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.