Teaching for Mastery: Supporting Research, Evidence and Argument
The NCETM’s work in advocating teaching for mastery, and in supporting Maths Hubs as they help schools implement such an approach, is largely based on the way maths has been successfully taught in East Asia, most notably, but not exclusively, Shanghai, China. An exchange programme between English and Shanghai teachers has informed effective pedagogic strategies for achieving mastery of mathematics. The striking performances of Shanghai, and other East Asian countries in maths has become well-established in successive international tests such as TIMSS and PISA.
For each of the key components of teaching for mastery identified below, we offer a key text, chosen for its relevance and accessibility (both in terms of readability and being able to access it for free online), as well as a list of further relevant reading - links in the main text will take you to the suggested key text and further reading. Much of this material features on a reading list for teachers on the Maths Hubs Mastery Specialists programme.
The component thought to be key to the success of the East Asian system, is the use of variation theory. Variation theory has several dimensions, including use of multiple representations of what a concept is, and what it is not. It is characterised by a carefully constructed small-step journey through the learning, paying attention to what is kept the same and what changes, in order that pupils might reason - make connections and build deep conceptual knowledge. Variation is applied to practice questions where attention is paid to the selection and order of the examples, often changing just one aspect whilst keeping others the same. The intention is to avoid mechanical repetition but instead to promote thinking to make connections. This is also known as ‘intelligent practice’.
Teaching for mastery is also characterised by:
- A belief that, by working hard, all children are capable of succeeding at mathematics. On this basis, children are taught all together as a class and are not split into ‘ability’ groupings. Carefully structured teaching, planned in small steps provides both the necessary scaffold for all to achieve, and the necessary detail and rigour of all aspects of the mathematics to facilitate deep thinking. The small steps are connected and concepts built, leading to generalisation of the mathematics, and the ability to apply it to multiple contexts and solve problems. It is expected that those that will achieve well on a particular topic may not necessarily be the same children that achieved well on other topics. An additional short session of 10 to 15 minutes is provided on a daily basis for any pupils who do not fully grasp the lesson content, in order that they 'keep up' with the class. Our experience shows that it is not always the same pupils who require this form of intervention and this boosts the self-belief of previously low attaining pupils.
- A focus on exposing the structure of mathematics and developing an understanding of how and why mathematics works. A key skill of the teacher is to be able to represent the mathematics in ways that provide access and insight for pupils. Concrete materials, contexts, drawings, diagrams, equations all play a role. These are discussed through opportunities for pupil-pupil and pupil- teacher talk, to develop reasoning, flexibility and adaptability in mathematical thinking.
- Memorisation and repetition of key facts (times tables and number bonds etc) as important aspects of learning, Evidence from cognitive science research suggests that learning key facts to automaticity ‘frees up’ working memory to focus on more complex problem solving rather than reaching cognitive overload trying to calculate simple operations. In terms of procedural fluency and conceptual understanding, one should not be prioritised over the other, but learning is most effective when the two are fully integrated.
- Teaching children precise mathematical language and insisting upon its use, to support children's ability to think mathematically. Having the language and using it, empowers children’s ability to think about the concept.
Teaching for mastery is in the early stages of adoption in England. We will be monitoring research and evidence of its effectiveness and adding to the final section, Recent Evidence and Research in England.
Notes and further reading
Some texts on this list may require either access through a library, or payment to get the full version.
Success of East Asian countries
This video summarises TIMSS international results for 2015. The report for England is here. Report for PISA International results 2015 here, and the report for England is here.
Sun, Xu Hua. "The structures, goals and pedagogies of" variation problems" in the topic of addition and subtraction of 0-9 in Chinese textbooks and reference books." Eighth Congress of European Research in Mathematics Education (CERME 8), Apr. 2013. 2013. This article explores Variation Theory drawing on examples from Chinese primary textbooks for the teaching of addition and subtraction of single digit numbers.
