I believe that if we are to adopt a teaching for mastery approach to maths teaching, consistent with the new National Curriculum, we must answer these questions. Many primary teachers have asked my NCETM colleagues and me these questions, and this blog explains our current thinking. We have been informed by the National Curriculum document itself, by teaching we have observed and by the textbooks used in regions and countries that teach maths very successfully, such as Shanghai and Singapore.

I also believe the answers to these questions have the potential to reduce teacher workload, as well as improving the mathematical learning of their pupils.

After describing the three over-arching aims of the curriculum - that pupils should become fluent in the fundamentals of mathematics, that they should be able to reason mathematically, and that they should be able to apply mathematics to solve problems - the National Curriculum programmes of study for mathematics state:

The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.

The NCETM has adopted the phrase intelligent practice to describe the type of practice that supports pupils to build conceptual understanding, at the same time as developing procedural fluency. We believe this is the type of practice all pupils need to develop sustained mathematical learning.

The paragraph in italics above, alongside the three aims, marks the new curriculum out explicitly as a mastery curriculum, as I have described in an earlier blog.

The sentence: ‘The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace’ strongly encourages whole-class teaching. This requires very careful design and sequencing of lessons.

The sentence: ‘However, when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage’, combined with ‘Those pupils who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on’, requires that pupils should not be raced through to new material before they are ready to build from their current learning, using what they know as a basis for progression. Computational fluency has been defined as the ability to carry out mathematical computations ‘efficiently, accurately and flexibly’ (Russell, 2000). I believe this definition can be extended to general mathematical fluency. Efficiency and accuracy can be achieved at a superficial level, creating a false impression that a pupil is ready to progress. For me, it is the ability to also use material flexibly, as well as accurately and efficiently, that indicates a pupil has achieved sufficient fluency for progression. I consider flexible use of curriculum material means being able to apply the curriculum content to reason mathematically and to solve problems, so meeting all three curriculum aims – only then has a pupil mastered the material.

The sentence: 'Pupils who grasp concepts rapidly should be challenged through rich and sophisticated problems before any acceleration through new content’, directly discourages acceleration through content, instead requiring challenge through ‘rich and sophisticated (which I interpret as mathematically deeper) problems’. Engaging with ‘rich and sophisticated problems’ involves reasoning mathematically and applying maths to solve problems, addressing all three curriculum aims. All pupils should encounter such problems; different pupils engage with problems at different depths, but all pupils benefit.

The NCETM is employing the phrase ‘mastered’ to indicate the stage at which a pupil has grasped an element of curriculum content well enough to build from that content, using it as a basis to progress to other areas of the maths curriculum that depend upon that content.

Secure mathematical understanding is developed through doing maths in ways that engage pupils in deep thinking. In Shanghai, carefully structured questioning, combined with exercises that employ variation, provide pupils with the opportunity to practice calculation whilst, at the same time, encouraging them to think about the relationships within the maths, thus deepening conceptual knowledge and helping them build mathematical connections - ‘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’ (Gu, 2004).

The questioning and exercises used by the Shanghai teachers enable pupils to master each piece of curriculum content as they encounter it, so that their learning is sustained over time, preventing the phenomenon often reported by teachers, along the lines of ‘I taught this topic last term (Last month? Last week!) and they all got it. Now we need it to progress to this new topic, it’s as if they’d never been taught it’. It’s not that they have never been taught it; it’s that they only learned it superficially - they hadn’t really ‘got it’. They were able to get some answers right, but their learning was not deep and they did not develop solid mathematical foundations they could build from. It is because maths is so interconnected, building continually on earlier ideas, that it is so important to ensure each concept (and any associated technique) is mastered to form a firm foundation for progression.

The issue of superficial learning impeding progress is relevant at all levels of maths education. When I taught A level Maths, even with students who had achieved top grades at GCSE, it was not uncommon to find students who, for example, did not appreciate that multiplying by 1/2 was equivalent to dividing by 2, or were not comfortable cancelling fractions or factorising calculations to simplify them (I could cite many other examples). These students had certainly ‘covered’ these topics within their GCSE course, and had succeeded at GCSE Maths, in the sense that they had achieved high grades, but they had not mastered the ideas. This greatly hampered their progress in the transition from GCSE Maths to A level Maths.

The teachers who visited us from Shanghai, through the China-England Mathematics Teacher Exchange programme, spending a month teaching maths in primary schools across England, were very skilled at questioning and challenging children to engage more deeply with maths within the context of whole class teaching. This questioning really challenges all pupils to think and reflect, including those who appear to grasp the ideas quickly, and helps all pupils to develop sustainable learning. When questioning pupils in class, as well as asking for the answer, the Chinese teachers almost always follow up by asking HOW the answer was obtained and WHY that method worked. The ‘HOW?’ and, especially, the ‘WHY?’ questions challenge the pupils to really think about the maths, encouraging them to develop abstract thinking skills and make mathematical connections so that they can articulate their reasoning – ‘The Answer Is Only the Beginning’ (Ge Fang 2007). These questions probe the depth of pupils’ understanding, enabling all pupils to be challenged within the context of whole class teaching because they can be answered at different levels of sophistication. All pupils benefit from such questions because they encourage pupils to engage with and understand concepts more deeply, rather than merely ‘getting all the answers right’.

