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A Learning Reflection of the Shanghai Mathematics Exchange by Melissa Molnar
A Learning Reflection of the Shanghai Mathematics Exchange
by Melissa Molnar
“The purpose of teaching is to not teach children. They can think for themselves. We are facilitators.” Professor Gu Lingyuan
But First Some Background
Woodhill has been a part of the London South East Maths Hub since Autumn 2016. Since then we have participated in a range of Department for Education funded programmes to support our implementation of a Mastery approach to teaching, one of them being the Mastery Specialist Programme. In 2018 I was selected to train to become a Mastery Specialist. This involved intensive subject knowledge training in the first year to support in embedding teaching for mastery within my own school.
I’m currently in my second year of the programme which involves supporting other schools in London South East to develop their schools’ teaching for mastery approach through an ongoing teacher research group as well as school visits to offer catered support. Additionally, in the second year of the programme, we are provided with training to become an accredited NCETM Professional Development Lead which enables us to provide this CPD at a professional level.
This level of investment from the NCETM in Mastery Specialists is what drew me to the program initially but I can honestly say the opportunities for me professionally have been endless since joining. The most rewarding being my recent trip to Shanghai in November.
Origins of the Shanghai Exchange Trip
There had been long standing concerns about the prior Maths curriculum in England, as well as the mathematics subject knowledge of many primary school teachers (Williams, 2008; Ofsted, 2008). In 2014, the UK government introduced a new curriculum with a mastery focus. The rationale for this originated from observation of high achievement in mathematics in East Asian countries, where there is not only a smaller gap in attainment between pupils, but pupils are up to three years ahead of UK pupils in their mathematics ability by age 15, according to Programme for International Student Assessment (PISA) tests (OECD 2012)(Boylan, M., Wolstenholme et al). Now there are a number of factors as to why this is, not just curriculum design, and I’m not here to comment on past curriculum or teaching practises. However, I do think there is merit in learning and observing what evidently is working well in education internationally and what parts of this can be incorporated into our curriculum and culture whilst still maintaining our own ethos and values in education in England. In order for us to implement this mastery curriculum in England, it was vital that we learnt from Shanghai practices, thus beginning the agreement with the Shanghai Municipal Education Commission to hold a teacher exchange.
My Experience of Shanghai Exchange Trip 2019
When I returned from my two week visit to Shanghai at the end of November 2019, the two most consistent questions posed to me were, “How was it?” or “How was your holiday?” How do you sum up a pedagogical, professional and culturally rewarding experience in one sentence? Beats me! How do you explain that whilst it was an amazing opportunity to be a tourist in between work, one that I’m extremely grateful for, it was also mentally exhausting and overwhelming? I have no idea.
Whilst the maths hub has provided opportunities to share my experience in Shanghai, I knew I needed another platform to share this experience. So here I am now, deeming myself as a newly found blogger. As condensable as I can manage, this is my full experience of Shanghai. It’s quite the journey, so make yourself comfortable!
Lecture from Professor Gu
On the first working day of the trip we were taken to Shanghai Normal University where we had a lecture from Professor Gu Lingyuan, a key figure in creating the Variation Theory. He is a leading professor of mathematics education in China and bases his theories on his well-founded research projects. His lecture has stuck with me still as one of the most valuable days of the exchange. I went into this experience with so much knowledge, or so I thought, about Mastery approaches to teaching. I recently attended a seminar from the maths hub in London in which speaker Emma McCrea, author of Making Every Maths Lesson Count, spoke about the ‘curse of knowledge’. Once we have it, it is very difficult to imagine what it’s like not to have it or to change this thinking. Professor Gu’s lecture challenged some of my previous knowledge and thinking it, which made it extremely impactful.
My 3 key learning points from his lecture:
- The importance of highlighting the key point in a lesson to learners
- The order and teaching sequence is critical in attainment
- Visual representation (which we rely so heavily on using mastery approaches) cannot replace understanding
Stress Importance of Teaching Sequence
In Shanghai, the level of research that goes into their curriculum is phenomenal. Professor Gu presented us with a research study on whether it’s best to teach Area or Perimeter first in the Grade 3 curriculum. The table below shows the initial arguments as to teach that unit of study first.
