"Based on what I’ve seen at primary level, I’m optimistic that teaching for mastery can have a major influence on our secondary maths teaching and learning."

Charlie's Angles

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

Teaching mathematics for mastery at secondary school

So far, the NCETM’s work relating to the teaching for mastery philosophy that my colleagues and I have been championing has focused mainly on primary school maths. It is being positively received and primary schools that are committing to the approach are reporting that, even at this early stage, the changes they have made to their practice are clearly having a beneficial impact.

My background is in secondary mathematics teaching and I believe strongly that a similar approach, suitably adapted, has the potential to support profound improvements in pupils’ learning in secondary maths. Over the coming months at the NCETM we will be focusing more of our work on secondary mathematics, whilst continuing to build on our primary maths work, and we will develop our understanding of how a teaching for mastery approach might be applied to help raise pupils’ achievement in secondary mathematics. The aims of the new curriculum and Maths GCSE are challenging and I believe teaching for mastery could be an effective way to achieve them.

This blog post gives my initial thoughts around mastery in the context of secondary maths teaching. As discussed in previous blog posts, a teaching for mastery approach in mathematics is underpinned by the belief that all pupils are capable of understanding and succeeding at school level mathematics. The approach involves the whole class accessing the curriculum together, with additional challenge provided through pupils accessing material at greater depth, rather than through acceleration to new material from later in the programmes of study: high attaining year 8 pupils shouldn’t be rushing onto to GCSE material, but instead should be going deeper into KS3 concepts. Carefully crafted lesson design and skilled questioning encourage deep mathematical thinking in all pupils, helping them to identify mathematical connections and steering them to develop mathematical reasoning and problem solving skills. Exercises and learning activities designed to provide intelligent practice enable pupils to develop conceptual understanding, at the same time as reinforcing their factual knowledge and procedural fluency. Damaging gaps in knowledge and understanding that impede progress are avoided by rapid identification and teacher intervention.

The American cognitive scientist, Daniel Willingham, has published an interesting article, Is It True That Some People Just Can’t Do Math? on teaching and learning mathematics. His work attempts to apply what we know about how our minds work to inform how we should teach, and his ideas are consistent with the teaching for mastery approach to mathematics outlined above: ‘Virtually everyone is capable of learning the numeracy content and skills required for good citizenship: an understanding of arithmetic procedures, algebra, geometry, and probability deep enough to allow application to problems in our daily lives’ and ‘For most topics, it does not make sense to teach concepts first or to teach procedures first; both should be taught in concert. Gaining knowledge and understanding of one supports comprehension of the other.’

In my previous blog post I highlighted how shallow, superficial learning can hold back even the highest attaining pupils when they make the transition from GCSE to A level in mathematics, and suggested that teaching for mastery could help prevent this, by ensuring pupils develop solid mathematics foundations, make connections and build a strong conceptual framework alongside procedural fluency. The damaging consequences of a superficial, exam-focused, rote learning approach to mathematics are highlighted in Jo Boaler’s book Experiencing School Mathematics, which examines in detail the very contrasting approaches to teaching mathematics in two English secondary schools. The book was published in the late 1990s, but it is still relevant to today’s secondary mathematics teaching, presenting interesting evidence about the effectiveness of ‘traditional’ and ‘progressive’ maths teaching, and also discussing gender issues, setting and ’mixed ability’ maths teaching – it’s definitely worth a look if you are not familiar with it. I think that a teaching for mastery approach is likely to prove an effective way to address many of the issues raised in this book because it requires that pupils engage actively with mathematical ideas and believes that all pupils can and should have access to deep learning in mathematics.

Another thought-provoking book relevant to secondary maths teaching is Malcolm Swan’s Collaborative Learning in Mathematics. Collaborative learning involves encouraging pupils to discuss carefully designed mathematical activities. These activities aim to expose misconceptions and encourage pupils to explain their thinking, challenging one another’s ideas and understanding – this paper provides a good summary. There is strong evidence that collaborative learning, focused on well-constructed tasks and activities, is very effective at engaging all pupils with mathematical thinking.

