Maths, Mindsets and Mastery

If we are to really improve maths education, and the ability of our population to be able to use mathematics confidently and effectively in work and life, we need to overcome the idea that ‘you’re either good at maths or you are not’, and that you are somehow born with a fixed ability to do maths. This view of mathematical ability is very damaging, but very widely held. I used to hold it myself. Two things changed my mind. The first is anecdotal, based on my own teaching experiences; as an enthusiast for maths, I want my students to enjoy it too and I discovered that if you could change students’ attitudes to maths, helping them to view it positively, they could make surprising progress, as shown by the following example:

As a lecturer in an FE college in the 1990s I was involved in teaching maths to adult students on ‘Access’ courses. These students wished to go to university but had not achieved well enough in their school qualifications to qualify. They ranged in age from early 20s to 60s. All Access students without at least a grade C or above in GCSE Mathematics (or equivalent), whatever their proposed future degree course, had to follow a level 2 mathematics course. Passing this course counted as a GCSE Mathematics pass for their university applications. At the start of the course we invited the students to rank their confidence in mathematics from 1 – ‘terrified’, to 10 – ‘relaxed’. We then taught them in two groups, the less confident in one group, the more confident in another. We split them up in this way so that the less confident students would not be intimidated by the more confident; the curriculum was the same for both groups. I taught the less confident group, most of whom were female students. They were an impressive group of people, many of whom had shown great commitment to pursue their educational aspirations, giving up jobs and disrupting their domestic arrangements to do so. They were clearly ambitious and intelligent, so why were they so hung up about maths? What I discovered was that they believed they could not do maths, largely because they had picked up from their parents and/or teachers and/or peers, from an early age, that they were no good at it. A large part of my job was to reassure them that level 2 Maths is not a high bar, and that they could succeed. It turned out that the majority of them had been taught maths as a collection of facts to be memorised and that they had not engaged with it because it seemed to make no sense – they had been taught procedures without developing conceptual understanding. They assumed the reason they were not succeeding was that ‘they could not do maths’, a view they felt had been reinforced throughout their educational experience, so they believed there was no point in trying to understand it. Many, but not all, of them were numerate at the start of the course, in the sense of knowing their times tables and being able to use written methods for arithmetic, but they did not understand that maths ‘makes sense’ and is a powerful tool for reasoning. Instead, they viewed it as a set of apparently arbitrary rules. Once this issue was overcome, almost all went on to succeed on the course, performing as well as those in the ‘confident’ group. Many changed their view of maths completely – a particular memory I have is of discussing proofs of Pythagoras’s theorem with them – they had just assumed it was a ‘rule’; the idea that it was a universal truth and could be ‘proved’ in satisfying and elegant ways (‘beautiful’ was one student’s memorable comment – a lady who went on, I think, to study for a History degree) was a revelation to them.

The second is based on international evidence:

The Pisa and TIMMS data indicate that many other countries achieve far better results in maths education than we do, and I am convinced that this is due, at least in part, to their more positive attitudes towards the subject. When I visited schools in Shanghai it seemed clear that teachers and students believed that success in maths was down to hard work, not ‘cleverness’. They don’t label some children as ‘mathematically able’ and others as ‘mathematically weak’ and they certainly do not believe that ‘you can either do maths or you can’t’.

Mathematical ability is a continuum, and I believe that the limit of mathematical potential for the vast majority of people lies well beyond that required to achieve a level 2 pass at GCSE, or become competent in the maths needed to succeed in everyday life and most areas of work.

I have recently read Carol Dweck’s excellent book ‘Mindset’¹ and recommend it strongly to teachers. Its key message is a very important one for mathematics teaching (and for teaching in general). She describes two belief ‘mindsets’, a fixedmindset in which a person believes their intelligence is fixed and cannot be developed, and a growth mindset in which a person believes they can develop their intelligence. The word ‘intelligence’ can be replaced by beliefs about different abilities, such as ‘artistic talent’, ‘skill at playing tennis’ or, for the purposes of this blog, ‘maths ability’. A basic summary of Carol Dweck’s message is that people with a growth mindset believe they can change and improve and so will work to develop their abilities, making them more successful, whereas those with a fixed mindset believe their abilities are fixed, so there is no point in putting in effort to developing themselves and this limits their success. In order to become more successful, people should develop a growth mindset, believing that that can change and improve. As teachers of maths, we need to develop a growth mindset in relation to maths in our students, so that they believe they can improve and so will work to do so. However, in order to develop this mindset in our students we must overcome the insidious cultural fixed mindset ‘you’re either good at maths or you’re not’, and many of us may need to overcome that mindset ourselves. Research has shown that teachers with a fixed mindset in relation to maths ability are inclined to comfort students who are having difficulty with maths, rather than offering them strategies for improvement, and that this has a negative effect on students’ motivation².

‘Mastery’ or a ‘mastery curriculum’ in maths are hard terms to interpret, but at the NCETM we are starting to define what we mean by ‘mastery’. The new National Curriculum programmes of study at key stages 1 and 2 are organised by years, suggesting the maths that students should learn in each school year, but with the flexibility for schools to customise this for their own programmes of study within each key stage. The new National Curriculum document states:

‘The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.’

I would describe this is a ‘mastery curriculum’ because the emphasis is not on ‘getting through’ the material, but on developing fluency and understanding, and the focus of differentiation to challenge those making fast progress should be through deepening their understanding, rather than moving on to new content. Maths as a discipline continually builds on itself, so developing firm foundations through a mastery approach seems very sensible, and it’s how maths is taught in high-performing jurisdictions such as Shanghai and Singapore.

How does this relate to mindsets? Well, the idea is that all students are exposed to the same curriculum, largely at the same time and ‘mastery’ implies developing secure understanding, so if a student is struggling with some aspect of the curriculum, the way to help them is to offer strategies to improve their fluency and understanding, not to sympathise with them for not being good at maths. The curriculum encourages this approach – it talks about ‘pupils who grasp concepts rapidly’ and ‘those who are not sufficiently fluent’, not ‘those who are more mathematically able’ and ‘those who are mathematically weak’. The curriculum does not assume students have a predestined ability in mathematics, and students will be far more successful at mathematics if they, their parents and their teachers don’t make this assumption. Professor Anne Watson sums this up nicely for me. She does not refer to students having difficulties with maths as having ‘low ability’, she describes them as ‘previously under-achieving students’.

Of course, there are individual differences in maths ability, as there are in just about every other ability, but with a growth mindset and a mastery approach to the curriculum, I believe vastly more students can succeed in school level mathematics and not find themselves restricted by a belief that they ‘can’t do maths’.

References:

1. Mindset, Carol Dweck, 2006, ISBN 978-1-78033-200-0
2. “It’s OK – Not everyone can be good at math”: Instructors with an entity theory comfort (and demotivate) students, Rattan, Good and Dweck, Journal of Experimental Social Psychology, 2012

You can view Carol Dweck giving a lecture about her work on Mindsets on YouTube.