If you were asked to show that 1.7 is greater than 1.35, how would you go about doing it? You might start by drawing a number line to show that 1.7 is further along the number line that 1.35, but such an approach is unlikely to convince most 8-year-olds. From their point of view, the ‘35’ is greater than the ‘7’, and you or I saying that 1.35 is here on a number line and 1.7 is there isn’t going to come close to winning an argument with the conceptual model that they’ve been building in their heads for the last five years that says that 35 is more than 7.
To convince most 8-year-olds that 1.7 is greater than 1.35 requires showing them what such numbers look like, not on a number line but using something more tangible. For me, this involves repurposing base-10 (or Dienes) blocks, so that the numbers become something that they can not only see but also touch. The pictures below, which represent 1.7 and 1.35, use hundred squares to represent one whole, with the tens representing tenths and the ones or units representing hundredths. Such an approach to ‘making’ decimal numbers helps children to understand that the ‘35’ is not actually 35 at all, but 3 tenths and 5 hundredths.
All of this brings us, somewhat belatedly perhaps, to one of the fundamental tenets of maths mastery: abstract concepts are made much more tangible through the use of both concrete objects and visual representations. This is sometimes referred to as the ‘concrete-pictorial-abstract’ approach, or, in the language of the National Centre for Excellence in the Teaching of Mathematics (NCETM), representations of mathematical ideas are used ‘to expose mathematical structure.’ It is important to understand that such representations are not only meant to be used with younger children but should be used to help children of all ages to understand mathematical concepts.
From here, any effort to pin down what exactly is meant by maths mastery becomes trickier, for the simple reason that there is no single agreed upon definition of what people mean when they talk about maths mastery. The NCETM lists ‘Five Big Ideas in Teaching for Mastery’, the first of which is ‘Coherence’. I started my teaching career before mastery became part of the vernacular, and I like to think that even in my first few years of teaching that the lessons I was planning were coherent and progressed in a carefully considered way through the curriculum. Furthermore, I think there are plenty of teachers who don’t teach maths who would be keen to claim coherence as fundamental to what they do too.
One of the issues with defining maths mastery is that the focus is very much on how maths is taught rather than what maths is taught. It would be perfectly possible to take a textbook or workbook associated with a mastery scheme and use it in much the same way as a maths textbook might have been used twenty years ago, or one hundred years ago for that matter. What matters then is not the content of the curriculum but the way in which that content is delivered.
So rather than a lesson starting with ten minutes of teacher instruction, after which the children are asked to put into practice whatever the teacher has modelled, a mastery lesson is more likely to begin with an exploratory activity, with children given the opportunity to work in small groups to discuss the problem and explore different ways of reaching the solution, often making use of concrete materials or visual representations. This gives the teacher the opportunity to observe the children and assess their current level of understanding. The emphasis is not on finding the correct answer but on the processes involved, with the exploratory activity followed by a whole-class discussion led by the teacher that considers different approaches and addresses common misconceptions, with the children sharing and explaining their ideas.
Through this discussion and analysis of different approaches and perspectives, it is hoped that children will gain a deeper understanding of the mathematical concepts at work. It is important to recognise that for this to work in practice it is essential that there is a classroom culture in place that allows for mistakes to be made, welcomes mistakes even, so that children feel comfortable sharing their ideas and understand that making mistakes and learning from those mistakes is an integral part of the learning process.
For me then, the essential elements of a mastery approach to teaching maths include: the use of concrete or pictorial representations; responsive teaching; and exploratory tasks/structured discussions that develop children’s conceptual understanding. As I have said already, however, there is no universally accepted definition of maths mastery, so this is very much my own working definition. It is also important to note that the research evidence for the effectiveness of mastery approaches remains rather thin. If nothing else, though, mastery approaches have helped to shine a spotlight on maths teaching and learning. For too long in this country, in contrast to many other parts of the world, maths has been thought of as being the preserve of the few rather than the many. Mastery approaches are helping to shift mindsets about maths, with children – and their parents – recognising that everyone can do maths. What I hope is that a more open, more visual approach to teaching maths that prioritises depth of understanding will result in children becoming self-confident mathematicians who leave school having enjoyed learning maths and with a richer appreciation of the subject.