Andrew Blair, a practising mathematics teacher and regular contributor to online debate about maths education research, recently blogged a critique of the NCETM’s support for teaching maths for mastery. Based on my previous blog post, he asserts that the second aim (second in order in the curriculum document – the three aims of fluency, reasoning and problem solving are all equally important!) of the National Curriculum, namely to ‘Ensure all pupils reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language’ is not compatible with a mastery approach to mathematics teaching, and that teaching mathematics for mastery treats pupils as passive learners. I certainly did not intend to give this impression. As anyone who had the exciting opportunity to observe one of the Shanghai primary mathematics teachers’ lessons during the recent exchange visits will know, pupils are certainly not treated as passive learners. Instead, they are challenged to explain their answers and to make up their own examples of how to apply mathematical concepts. Exercises and tasks are designed so that pupils can engage with them through thinking critically about mathematical situations, actively developing their own understanding. I think many of the tasks on Andrew’s website Inquiry Maths, and the example he offered in his blog post, are excellent examples of tasks that engage pupils in mathematical reasoning and in developing their own conceptual understanding, whilst also reinforcing their factual knowledge and procedural fluency. These tasks could easily and successfully be used by a teacher committed to teaching for mastery.

I believe that carefully designed collaborative learning activities and inquiry tasks can be used as intelligent practice within the well-designed lessons that are central to a teaching for mastery approach to mathematics. However, I think a reflection from one of the Chinese maths teachers involved in the primary teacher exchange is especially relevant to their use and explains a key aspect of teaching for mastery (I paraphrase!):

The teacher holds the string of a kite and lets it unravel so that the pupils can go their own way to explore and make their own sense of the mathematics, but the teacher must also reel the kite back in on a regular basis, to ensure that all pupils have made the most valuable connections and developed correct understanding of the mathematical ideas.

The Chinese teacher then observed that teachers sometimes forget to reel the kite back in! For all pupils to develop the conceptual understanding of mathematical ideas required for sustainable learning, regular ‘reeling in’ is vital. This is a key feature of teaching for mastery and without it pupils may fail to make the key connections that are important for successful progress.

The NCETM will be working with the Maths Hubs on a secondary maths teacher exchange programme with Shanghai in the coming academic year, focusing on maths teaching at the KS2/3 transition. This should provide powerful evidence about how we can develop teaching for mastery at secondary level (I’ll have more to say about this in future blog posts, as the exchange progresses). Based on what I’ve seen at primary level, I’m optimistic that teaching for mastery can have a major influence on our secondary maths teaching and learning, enabling many more pupils to engage positively with maths, enjoy learning it and develop a sustainable understanding of mathematical ideas. This will prepare them well for using maths with confidence in their future life and work, as well as helping them to succeed in their Maths GCSE.

References

‘Is it True That Some People Just Can’t do Math?’ Daniel Willingham, American Educator, winter 2009-10
‘Experiencing School Mathematics’, Jo Boaler, 1997. ISBN 0335199623
‘Collaborative Learning in Mathematics,’ Malcolm Swan, 2006 ISBN 186201311X

If you find Daniel Willingham’s article interesting, you might also find his book ‘Why Don’t Students Like School’, 2009. ISBN 9780470591963 – a worthwhile read, though it is general, rather than maths specific.

What are your thoughts about this? We've started some discussion threads in the Maths Café forum - log on and join in the debate.

13 July 2017 12:45 Very pleased to discover this article. I will be a trainee this September (2017) and I would very much like to focus on what Secondary Maths can learn from Primary Maths, especially through collaborative work. It's been 30 years since I was at school and teaching methods in secondary look exactly the same while primary is unrecognisable.

