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Mastery in mathematics: What it is and why we should be doing it

Created on 03 October 2014 by ncetm_administrator
Updated on 03 October 2014 by ncetm_administrator
" I think it may well be the case that one of the most common ways we use differentiation in primary school mathematics… has had, and continues to have, a very negative effect on the mathematical attainment of our children at primary school and throughout their education. "

Charlie's Angles

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

Mastery in mathematics: What it is and why we should be doing it

Approaches to differentiation; defining a ‘mastery’ approach; the ‘England-China Mathematics Education Innovation Research Project’

I’ll be controversial: I think it may well be the case that one of the most common ways we use differentiation in primary school mathematics, which is intended to help challenge the ‘more able’ pupils and to help the ‘weaker’ pupils to grasp the basics, has had, and continues to have, a very negative effect on the mathematical attainment of our children at primary school and throughout their education, and that this is one of the root causes of our low position in international comparisons of achievement in mathematics education.

If my suspicion about the damage caused by current practice in differentiation in many maths lessons is correct, we should do something about it. However, I do recognise that an individual school’s interpretation of differentiation is rarely as black and white as I paint it below, and I know that many primary teachers put a great deal of thought and effort into developing differentiation models for maths teaching. For that reason, we should examine the evidence very carefully and carry out serious trials to help determine whether a different approach will improve children’s mathematical learning.

Put crudely, standard approaches to differentiation commonly used in our primary school maths lessons involve some children being identified as ‘mathematically weak’ and being taught a reduced curriculum with ‘easier’ work to do, whilst others are identified as ‘mathematically able’ and given extension tasks. This approach is used with the best of intentions: to give extra help to those who are having difficulty with maths, so they can grasp key ideas, and to challenge those who seem to grasp ideas quickly. It sounds like common sense. However, in the light of international evidence from high performing jurisdictions in the Far East, and the ‘mindset’1 research I referred to in my last blog, I’m beginning to wonder whether such approaches to differentiation may be very damaging in several ways.

For the children identified as ‘mathematically weak’:

  1. They are aware that they are being given less-demanding tasks, and this helps to fix them in a negative ‘I’m no good at maths’ mindset that will blight their mathematical futures.
  2. Because they are missing out on some of the curriculum, their access to the knowledge and understanding they need to make progress is restricted, so they get further and further behind, which reinforces their negative view of maths and their sense of exclusion.
  3. Being challenged (at a level appropriate to the individual) is a vital part of learning. With low challenge, children can get used to not thinking hard about ideas and persevering to achieve success.

For the children identified as ‘mathematically able’:

  1. Extension work, unless very skilfully managed, can encourage the idea that success in maths is like a race, with a constant need to rush ahead, or it can involve unfocused investigative work that contributes little to pupils’ understanding. This means extension work can often result in superficial learning. Secure progress in learning maths is based on developing procedural fluency and a deep understanding of concepts in parallel, enabling connections to be made between mathematical ideas. Without deep learning that develops both of these aspects, progress cannot be sustained.
  2. Being identified as ‘able’ can limit pupils’ future progress by making them unwilling to tackle maths they find demanding because they don’t want to challenge their perception of themselves as being ‘clever’ and therefore finding maths easy. A key finding from Carol Dweck’s work on mindsets1 is that you should not praise children for being clever when they succeed at something, but instead should praise them for working hard. That way, they will learn to associate achievement with effort (which is something they can influence themselves – by working hard!), not ‘cleverness’ (a trait perceived as absolute and that they cannot change).

I’m not going to address differentiation in secondary school maths teaching directly here as I plan to make that the subject of a future article in this blog, but I do think much of what I’m saying here also applies at secondary level.

Countries at the top of the table for attainment in mathematics education employ a mastery approach to teaching mathematics. Teachers in these countries do not differentiate their maths teaching by restricting the mathematics that ‘weaker’ children experience, whilst encouraging ‘able’ children to ‘get ahead’ through extension tasks (terms such as ‘weaker’ and ‘able’ are subjective, and imply that children’s ability in maths is fixed - I think they are very damaging and we should stop using them – many teachers already have, but many still use them). Instead, countries employing a mastery approach expose almost all of the children to the same curriculum content at the same pace, allowing them all full access to the curriculum by focusing on developing deep understanding and secure fluency with facts and procedures, and providing differentiation by offering rapid support and intervention to address each individual pupil’s needs. An approach based on mastery principles:

  • makes use of mathematical representations that expose the underlying structure of the mathematics;
  • helps children to make sense of concepts and achieve fluency through carefully structured questions, exercises and problems that use conceptual and procedural variation to provide ‘intelligent practice’, which develops conceptual understanding and procedural fluency in parallel;
  • blends whole class discussion and precise questioning with intelligent practice and, where necessary, individual support.

