Greetings from Shanghai!

I am writing to you from China’s largest city, home to more than 24 million people, Shanghai.  The weather here is wet and cold (10oC) today – much like home – but it is also changeable too.  Tomorrow it may be warm and sunny!  The city is located on China’s central coast, making seafood especially popular here.  At eight hours ahead, it is now bed-time in Shanghai but you will just be sitting down to lunch.


We arrived on Sunday 6th November, after a 15 hour flight, including an unscheduled landing in Beijing for 4 hours due to heavy fog in Shanghai.  It was a long and tiring journey but one worth making: I have learnt a great deal already!

I am here with 69 other teachers from the UK, representing our government, to learn more about how schools in Shanghai teach maths.  For many years now, Shanghai has been globally recognised as the world’s leading jurisdiction in maths teaching. So we want to know more about what makes their lessons so fun and memorable.  However, this is a two-way exchange.  Shanghai teachers are also keen to learn about schools back in England, they are very interested in our Computing and PE lessons in particular.  

St Joseph’s is very lucky because we will be welcoming two teachers from Shanghai back to our school at the end of November.  Miss Zhao (pronounced “J-oaw”) who will be teaching in Year 6 and Mr Jiang (pronounced “J-ong”) who will be working in Year 2.  We are only of only 35 schools in the country to receive this great honour and I am very much looking forward to introducing them to you.


Monday 7th November 2016
The Welcoming Ceremony

On Monday we were officially welcomed to the city by some important men from the Education Ministry of China.  They explained that children in China don’t start school until the age of seven, which is very different to England.  They went on to describe how most primary schools in Shanghai have between 1-2 thousand pupils (very large by our standards).  Interestingly, lessons in Shanghai tend to be 35 minutes long, which is short compared to ours, but children have eight separate lessons every day.  

The Minister of State for Education also explained that teaching is seen as a very noble/profound career choice.  He said, “Teachers engineer the soul.”   In response, the universities in Shanghai have departments dedicated to the “cultivation” of teaching pedagogy.  It has been long established that lessons must be “meticulously crafted” in order to impart knowledge effectively, ensuring no misconception is left to chance.  They are preempted and built into lesson structures.  Therefore, teachers must not work in isolation but develop lessons very carefully in collaboration with their peers.  Teachers here use carefully designed textbooks (nationally adopted) to support their learning journeys and ensure they only ever take small, considered steps when introducing new learning. Having such large schools allows teachers to share expertise and affords them time to focus on the tiniest aspect of a lesson’s design.  

Clare Fowler, the DfE’s head of maths policy, got a knowing look from the seventy teachers in the room when the Minister of State explained that the Shanghai education policy hasn’t fundamentally changed in 30 years!  He described how the city’s on-going five year education plan is intended to make small changes but not too much and not too quickly.  He said, “Don’t not change, but don’t change too much.  Rome wasn’t built in a day but don’t knock down what you’ve built – just refine it.”  

One of the city’s recent policy changes was the introduction of the title, “Professor Teacher” which recognises those teachers who have made a significant contribution to the practice and philosophy of their colleagues.  In China, teacher progression is marked by different teacher titles: Junior, Senior, Master and Professor.  They are awarded these titles based on the number of hours of professional development undertaken and the number of teacher research groups contributed to.  Successful teachers here are researchers, who observe and reflect on what makes learning successful.  When discussing a lesson, there are no judgements or standards, groups of teachers (between 5 – 30 at one time) discuss the questions and activities used and which had the greatest impact on learning.  There is no Ofsted equivalent and no lesson judgements!

Teachers in Shanghai, even in primary school, teach only one subject.  Miss Zhao and Mr Jiang are primary maths teachers. They teach 1 – 2 maths lessons each day and spend the rest of their time catching up with children who need more help, watching other teachers’ lessons and planning and preparing their wonderful activities for the next day.