26 November 2015 02:35 I have read Charlie's blog with interest as being one of the primary schools which have embraced the mastery approach, taken part in the research trial into the Singapore style of teaching and use of textbooks (evidence seen in the NCETM end of year 1 report for those interested - we of course have our own in school data to support these findings), having observed teachers from Shanghai teach in both primary and secondary schools over here .. and delivering CPD across a range of schools around the region ....... but most of all, teaching daily following the mastery approach as well as being the maths leader in our large primary school.

I can't say it has been easy or that there haven't been challenges along the way, or that initially it didn't add to teacher workload, but it has certainly been effective. The NCETM and the Maths Hubs are working closely with primary schools who have trialled this approach,so they are aware of the impact it is having.

When I gave the Y1 NCETM test to our children at the beginning of Y1 last year, followed by the pupil interviews, I was not surprised by the low scores, because it was so challenging. I was however, shocked by the high scores just a few months later as children at the end of Y1 tackled problems they'd just not have had a clue about previously. This year I'm lucky enough to be teaching in Y2 and teachers from many other schools have visited us to observe maths lessons - the depth of children's understanding is undeniable.

Split lessons and same day catch up sessions sound tricky, but once you've got over the timetabling issues, they work well and make a huge difference. I wouldn't do it any other way now.

And teacher workload - it's getting easier this way as the changes are becomming embedded.

Skilled workforce? We have invested heavily in CPD for our teachers, especially in mathematics, so yes all our teachers are skilled - they all have additional qualifications in mathematics. The Maths Hubs have been supporting many schools in up-skilling their workforce through the PD Lead and SKE/TSST programmes, which have been supported by the NCETM and various universities.

Behaviour - well I know I work in a primary and not a secondary, so I am aware it is ver y different, but behaviour at our school is excellent - not a learning moment is wasted through poor behaviour as it is tackled immediately and consistently, but also because the children are so engaged and motivated - they really want to learn!

We've begun to work with some of our local secondary schools and share our teaching approaches with them.

Please don't dismiss 'mastery' as a fad or just another initiative thrust upon schools - it makes sense and speaking from experience, it works. I wish my 3 children had gone through schools which had used this approach as it would have meant I didn't have to teach them maths myself at the end of a day of work.

Yes teachers work long hours ..... I start at 7am and leave school at 7pm and then work in the evenings and weekends .... but this isn't a job is it ... I see it as a vocation. If you just want a job and a pay packet at the end of the month, don't go into teaching, but if you want to see the rewards for your labour, it is the best profession to be in. Please go and see teaching in a school where the mastery approach has been adopted before judging it.

07 October 2015 16:12 The damaging consequences of a superficial, exam-focused, rote learning approach to mathematics are highlighted in Jo Boaler’s book…..”

As usual the exam system is being criticised, I remember being at a presentation by Malcolm Swann who was demonstrating his Mathematics Assessment Project. When asked how is the student’s learning assessed he was unable to provide a coherent answer. You cannot ignore the examination system unless you are prepared to provide an alternative that is acceptable to employers globally. Stop ignoring the fact that most children are sent to school to perform the best they can in external examinations. From my experience Ofsted inspections are driven by data and that data is derived from the exam results. This leads to schools judging teachers solely by exam results, which unfortunately is largely determined by the type of school, the catchment and the set.

Furthermore, collaborative learning in mathematics is an ill thought out impossible dream. The discipline in UK schools is so poor that this approach is rarely going to work. The teachers do not have the time, energy and sometimes knowledge for this to work. There simply is not enough time in the school day to attempt this approach. At best students get 3 to 4 hours maths teaching per week, my experience was 2.5 hours at KS4. The only sensible option was to use a text book.

This leads me onto the latest fashion of quoting Shanghai maths. Why are the Asian counties so successful on the PISA rankings (which is an examination), it’s because the students behave themselves (watch “Are our kids tough enough”), they have more maths teaching time (10 hours was quoted in aforementioned programme), the teachers are well paid, are respected in society and have more non-contact time. When children have a longer school day in which they spend 2 hours in supervised conditions to complete homework, of course teachers can intervene. To talk about intervention in the context of our education is extraordinarily unhelpful. The end result is that well-meaning, hard-working, dedicated teachers are lead to believe that they are not good enough.