Colleagues at the NCETM and I have produced this short paper: ‘Mastery approaches to mathematics and the new National Curriculum’ , which defines what we mean by mastery, links it to the National Curriculum, and highlights its implications for the professional development of teachers. This work is supported by the Department for Education, which is keen to see how mastery teaching can raise achievement in schools. This video clip of an English year 2 primary class learning how to add fractions shows mastery teaching in action.

A major element of the NCETM’s leadership and development of mastery teaching is through the DfE-funded ‘England-China Mathematics Education Innovation Research Project’, involving more than 60 teachers from England shadowing primary mathematics teachers in Shanghai (the English teachers are in Shanghai as I write this) to observe mastery teaching in practice, followed by the Shanghai teachers coming to England to exemplify mastery teaching in our classrooms and to support the English teachers in making sense of and trying out a mastery approach to their maths teaching.

This project is being run through the NCETM’s Maths Hubs initiative. Testing out new ideas in the classroom to gather evidence of how effective they are, before advocating which should be adopted more widely, is a key role of the Maths Hubs. The English primary school teachers involved have embarked on this project with great enthusiasm. They have a strong desire to learn as much as they can about how maths is taught in Shanghai and want to use what they learn to develop their own teaching back in England to improve their pupils’ learning.

The project will help us to develop how we use the mastery approach to maths teaching in our primary schools, to improve maths education and the mathematical futures of our young people. It also provides a brilliant opportunity to develop close working relationships between the English and Chinese teachers involved, so that they can learn from each other, to the benefit of teachers and children in both England and Shanghai.

It might also lead us to start moving away from the practice of dividing primary maths classes into different tables, with harmless sounding names, but names which nevertheless don’t fool even the pupils on the ‘red’ table!

It will not be quick or straightforward to improve the learning of our lower attaining pupils, narrowing the wide gaps between pupils’ mathematical attainment that currently exist in our classrooms, but we must be committed to doing so. I believe that mastery teaching will – with time and effort – enable us to achieve this.

1. ‘Mindset’, Carol Dweck, 2006, ISBN 978-1-78033-200-0

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30 August 2019 15:47
Reading the mixed age group post below was interesting as I was lucky enough this year to visit a yr5/6 class where this was done really well. The teacher sometimes taught the two age groups seperately and sometimes together. The independent practice provided was always appropriate to the age group. Comments from the children showed how well this worked. Yr 5 enjoyed learning from and with their peers and yr 6 found the revision of previously learnt material useful in consolidating their knowledge.
By lstaples
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17 March 2019 06:31
This comment has been reviewed during our moderation process and blocked
because it breaches our terms and conditions

12 August 2016 08:14
Which top ranking countries use a "mastery" aproach? Do any of them have a culture similar to the UK's? Or one to which we should aspire? What is their attititude to educational inclusion? Where are the case studies proving the mastery aproach works in EYSF/KS1?
19 February 2016 08:39
I would like to clarrify I suggested 2 x 30/35 min lessons plus 2 x 30/35 min intelligent practice time. This is not half the time pupils in singal aged classes are getting if they follow the Shanghai model. Shanghai lessons are much shorter because this is the optimum time to sustain children's concentration.

The second model is to teach them together for a period of time and then group teach. Teaching for mastery is not just about structure but the underlying principles of going deeper, developing fluency and reasoning,making connections spending longer time on topics etc.
18 February 2016 21:56
I am maths subject leader in a small primary school with mixed age classes. I have been following a mastery approach since last spring and think that it works really well as an approach. The trouble I have found with covering fewer things in greater depth is that it takes more time and I have been starting at the simplest point to ensure I encompass those children who haven't yet grasped concepts at the most basic level.

I have a y2/3 class and find that I often run out of time to cover much of the y3 curriculum. With the interim assessment frameworks requiring pupils to achieve all assessment points to be considered at the expected level, I know that most if not all of my class (even my year 2s) will not achieve this as I am going to run out of time to adequately cover certain areas such as time.

