Mastery and understanding mathematics

In the context of the challenge to raise standards in mathematics in schools in England the word ‘mastery’ has recently become prominent in the vocabulary of the English mathematics curriculum (NCTEM, 2014, www.ncetm.org.uk/public/files/19990433). It is reassuring to note that the way in which the word ‘mastery’ is being used is entirely consistent with the approach to children’s learning of mathematics that I have promoted in my own writing.

Mastery is seen as children developing fluency in mathematics alongside a deep understanding of mathematical ideas and processes. So, for example, teaching approaches for mastery should ‘foster deep conceptual and procedural knowledge’ and ‘exercises are structured with great care to build deep conceptual knowledge alongside developing procedural fluency’ (op.cit.). This is a key principle in teaching mathematics to young children: that mastery of the subject is not achieved simply by repeated drill in various procedures. Instead, the focus is on the development of understanding of mathematical structures and on making connections.

Making connections in mathematics – a recurring theme in all my books – ensures that ‘what is learnt is sustained over time, and cuts down the time required to assimilate and master later concepts and techniques’ (op.cit.). Nearly all mathematical concepts and principles occur and can be applied in a wide range of contexts and situations. Because of this, the deeper understanding central to mastery in mathematics is facilitated by a wide variation in the experiences that embody mathematical ideas.

For example, mastery of the 5-times multiplication table by Year 2 children is not just a matter of memorizing a chant that begins ‘one five is five, two fives are ten …’ – although that is part of it. It would also involve, for example:

·       connecting each result in the table with a collection of 5p coins and the total value;

·       articulating the pattern of 5s and 0s in the units position in the odd and even multiples of 5;

·       explaining how to get from 4 fives to 8 fives by doubling;

·       explaining how to get from 6 fives to 7 fives by adding 5;

·       counting in steps of five along a counting stick;

·       knowing that, say, ‘3 fives are fifteen’ is what you use for the cost of 3 books at £5 each;

·       constructing patterns with linked cubes that show 1 set of five, 2 sets of five, and so on;

·       filling in the missing number in number sentences like ‘6 × □ = 30’.

To teach for this kind of mastery teachers themselves need a deep structural understanding of mathematics, an awareness of the range and variety of situations in which a mathematical concept or principle can be experienced, and confidence in exploring the connections that are always there to be made in understanding mathematics. Any teachers looking for this? I can recommend one or two books.