‘Can do’ Maths – why effort trumps ability

Getting stuck on a maths problem causes some children to give up, but we need to help learners shift from thinking ‘I can’t do this’ to ‘I can’t do this yet,’ says Mike Askew...

I blame Miss Nixon for my disliking history. (I’ve not changed names to protect the guilty.) My first year in secondary school was a long time ago, but I still remember my arm aching from her hourlong dictations given while patrolling the room, pointing out incorrect spellings and tutting over poor penmanship.

To this day I avoid reading history, but I don’t believe I’m missing the ‘history gene’. Instead, I believe that Miss Nixon spectacularly failed to engender in me a productive disposition towards history: a disposition that would have encouraged me to see history as an engaging, meaningful discipline, believe in my ability to learn it and regard the subject as something that could enhance my life. Historians are made, not born. So are mathematicians.

In this article, I argue that we have a duty to help primary school children develop productive dispositions towards mathematics, so they in turn will see themselves as able to learn maths and - even if they do not wish to study the subject beyond school – will at least not regard it as a waste of time.

Productive dispositions

As I discussed in the first article in this series (Right from the start, issue 6.1, tinyurl.com/tprightstart) ‘productive disposition’ is a core proficiency in mathematics, originally defined as an ‘habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy’ (Adding it up, National Academy Press).

Let me unpack this definition.

‘Habitual inclination’ echoes what I have argued above, that, like any habit, we are not born with a productive disposition. It grows through practice.

But not any old practice will do. While core fluency skills (for example, knowing your multiplication bonds or being able to use a ruler accurately) will benefit from short bursts of practice, the habitual practices that support developing a productive disposition are those that involve ‘sensible, useful and worthwhile’ mathematics. And that comes through problem solving and reasoning, as I ’ve argued throughout these articles.

A ‘belief in diligence and one’s own efficacy’ is influenced by how we think and talk about mathematical ‘ability’ and in particular how we understand the interplay between ability and effort.

FAST AND SLOW THINKING

The psychologist Daniel Kahneman in his recent book, Fast and Slow Thinking, presents a wealth of evidence that our minds operate on two systems: one which is fast and intuitive and a second that is slower and deliberate.

Fast thinking, Kahneman argues, is clustered around intuition, creativity, and is linked to being in a good mood. Slow thinking is clustered around analytic reasoning, increased effort, and results in a mood of sadness. He illustrates this difference with a number of examples from mathematics. If his thesis is correct (and his evidence is compelling) this raises all sorts of questions about the ‘mood’ of mathematics classrooms, and how rather than trying to make mathematics ‘fun’ we perhaps need to be open with learners that the satisfaction comes from the hard thinking.

Effort, not ability

In the UK we often talk about mathematical ability, especially when accounting for why some children are more successful in learning school maths. Across the globe, however, in places like Japan, Singapore and Korea, the emphasis is much more on effort than ability. Teachers there believe and expect that most children can succeed in the mathematics they are expected to cover by the end of primary school. (Children who display a particular aptitude for the subject develop this further at secondary school.)

One drawback with a focus on ability (or its sister ‘potential’) is the accompanying metaphorical assumption of ‘amount’. Learners have a lot (high) ability or not very much (low) and this is largely ‘fixed’. This may be sound in describing physical ability – no matter how hard I train there will always be a limit to how high I can jump – but is that necessarily the same with stretching my mental maths muscles?

The American psychologist, Carole Dweck, has devoted her lifetime’s work to looking at how viewing ability as ‘given’ and ‘fixed’ impacts on attainment and has consistently found negative results: when learners who view their ability as ‘fixed’ find themselves hitting a mathematical wall – meeting fractions or algebra being the point at which this happens for many – they interpret this as an indication they have reached their mathematical limits.

Dweck’s work has also shown that, in contrast, fostering a ‘growth’ view of mathematical ability can have dramatically different effects, encouraging further and deeper learning.