We also listened to a fascinating lecture from Professor Gu, lovingly known here as, “The Father of Maths Education” because he founded the idea of variation – over thirty years ago.  Variation is a way of learning about maths by looking at one topic from lots of different perspectives – this helps children to master each topic before moving on to something more challenging.

Professor Gu also explained that it isn’t about just one lesson, it is about a series of carefully crafted sessions that are taught in a coherent fashion.  The series of sessions helps to build connections because tiny steps in learning give children more confidence and depth before they approach the most challenging part of the topic.  E.g.  When preparing for long division, children spend five or six sessions looking at how numbers can be partitioned and divided (the distributive law) in order to understand how the long division algorithm works.  So 72 divided by 4 = (40 divided by 4) + (32 divided by 4).  

It did lead me to think that perhaps, unwittingly, we take much bigger steps in a series of lessons than we ought too.  We need to be more knowledgeable about the tiny, deeper steps that contribute to understanding the larger objectives in our national curriculum.  Teacher subject knowledge leads to more effective planning and preparation.  I then stopped thinking to enjoy lunch.

Tuesday 8th November 2016
Aiju Primary School – Shanghai

Aiju is a “small” primary school (with 1,100 students) that specialises in expressive arts i.e. music, dance and art.  Parents send their children to this school if they want them to study these creative subjects and have the opportunity to participate in performance activities at a local and national level.  The school building is vibrant and inspiring – with children’s artwork on display in the main lobby.  

There are forty children in each class and they all sit at individual tables, arranged in pairs, facing the teacher.  There is no carpet work in Shanghai.  The children greet and bow to the teacher before starting the lesson – this sign of respect shows that children understand the status of the teacher and the need to pay careful attention to the following lesson.

However, that doesn’t mean that lessons in Shanghai are quiet; far from it!  I observed one lesson about classifying a collection of shapes.  It was taught to Grade 1 (our Year 3).  The children were talking about and moving the shapes around – discussing and debating with their partner the best way to categorise the shapes: by shape, by colour or by size.  The teacher then asked children to share and compare their ideas before teaching them how to classify groups of objects in different ways at the same time, like in my diagrams.  Using the diagrams, children explored the different ways shapes can be grouped.  They used their new understanding about the layout of groups to identify missing shapes in the teacher’s “Who’s missing?” game.  

Although I began the lesson by wondering how this could possibly be classed as geometry, there was no discussion about sides and corners, it was later explained to me that this is in fact a first look at statistics!  Children know what squares, triangles and circles are, they were looking at how to represent classification and how to show different categories in charts and diagrams.  It struck me that the array diagram is intrinsically linked to the skill of reading axes on graphs and recognising two variables.  It was remarkably subtle but will help children much later on to read more complex charts and graphs if they begin representing simple categories in this way now.  

The classroom we watched the lesson in is three times the size of a standard classroom and two thirds is given over to seating for teacher observers! – The science of teaching is alive in Shanghai.

After school was a chance to enjoy some sightseeing…

Wednesday 9th November 2016

Another inspiring maths lesson today.  Grade 3 (our Year 5) had been preparing for long division this week and Miss Lee, their teacher, wanted us to see the first written method lesson because it is a very tricky one to teach.  I was pleased that she chose it because it is one skill that many children in England find challenging too.


A young boy called Lucas showed me how they had been preparing in the days before by using their knowledge of times tables to help understand the distributive law more deeply e.g. 72 ➗ 4 =40 ➗ 4 + 32 ➗ 4.  This is important in order to understand what all the digits in a long division calculation actually mean…

Again, I was astounded by the importance of knowing the steps that come before a procedure deeply myself.  Without a profound knowledge of how to build up to an efficient method, we cannot help children to understand the meaning of the calculation they are doing.  If it is treated merely as a procedure, some children can quickly grasp it but not understand it whilst others struggle to remember all the steps in the correct order.  However, by spending quality time preparing the “anchoring knowledge” beforehand, the procedure is easier to remember because children understand what it means, and they are more likely to be accurate because they appreciate the value of the digits used to make the procedure efficient.  A very eye-opening experience!