So, what are the answers to the two questions posed in the title of this blog?

Well, my NCETM colleagues and I believe both answers relate to DEPTH.

Meeting the needs of all pupils without differentiation of lesson content requires ensuring that both (i) when a pupil is slow to grasp an aspect of the curriculum, he or she is supported to master it and (ii) all pupils should be challenged to understand more deeply.

This can be achieved by:

1. Ensuring that any pupils having more difficulty in grasping any particular aspect of curriculum content are identified very rapidly and provided with extra support to help them master that content before moving on to new material.
   
   Same day intervention can provide the necessary support to secure learning before the next lesson. This requires rapid formative assessment and mechanisms for enabling pupils to access support as soon as the need has been identified. Some of the primary schools involved in the China-England Mathematics Teacher Exchange programme are already piloting ways to achieve this and are reporting immediate significant benefits for their pupils. The NCETM, working with the Maths Hubs, will publish case studies of how schools are doing this during the summer term.
   
   
2. Incorporating skilful questioning within whole class teaching.
   
   The success of teaching for mastery in the Far East (and in the schools employing such teaching here in England) suggests that all pupils benefit more from deeper understanding than from acceleration to new material. Deeper understanding can be achieved for all pupils by questioning that asks them to articulate HOW and WHY different mathematical techniques work, and to make deep mathematical connections. These questions can be accessed by pupils at different depths and we have seen the Shanghai teachers, and many English primary teachers who are adopting a teaching for mastery approach, use them very skilfully to really challenge even the highest attaining pupils. As pupils’ mathematical education continues, they experience deep, sustainable learning of increasingly sophisticated mathematical ideas. Shanghai teachers sometimes emphasise challenging questions by saying “dong nao qing”, meaning (I think!) “Please use your head!”, to make it very clear that deep thinking is expected. Evidence from the exchange visits suggests that pupils really enjoy such questions and are inspired by them - when discussing the Chinese lessons, one higher attaining child commented “I like the Chinese lessons, I need to think very hard because I know Miss Lu will ask me to explain why and I will need to have a good answer!”

Recording progress without levels requires recording evidence of depth of understanding of curriculum content, rather than merely showing pupils can ‘get the answers right’.

The NCETM, working with other maths experts and primary maths specialists from the Maths Hubs, is currently producing guidance on how to do this for the primary maths National Curriculum. For each curriculum statement, the guidance will show how to identify when a pupil has ‘mastered’ the curriculum content (meaning he or she is meeting national expectations and so ready to progress) and when a pupil is ‘working deeper’ (meaning he or she is exceeding national expectations in terms of depth of understanding). We believe this guidance will support primary teachers to record their pupils’ progress in a way that really supports their mathematical development, consistent with the aims of the new maths curriculum. This guidance will be published in the summer term.

The classes I have observed in our primary schools through the China-England Maths Teacher Exchange Programme and the ways in which the schools involved have developed their teaching after the Shanghai teachers returned home, combined with a ‘growth mindset’ view of people’s ability to succeed in maths, have convinced me that a teaching for mastery approach to maths teaching can really improve mathematical learning for our children here in England.

My NCETM colleagues and I are continually learning from the experiences of schools that are implementing a teaching for mastery approach to maths teaching. We will continue to promote the approach and to support schools and teachers who wish to adopt it. Key aspects of this support will be helping primary schools to address how to teach maths without differentiation of lesson content and how to record progress in maths without levels, both of which I believe will improve pupils’ learning. What’s more, if we get things right, this might even reduce teacher workload because you’ll only need to prepare one, whole class, lesson rather than several ‘differentiated’ variants, and recording progress should also become significantly simpler.

References

Russell, Susan Jo. (May, 2000). Developing Computational Fluency with Whole Numbers in the Elementary Grades. In Ferrucci, Beverly J. and Heid, M. Kathleen (eds). Millennium Focus Issue: Perspectives on Principles and Standards. The New England Math Journal. Volume XXXII, Number 2. Keene, NH: Association of Teachers of Mathematics in New England. Pages 40-54.

Gu L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In Lianghuo, F., Ngai-Ying, W., Jinfa, C., & Shiqi, L. (Eds.) How Chinese learn mathematics: Perspectives from insiders. Singapore: World Scientific Publishing Co. Pte. Ltd. 309 – 347.

Meg Schleppenbach, Michelle Perry, Kevin F. Miller, Linda Sims, Ge Fang (April 2007) The answer is only the beginning: Extended discourse in Chinese and U.S. mathematics classrooms. Journal of Educational Psychology (Impact Factor: 3.08). 04/2007; 99(2):380-396. DOI: 10.1037/0022-0663.99.2.380.