Deep Mastery

Like almost everyone else in Maths education, I have recently decided to broach the ‘mastery’ word.  It does feel a bit odd to do so, given that it’s been around at least since at least the 60’s, and I’ve never felt a need to use it before – despite having been an advocate of the kind of learning that relates in some ways to the current usage of the word.

It also feels uncomfortable because of the associations conjured up by the word (slave-master, schoolmaster, Masters’ degree), which in my mind are often taken to be male and sometimes bossy.  But – thankfully – this doesn’t seem to be the way people are taking it.

I won’t be doing a synthesis or critique of the ways in which people are using the word in mathematics education currently. Rather, I will refer to one source: Helen Drury’s book How to Teach Mathematics for Mastery’, published by OUP. This is for three reasons:

  1. Helen was using the word before it became an on-trend and politically-charged buzzword,
  2. Helen’s thinking is informed by sustained experience working with over 500 schools,
  3. Most importantly, Helen uses the term ‘mastery’ in a deep sense.

Let’s take a look at Helen’s definition:

“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.”

This is a definition of learning—which is what the word was about when it first came up in the 60s—and not a definition of teaching. But, there are still some clues about some components of teaching for mastery that are uplifting to the mathematical soul:

Clue 1: exploration

Clue 2: clarification

Clue 3: over time

Clue 4: independently apply

Clue 5: unfamiliar situation

She is not writing about ways to devise a step-by-step ‘smoothed’ curriculum, but rather about what constitutes deep, sustained, connected and coherent mathematics. Here the final ‘step’ is not fluency, or merely readiness for a next step, but rather the ability to recognise where and how to use the concept or skill.


Here is a mathematical problem that might be unfamiliar for you, but which uses some concepts and skills that you’ll have likely explored and independently applied over time.

 

  1. Write down all the natural numbers that are less than or equal to some number n. (note, it’s worth reading to the end before choosing your n.) 
  2. Now, pick from your list all the pairs of numbers whose total is greater than n.  Do not use repetitions such as 3 + 3, and note that (2, 3) is regarded as the same pair as (3, 2).
  3. Now pick out all of these pairs that are relatively prime and abandon the rest.  Now for each of these pairs, write down the unit fraction whose denominator is the product of the numbers in the pair.
  4. Add up all these unit fractions.  The answer should be a half. You might like to wonder why.

 

I am sure that deep mastery of some mathematical concepts, as well as some skills and insights, are necessary to tackle this task with any confidence.  On the face of it, you may think that it’s simply an exercise in following instructions, but a second look might reveal important subtleties that a student could miss: for example the distinctions between > and ≥;  or < and ≤.

There also needs to be precision around the definition of ‘relatively prime’ – I notice in classrooms that there are some doubts around whether 1 can be a factor of a number or not (it always is, but is not itself a prime).  Plus, two numbers being relatively prime does not necessarily mean that each of those numbers is itself prime.

Systematic working is also seemingly important for this problem, to ensure all number pairs are included or excluded as appropriate.

Finally, there’s the technical required in adding all of those unit fractions, which can become unwieldy fairly quickly.

I personally started with n = 9 and, whilst being determined to stay in the world of fractions so that I could keep track of the components, I had to take some strategic decisions about when and why to use a calculator. Similarly, in easier cases, I had to decide when and how to use the commutative and associative laws.

When it then came to ‘wondering why’, I was in fact glad I had stayed with fractions. The lesson here is that, whilst creating an algorithm or using a quick hack to do the hard work is an option, it might not help the critical analysis that comes at the end. Furthermore, when it did come to ‘wondering why’, I also went back to exploring a more simply example with a smaller value of n.


Looking back at Helen’s definition of mastery, I can see that this task could not only contribute to mastery of several familiar concepts and skills, but it could also be a way of exercising such mastery. It is an excellent opportunity for using several different skills to explore, in depth, something that was to me quite amazing and intriguing.

Now if you are still reading, and have done the task, you might want to know that the proof was set in the 3rd All Soviet Mathematics Competition in 1969, but in symbolic language. It used number pairs pq where the first number p must be smaller than the second number q. That is, 0 < p < q n. (You may find it interesting, however, to consider the result of removing this stipulation and allowing duplicated pairs.)

You may find that this problem is at too high a level for all your students (particularly when presented in symbolic language), but the real point is this: what mathematics have you mastered that enabled to you to tackle it, what does ‘mastered’ mean, and how did you come to master it?

Above all (and this is certainly brought out in Helen’s book), the willingness and capabilities of mathematical problem solving, based on secure knowledge and experience, were what I ultimately brought into play in tackling the problem myself. And that’s what I call mastery.

 

Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications.  Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent. 

 

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