Book Review: Lisa Coe, Development Lead
Exploring teaching for mastery in maths in literature
With the release of Mark McCourt’s book, Teaching for Mastery, and the term mastery being used more widely both within and outside of mathematics education, new discussions have emerged over what it means to teach a mastery style of mathematics and, if the definitions of approach differ, whether there is a right approach.
Let’s begin by looking at the starting points for both Helen Drury, and, by extension, our Mathematics Mastery (MM) programme, and Mark McCourt.
East vs. West
In How to Teach Mathematics for Mastery: secondary school edition, Helen Drury largely draws inspiration for the mastery approach from high performing jurisdictions such as Shanghai and Singapore. While recognising the challenges of application of this in the UK and drawing on other sources within the MM programme, the root of the approach draws on these educational regions. What is it that these educational systems do which the UK can learn from?
Drury goes on to define what teaching for mastery means: “To teach for mastery is to teach with the highest expectations for every learner, so that their understanding is deepened, with the aim that they will be able to solve non-standard problems in unfamiliar concepts.”
Mark McCourt focuses on American research and evidence, drawn from the Winnetka plan – an approach to schooling developed by Carleton Washburne from 1919 until 1943. In brief, the system was an individualised approach to learning with distinct steps in a ‘logical and tested sequence’ which, unless pupils successfully completed a test, they could not move on from. The pupil would progress ‘at a pace unique to themselves’. Teacher intervention would be focused on what the individual pupil needed and would be immediate – closing gaps and avoiding misconceptions.
It is within this sphere of understanding of a mastery approach that McCourt provides his definition of a ‘mastery model’ taken from Guskey (2010) which includes a focus on pre-assessment, formative assessment, effective teacher instruction and intervention and enrichment activities.
While the definitions differ on the surface, they are intertwined. The MM programme incorporates all of the elements McCourt suggests makes a mastery model of teaching, and implementing McCourt’s definition means teachers set high expectations and will deepen understanding.
Application of McCourt’s mastery model will lead to the teaching for mastery suggested by Drury.
Sequencing a mastery curriculum
McCourt makes the following claim:
Schools that have schemes of work based on a pupil’s age (rather than where that pupil actually is, mathematically) continually serve up the wrong mathematics to their pupils. I suggest that the overwhelming majority of pupils … are being ‘taught’ the wrong mathematics.
This is in contradiction to the MM approach, organised into year groups from Reception to Year 11 and, in fact, in contradiction to the National Curriculum, ordered into year groups or Key Stages. However, McCourt emphasises the need for a logical progression in ideas, and this is exactly what the MM programme provides – a careful construction of ideas, revisited and built upon, over time. The MM programme is designed to close the attainment gap, meaning that, over time and with interventions as appropriate, pupils are able to access the same mathematics. The key difference is that the MM approach works within the current system of the National Curriculum and assessments.
Are the approaches defined by Mathematics Mastery and McCourt’s Teaching for Mastery so very different?
McCourt emphasises throughout ‘Teaching for Mastery’ Bloom’s statement that all pupils can learn well given the right time and conditions, and this is echoed in the Mathematics Mastery programme’s focus on success for all.
Furthermore, the MM programme is based on Drury’s Dimensions of Depth – three underlying principles required in mathematics education for pupils to meet the following definition of mastery:
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 they can independently apply it to a totally new problem in an unfamiliar situation.
While McCourt argues that one cannot master a mathematical concept, as they are infinitely broad, but can become more expert in it, both he and Drury promote the importance of language and communication, conceptual understanding and mathematical thinking.
For example, in Part IV of ‘Teaching for Mastery’, McCourt stresses the importance of multiple representations, identifying research that shows ‘pupils who were able to recognise the same relationship presented in different models…were more likely to perform well on later standardised tests.’ As with the Mathematics Mastery approach, McCourt stresses the need to move back and forth between multiple representations, including language.
Teacher subject knowledge
In Teaching for Mastery, McCourt regularly stresses the importance of subject and pedagogical knowledge of the teacher. For example, he argues that in order for all pupils to attach meaning to a new idea, teachers should be familiar with multiple ways in which to represent this idea, and have a deep understanding of how to connect this with what pupils already know. Again, teacher subject knowledge is promoted in Mathematics Mastery – both approaches recognise the importance of upskilling teachers.
So, both approaches to mastery emphasise a need to connect new or novel ideas to those pupils already know, and that these concepts develop over time to move towards mastery or a more expert understanding.
This more expert understanding is attainable for all pupils. Both focus on the importance of developing these skills and ideas to make them “stick” through making connections and embedding the ideas in long term memory.
While I feel that Drury is more pragmatic in acknowledgment of mathematics to be learned within year groups, defined by the National Curriculum, both the Mathematics Mastery approach and Mark McCourt’s vision promote the same ideal – all pupils can learn well, with information provided in carefully thought out, logical steps, over time.