A couple of important observations: in Shanghai, all teachers and all students write the written method in exactly the same way, even down the the nuances of how they draw the symbol.  It is this strength of coherence that makes the whole system so effective and ensures that misconceptions and miscommunications are less likely to occur in the first instance.  It was also refreshing to see Miss Lee invite us to the hardest lesson to teach – she wanted our help identifying exactly which questions and activities were effective.  She wanted to unpick the examples with us to see if there was a better way of presenting and explaining them to the children.  We did not judge the lesson; we discussed the learning.

Thursday 10th November 2016
Another remarkable morning of maths observations!  We began in Grade 2 (Year 4) as they prepare for the formal written method of short multiplication.  They began by chanting their times tables, led by a class prefect as they clapped and sang.  It was a very inspiring sight.  The teacher, Miss Daisy, then reviewed with children how to use a simple picture problem (5 groups of four apples) to write a multiplication sentence – agree with children that, “4 is the value of each group, 5 is the number of groups – five groups of four apples equals 20 apples.”

Miss Daisy then very cleverly helped children to recognise the multiplicative relationships in this picture…

5 X 4 + 3 X 4 = 8 X 4 = 32.  She reviewed again with children what each number actually means – “What does the 5 mean?  What does each 3 stand for?  Why isn’t it 8 x 8?”

Next, Miss Daisy shared with children this image and ask them to write down everything this tells us about multiplication.  Why not try it yourself?

Some of the children’s answers included:

2 X 8 + 1 X 8 = 3 X 8 = 24

4 X 4 + 2 X 4 = 6 X 4 = 24

Finally, Miss Daisy asked children to think and write about these two examples…tomphoto5

2 x 5 + 5 x 7 =


5 x 8 + 8 =

Have a think about them yourself.  Did you recognise these links?

2 x 5 + 5 x 7 = 9 x 5


5 x 8 + 8 = 5 x 8 + 1 x 8 = 6 x 8


Notice that Miss Daisy wasn’t interested in the answer to the calculation, she wanted to know what other links and connections could be made from these questions.  

tom-photo5Some reflections on this lesson:  Firstly, WOW! – It was inspiring that, even in Chinese, this lesson was so carefully prepared and clearly delivered that I could confidently understand every step.  Secondly, it is clear that teachers need to understand a great deal about the maths that comes before written methods in order to help children master written methods too.  Thirdly, one child in the lesson was clearly confident and answered each question very quickly (and accurately).  However, the teacher was not concerned about this, she simply prompted him to keep looking and think even more deeply about the relationships in the very same questions.  She did not provided anything different or harder.  The boy was genuinely gifted but he wasn’t bored, he was enjoying looking for even more connections and relationships in these tasks.  It brought to mind a phrase I heard on my training last year, “A child can be clever or bored, but they can’t be both.”

After a short break, we observed Miss Zhao teach a lesson to Grade 4 (Year 6) on the laws of the operations (commutative, associative and distributive – look them up if they are not familiar, it is utterly fascinating!)  To begin, Miss Zhao reviewed some homeworks the children had prepared as part of their revision of these rules, which they already knew.  Some examples are below…

tomphoto7I was struck by how colourful and personal they were.  We often have a preconceived idea of education in Asian jurisdictions as being rote and involving endless practice.  This is not the case here in Shanghai.  Children are allowed to make their work bright and bold, in order to emphasise meaning and help them remember important facts.

Surprisingly, Miss Zhao praised homeworks that were concise. She did not want endless details and examples about the laws of the operations, she liked the homeworks that chose the one best example and the most precise notes.  She told some children that they were doing too much!  

