Mastery – what it is…and what it isn’t!

Blog: Ian Davies, Director of Secondary

Mastery – what it is…and what it isn’t!

Posted: 5/06/15

Here’s a claim I often make:  I have mastered addition.

Sometimes when I say this, some people are not as impressed as I hope they will be.  Some people are a little confused that someone who has been working in mathematics education since 1987 would bother to state what appears to be an obvious truism, but they don’t understand my meaning.  I don’t just mean I know my number bonds.  I don’t just mean I’m ‘good at it’.  I don’t just mean that as well as being able to perform the procedure(s) of addition efficiently, reliably and accurately that I also understand how all of the many methods available to me work.  All of this is true, but it’s much more than that…

Some examples:

  • I can represent additions in many different ways and make an informed choice as to the best way for any particular problem
  • I can foresee some of the consequences of what I’m doing – the answer’s going to be over 1000 so it won’t fit in the box so maybe I won’t even bother doing the exact calculation at all
  • Even in very unfamiliar situations, I know when addition is the right operation to use – and when it’s not
  • I can interpret my results – again even in unfamiliar contexts
  • I know and can do the inverse operation of subtraction to a pretty high level too
  • Faced with different situations, I know which approach to performing an addition – counting on, compensating, a formal written method, getting out my calculator etc.
  • I know that if I’m suddenly working in modulo 12 – as I often do in time calculations – then 8 + 5 = 1 is more useful than 8 + 5 = 13

…and so on!

Confession time

I often follow up my bold claim about addition with this confession:  I am yet to master integral calculus.

Again, I’m not ashamed to admit this.  Even though, by many people’s standards, I am prepared to flatter myself that I’m ‘pretty good at’ integrating and I know the link between integration and differentiating etc. there a whole lot of functions out there that I cannot even begin to imagine how to integrate.  More than that, there are a plethora of situations and problems where I wouldn’t be entirely sure if integrating would even be the right approach to take.  Our Director Dr Helen Drury says, “in mathematics, you know you’ve mastered something when you can apply it to a totally new problem in an unfamiliar situation”¹. So I confess; despite those years of successfully teaching Further Maths, and to the potential horror of my students – I have not mastered integration!

¹ Drury, H. (2014) Mastering Mathematics. Oxford University Press, pp8.

With a bar that high, surely you can’t expect “mastery”?!

That’s right.  I don’t!  What we at Mathematics Mastery are striving for is to put our students on the road to mastery – the same road that we teachers and other educators are still on.  We aim to give them the skills they need to make sensible choices and to use their knowledge to tackle problems in all sorts of situations.  We also aim to develop their resilience for when the road gets tough, which it does for all of us at many points on the way.  Helen summarises what we’re aiming for like this:

“A mathematical concept or skill has been mastered when, through exploration, clarification, practice and application over time, a person can represent it in multiple ways, has the mathematical language to be able to communicate related ideas, and can think mathematically with the concept so that they can independently apply it to a totally new problem in an unfamiliar situation”²

² Drury, H. (2014) Mastering Mathematics. Oxford University Press, pp9.

But don’t knowledge and procedural fluency come first? What about working memory?

There is a real need for a balanced approach here.  Of course having key facts at your disposal is very helpful when it comes to solving problems, especially when in an unfamiliar context but the need to be flexible and adaptable is too. Also helpful is being able to use what you do know to get you facts you don’t know.  I remember when learning my tables that it took a while to get 11 x 12 to “stick”, but I was so confident with 10 x 12 = 120 and I knew how to get to the answer pretty quickly by adding on adding another 12.  It was my understanding of the structure and not just my knowledge of the facts that helped me out in tables tests.  That’s what set me on the road to mastery of the times tables!

Our Mathematics Mastery curriculum encourages ‘intelligent practice’ to enable students to develop conceptual understanding alongside procedural fluency.  We use multiple representations to support this understanding and to encourage students’ reasoning.  We’re also solving problems from the very start of the curriculum journey, not seeing it as some ‘add on’ that can only be considered when all the facts are in place.  The challenge is developing these skills and understanding concurrently and that’s what we’re working with the teachers in our partner schools.

An example

I like to include some maths in my blogs – it’s about doing maths, not just thinking about it!  Don’t look ahead down the page, just have a go at this calculation:

36 x 175 ÷ 63

I’m sure there’ll be a variety of approaches:

  • Perhaps use commutativity to start with 175 x 36 as a long multiplication before tacking the division.
  • Perhaps change x 36 to x 3 and then x 12 so you’re using times tables you “know”
  • Perhaps ÷ 7 and then ÷ 9 instead of ÷ 63
  • Perhaps change the order to have 36 ÷ 63 x 175, rewrite as a fraction and do some cancelling

There are many ways (my favourite is factorising to 4 x 9 x 7 x 25 ÷ 9 ÷ 7 which reduces to 4 x 25 = 100.  It still surprises me that that’s the answer!) and part of the joy of mathematics, and the joy of journey to mastery is seeing the links between them all – appreciating the flexibility and variety and not being confined by an inflexible rule or procedure.  For us, confinement to procedures and rules is what mastery isn’t!