## Myth #3: Mastery requires repetitive practice

The ‘mastery’ word is being much used, and rarely with exactly the same meaning. I say more on this here, but in this post I want to focus on one particular way it is sometimes used – to describe repetitive (boring, mindless) practice.

Some people have come to believe that mastery of a procedure requires repeated practice at that procedure. This can lead to the misconception that a mastery approach to mathematics requires repetitive practice with little variation in the items practiced for any particular procedure.

This is not the case!

Teaching for mastery means teaching so that learners experience success in all three aims of the National Curriculum – fluency, reasoning and problem solving. None of these will be achieved through mindless repetitive practice. Not even fluency.

In fact, I think ‘fluency’ risks being undervalued. If we were aiming for fluency with Spanish or Chinese, people might consider us over-ambitious, but somehow mathematical fluency has become associated with mechanical, routine performance.

Para 239: …we need to distinguish between ‘fluent’ performance and ‘mechanical’ performance. Fluent performance is based on understanding of the routine which is being carried out; mechanical performance is performance by rote in which the necessary understanding is not present. Although mechanical performance may be successful in the short term, any routine which is carried out in this way is much less likely either to be capable of use in other situations or to be retained in long term memory.[1]

If practice is just repeating the same procedure with different numbers, apparently chosen at random, then it has no purpose.

In China, procedural variation is used to promote deep understanding of mathematics.

The types of practice promoted by mastery in mathematics include use of a concept or procedure in a variety of contexts.[2]

In designing [these] exercises, the teacher is advised to avoid mechanical repetition and to create an appropriate path for practising the thinking process with increasing creativity.[3]

Variation theory tells us that by systematically changing significant aspects of a task, keeping the rest fixed, we can focus the students’ attention on those aspects and learning with understanding can result.

The emphasis in making such variations is not to develop speed but to develop an awareness of pattern, leading to conjecture, generalisation, explanation and deeper understanding.

Systematic changing of aspects of a task is key to designing effective practice activities for students.

Control of dimensions of variation and ranges of change is a powerful design strategy for producing exercises that encourage learners to engage with mathematical structure, to generalize and to conceptualize even when doing apparently mundane questions.[4]

[1] Cockcroft, W. H. (1982) Mathematics counts : report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the chairmanship of W.H. Cockcroft, London, HMSO.

[2] Lai, M. Y. & Murray, S. (2012) Teaching with procedural variation: A Chinese way of promoting deep understanding of mathematics. International Journal of Mathematics Teaching and Learning.

[3] Gu, L. (1991) Xuehui Jiaoxue [Learning to teach]. Beijing, China: People’s Education Press.

[4] Watson, A. & Mason, J. (2006) Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8, 91-111.

## Myth #2: Mastery is a level

This myth is a funny one. Right from the off the teachers I was working with on Mathematics Mastery were absolutely clear that mastery was for all.

A mastery curriculum is one where all pupils learn what is expected.

In high-performing countries the intention is to provide all learners with full access to the curriculum, enabling them to achieve confidence and competence – ‘mastery’ – in mathematics’.

Everyone is entitled to:

• deep and sustainable learning
• learning that can be built upon
• learning that can be reasoned about
• learning that’s connected
• conceptual and procedural fluency

The confusion may arise from a DfE consultation on primary performance descriptors testing, in which they baffled and bemused us all by suggested four levels of achievement for the end of key stage 1:

So it makes no sense to think of mastery as a level that some learners will reach and others won’t.

Mastery should be the aim for all.

• Below national standard
• Towards national standard
• At national standard
• At mastery standard

There was so much wrong with this I didn’t know where to begin in responding to the consultation. Maybe it was a blessing in some ways, as it triggered a national debate about whether ‘mastery’ was a performance level above national expectations, or something all should strive towards. (And a general consensus that it was the latter).

But not only is ‘mastery’ not a level that only some learners will reach, it is a level of competence that arguably no learners will ever actually reach.

You never achieve mastery!

I get quite upset when I see textbooks and other classroom resources with ‘mastery checks’ – the implication being that by answering five straightforward and familiar questions correctly, you demonstrate mastery of a concept or skill. Of course you don’t.

As I say in Mastering Mathematics:

“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 an infinite continuum – there are always more interesting ways to represent an idea, there’s more language and more complex communication, and mathematical thinking can surely always be deepened? As for new problems and unfamiliar situations; new problems arise all the time, and you can never claim to have considered every possible situation.

