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"How can we meet the needs of all pupils without differentiation of lesson content? How can we record progress without levels? … I believe the answers to these questions have the potential to reduce teacher workload, as well as improving the mathematical learning of their pupils."

Charlie's Angles

Thoughts on topical issues of mathematics education from the NCETM’s Director, Charlie Stripp

How can we meet the needs of all pupils without differentiation of lesson content?
How can we record progress without levels?

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:

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.

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.

04 October 2015 10:54 This post - https://timdracup.wordpress.com/2015/09/24/more-on-mastery-depth-and-high-attainers/ - explains why the NCETM's emphasis on 'depth instead of pace' is not fully consistent with the National Curriculum's emphasis on 'depth before pace'. It reviews the NCETM's 'assessment of mastery' materials in this light and considers the read-across to 2016 tests and statutory teacher assessment.

15 July 2015 09:41 Where is the assessment guidance that is to be published during the summer term? Have I missed something? Doesn't seem easy to find.

12 May 2015 15:21 Is mastery the beginning of a new revolution in maths learning or is it just another great idea that will not take off? The ideas behind teaching for understanding and problem solving (which of course requires the ability to reason) have been around since the eighties for instance just look back at the Shell centre resources on problems with patterns and numbers. So why over 30 years on did these great developments in maths teaching not fully take off? If we know the answer to this question and learn from our mistakes surely we will have a chance to make this fundamental and necessary shift in maths teaching actually fully develop and turn out very numarate, confident youngsters who are much better prepared for the society in which they will live and work.

Why will it be different this time? Having jumped on the Maths Hub bus with two hands I feel that they have a crucial role to play because at their heart are maths teaching practitioners who really understand the opportunities (and pressures) that exist. Being the maths lead for the YR maths hub I would welcome thoughts on any lessons that need to be learnt so that I can make proper use of this rare and privileged opportunity to make a difference.

29 April 2015 23:44 Thanks for your response Debbie and I don't think anyone could argue about the importance of ensuring that all children have the opportunity to reason mathematically and this is easily done by developing teachers' questioning skills in lessons. However at some point mastery of some (age appropriate) 'sophisticated' aspects of mathematics requires us to remember certain things. In order to reason one must be able to articulate one's thinking and this is where the pupils with poor auditory and visual memory are unable to make progress because the cannot fluently retrieve the vocabulary or the images that will support their thinking. We are taking about a minority of pupils in our classrooms here but they are the ones that the teachers are rightly concerned about (even if they fall within the "acceptable 15% below floor" that the outgoing goverment have suggested will be allowed), because the cannot see how these children, even with additional support will keep apace in a class that the mastery approach suggests is possible to do. I do think it is worth giving this the best go we can in the circumstances that our Engish classrooms present us with and there is much yet to learn so I am certainly not suggesting that we fall at the first hurdle but NCETM now have an important job of convincing sceptical teachers who teach pupils who find learning maths difficult that there are ways to fix their difficulties by presenting case studies of what is working so that not just the pupils in 3 form entry primaries in the leafy suburbs respond well to this but that it can be appropriate for all class wherever you teach. And of course this isn't going to happen over-night so schools need breathing space from stakeholders while the approach for mastery is embedded.

27 April 2015 11:52 I have read the comments with interest. Here are a few comments from me re the points raised above:

*In relation to national assessment the term mastery equates to 'national expectations'. A key principle underpinning a mastery curriculum is that all/the majority of pupils can and will achieve.

*The countries who adopt a mastery approach do not have mixed aged classes, so there are no existing models and we will need to create our own. There are schools that are doing this, as the one outlined above and it is important that we share these as they are developed. Another model with a class of two age groups might be to teach two 35min maths lessons where the group that isn't being taught engages in follow up practice from their previous lesson. The key thing is that sustainable learning is developed through spending longer time on topics, going deeper, developing both conceptual and procedural fluency.

*Re struggling learners, who maybe have poor memory skills; the key here is to develop their reasoning skills and their ability to make connections. When you reason and make connections in mathematics, you cut down on the amount of mathematics to be 'remembered'. This is a feature of teaching for mastery where time is taken to explore the mathematics and look at it in different ways, through the application of variation. Conceptual variation gives access to multiple representations of the same concept, giving children the opportunity to make connections and generalise the mathematics. Once children have generalised they have secured one piece of mathematics that can be applied to multiple contexts. A mastery approach to teaching will teach all children to think and reason in mathematics. One teacher applying a mastery approach and involved in the textbook project recently said our children used to be passive learners, but now it's as if they have woken up to mathematics, they are thinking and reasoning and coming up with their own ideas.

24 April 2015 14:48 Mastery approaches are not unique to Singapore. Richard Dunne crafted a Mastery Curriculum employing a concrete Learning System for mathematics with a teaching cycle for whole class teaching that differentiates by assistance. He devoted his lifetime to researching his approach with deep underpinning philosophy and consistent pedagogy. It is called 'Maths Makes Sense' (Published by OUP). Richard died just before the new NC (which he was an advisor to) went out for review and then became statutory. As Director of Richard Dunne Maths Training I know many MMS schools who are showing superb closing gap data and significant, sustainable gains in maths progress for all children. Advocate schools hold 'Open Mornings' for visitors to view the maths in their schools and talk about their experiences. Interestingly, the NCETM have never taken up the offer of visiting one of these schools where the program is established to gain a picture of alternative Mastery teaching where text books are not a necessary component and where children are excited by maths and teachers' subject knowledge is supported by outstanding PD provision. Visit www.richarddunnemaths.com to see for yourselves or 'Google' to read some of Richard Dunne's research into learning from Exeter University - he really had been advocating mastery without the need to differentiate lesson content from before the old Numeracy strategy - just back then his ideas were regarded as too 'left field' I imagine to be taken seriously. They are now of course 'fashionable'!

