Teaching mathematics for mastery at secondary school

As I explained in my previous blog post, following the success of the teacher exchange the NCETM ran with primary teachers from England and Shanghai in the autumn and spring terms of academic year 2014/15, the NCETM is keen to explore what we in England might learn from Shanghai that could support our maths teaching at secondary school.

During this autumn term the NCETM managed an exchange between England and Shanghai for secondary teachers, focusing on years 7 and 8. In September around 70 English teachers, 2 per Maths Hub, usually from the same secondary school, visited Shanghai for a week to observe lessons at both upper primary and early secondary (ages 10 – 13), led by Robert Wilne, the NCETM Director for Secondary and three NCETM Assistant Directors, Ione Crossley, Pete Griffin and Andy Tharratt. In November, a similar number of Shanghai teachers came to England to teach for 3 weeks in the secondary exchange schools across England.

The key principles of Shanghai maths teaching seen during the primary exchange were carried forward into secondary mathematics and there is a high level of continuity of teaching style across the primary/secondary interface. The importance of using representations and examples to reveal structure, the insistence on precise mathematical language and clear reasoning from the pupils, and a determination to engage the whole class in all of the mathematics being discussed, were all features that we had previously observed in primary maths classes and that were still prevalent at secondary level.

After their return to England the NCETM Assistant Directors made this short video to record their immediate reflections on their experiences of Shanghai maths teaching, which I think is well worth watching. It includes their thoughts and observations on Shanghai lessons, use of textbooks and ‘teacher research groups’. In Shanghai, maths teaching is underpinned by the use of a common textbook, and regular meetings between teachers to analyse their teaching and share ideas and experiences (a ‘teacher research group’) are a standard part of every maths teacher’s working week.

The maths teachers that visited Shanghai were similarly enthused. Their experience was similar to mine when I visited Shanghai schools to observe Maths lessons in 2014. Even though we had told them to expect to see some excellent maths teaching, the stereotypes of what we expect Chinese teaching to look like - learning by rote, little or no discussion – are so strong that our teachers were still greatly surprised, as well as impressed, by what they saw. More importantly, they believed they could learn from the maths teaching they saw in Shanghai to develop the maths teaching in their schools back in England and improve their own teaching. Several indicated it was the best professional development they had ever had and they were excited at the prospect of the Chinese teachers coming to work in their schools in November.

The Chinese teachers that came to England were observed teaching in the host schools by over 1500 English maths teachers from surrounding schools. Before I talk in detail about the lessons themselves, I must acknowledge the bravery and commitment of the Chinese teachers. They came thousands of miles from home to teach maths to children they had never met before, in a completely different language to their own and were critically observed by us while they did it!

Whilst the Chinese teachers were here I observed two lessons at the same school, along with about 10 other teachers from different schools in the area. The day was very well planned by the host school. There was a session at the start to give the background to the exchange and help the observing teachers to know what to look for, and then, later, really useful discussions organised with the Chinese teachers to analyse the lessons after they had been observed.

One of the lessons I observed was on reciprocals of integers and rational numbers, and the other on a property of division. I was impressed by both lessons – both were very well structured and stimulated the pupils to engage deeply with the mathematics. There is a link to the slides used by the teachers in these lessons at the end of this blog post. I’ve put some notes on the slides giving some of my reflections and have included some additional slides on the ‘Division of Fractions 1’ lesson, anticipating how it will be further developed into teaching division of fractions.

In both lessons pupils showed they were making progress by grappling with difficult ideas and by answering a small number of precisely focused deep questions – no repetitive “drill and kill”.

Key features of both of the lessons I observed were:

• Student-focused but teacher-led, with structured teacher input (5 minutes or so at a time), interspersed with purposeful pupil activity on tasks employing procedural variation to encourage deep learning, along whole-class discussion of difficult or crucial points.
• Whole class teaching, with no differentiation of activities or content: all pupils engaged with the same mathematical ideas.
• Very carefully designed and sequenced lesson content
  • concentrating on one key mathematical idea, and going deeply into this one idea, rather than darting about on the surface of three or four different ideas
  • using well-prepared resources (based on Shanghai textbooks)
  • deploying short, precise steps with clear logical progression to develop the mathematics and help ensure it is fully understood by the whole class
  • involving dialogue between the teachers and the pupils to draw out key mathematical ideas and address difficult points
  • incorporating pupil discussions, particularly in pairs, at key points to encourage deep engagement with the mathematics
  • insisting that pupils use precise mathematical language – e.g. ‘product’, ‘quotient’, ‘numerator’, ‘denominator’, ‘reciprocal’ – and express themselves clearly and concisely
  • using exercises that (i) enable the teacher to check the pupils have grasped the key points and can draw on them quickly (i.e. that the pupils have procedural fluency) and (ii) provide intelligent practice, helping pupils to develop deep conceptual understanding
  • emphasising the structure of the mathematics with results generalised algebraically and these generalisations discussed – by ALL pupils, not just by the “rapid graspers”
  • investing time to establish the important mathematical definitions, e.g. ‘reciprocal’
  • employing deep questioning, not just focused on ‘getting the answer’ but also requiring the pupils to articulate how they worked out the answer and why they believe their method is correct.

The pupils seemed to respond well to this style of teaching, really engaging with the mathematics and focusing on understanding ideas and connections, rather than following an algorithm to answer a question.

Our secondary maths teachers who were involved in the KS3 exchange have witnessed the Shanghai teaching methods at first hand both in Shanghai and in their own schools, and appreciate their power. They are keen to try to implement what they have learned. You can see several of them giving their personal reflections on the exchange on these short video clips.

Over the rest of this academic year and into next year we will see how the exchange schools, and others, can use the experience of the KS3 exchange to develop their teaching. The teachers involved certainly believe that what they have learned has great potential to help develop their secondary mathematics teaching. We intend to provide examples of how their teaching has developed as a result of the exchange, and how pupils’ learning has been deepened and made secure.

I am increasingly convinced that what we are learning from the Shanghai teachers can have a really positive impact on secondary school maths teaching in England, but the structural and systemic changes needed are far-reaching. Shanghai maths teaching is underpinned by expert and knowledgeable specialist maths teachers (and teacher training that ensures this), excellent textbooks that are informed by research and focus on developing deep mathematical knowledge and understanding, and regular (weekly, at the least) meetings of ‘teacher research groups’ that involve teachers sharing and discussing their lessons and constantly working to improve them. To emulate this will require an expert specialist teacher in every maths classroom, together with significant shifts in the way we develop textbooks and other teaching and learning materials, and in the way maths teachers work together and support each other. The NCETM, alongside the Maths Hubs and others across the maths education community, is already considering

• how what we are learning can be incorporated into initial teacher training and professional development in mathematics;
• how we might change the way maths teaching and maths teacher development is organised in our schools, so that we can incorporate more teacher collaboration and reflection;
• how we can improve the development of textbooks and other teaching and learning resources;
• how we can develop lesson design and share good practice effectively within and between schools.

It’s a long-term challenge, but we are taking the first steps.

Links to Shanghai teachers’ lesson Powerpoints:

• ‘A property of division’
• ‘Division of Fractions 1’