Dweck’s recommends nurturing a growth view by praising learners more for the efforts they put into doing the mathematics, than for how much they get right. We need to help learners shift from thinking ‘I can’t do this’ to ‘I can’t do this yet’; to encourage, in all learners, a ‘can do’ attitude. Developing an ‘I can’t do this yet’ disposition means being comfortable with getting stuck on some mathematics. This runs counter to many children’s experiences in mathematics lessons where a measure of being good at maths is how quickly you can get to the answer.

It is important to talk with the children about how it is OK to be stuck, and to share strategies for dealing with this that do not involve going immediately to the teacher.

One class drew up a list of things to do if you are stuck that included:

>Draw a picture or diagram
• Try and make the problem simpler (change the numbers)
• Look in a maths dictionary
• Ask a friend
• Read a book for five minutes

They made a poster of these suggestions, which they referred to in lessons. Part of engendering a ‘can do’ attitude means being open with children about how learning some maths is challenging and will take effort, rather than teaching it in such a way that each small step seems effortless. Such a smallincrements- little-effort approach can backfire – when effort is required, learners again misinterpret that as indicating the limits of their mathematical ability.

Collective benefits

An individual learner’s beliefs and efforts are important in developing a productive disposition, but research is showing that how we identify with groups can also have a powerful impact. For example, research with secondary aged learners randomly split Asian girls going into a maths test into two matched ability groups: one group subtly had their attention tuned into the fact that they were all Asian, the other group into the fact that they were all girls. The girls in the former group scored better on the test than those in the latter.

The researchers argue that this was because the girls attuned to their ‘Asian-ness’ enacted the Asians-are-betterat- maths stereotype while the ‘girls’ group enacted the girlsare- not-so-good-at-maths stereotype of (for which, incidentally, there is no evidence).

There are similar findings for other groups and although these studies involve older learners, it is reasonable to think the same could apply to primary pupils. Developing a ‘can-do’ disposition towards mathematics depends on seeing yourself as a member a ‘can-do’ group. It also relies on teachers having high expectations of everyone and not labelling children.

These research findings indicate the importance of creating a sense of ‘mathematicians’ being a group that learners want to be a part of. Positive images of mathematicians are important, and this may mean sharing stories of mathematicians that counter-act the negative geek, boffin or madman images presented in the media. It also points to the importance of creating a positive classroom culture where everyone is viewed as being a mathematician.

Becoming mathematicians

Talking about children becoming readers, or becoming writers, is commonplace in primary schools: talking about them becoming mathematicians is less common. We talk instead about learning maths, suggesting that the learner and the mathematics are separate things: the mathematics is ‘out there’ and the learner has to ‘take it on board’. Separating the learner and mathematics can reinforce seeing maths as a bitter pill.

We need instead to shift our attention to what mathematicians do. And what do mathematicians do? They solve problems, look for patterns, reason through answers and take delight in what they create through these processes.

WORKING WITH COLLEAGUES
PREPARE A NUMBER OF ‘STRONG POSITION’ STATEMENTS, FOR EXAMPLE:

• ‘Many people are just not cut out to learn mathematics.’
• ‘Most mathematics is not interesting to most people.’
• ‘The only way for children to learn mathematics is to be in ability groups.’
• ‘Most of the mathematics curriculum could be removed, an no one would suffer.’

Working in pairs, one person selects one of these statements and decides whether to support the statement or argue against it. It does not matter whether or not they actually agree with the statement – they just need to take a strong position. The activity is usually more enjoyable when arguing for a position you do not agree with.

Their partner, for a few minutes, interviews them about the statement – why they agree or don’t agree with it, and what evidence they have for their position. (You are allowed to make this up – the point is to be playful with the ideas.) Pairs swap over, the person interviewing taking up the statement and adopting the exact opposite position to the one their partner has just taken.

Once pairs have worked with a few statements, discuss the statements as a whole group. The point is not to reach a definitive answer, but to explore how many of our assumptions are based on beliefs that may not be substantiated. And as beliefs they can be changed.