I contacted Debbie Morgan in the summer about this predicted problem and she suggested teaching year groups separately, which means half the time per year group single age classes get and therefore still not covering the curriculum in sufficient depth; or teaching them together, which I like but has resulted in the issues raised above.
By Oirecocobean
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27 November 2015 08:12
Hi there

find this blog really interesting. I've been doing alot of research recently on Mastery as I'm supporitng a school. Just wondering if there is any advice out there with regards to planning/models/templates/examples etc?
28 May 2015 09:15
At the NCETM we are aware that what might be called a 'pure' mastery approach needs modification for mixed age classes, and, during this developmental stage of teaching for mastery more generally, we are gathering evidence from around the country on how elements of the approach can successfully be integretaed to mixed-age teaching.
26 May 2015 10:05
Could you please clarify how this approach will work in a mixed age class as I currently teach Reception, Year 1 and Year 2 in a class of 20 children.
By cheryltom
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27 February 2015 15:47
Interesting blog on the EEF's report on 'Maths Mastery'

By Bambol
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16 December 2014 10:36
The benefit of having a well planned and illustrated common text from which to teach maths is probably not in dispute. What we need reassurance on is whether such an outdated medium is capable of engaging 21st century pupils, especially 5-7 year olds who have come to expect inspiration from the interactive whiteboard. Are Singapore pupils working purely from textbooks or do they have access to an e-version that can be displayed on the IWB?
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02 December 2014 21:55
While the problem of differentiation has been rightly identified, I am not convinced of the soution - when the same lessons are delivered with same mediums of instruction, it will work only when all children learn the same way. Unfortunately, this is not the case. Some children learn by reading, some by listening, some by imagination, and some by doing. In most cases, one or more or all of these are involved. Learning is a personal process - I am not enirely sure how a Chinese approach (where children are culturally trained not to question, but to accept whatever they are being told) will fit into a nation where Children are actively encouraged to think critically.
By notprathap
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15 November 2014 12:57
Excellent article - just what I am looking for to support my reasoning behind stopping rapid acceleration of the curriculum but instead developing more complex problem solving and investigative work to deepen understanding
By cpitois
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14 November 2014 10:36
There is a company that has published 'Singapore' maths textbooks for the UK which follows the UK national curriculum and relates to the above blog. My school is beginning to use them - mixed with a good amount of CPD - they are very effective. http://www.mathsnoproblem.co.uk/ I am also aware that OUP will soon be publishing textbooks of a similar nature.
By jhannay
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14 November 2014 05:35
Is the new Mastery approach to teaching Maths centered around pupil talk? I am slightly worried that adopting different cultural practices which advocate whole class teaching and the use of textbooks, may distract from what we all know is at the heart of excellent Maths teaching in this country.

Andrew Murphy (MAST teacher)
By nathalie
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13 November 2014 23:05
I would like to echo the above point. NCETM seems to rapidly be becoming the Government's mouthpiece.

"Our low position in international comparisons"? This would be PISA? A deeply flawed comparison as any examination of their research methods would reveal. And even under PISA we are at broadly similar levels to most similar European nations.

I don't entirely disagree with much of the above blog, but am becoming increasingly suspicious about the enthusiasm with which NCETM promotes the Government's position now, especially given the widespread criticism of the NC from most of the maths 'community' upon its publication.

A little more critical evaluation is needed soon or you will lose credibility with the profession.
By bears1404
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13 November 2014 19:37
Mastery is an interesting idea, but it is also subtle and problematic. Continually trumpeting the government's line means NCETM is in danger of losing credibility.

Take this quote from the latest maths programmes of study:

The expectation is that the majority of pupils will move through the
programmes of study at broadly the same pace. When to progress should
always be based on the security of pupils' understanding and their readiness
to progress to the next stage.

