Making Sense of Mastery in Mathematics
I recently had the pleasure of attending a mathematics course presented by the amazing Tara Loghran, an independent mathematics consultant. Tara’s engaging seminar demystified what is meant by the terms ‘mastery in mathematics’ and she offered plenty of simple but effective strategies for primary teachers to promote numerical fluency. The aim of this post is to share some of my learning about mastery in mathematics from Tara’s seminar as well as other sources.
What is mathematics mastery?
Simply put, mathematics mastery means covering fewer topics in greater detail and sees greater emphasis placed upon:
- The use of objects (such as numicon) and visual representations of mathematical concepts.
- Promoting pupil’s understanding and use of mathematical vocabulary.
- Challenging pupils to apply their knowledge and to deepen their understanding of maths through rich problems and puzzles.
Dispelling the myths about mastery in mathematics.
It seems that a number of misconceptions surround mastery mathematics. Perhaps most common among these is the idea that a mastery approach would see the end of differentiation as teachers begin to teach their classes at ‘broadly the same pace’. This is simply incorrect. A mastery approach does not rule out the need for teachers to tailor teaching to the needs of pupils. What is does mean however is that teachers should avoid having three different concepts being taught to different ability groups at the same time as I have often attempted in the past. Instead, teachers should ensure lessons have a greater emphasis on whole class teaching with fewer concepts being taught. Allow me to elaborate…
Let’s say you’re teaching children about number bonds. Let’s also say the children are beginning to gain some understanding of multiples of ten. All pupils should be taught how to spot number bonds to 10 (perhaps using tactile apparatus) and they should all be given plenty of opportunities to apply this knowledge and to apply it in relation to multiples of ten.
For example, this could involve an activity like whereby an assortment of random numbers are presented to children (different numbers for each ability group) and children could be challenged to find ways of using the numbers to make multiples of ten. Allow me to give you an example…
Lower ability pupils could be presented with the numbers 7, 9, 4, 3, 6, 12, 13, 19, 1
More able pupils could be presented with the numbers 70, 49, 24, 36, 21, 31, 19, 94
ALL pupils could then be challenged to find ways of making multiples of ten. This is nice and open ended and gives the lower ability pupils the opportunity to practice spotting those bonds to ten (e.g. 13 + 7 requires knowledge of 3 and 7 as a bond to ten as well as the knowledge that the answer will be a multiple of ten). Meanwhile, more able pupils are using similar knowledge of bonds and multiples of ten but are applying this to numbers beyond twenty. This brings me to a crucial point…
Challenge in maths is not necessarily about using bigger numbers though!
It’s more about getting pupils to use vocabulary, to reason and to find ways of working systematically. The latter is particularly important in extending the level of challenge in the above activity example. Here, inviting pupils to find all the combinations of numbers that make multiples of ten offers significant opportunities for pupils to reason and to find ways of working systematically. Make sure that pupils are afforded such opportunities. I certainly will be!
This brings me to my next point…
The way to go is to let pupils get on with it!
Of course, I don’t mean this quite as literally as it may sound. What I mean is let pupils explore and play with number and mathematical concepts in a way that is structured. Present children with challenges like the one above and see what they come up with. Then, highlight areas for improvement by teaching them and offering your insight as the expert. Rinse and repeat with different activities and always aim to deepen their understanding. As Tara Loghran explained, this is the essence of the celebrated Shanghai model of maths teaching.
Be clear in what is fluency looks like.
Let’s come back to the example in blue above. If a child needs to add a pair of numbers by using a number line or a hundred square to see if it produces an answer that’s a multiple of ten the child lacks fluency in the number bond concept. Why do I say this? Because fluency is being able to do something more efficiently and without having to rely on long winded strategies to get a result. It forms part of your thinking that is less laboured and more automatic. Like driving. At first you have to think about every little detail – gears, speed, checking mirrors, etc. Eventually, it becomes more of a second nature – you become fluent in driving. Maths is not dissimilar.
So what would fluency look like in the above example? It might look/sound something like this:
“I can instantly see that pairs of numbers where the ones (units) form a bond to ten are the ones that give an answer that is a multiple of ten. For example, 24 and 36. 4 and 6 make ten so I know that the answer will be a multiple of ten.”
Of course, a child might not put it quite like that but you get my drift! The reasoning is there as is the vocabulary and knowledge and skills (bonds to ten) have been applied.
As primary maths teachers, we must promote mastery of mathematical concepts through the use of concrete models to illustrate abstract concepts. We must provide students with ample opportunities to use mathematical vocabulary and to use this vocabulary in their reasoning. We must give them opportunities to explore ways of working systematically and we must come away from only teaching maths mechanically and the falsehood of extending learning by giving them bigger numbers to use in column addition. Not that I’ve ever done that of course…
Coming soon: Songs for Teaching will offer a number of other exciting maths activities to use in class. Watch this space!