Next, she challenged children to apply their understanding of these laws to very complex calculations.  She did not want to know the answers to the calculations, she wanted children to show how the laws could make the calculations easier to answer.  
A brief reflection: shocking to see how effective a truly deep appreciation of maths can be.  Children in China don’t start school until the age of 7 (our Year 3).  They begin by learning numbers 0 – 10 and counting.  So lessons in Grades 1, 2 and 3 often appear far simpler than ours.  However, because the maths curriculum is so carefully designed, children are constantly making links and connections in everything they do, so by Grade 4 (Year 6) all children are working confidently with more complex ideas and their understanding of maths is deeper.  They quickly then overtake our curriculum and can start to answer very complex puzzles independently by age 12 – puzzles more akin to GCSE work in our country.  

tomphoto10Our afternoon involved a traditional Chinese painting lesson – using watered ink on blotting paper to create different tones and effects.  The theme was hedgehogs!  We also enjoyed learning to use the four traditional Chinese string instruments: the erhu, the pipa. the guzheng and the dulcimer.  The sounds of these instruments is very soothing and peaceful – the strings make a harmonious sound that is easy to listen to.

tomphoto9    tomphoto11

Friday 11th November 2016
Our last day at Aiju Primary school – very sad to leave having made such good friends with Miss Zhao and Miss Daisy.  They invited us to watch the morning exercise routine.  This is an activity that the children do every single morning: 8:15 – 8:30am.


It was amazing to see all the children moving in unison as they followed the routine. 1,100 children all exercising together on a single playground!  However, it wasn’t led by a teacher, it was lead by the children themselves.  A very impressive sight!

Our first lesson today was with Grade 1.  They were learning about the relationships between parts and wholes, using addition and subtraction sentences to express the different things we know about how three numbers can work together.  


Miss Daisy then used variation to help children understand this relationship more deeply – testing them to see if they knew when the relationship wasn’t being expressed correctly and using missing digit problems to encourage children to reason about what might be missing and why.

Finally, she challenged children with a problem to see if they could pull out of a selection of numbers all the relationships that existed within them.  Children had good fun finding, expressing and discussing these relationships.  Why not have a go at finding all the number sentences within them too?

You might be surprised to hear that lessons here are very lively – children have many opportunities to talk about and debate their understanding.  Very often children will lead their peers in the role of the “Little Teacher” and when a child says something worthy of praise, all the children clap and cheer their understanding.  They celebrate their learning.  It is a truly lovely and wholesome atmosphere.  

It is true that parents here “hothouse” their children – giving them many hours of extra homework to ensure they never fall behind. Nevertheless, the teachers don’t see this as healthy of beneficial.  Miss Zhao explained, “They learn some things outside of school and they think our lessons are sometimes easy.  However, they have not learnt the true knowledge outside of school – there is still far more to learn about my lessons.  What children do outside of school is not important, it is not the true knowledge.  Only I can give them the true knowledge and then they will understand properly.  So if a child tells me, “This is easy.”  I will give him a very difficult question to show him that he does not really understand.  Then he will see that he does not understand and so he must pay closer attention to my lesson.”

Saturday 12th November 2016

A wonderful day’s sightseeing in the city.  We were kindly hosted by our Shanghai colleagues who were eager to show off the city’s culture and history.  Starting at the newly refurbished Natural History Museum, we made our way through thousands of years of evolution as we examined how livings things have changed in our world since the dawn of time itself.

Moving across the city, we stopped of at a Buddhist Temple to learn more about the religion and culture of the Chinese people. Buddhism is the most popular religion in China; some people describe it as a way of life, rather than a religion.  At it’s heart, Buddhism recognises that in order to live a happy and prosperous life, one needs to learn how to be content and peaceful in your own life first.  The temple itself is a stunningly beautiful yet simple building – a peaceful haven away from the hustle and bustle of the city.

tomphoto17        tomphoto19

Finally, as day turned to night, we enjoyed a river cruise before sampling some of the most unusual and mouth-watering delicacies that Shanghai has to offer.  Deep fried bean curd being a particular favourite with the locals!