This could be intepreted as suggesting that maths learning takes place in small, linear increments and that students should never be challenged, or thrown in at the deep end, or have to cope with being confused. Another, quite different view of learning, is that it is about creating, developing and continually modifying a network of ideas. It is not straightforward to reconcile this with a mastery view.
By lukeandjim
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30 October 2014 18:59
The Mastery Approach is clearly going to make an impact, but has anyone spoken to Ofsted about this? The Mastery Approach carries with it a change to the way differentiation is used in maths and a change to what we mean by differentiation as emphasised by the quotation at the top of the blog. It is still common to read in inspection reports that differentiation isn't good enough because the children are doing the same work in maths. How can schools develop a new approach to differentiation when the inspectorate has set expectations? Will this change? How will schools know it has changed? Headteachers will be wary of changing differentiation unless Ofsted teams accept it as valid.
By JLPearson
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29 October 2014 15:21
Thankyou. Thankyou, thankyou, thankyou. This is exactly what I think and feel about 'differentiation'. It is extremely helpful to have someone who holds a position such as yours articulating these ideas. I am about to start designing a research project with at least one, hopefully more, local schools into this area. Crucially, we hope to start building a local, relevant evidence base for the changes in practice I am recommending colleagues make. I hope that, in the long term, it will enable them to feel more empowered to truly meet the needs of the children they teach and hits all the buzzword agendas especially 'closing the gap' and 'ensuring all pupils make progress'. You have renewed my flagging spirits!
By SParkes1407
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19 October 2014 20:07
Is this Singapore Maths? Where are the textbooks referred to in the Mastery paper. I can understand why there may be hesitation in the endorsement of one particular set of textbooks but is it safe to say that none of those currently on the market come close to delivering on the vision outlined in this paper? As a newly appointed Maths Co-ordinator in my primary school I am currently investigating the direction our school should now take to achieve the increased age related expectations timeously. There are of course significant budgetary constraints involved with buying into schemes of work and textbooks. I am aware of the Maths Enhancement Programme currently freely available on the CIMT website. As a significant shift in teaching styles and materials is apparently what is required some concrete pointers would be gratefully received.
By Rebechan
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15 October 2014 22:45
Very good article; but it comes with implications for inclusion as time moves on and "little Jenny" fails to keep up with the majority: as all (to take it to an extreme) prepare to take the "Higher" paper at Key Stage 4, "little Jenny" (suffering from seemingly randomly occuring dysclaculia and dyslexia) requires a TA who has an awesome amount of mathematical (and pedagogical) knowledge to "bridge the gap" between where the teacher/class is and where she is. It also requires that TA to be able to focus her time exclusively on "little Jenny"; which (best will in the world), stretched money and resources won't allow. Of course, one might argue the teacher ought to brief the TA and prepare resources appropriately. But the TA works 9am to 3pm (strict) and is sufficiently "in demand" to be tough to find time to brief on the nuarances of dyslexia and dyscalculia; and the teacher has a mixed ability class of "well north of" 15 to teach; she'll do her best; but (politics, diplomacy & PR aside) we do wor in the "real world". So does "little Jenny" languish? Or does she get "disappeared" to a separate intervention class or indeed special school? Policy-level decisions; not comfortable ones. I look forward to your article about mastery in non-selective state secondary schools with interest; mainly because I agree that having all aim for the highest level is most likely to motivate the bigger majority (indeed I have had recent experience of this working). However, there will be a very small minority who are in the class for reasons of social inclusion who may not get as much from this technique as the others; and they may be vocal and have vocal parents (of a type the PRC is less familiar with at present - Hong Kong aside).
By BW_2012
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15 October 2014 02:09
Thanks for that excellent summary of mastery teaching, Charlie. I will be passing on the link to teachers in Australia.

You may be interested to know that there is a renewed emphasis on maths pedagogy in Australia as well (we also have a new National Curriculum) - with a focus on 'explicit teaching'. We don't lessen the importance of differentiation, rather we place it withing the explicit teaching model. Interested teachers can see more on explicit teaching, especially in relation to interactive teaching strategies on an IWB, at this link: http://schoolcentre.s3.amazonaws.com/content/22600/file/Explicit_Teaching-Learning_Model.pdf

Peter Nowland (peter@schoolcentre.co.uk)

Director, SchoolCentre
By hopalong
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14 October 2014 12:17
Inspiring - thank you. I recommend reading the paper Mastery approaches to mathematics and the new National Curriculum as well.

I really thought I was retiring completely at the end of last year, but I am doing a little bit more. So I checked out the NCETM again, to keep up to date - and what an uplifting experience that has been this time. I think that your paper Mastery approaches to mathematics and the new national curriculum offers the beginnings of a valuable framework for teachers and other mathematics educators. Problem solving and reasoning are at the core of mathematical activity; easy recall of number knowledge and facility with procedures are necessary functional skills and essential prerequisites for freeing up processing capacity to support problem solving and reasoning. You offer the beginnings of a way of integrating these coherently. And the argument presented here about differentiation follows from the principles of the Mastery approach (as you are outlining it). This is what is important and inspiring - that the core ideas give rise to the justifications for judgements about pedagogic choices.
Best wishes and good luck with all this.
Bob Davies b.davies@bathspa.ac.uk
Module Leader for Learning in Mathematics at Bath Spa University (for just a bit longer).
By BobDavies
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