Mastery is simultaneously:

• something that every teacher should aspire to for every pupil, regardless of starting point
• an unachievable lifelong quest.

Mastery is not a level.

## Myth #1: Mastery has a single definition

Mastery Myth #1: Mastery has a single definition

When Tom Bennett asked me to present at researchED, it seemed a great opportunity to bust a few of the myths around ‘mastery’. Here are the five myths I chose to challenge:

• Myth #1: Mastery has a single definition
• Myth #2: Mastery is a ‘level’
• Myth #3: Teaching for mastery requires repetitive practice
• Myth #4: Teaching for mastery means no differentiation
• Myth #5: Mastery is new

They all attracted considerable interest on Saturday (Myth #5 seems to have had the most attention on Twitter) so I’m going to attempt a bit of written mastery myth-busting here.

Myth #1: Mastery has a single definition

Way back in 2011, the lovely folk at the EEF gave Ark funding, and I was thrilled to find myself with the opportunity of starting a national partnership of schools to transform achievement in mathematics. But what to call it?

As you can imagine, this was the topic of considerable debate, and when we finally alighted on ‘Mathematics Mastery’ we had a few concerns:

• Was it a bit masculine? Did the ‘master’ aspect suggest that success in maths was not for girls?
• You don’t really ever ‘master’ maths; was it odd to launch a partnership programme with an unachievable aim?

One thing we didn’t do was check the academic literature for references to mastery – if we had, we’d have discovered that Benjamin Bloom did a lot more than just his famously triangular taxonomy. In the late 60s and early 70s, he used the term ‘mastery’ prolifically to refer to a cyclical approach to teaching and learning[1].

By 1990 a fair few studies into the effectiveness of mastery learning had been carried out, mostly in the US, and a meta-analysis of 108 concluded that it raises achievement. The effects appeared to be greater for lower achieving students, though they were enormously variable.[2]

The term ‘mastery’ has also been used by Carol Dweck to describe a mindset where learners seek to improve and develop, to acquire new skills and master new situations, rather than being preoccupied by proving their ability to others or avoiding negative judgments.[3]

A year or two later I had a very memorable conversation with Jeremy Hodgen (@JeremyHodgen Nottingham University, formerly King’s) which looped along the following lines:

Me: When we say mastery we don’t mean Bloom’s mastery
Jeremy: But mastery is Bloom’s mastery
Me: We use it to describe an approach with high expectations for all, and more time for each topic. But we don’t make pupils stick with a topic until they get 80% on a test.
Jeremy: Then why on earth did you call it mastery?

Note to self – in the event you find yourself naming an organisation again, check the research literature first.

Actually, I could argue that there are more similarities than differences between Bloom’s and our use of ‘mastery’. And there are certainly many similarities between my use of it with Mathematics Mastery, and its use by organisations such as the NCETM.

What do mastery approaches have in common?

An emphasis on success for all – a commitment that all pupils can and will succeed.

Belief that this can be achieved by developing conceptual understanding, with a focus on mathematical structures.

• keeping the whole class together
• teaching less in more depth
• not moving on until ideas are understood
• promoting understanding through a variety of representations.

However…many teaching approaches advocate all of the above and do not use the term mastery; many approaches that call themselves ‘mastery’ don’t advocate all of the above.

Important post script

I knew I’d been inspired by NAMA’s ‘Five Myths of Mastery in Mathematics’, though I did pick some slightly different myths to bust. Many thanks to Andrew Jeffries for pointing out just how aligned my presentation was to the NAMA document – and even more thanks to NAMA!

[1] Bloom, B. S. (1968) Learning for Mastery. Instruction and Curriculum. Regional Education Laboratory for the Carolinas and Virginia, Topical Papers and Reprints, Number 1. Evaluation comment, 1, n2.

Bloom, B. S. (1971) Individual Differences in School Achievement-a Vanishing Point: A Monograph. Aera-pdk Award Lecture Annual Meeting American Educational. Research Association New York February 6, 1971, Phi Delta Kappa.

Bloom, B. S. (1971) Mastery learning. Mastery learning: Theory and practice, 47-63.

[2] Kulik, C.-L. C., Kulik, J. A. & Bangert-Drowns, R. L. (1990) Effectiveness of mastery learning programs: A meta-analysis. Review of Educational Research, 60, 265-299.

[3] Dweck, C. S. (1986) Motivational processes affecting learning. American Psychologist, 41, 1040.