CfEM blog: What is mastery?

Blog by Eddie Playfair, Senior Policy Manager at the Association of Colleges (AoC), for the Centres for Excellence in Maths (CfEM) programme.


The idea of mastery learning can be traced back to the work of psychologist Benjamin Bloom and his 1971 book ‘Mastery learning: Theory and practice’. In England, much of the recent work has been done by the National Centre for Excellence in the Teaching of Mathematics (NCETM), itself influenced by practices developed in Shanghai and Singapore.

NCETM emphasises 5 fundamental ideas which underpin mastery:

  • Coherence: connecting new ideas to concepts that have already been understood. Once understood and mastered, these ideas are used to support coherent incremental steps in learning.
  • Representation and structure: representations are designed to expose mathematical structures and enhance understanding, with the aim that students can eventually do the maths without resorting to the representation.
  • Variation: varying the way concepts are presented in order to deepen understanding and promote mathematical thinking.
  • Mathematical thinking: Ensuring students think, discuss and work actively on mathematical ideas rather than receive them passively.
  • Fluency: rapid and confident recall of both facts and procedures and movement between different contexts and representations which facilitates further learning.

In this first year of the Centres for Excellence in Maths project, we have agreed the following key principles to inform our approach to mastery teaching. We felt that it should:

  • Value and build on students’ prior learning.
  • Develop the understanding of mathematical structures.
  • Develop both fluency and understanding of key ideas.
  • Prioritise coherence and connections.
  • Develop a culture where everyone believes that everyone can succeed.

A mastery curriculum should create a connected and purposeful pathway through the curriculum alongside coherent steps in progressing understanding. Both the steps and the coherence are important.

Mastery teaching aims to enhance understanding and motivation, including the use of introductory problems and dialogic learning and the development of fluency and conceptual understanding with an emphasis on mathematical structures.

We need to consider how to create resources suitable for a variety of settings, which topics should be taught and how they should be grouped, how much guidance and flexibility to give teachers and how to equip them with alternative strategies.

The use of manipulatives and representations needs to be based on a clear rationale, make the link with the corresponding mathematical idea and help to build understanding of the underlying maths rather than simply being a new procedure. There is also a place for the use of introductory problems and dialogic learning.

At the core of the mastery approach is the belief that all students are capable of being successful in maths. FE students who have not yet succeeded will certainly need more support if we are to foster this belief.

“Students and their teachers can have different beliefs about their intellectual abilities. Some believe that they are basically fixed…others believe that they can be cultivated and developed through application and instruction. They do not deny that people may differ in their current skill levels, but they believe that everyone can improve their underlying ability.” Carol Dweck, ‘Self Theories’ (1999).

In summary, achieving mastery means that students know facts, can apply facts, have a depth of understanding and are fluent in concepts and procedures which they have understood and which allow them to solve problems.

Our national research studies in the 2019/20 academic year will be a great opportunity to evaluate how post-16 GCSE retake students can best be supported to develop genuine mastery in maths.

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