In the study, they decided to have a novice teacher, with two years of experience, teach perimeter first and an experienced teacher of 17 years teach area first. Then swap the novice and experienced teacher in teaching the other unit first. Below are the results of the data comparison.
The key Issues they found when they taught area first were that the teacher spent lots of time and effort, while students only remembered the formulas, and got confused between the concepts of perimeter and area. Even when taught by an experienced teacher, children made improvements in their skills but the efficacy was not obvious (Lingyuan 2019).
This was eye opening that is wasn’t a result of how the teacher taught or experience of the teacher which affected the children’s outcomes. It was the fact that children didn’t have the basis knowledge of perimeter first before applying this to area. The tables clearly show that in every situation, pupils who were taught perimeter first had better attainment in both perimeter and area.
Visual Representation Cannot Replace Understanding
In teaching with a mastery approach we have long been stressing the importance of using a concrete, pictorial and then abstract approach with the basis that using physical materials and pictures will help form the concept for the children, thus leading them to solve the maths abstractly. I have always stressed this structure with teaching staff that we must follow a CPA approach. However, in the past I never thought about stressing the careful consideration of which images and representations to utilise and why. If we were using concrete and pictorial representations, then we were winning right? Children will understand better? In the image below, you can see the visual representation used to show the concept of adding fractions with the same denominator has actually created a misconception for a child.
Some pupils thought that the representation was showing 6/10 because there were 10 parts and 6 of them were shaded. Is the student wrong or have we not highlighted a vital learning point in using this representation? This misconception is a result of two reasons; one being we mainly focus on using an area model to represent fractions. Other models such as volume, mass and length are often not represented within fraction units. The second reason, which brings me to my third take away from Professor Gu, is that the teacher didn’t highlight the key point of understanding of the fraction unit being fifths. Perhaps there wasn’t an emphasis on the concept that fractions are a unit just as a unit of counting in twos or fives, we can also count in fifths or sixths, etc.
Highlighting the Key Point
As stated above, the key point when adding fractions is the understanding of the fraction unit and the importance to emphasise this for understanding. Even children at a reception level have the conception that only two objects with the same unit can be added together. For example, when they are asked what is 3 horses plus 5 cows they say it equals 8 animals. But this point needs to be highlighted and related to a real life context more often to truly understand this in relation to fractions.
School Contexts In Shanghai
Whilst in Shanghai I attended two different primary schools. Minhang District Hua Yuan Primary School with about 900 pupils and deemed as a ‘small’ school, showcased the wider curriculum as being just as vital as Maths. All subjects were well resourced and taught by extremely skilled teachers in their specified subject areas. Additionally, to Maths, they wanted us to experience the whole curriculum so we were able to observe lessons in calligraphy, kungfu, woodworking, drumming, swimming and 3D printing design lessons. The curriculum at the school was very enriching and there was a strong ethos for a love of learning and being the best.
The second school I attended was Gao’an Road No. 1 Primary School which was a large school of 1800 pupils and was viewed as a highly reputable school. The school has been visited many times by Debbie Morgan (NCETM Primary Lead) and our partner teacher within the school, who is newly qualified, was competing in an esteemed teaching competition.
Over the two weeks, all Mastery Specialists had to meet after school was finished at the hotel and discuss and feedback on the learning for the day. Each day the meetings had a focus. This time allocated to listen and reflect on each other’s experiences was just as valuable as being in a school. Some specialists had more English translation then others during lessons and so this allowed us to pick up key findings we may not have understood or share common threads, concepts etc which was extremely useful. Our discussions and topics focused on the questions below.
Mathematics lessons in Shanghai are 35 minutes with the key concept always in mind. If pupils wanted to veer off the key concept, the teachers would respond to this but pull focus back to the key concept relatively quickly. Lessons heavily focused on children exploring a problem or question without specificity on how to solve it at first. This challenges the practise of always following an ‘I Do, We do, You do’ model with a strong emphasis on getting the children to think straight away and little teacher input.
Teachers would then draw out the concept or method they would want the pupils to use through the pupil’s responses and strategies. This was best displayed through their use of high-quality visualisers and the pupils always showing and explaining their methods. Precise questioning and pre-empting pupils’ responses was key in moving pupils through the lesson and getting them to understand the concept of the lesson. In every lesson I observed, there was always at least one child who used the method or strategy the teacher wanted to showcase. They think how a child might think as they are planning a lesson.