Monday 14th November 2016

tomphoto20Our first morning at Xiangyang Primary school.  Mr Jiang took us on a tour of the school, which is a bright, colourful building.  There is a Duplo Wall for children to play with and be creative at break times and they have special classrooms for music, art and science.

The outside of the building is decorated in an extensive mosaic that the children themselves designed many years ago.  Xiangyang (pronounced “Shung-yang”) is a very large primary school with over 1,800 students and eight classes in each year group!  Their football teams are famous in the city as the current champions and everyone is very proud of them.  Any child who does not work hard and complete their homework is not allowed to represent the school at football!


You will be fascinated to learn that children in Shanghai run down the corridors and play in the stairwells.  Their free time is their own and they are allowed to roam the school as they wish.  They have eight 35 minute lessons each day and these are split up by 8-10mins break between them plus one hour for lunch.  Children have a meal delivered to their classroom and they are expected to eat and clean up after themselves.  They do not have a strict uniform policy and teachers themselves dress very informally.  It is believed that you must be comfortable in order to learn (and teach) well.  

All classrooms in all schools have a very similar standard of equipment: some form of projection screen, a top of the range visualiser, PPT is the preferred software for most teachers and there is abundant white/blackboard space.  The set-up in almost every classroom in the country is identical!

We were treated to a Grade 5 lesson by Miss Anne today.  She is an experienced teacher with two teenage daughters.  I was asking her about how maternity leave works in Shanghai… She explained that most mothers in the city have to got back to work: it is a very expensive place to live.  Teachers take six months off work and, when they return, grandparents take on the role of the child’s parent during the day.  This can cause all kinds of tensions at home but it is generally accepted as the norm. However, in order to support mothers back to work, the Government allows flexible working times for women who return.  For the first year after returning to work, women are entitled to a later start and earlier finish than their colleagues.  This is seen as important, especially for mothers who wish to continue breastfeeding their young child.

tomphoto21Mr Jiang invited us to observe his Grade 2 lesson, where children were learning to understand division with remainders.  He began be recapping what division actually is; the sharing or group of equal parts of a whole.  He helped children to understand that some whole numbers cannot be split into equal groups and so a remainder is left over.  However, the remainder is always smaller than the size of the group.  He showed the children how this is directly related to multiplication:

16 ÷ 5 = 3 r1      so 16 = 5 x 3 + 1

 Another example:

7 x 4 + 2 = 30 so 30 ÷  4 = 7 r2

After lunch, we joined Grade 1 for an art lesson – creating symmetrical dragon flies using folding and cutting skills.   They left me some of their work, to share with you back home, as a gift.  Everyone here is extremely kind and welcoming; they are fantastic hosts!


Thursday 17th November 2016

We have continued to observe and be inspired by some fantastic maths teaching this week. In this Grade 1 lesson, children were looking at addition by bridging ten. I was amazed at how clearly the teacher helped children to recognise the need to use bonds to ten to help find the solution.

Children are very good at recognising the value of numbers to ten here – just looking through their textbook it is clear that the very first thing they do in school is learn about the value of number. By recognising that, for example, five is the whole and the whole is made from 5 + 0, 4 +1 and 3 + 2 etc, children can use their prior learning to answer questions like 9 + 5 with ease.

The teachers here use a selection of carefully crafted diagrams and number sentences to help children see how to break the smaller number into parts and then annotate their calculations to help demonstrate their thinking.

Once they have understood the concept deeply, they are ready to answer more challenging and complex questions with ease.

Without doubt, well thought out textbooks are used consistently here to ensure that every child receives exposure to the most valuable questions and to help every teacher to recognise the tiny steps that are needed to build up to a more complex calculation. I have purchased some while out here (they’re only £1 each!) so that you can see what I mean.

