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.

"By George, she's got it!"

These are the words of Professor Higgins in My Fair Lady when Eliza shows significant progress in her learning! We use a lot of different idioms to indicate that we understand something or do not understand something. For example, for understanding, we might say things like:

• Oh, I see! 
• Now it’s clicked. 
• Everything is falling into place. 
• I think I've cottoned on! 
• I get the picture. 
• My eyes have been opened! 

And for not understanding, we have phrases like these:

• Sorry, I don’t get it. 
• I’m still in the dark. 
• I can’t see the sense in that! 
• That's as clear as mud! 
• You're talking double Dutch! 
• It just went over my head. 

What's your favourite way of saying 'I understand' or 'I do not understand'?

Expressions like those above reveal a number of insights into the nature of understanding. They show, for example, the importance to us of understanding things, of making sense of them, rather than just learning by rote. We can see within the expressions used a clear sense of closure, of things fitting into place, a sense of relief almost, when we 'get it'. And, by contrast, phrases that indicate not understanding reveal a sense of frustration. Understanding or not understanding is as much an emotional experience as a cognitive one!

Then we might notice that understanding brings clarity and light to the learner. Many of the phrases we use for understanding talk about 'seeing' something, having our eyes opened, or, by contrast, being in the dark.

Finally, these expressions for understanding support the idea that learning with understanding is all about making connections: connecting some new experience with other experiences or existing understandings. So, we talk about understanding as though it feels like things 'clicking' (like two Lego pieces fitting together), or 'falling into place' (like a piece in a jigsaw connecting with other pieces and suddenly making sense), or 'getting the picture'. And when we don't understand, the words 'go over our head', so we have a sense of not being able to connect them to anything and embed them in our minds. 

To learn that mathematics can be learnt with understanding in the ways indicated by these idioms is the most important thing for children to learn about this subject by the time they leave primary schools.

Learning the facts of multiplication

Some of the press who are particularly supportive of Michael Gove's reactionary, dreary and unrealistic curriculum proposals deliberately misrepresent those of us who oppose them. We are accused of not wanting to teach children facts. For example, it seems that we don't think it's necessary for children to learn their multiplication tables by heart, and so on.

Let me put the record straight.

Of course, we want children to memorise their multiplication tables. And the more multiplication results they memorise the better. If this is helped by chanting the tables in the traditional way, then that's fine by me. This is precisely what I say in my books! But there is a distinction to be made between learning these results with understanding and learning them by rote. Rote learning usually refers to learning which is not connected and therefore not meaningful to the learner. Results such as multiplication tables can be memorised by rote, but usually the  learning is less secure and it is always less useful and less transferable. In fact, it is well established that using patterns and connections reinforces memorisation.

So, for example, I would want pupils to use the fact that 3 × 4 = 12 to get 3 × 8 = 24 by doubling, and then to get from this to 6 × 8 = 48 by doubling again; and to see the connection between 3 × 7 = 21 and 6 × 7 = 42, and then by adding another 7 to get 7 × 7 = 49.

Then they can explore the simple patterns in the 5-times, 10-times, and 9-times table. They can find which numbers don't turn up as results in any of the tables (apart from their own). They can learn about commutativity and exploit it. They can learn how to use what they know to work out what they don't know.

Then the children need to connect these results with everyday situations, with rectangular arrays, with areas of rectangles, with the informal and formal language of multiplication, with steps along a number line, with corresponding divisions results, and so on. They need to explore questions like: why do many medicines come in packets of 28 tablets?

Yes, memorise the results, but also explore, exploit and enjoy all the connections. In this way the tables begin to make sense. The 'facts' can be learnt with understanding and pleasure.

For calculations with 2- and 3-digit numbers (and larger) we need know no more than the tables up to 10 × 10. But, bizarrely, presumably because they learnt them at school when there were 12 pence in a shilling and measurements of length were done in feet and inches, Mr Gove and his crew are prescribing that tables must be learnt up to 12 × 12.

There's no harm in children learning by heart the 11- and 12-times tables, but it is impossible to justify picking out these tables rather than any others and making it mandatory that children should memorise them. It would be easier to justify learning the 28-times table, so that those who go into healthcare will be good at handling tablets that come in packets of 28. I like to get Year 6 pupils to explore the 37-times table and to discover the delightful pattern in there: 3 × 37 = 111, 6 × 37 = 222, 9 × 37 = 333, and so on.

So, let me repeat! Give them the facts, yes. But give them understanding and enjoyment and purpose in learning as well.