Concrete equipment was used when it was purposeful to draw out the concept and taken away as soon as it wasn’t necessary. Most independent practise of the skills was completed in homework, although I’m still unclear as to if this was done at home or during lunch time or both.
Teachers’ knowledge of the curriculum is superb! I definitely felt overwhelmed at times and almost embarrassed that the maths, at times, was way over my head, especially in many algebra lessons I saw in grade 5. I definitely felt out of my element! I’ve taken this as a learning point for myself that my next step is to familiarise myself more with the secondary maths curriculum in England.
This incredible subject knowledge the teachers have is partly due to the fact that they specialise in the subject, so they only teach Maths. They also follow their classes up each year. So, if they teach a class in grade 1, they will teach them maths throughout primary school when they finish in grade 5. This is an important takeaway point. How well do upper key stage teachers know the secondary curriculum or how well do early years practitioners know the upper key stage 2 curriculum? I’m quite confident there are some amazing practitioners who do, but I know I still have a lot to learn myself.
Shanghai teachers consistently used textbooks to help plan their lessons, but I saw no examples of the textbooks themselves being used in a lesson. This challenges the belief of how textbooks are meant to be used in England. I find that some schools stick rigidly to textbooks and schemes whereas in Shanghai it was more so a tool to support the teacher and to inform planning of their lessons, not strictly followed.
I found it fascinating the way the curriculum was designed in terms of fractions not being introduced until grade 3 and multiplication not being introduced until grade 2 at all. In terms of number units in grade one, the only focus was addition and subtraction. They still teach other areas of measure etc in grade one but it made me wonder what impact on our pupils’ attainment this could have if we had more time to focus on addition and subtraction and hold off from multiplicative concepts until this was mastered? This brings me back to Professor Gu’s point on stressing teaching sequence. Yes, there are several underlying factors why Shanghai pupils are attaining at a higher level than English pupils, many of which are out of our control as teachers. But I do think there is a piece of work in researching and trialling teaching sequence within Mathematics units within our own curriculum.
Pedagogy and Professional development
If the children don’t understand a concept, the teachers don’t question their ability to teach but question what was missing from the lesson. Critiquing and perfecting lessons to ensure all learners achieve understanding is highly valued. Teachers meet weekly in teacher research groups within their own schools to discuss lessons prior and post being taught to discuss what can/could they change about the lesson to ensure the children will understand. Collaboration is key. Consistently, lessons and units are planned together and when something doesn’t go to plan, it is adjusted for the following year. So to soothe my own ego, the amazing lessons we were observing had most likely been taught many times before and perfected to get to where they were.
Schools can also work collaboratively with other schools and other educators on specific projects and peer observation consistently is part of their practise. Some lessons I observed were part of a trial to redesign a division unit of work in grade 3 and 4. This is because the teachers found the textbook sequence wasn’t quite right for their pupils. Perhaps it worked well for a previous cohort but it wasn’t working for their current cohort. This was relieving to hear that even in Shanghai, the pupils from year to year, have different abilities! Phew!
The most rewarding part of seeing this study in action was listening to the feedback from some of the lessons and how this was informing what would be taught next. It is important to note that the Maths teachers had time in the school day to have these research meetings as well as daily planning time within the work day. It was a normal part of their everyday practise and whilst we don’t have the capacity to do all of this for a number of reasons in England, I do think there is value in ensuring more collaboration, not just in maths, but in all subjects.
Bansho is a method of teaching developed in Japan that focuses on teaching through problem solving. It allows students to see connections and progressions of the thinking involved when developing strategies to solve a problem. Representations and structures, often animated, are shared on IWB while the story of the lesson, number sentences, steps, key vocabulary and sentence stems are recorded on a blackboard.
This was new learning for me as all of the Shanghai Showcase lessons that I had seen in England in the past hadn’t incorporated the writing/recording aspect as much. It was heavily reliant on the PowerPoint so it was interesting to see how both aided each other and when they used one over the other and why.
