An important component of the overall Teaching for Mastery Programme is the annual exchange of teachers between schools in England and those in Shanghai. Every school year since 2014-2015 a group of teachers in England, usually two per Maths Hub, has taken part in this exchange. The format is broadly as follows: teachers from England spent a week or two observing how maths is taught in Shanghai schools, and later in the school year host partner Shanghai teachers at their own schools in England for a fortnight or more. When the Shanghai teachers are here, their lessons are observed by teachers from neighbouring schools in every Maths Hub area, and post lesson discussions take place, in which the lesson design and delivery is un-picked in detail. In this way, across the country, several thousand English school teachers have been exposed to this version of teaching for mastery in this way.
So far, there have been three exchanges between primary teachers and one with secondary maths teachers.
UK-China Education Research and Innovation Project Exchange Programme
Stratford School Academy Maths teachers, Ms. Mahroof and Mr.Kibria visited Shanghai for a fortnight along with 68 other teachers from across the UK to visit Chinese teachers as part of the Maths Hub National Collaborative Project. The aim of the project was to develop a deep understanding of how maths is taught for mastery in the equivalent of KS3 classes in Shaghai; to develop the knowledge, skills and attitudes to attune and align with those of their Shanghai counterparts and finally, to implement successfully some of the Shanghai structural and pedagogical approaches in our KS3 classes back at Stratford School Academy.
Teaching for mastery ensures that students have a deep conceptual understanding, rather than just procedural fluency; i.e students understand WHY as well as knowing what to do in a situation.
“I was fortunate enough to be part of this nationwide project and was really impressed by some of the aspects of the Shanghai approach of teaching Mathematics concepts.
One of the main aspects was how well they train their students by encouraging them to do the higher order thinking skills by emphasising on only one concept in a lesson and exploring different types and levels of questions, clarifying misconceptions before they are met. These are achieved by working alongside the school’s Maths teachers in a carefully planned and supervised programme called Teacher Research Programme. We are hoping to research elements of the Shanghai approach to teaching Maths that may be implemented in English school settings. ” Ms Mahroof- Head of Maths.
“It was a unique once-in-a-lifetime opportunity to observe and co-plan lessons, provide feedback and discuss and share best practice of mathematical teaching and learning, with Shanghai teachers who are part of the best educational system in the world.(Shanghai are ranked 1st in the world for Maths, Science and Reading, Pisa 2012). It’s been a truly eye-opening experience.” Mr Kibria.
Stratford School Academy’s Maths Department then hosted two maths teachers from Shanghai, for the return exchange part of the project. Over 90 teachers paid a visit to the school to watch lessons and take part in pre and post lesson discussions.
Click on the link for a news item from the Newham Recorder.
Impact of the exchange
The Maths Department has been collaboratively planning with its teachers in order to gauge the sequential thinking behind their lesson planning. This meant that as a school we began to consider how we would adjust our lesson design template in order to give weight to some of the key teaching approaches. The “away” and “home” legs of the exchange have resulted in greater collaboration/joined up working with local schools.
Conceptual understanding/ reasoning
Students have been exposed to a variety of representations of the same concept/ method, and hence have a deeper understanding of a new concept. (Algebra)
How do you know?
When Musy was explaining the expanding brackets, she introduced the topic with finding the area of a garden. This was a good introduction as the class already knew how to find area of rectangles with numeral sides. The lengths were then changed to algebraic expressions and students were immediately able to make the link. The abstract questions all have had visual representations beside them which embedded the conceptual understanding and further linked the two together. A lot of eureka and “ohh” moments took place!
Another example, was when Musy introduced Algebra. She started off with a box of random items and asked students to estimate/guess the number of items in the box. No student could give the exact answer so she called it n, and it was a powerful way of introducing generalisation.
Pupils can see the purpose behind using a letter to represent an unknown.
Pupils are becoming more confident with dealing with certain questions as they now have a variety of methods that they can draw upon and can really understand why a certain concept holds.
For examples -Expanding brackets
“Area of rectangle method”
Stratford School Academy continue to implement mastery within their mathematics curriculum. All pupils in KS3 are now taught in a mixed ability setting with teachers teaching KS3 or KS4 only. Collaborative planning on a weekly basis has meant that the department are discussing mastery principles together and planning lessons which are consistent in resembling the tenets of mastery pedagogy adopted from the exchange. Regular intervention is now in place to address misconceptions of pupils who are struggling. If you would like to get in touch with Stratford Academy, please contact Sajida Mahroof, Head of Department at: firstname.lastname@example.org
In September 2017, Ian O’Connor (Mastery Specialist from Hillyfield, Waltham Forest) and Max Lawson (Mastery Specialist from London Fields, Hackney) went to Shanghai along with other Specialists from across the country. Here’s what they had to say:
After completing my Mastery Specialist training in July I felt I had a reasonably good picture of what constitutes Mastery but two weeks in Shanghai cemented that understanding immeasurably and also made me realise how good our NCETM led residentials had been. It’s one thing to be told something but there’s always that element of doubt until you see it done exactly as you have been told !
The fluency of the children, in Shanghai, in their times table facts is striking as is their place value knowledge. This was demonstrated in many of the lessons I watched and enabled the children to manipulate figures rapidly and accurately without the distraction of struggling with times table facts or being hindered by place value understanding. In an afternoon history/maths lesson about the Great Walk of the Chinese army in the 1930s when retreating from the Japanese, it was incredible to see the children multiply and divide with ease when presented with facts about the march . For example they were told that every day ‘x’ number of soldiers died and were asked “How many died in say 9 months?. Our school has 1200 pupils. How many of our schools would that be that died each month?”
