## Exploring mastery in secondary maths teaching

The process of mastering maths is more easily imagined when pupils are starting from scratch: building mathematical knowledge from a base of almost nothing. Partly for that reason, the work in this field by the NCETM and Maths Hubs has been concentrated in primary schools.

But the objective of achieving deep understanding of maths is just as valid, of course, in secondary schools, even though pupils are mid-way through their journey of mathematical learning, some with vastly more accumulated understanding than others.

During the school year just coming to an end, a group of around 100 secondary maths teachers - a couple or more from every Maths Hub - have been working together on the secondary Mastery Specialists programme. The aim has been to explore ‘what mastery means’ in the secondary maths context, and try out some teaching, and teacher-collaboration ideas.

Here’s a snapshot of what some of the teachers have been trying in their own classrooms. And you can hear them speak for themselves in these short video interviews.

**More time for ‘productive struggle’**

*Nick Wong, Lead Practitioner, Longsands Academy, St Neots, Cambridgeshire*

"The biggest thing we’re doing is spending more time doing rich, long problem-solving tasks, and that teachers are feeling comfortable allowing time in class for that ‘productive struggle’ so that students hopefully play around and explore their maths a bit more and apply it to new problems. ‘Productive struggle’ means that students are having to think hard about what method they use: how to get started and progress through a problem. They’re not just doing the same procedure several times over."

**One simple, focused target per lesson **

*Irina Tolchenova, Lead Practitioner, Nobel School, Stevenage*

"The one thing I have changed is to teach only one, really, focused and specific target per lesson. For example, if I planned a Year 10 lesson on using the quadratic formula, I would now have one lesson to recognise the coefficients *a*, *b* and *c*. So, I would change the positions of the terms, and I would target misconceptions, for example when the *x* squared term has no coefficient, children think that *a*=0 rather than 1. The next lesson would just concentrate on the discriminant."

**More use of pictorial representations**

*Jose Salinas, Head of Maths, Heathside School, Weybridge, Surrey*

"For me the main difference in my lessons has been (increased use of) pictorial representations of the topics I’m trying to get across to students. So there’s a lot more bar drawings and diagrams. And I find myself, when students ask me for help with a question, saying ‘Draw me a picture. Let’s see what it looks like.’ And then we take it from there."

**Deeper mathematical discussions**

*Teresa Booth, Head of KS3 Maths, Tomlinscote School, Frimley, Surrey*

"I’ve tried to make sure that I tighten up my use of language and be very precise mathematically. Then, modelling that, I require that of the students as well. So I don’t just accept an answer any more. I want them to explain what that answer represents and the from that, there would be a question about ‘How have we got to that?’ and ‘Why have we done that?’ ….and ‘What does that tell us?’ …and exploring lots more around the problem, rather than thinking that the objective is just to come up with a number that is ‘right.’"

**Emphasis on visual representations**

*Christian Walsh, Responsible for teaching and learning in Maths, Tewkesbury School, Gloucestershire*

"Putting the emphasis on visuals to a far greater degree has enabled me to teach much more powerfully. For example, with fractions this year, I spent a lot longer on adding and subtracting fractions just with diagrams, and the students understood it much more powerfully. Rather than saying ‘Here’s the method; let’s get the common denominator and then let’s fly through 20 questions,’ it was much more ‘Why do we need a common denominator; let’s look at this picture,’ with students drawing more pictures in their books. The understanding there was so much more powerful, in a Year 8 class that wasn’t the best, than anything I’d seen previously."

**Slower pace develops confidence to tackle unfamiliar problems**

*Terry Butler, Maths Teacher, Lipson Cooperative Academy, Plymouth*

"The main thing is the pace of the learning in my lessons. There’s is a lot more discussion and time taken to understand what’s going on. Gone are the days when it’s question, question, question. It’s now one question which could take three lessons worth of work to get through. There’s no point whizzing through the curriculum just to say you’ve ticked the boxes. They have to understand the mathematics.

"Now students are willing and able to attack any problem I set them– just because we’ve built the foundations quite strong. The discussions that we have, the way they praise each other’s efforts, the way that they spot different ways of doing things. I don’t praise the answer any more, I ask them, ‘How did you get there? How does that work?’ I try to confuse them as well. They are willing to get stuck now – they didn’t used to be."

**Variation theory**

*Garry Potter-White, Lead Practitioner for Maths, Purbeck School, Dorset*

"The main thing for me in my classroom is the introduction of variation theory, so I like to try to bring in lots of diff ways of posing questions but still focusing on the same concept…. We tend not to rush on and move on to making the concept more difficult but sticking to the same concept and seeing it in different sorts of ways. And maybe bringing in previous knowledge and different ideas into that one concept that I’m teaching. Students like to be remined of skills they’ve covered previously."

**Narrowing the curriculum in KS3**

*Linda Greaves, KS3 Maths Coordinator, Priory City of Lincoln Academy*

"We’ve narrowed the curriculum on Years 7 and 8 to Number and Algebra, to make sure that by the time they enter Year 9 they are GCSE-ready in Number and Algebra, and then we can build the Data and Measures in later. Because of that, they are much more confident and fluent in Number and starting to make links between different concepts."

**The answer is only the beginning**

*Andy MacDonald, Head of Maths and Computing, Marling (Grammar) School, Stroud, Gloucestershire*

"We’ve made more of the idea of the answer being just the beginning. There are two strands to this. First, the idea that multiple solutions to the same problem are better than the same solution to five highly similar problems. And second, giving students the answers and making them communicate and write the journey of getting to the answer. This makes the answer almost the trivial part of the task, rather than the main achievement, and this means their sense of achievement comes from communicating the reasoning rather than getting the answer. That’s had a significant impact on my practice."

**Engaging with the pedagogy to support teaching for mastery changes**

*Peter Mattock, Head of Maths, Brockington College, Enderby, Leicestershire*

"The programme has given me that grounding in the background understanding and the research, that as a teacher, is so hard to come by, and so hard to find time to assimilate. I have had time to take a step back and engage with the writing and what the people are saying about teaching for mastery, and then been able to identify those elements that already exist in my own practice, and those I would like to develop further. This has been the major benefit for me. As head of department, I’ve found that being exposed to some of the background materials, resources and articles, which I’ve shared and discussed with my department, has meant that other teachers have been able to take that into their practice with more assurance and confidence."

Thanks for this article; it's nice to know some of the things I'm doing (in Australia) are on the right track. Posing the question; why is the answer...? can lead to good thinking practises, and thinking in pictures is very useful. also the idea of "playing" with questions when stuck, trying different things; I hate it when my boys leave a question blank.