Visualisers are used frequently to compare and contrast pupils’ learning. Not just any visualiser, they were very high tech and it was such a useful tool for the children to see each other’s learning. What an incredible learning resource to have! How do I get my hands on some of those?
Shanghai Pupils as learners
Attitude Towards Learning
Pupils love school and Maths! They want to be challenged and get excited when they are faced with a difficult problem. They couldn’t understand my question when I asked about not coming to school. They mostly answered with ‘why would you not want to go to school?’
Behaviour during learning time vs. break time was interesting. There were 10 minute breaks between every lesson and it seemed consistent across schools that these weren’t monitored and the children were ‘free’ to do as they pleased and go where they wanted during these 10 minutes. The first time I saw a break start I was absolutely shocked! Children could be seen running in the corridors, outside on the playground or just playing somewhere in the classroom and the teachers just got on prepping for their next lessons as if it was normal behaviour. Suddenly, music would start to play through the intercom and it was as if break had never happened. Children would walk silently to their next class and immediately be ready to learn. I found this absolutely fascinating! They all knew what it meant when learning time was to begin but also great to see them just behaving like normal children when their learning was finished.
There was a definite drive to succeed from all pupils and you could tell this is just the cultural norm. Education is highly valued and it was quite common that the grade 5 children spoke English well, sometimes better than some of the teaching staff. This is because of the outside of school tutoring the parents enlist their children in as well as how much involvement parents have in ensuring their children practise maths outside of school. The expectation is that children know all of their number bonds within 20 before they begin grade 1 and that all pupils know their times tables up to 10 x 10 before they begin grade 2. I saw some of the tests the teachers gave the children…I think I would have struggled, even if it was all in English.
It was clearly evident that learning through mistakes is common practise, not just for the children but also for the teachers. Teachers didn’t seem phased by feedback. They knew it wasn’t personal as so many of the lessons were crafted collaboratively. If a part of the lesson wasn’t spoken about, it was assumed it was good and in most feedback sessions it jumped right into what could better the lesson next time.
During lessons, children wanted to be chosen as having the correct method or answer, especially so when ‘English teachers’ were watching. We did see some minor disappointment when a child was wrong, however it just seemed like common practise, this is how we learn. I loved this aspect of lessons. I think we can all take more opportunities to value learning mistakes. It is what drives Mathematics teaching and draws out concepts so clearly to pupils in Shanghai. It also gives more positive light and acceptance of mistakes and taking risks, something I’ve had to learn the hard way, rather than being my own mindset.
My Key Learning Points
I know...it’s been quite the ride with this blog, but thanks for coming with me on this journey! Whether you’ve made it this far or you’ve skipped to the end to my key learning points, hopefully there’s something for everyone to take away from my incredible professional development opportunity or as some still deem ‘holiday. I’ll take any description as I’m so grateful for it!
To conclude my key points would be:
- Collaboration is key. Even more so then I initially thought. You need outside perspective and I think I would like to create more opportunities for peer planning/observation within my own practise and schools.
- Subject knowledge is absolutely vital – not just skills, but in terms of progression of a subject. What is the starting point and end point and even further beyond to secondary school? How would this knowledge impact our teaching? This could and can be applied to every subject.
- Letting the children think for themselves and driving the learning through their mistakes. I had the opportunity to plan and teach a lesson at the end of my trip and my feedback consistently was ‘stop thinking for the children’ or ‘you haven’t given them time to think for themselves.’ ‘Stop telling them how to think!’ Teachers should be facilitators of learning and thinking and have the courage to not be afraid of the children not understanding. Let’s start to take more risks!
“Students get smarter with the help of the teacher by themselves.” Professor Gu Lingyuan
Boylan, M., Wolstenholme, C., Demack, S., Maxwell, B., Jay, T., Adams, G. and Reaney, S. (2019) Longitudinal evaluation of the Mathematics Teacher Exchange: China-England - Final Report. Sheffield Hallam University: Department for Edducation
Gu Lingyuan, (2019) Stories about primary school mathematics classroom: Shanghai Academy of Educational Science, School of Mathematical Sciences, East China Normal University
Ofsted, (2008) Mathematics: understanding the score. London: Ofsted.
Williams, P. (2008) Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools. Nottingham: DCSF Publications.