The speed at which the children were able to calculate these figures accurately was amazing. The maths purpose of the lesson (Year 6 class ) was to develop a number sense for large numbers which it did amazingly well. But the fluency of calculation by all children in the class was astonishing.
When watching a lesson on multiplying a two digit by one digit number what really struck me was how one example was worked with (42×3) to show the journey through the lesson with visual representations of tens and units; jottings to show the distributive law; expanded column procedure and the compact column method. At each stage highlighting explicitly, via questioning, the similarities and differences between the methods but overwhelmingly concentrating on the conceptual understanding.
In all lessons I was struck by the preciseness of the language used by the teacher and children and the precise questioning drawing out discussion from the children. Also we saw the use of visualisers in each lesson to discuss children’s methods and highlight that there are multiple approaches to solving the same problem. Children regularly astonished me with their rapid grasp of challenge questions- the ‘Dom Nao Jin or ‘use your head’ at the end of most lessons.
In a lesson on unit fractions to a Year 4 class the children were not explicitly given the objective of the lesson but halfway through the lesson were asked to come up with a stem sentence of what they had learned. After several children expressed their stem sentence the teacher chose the most appropriate and they practised “ I say, you say we all say” . This was a common feature of lessons and far from being rote learning I consider it an aide memoir to the conceptual learning that they had just undertaken.
Also very clearly demonstrated in this lesson was conceptual variation as standard, non-standard and non-concept. So ¼ shown as a quartered square(standard); a quarter shown as two triangular eigths within a square(non-standard) and finally a non-quarter – a large square split into four non-equal parts with a small square highlighted .
On a personal note I would just like to mention that the hospitality that we received from our Chinese hosts was overwhelming and both Max and I endeavoured to return that hospitality when our hosts became our guests.
It was a truly unforgettable experience and one that will impact on both my own teaching and sharing with my teacher research group.
The Shanghai-England Mathematics Teacher Exchange 2017 – The return leg
Max Lawson – Mastery Specialist based at London Fields Primary
In early November London Fields Primary School were delighted to host 2 maths teachers from Shanghai as part of the return leg of the exchange; Chen Chen and Xiaoyi Chen from Guang Biao Primary School and Tianshan No1 Primary School respectively. Over two weeks they taught maths lessons daily in year 2, where there was an addition and subtraction focus, and year 4 where the children learnt about fractions.
Not only was this a great opportunity for teachers from our own school to observe aspects of teaching for mastery but many teachers from across the local area and neighbouring boroughs joined us as part of Teacher Research Groups or through our showcase events.
Year 4 – Fractions – Chen Chen
Initially in year 4 the early lessons appeared to start simply. In Shanghai children are not introduced to fractions until year 4. The first lesson focused on the language of part and whole, before tackling both the area and discrete models of unit fractions in lessons 2 and 3. Each lesson had a single clear learning point that the children revisited again and again.
However this was not at the cost of depth or challenge. For example, although the initial lesson solely focused on the language of parts and wholes, by the end of the lesson children were able to articulate the distinction between wholes that are made up of one object and wholes that are composed of several. They could list a series of counter-examples when confronted with the incorrect assertion that a whole can never be a part. They were able to talk about how an incomplete sentence such as ‘1/2 an apple is the part’ is incorrect because a part can only be talked about in these terms when it is in relation to a whole.
Through a series of carefully selected and conceptually varied examples, Chen was able to develop a deeper conceptual understanding that children were then able to build on in subsequent lessons.
“From observing Chen teach her lesson on fractions, I have learnt that it is important for children to see concepts in lots of different ways. It was hugely beneficial to see the use of standard, non-standard and non-concept models that we have looked at in staff meetings and the impact it had on children’s understanding” Year 3 Class Teacher
Year 2 – Addition and Subtraction – Xiaoyi Chen
In year 2, Xiaoyi Chen focussed first on children developing children’s fluency with basic number combinations before looking at how these could be applied to bridging through 10. Children learnt a rhyme about the combinations to 10 and were required to use the lyrics to justify their answers to calculations. Emphasis was placed on children developing a rapid recall of these combinations when answering related questions.
However this was not rote learning. Each example was carefully chosen to highlight an important feature of the structure of the mathematics, an error the children may make as a result of a misconception, a difficult point of learning or an important generalisation the teacher wanted the children to make.
Children were required to reason throughout the whole course of the lessons as a result. Their understanding of the structure of the mathematics was deepened and the practice work they completed not only consolidated their understanding but moved it on. Often Xiaoyi Chen would talk about developing children’s ‘observational skills’ by carefully selecting sequences of calculations that would illustrate mathematical structures or patterns.
“After visiting a number of year 2 lessons I am beginning to understand the importance of carefully selected small steps within lessons. I have also seen how each lesson needs to be carefully linked to and built upon the previous.” Year 2 Class Teacher
Across all of the lessons in both year groups, one of the key elements that we observed are now looking to develop is the clear focus on children speaking in full sentences and using standardised, precise language models to talk about the mathematics they are learning. Beginning as simple stem sentences, these linguistic structures built over the lessons to eventually give children a scaffold to talk about their conceptual understanding at a much deeper level.
“As a result of the exchange I have learnt the importance of giving children sentence structures in maths lessons and requiring them to use precise mathematical language.” Year 5 Class Teacher
As a school a further point of reflection is the importance of the coherent conceptual mathematical journey that characterises all lessons following a teaching for mastery approach. Maintaining a single key learning point in any one lesson, carefully linking the steps within that lesson and designing lessons that build upon and link to each other coherently are all key components of the approach we observed. Coupled with this, the careful and deliberate use of procedural variation to move children’s understanding onto the next stage is also a key aspect of teaching for mastery we want to develop further as a result of the exchange.