The vast majority of maths teachers are dedicated, hardworking and enthusiastic who are doing their level best under extraordinarily difficult circumstances. It may come as no surprise to you that I have left the profession and am earning a living as a private tutor passing on my knowledge and using text books for practice.

07 October 2015 16:12 “It is being positively received and primary schools that are committing to the approach are reporting that, even at this early stage, the changes they have made to their practice are clearly having a beneficial impact.”

What evidence do have that this is true? How did you measure this? What was the size of the sample?

“I believe strongly that a similar approach.” “I believe teaching for mastery could be an effective way to achieve them.” “..the belief that all pupils are capable of understanding and succeeding at school level mathematics.” “I believe that carefully designed collaborative learning activities a…”

This type of phrasing is often used in sales and is referred to as weasel words. Mathematics is the one subject where evidence based logical arguments should be used.

“…high attaining year 8 pupils shouldn’t be rushing onto to GCSE material, but instead should be going deeper into KS3 concepts.”

The GCSE consists of topics taught in KS2, KS3 and KS4, this sentence is nonsense. There simply isn’t a distinction between KS3 maths and GCSE maths other than the GCSE contains a small number of more advanced topics.

“Damaging gaps in knowledge and understanding that impede progress are avoided by rapid identification and teacher intervention.”

So we have an underpaid, overworked and, in many schools, an under qualified workforce trying to teach large classes where discipline is no longer a priority. This people are going to be able to provide rapid identification and teacher intervention. Are you being serious? As my late friend and colleague used to say “You are in the tallest ivory tower in the middle of Fantasy Island.” If you really wish to see improvements in the attainment of mathematics you cannot continue to promote constant teaching revolutions ignoring the environment in which students are being taught. Quite simply the only way for teachers to manage teaching 200+ students per week is by basing lessons on a text book and spending 10 minutes per lesson planning. Just take a full-time secondary teacher’s timetable and set up a spreadsheet to measure how many hours per week you expect a teacher to work. You will very quickly exceed the European Working Time Directive limit of 48 hours. The average secondary school teacher works on average 56 hours per week. So until the day that this fact is no longer ignored there will be little progress in the improvement in mathematics learning in this country.

“The American cognitive scientist, Daniel Willingham………His work attempts to apply…..”

In the UK we have many bodies and Universities who study education and produce reports on mathematics education. I wonder why you have to resort to an American cognitive scientist. Havind read his article there really isn’t anything ground breaking in it, certainly nothing more than Stemp pointed out in 1971. What are the views of the De Morgan Forum, ACME, the Universities of Cambridge and Exeter etc.

01 June 2015 17:36 I agree with Daniel Willingham when he emphasises the importance of relevant context in helping students to understand concepts (such as halving a cookie rather than a book). It is certainly true that teachers from Shanghai think very carefully about using contexts that are relevant and helpful to conceptual understanding in the images that children discuss. (for example- a picture with small bikes and big bikes allows an understanding of the amalgamation of the whole picture into a general group of bikes) However, I disagree that manipulatives "distract from their purpose". Manipulatives (for example, base 10) help to expose the multiplicative nature of the number system and could support an argument, and therefore understanding, about why 0.015 is not bigger than 0.05, even though "15 is more than 5". I do acknowledge that the idea is to develop conceptual understanding and the ability to generalise (or abstract) is the aim, not a dependence on kit.

Very pleased to discover this article. I will be a trainee this September (2017) and I would very much like to focus on what Secondary Maths can learn from Primary Maths, especially through collaborative work. It's been 30 years since I was at school and teaching methods in secondary look exactly the same while primary is unrecognisable.