Watson, Anne, and John Mason. "Seeing an exercise as a single mathematical object: Using variation to structure sense-making." Mathematical thinking and learning 8.2 (2006): 91-111. This article looks in depth at use of Variation Theory in written exercises given to learners and explains how exercises written using Variation Theory can support learners in perceiving relationships and structure rather than simply practising a mechanical procedure.
GU Lingyuan HUANG Rongjin MARTON, Ference. "Teaching with variation: A Chinese way of promoting effective mathematics learning." Ch.12 p309 of this book: Lianghuo, Fan, et al., eds. How Chinese learn mathematics: Perspectives from insiders. Vol. 1. World Scientific, 2004. Marton researches more and less successful instances of learning in various collaborations. In this paper, he collaborates with Chinese researchers to ‘unlock the paradox of the Chinese learner’ with variation theory as the key.
Askew, Mike. Transforming primary mathematics: understanding classroom tasks, tools and talk. Routledge, 2015. Chapter 6, p75 discusses Variation Theory (the online version is not complete). Askew provides a readable account of how using Variation Theory can turn mathematics from a set of procedures to a ‘mindful activity’
All Children are Capable of Succeeding at Mathematics
Dweck, Carol S. "Mindsets and math/science achievement." (2014). Dweck is a cognitive scientist, known for her work on ‘Growth Mindset’ – the belief that ability is not fixed, but something that can be worked upon. Here, she applies her theories to maths and science in particular.
Watson, Anne, Els De Geest, and Stephanie Prestage. "Deep Progress in Mathematics." University of Oxford (2003). Project where a handful of teachers looked at ways to approach teaching bottom sets removing assumptions about ‘can’t’ (also available here).
Boaler, J. (1997) Experiencing School Mathematics: Teaching Styles, Sex and Setting. OUP. This text has since been revised in 2002 for the American audience, but the original is still in print, though not available online.
Cambridge Mathematics Espresso 5 summarises recent research into ability-grouping in Maths.
The Education Endowment Fund considers the impact of setting/streaming in its Teaching and Learning Toolkit.
Structure of Mathematics
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates. This research compares the ‘profound understanding of fundamental mathematics’ of teachers in China and the U.S., and how they would use this understanding to inform their teaching. The results demonstrate the deep structural understanding held by Chinese teachers that helps inform their teaching strategies. Summarised and reviewed here - book is in print, but not available online
Memorisation and Repetition
Willingham, Daniel T. "Is it true that some people just can’t do math?" American Educator 33.4 (2009): 14-19. Evidence from cognitive science of cognitive overload. that learning facts to automaticity frees up working memory, so avoids cognitive overload. Argues that procedural fluency and conceptual understanding should be taught in tandem.
Cambridge Mathematics Espresso 10 asks Why is working memory important for mathematics learning?
Baroody, Arthur J. "Mastering the basic number combinations." Teaching Children Mathematics 23 (2006): 22-31. This article argues that basic number fluency is best achieved by teaching it alongside conceptual understanding. Includes classroom suggestions and examples.
Dahlin, Bo, and David Watkins. "The role of repetition in the processes of memorising and understanding: A comparison of the views of German and Chinese secondary school students in Hong Kong." British Journal of Educational Psychology 70.1 (2000): 65-84. This research gathers and compares the experience of learning through repetition and memorisation, of German and Chinese students studying in Hong Kong. Results suggest that the Chinese students place more emphasis on the role that repetition plays in understanding than their German peers. Full text requires payment.
Recent Evidence and Research in England
A guidance report from the Education Endowment Foundation Improving Mathematics in Key Stages 2 & 3, published in November 2017, endorses many of the components of teaching for mastery. Links between its recommendations and a teaching for mastery approach are made explicit in this blog by Professor Jeremy Hodgen, Chair of Mathematics Education at the UCL Institute of Education, who led the evidence review.
A report by the Fair Education Alliance looks at schools with good outcomes for disadvantaged children in maths. Investigating 20 schools and Early Years providers, it pinpoints the factors in their success. Many of the schools cite teaching for mastery as a key factor.