23 April 2015 21:10 A good read and helping to further my thinking more in the drive for a mastery approach.

I echo the thoughts of Laurie. I would be interested to hear how intervention can be used to support those children to 'prevent the gap'. I am particularly focusing on the Year 1 cohort at the moment to monitor how they are achieving all of the objectives so they move to year 2 at broadly the same level of understanding. Having a clear understanding of what this might look like would be useful.

I am also interesting in my SENCo role about what makes effective intervention for children - particulary for those with a poor working memory. It is important that this is caught as early as possible.

23 April 2015 04:20 Thanks Charlie. Another very thought-provoking blog. I would very much like to see some examples of what this represents as rich engaging activities at key stage 3 and 4. e.g. A few months ago you wrote about a Ben Nevis temperature and elevation problem solving question using proportional understanding and rates of change (I am devleoping a dynamic model of this situation). More descriptions of exemplar activities like this please relevant to the above article will enable us to integrate swiftly such teaching ideals. Matt Dunbar

21 April 2015 22:38 I am keen to implement such an approach and currently use many of the 'low threshold, high ceiling' type tasks in order that all children can have the necessary challenge on the same activity. However I currently have a mixed year3/4 class and from September it will be years 3/4/5 so also very keen to listen to advice regarding taking a mastery approach to teaching the new 'year based' curriculum.

21 April 2015 22:32 I completely agree that we should focus less on pace and more on depth. I have found same day interventions with small groups very valuable and where possible teacher led.

21 April 2015 20:59 I am interested in learning more about this approach and, in particular, how this would work with mixed year groups. I currently teach a 3 stage class, any advice would be welcome.

21 April 2015 19:45 I read the blog with interest and agree with all of the basic principles espoused. However, I am confused by one aspect. Not so much confused actually, more dealing with the different messages we are all getting. It revolves around the distinction (or otherwise) of the terms National Standard and Mastery. At a conference in London I asked the DfE keynote speaker about the L4B 'myth' I.e. the new National Standards (yr 6) being roughly equivalent with an old L4B. The answer was clear and categorical... National Standard would be roughly equivalent to old L4B (and therefore roughly 100 on the new scaled scores) and Mastery would be something that sits above this standard. The blog here seems to suggest that Mastery is the approach to securing the National Standard. I do like this latter idea better, but am also worried that the profession is receiving too many mixed messages. It would be interesting to hear Charlie's views on this.

21 April 2015 18:49 I have a mixed age class (Y1/2) and was involved in the Shanghai exchange. Whilst for the three weeks that the Shanghai teacher taught in my class, we put all the Y1s/Y2s together from two of our classes, I am now back with my mixed age group. I am trying to implement my own interpretation of a 'mastery' approach. It is certainly more challenging than teaching a straight year group, but I am managing to keep the class more together than I used to. I inevitably do differentiate to meet the needs of all my pupils (and I do have very low and very attaining pupils in each year group) but we are generally all working on the same concept, and not moving beyond Y2 objectives (which actually makes life much easier). The ideas for 'intelligent practice' offer some support, eg use of missing numbers, or expecting higher attainers to use more sophisticated reasoning skills. Also, the expectation of the precise use of mathematical language (even if not all children can use it, at least they are hearing terms used correctly) and explanation of the 'how' can easily be used in any class.

It would be great, NCETM, if we could continue to consider and support the achievement of mastery within mixed year groups. In rural counties there are many! I am often asked by colleagues in other schools locally how they can manage this in their classes.

20 April 2015 20:47 How should a mastery curriculumbe taught to children in mixed age classes? Mixed age classes could be 2 year groups, or 3 or even a whole KS2 class.

19 April 2015 21:08 I am a convert to the mastery approach. I do hear from teachers however that they cannot see how this will work for those children who really struggle. I recently provided a demonstration lesson on fractions with a small group of 'low attaining' Y3 pupils. These pupils struggled to repeat back more than three words in sequence because of their poor auditory and short term memory skills. This was a significant barrier to their progress in mathematics which relies on learning, remembering and using subject-specific vocabulary. I would be interested to hear from schools about the interventions they are using with children experiencing similar challenges to their learning of maths in their drive to 'prevent the gap' .

Further examples of Intelligent Practice using procedural and conceptual variation can be found this months primary magazine PM4 https://www.ncetm.org.uk/resources/46773

Exemplification of deep learning will be most welcome and an assurance from the NCETM that any use further use of mastery in relation to maths by the DfE (and I am referring here to the draft performance descriptors) is in harmony with your definition here and elsewhere on the website as schools who are trying to be one step ahead are interpreting mastery as 'better than age expectation'.

It would be fabulous to have video clips of teachers using variation to add to the suite of NC support Videos that are already available, to help teachers grasp what this looks like in practice.

19 April 2015 13:49 Hopefully mathematical learning can be improved using this approach alongside developing more positive attitudes to maths.

However Charlie (& team), I'd like to hear your thoughts on what the 'mastery approach' looks like in a class with two or sometimes three year groups of children. There are many primary schools here with mixed age classes.

17 April 2015 21:19 Timely following a 'should I move them on' question from a junior colleague. Need to convince some that depth is as important as breadth. Experienced the GCSE to A level issue with my son who will unfortunately drop maths at thend of y12 as a consequence.

This post - https://timdracup.wordpress.com/2015/09/24/more-on-mastery-depth-and-high-attainers/ - explains why the NCETM's emphasis on 'depth instead of pace' is not fully consistent with the National Curriculum's emphasis on 'depth before pace'. It reviews the NCETM's 'assessment of mastery' materials in this light and considers the read-across to 2016 tests and statutory teacher assessment.