarter of the total number of kiwi fruits in the box. How many kiwi fruits were in the box that Ruksana bought?

Craig bought a bag of cherries. On the way home, he ate 1/4 of the cherries in the bag. When he got home, Craig counted 12 cherries left in the bag. How many cherries were in the bag to begin with?

Counting sheep

Finally, here’s a challenge that can be offered to learners of many different ages and levels of attainment. For some, the challenge will lie in setting up a model of the situation and working systematically. For others it might involve extending the problem and asking ‘What if?’. (E.g. ‘What if there were a different number of sheep ahead of Eric?’ Or ‘What if Eric jumped over three sheep each time?’.)

Eric the sheep is in line to be shorn. There are 50 sheep in the queue ahead of Eric. Being a bit impatient, every time a sheep at the head of the queue moves aside after being shorn, Eric jumps over the two sheep immediately in front of him. How many sheep will be shorn before it is Eric’s turn?

¹ Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press. Available to download free of charge fromnap.edu

About the author

Mike Askew is professor of mathematics education at the University of Witswatersrand, Johannesburg as well as freelance, teacher, researcher and writer about primary mathematics (mikeaskew.net).

Helping children to persevere