Another interesting observation is the design of the lesson structure out here. Almost without fail, there is a simple pattern to how a 35 minute lesson is delivered:

Stage 1: Quick rehearsal of basic number facts (usual child-led) before the lesson begins – it’s used as a kind of transition/settling activity.

Stage 2: Review a mathematical concept that children already know deeply – build their confidence and set the baseline by ensuring that all children are starting from the same point.

Stage 3: Give children a feeling of the new concept (which is related to stage 2 but take the children a step further). Allow them to discuss and solve the problem in their own way, without explicitly teaching them anything yet.

Stage 4: Draw together the children’s observations and solutions to teach them and generalise about the new learning. Giving them further examples to help make links.

Stage 5: Use carefully prepared variation questions to support children in making the generalised principle stronger and take it deeper still.

Stage 6: Summarise the understanding gained by reviewing and marking the questions answered.

Stage 7: Take a break.

Stage 8: Rehearse the skill again to consolidate it, as a homework task.

Friday 18th November 2016

What a remarkable couple of weeks! With mixed feelings, it’s time to return home.

Tom Collins from St Joseph’s Catholic Primary School in Guildford and Nicci Bailey from Hook Junior School have been selected to take part in a prestigious teacher exchange, focusing on mathematics,  with a partner school in Shanghai.

Tom and Nicci are among 70 expert mathematics teachers from schools across England taking part in an exchange as part of the Maths Hubs programme. St Joseph’s Catholic Primary School and Hook Junior School are working with the Surrey Plus Maths Hub.

The purpose of the exchange is to further develop the understanding and implementation of mastery approaches to teaching maths in St Joseph’s Catholic Primary School and  Hook Junior School, and in local partner schools that Tom and Nicci will be working with during the school year 2016-17.

This is one part of the Teaching for Mastery Programme, run by the National Centre for Excellence in the Teaching of Mathematics (NCETM) in conjunction with the Maths Hubs programme.

In July this year, the School Standards Minister, Nick Gibb, announced £41m of funding for the Teaching for Mastery programme over the next four school years.

The group of teachers, all of whom completed the Mastery Specialists programme within their Maths Hubs last year, will be visiting Shanghai in the first half of November this year, and hosting their partner teachers from Shanghai in English schools in late November/early December.

This is the third teacher exchange with schools in Shanghai since the Maths Hubs programme was launched in autumn 2014.

FREE lesson breakdown & textbook mapping resources

We’re really excited to be approaching the end of term with an announcement about some exciting new resources that the Surrey Plus Maths Hub has developed to complement resources from the White Rose Maths Hub.

Surrey Plus and White Rose have a shared passion for supporting you to use a teaching for mastery approach. Our teams have worked together to launch a new range of resources that provide a lesson breakdown and map objectives to some of the major textbooks that many of you are using.

They are designed to be used in conjunction with the White Rose Maths Hub’s Schemes of Learning, and provide a useful guide to indicate progression and pace in line with National Curriculum objectives. Documents are now available for Years 1 – 6 for the Autumn term, and can be found by clicking here.

The Spring and Summer term lesson breakdowns and textbook mapping will be released in the Autumn term, and mapping to Abacus books will be produced asap.

As a hub, we are very keen to work collaboratively with a wide range of experts, and we are very excited to announce this brand new resource. As hubs, we hope that you find them useful, and as always, please feel free to contact us with any feedback or comments.

SurreyPlus White Rose





TC 2

Depth NOT acceleration

TC 1The old national curriculum, measured in terms of levels, encouraged undue pace. Children were accelerated onto more complex concepts before really mastering earlier ones. Imagine the cubes to the left represent the building blocks of maths. As you move on swiftly from one topic to the next the tower gets taller and taller, until eventually it comes crashing down. By moving on before a child is ready, before they fully understand the concept, you are helping them build a tall tower, which at some point will become unstable.

TC 2

The new national curriculum encourages the study of fewer skills in greater depth in order to achieve mastery. By taking the same cubes and arranging them differently, the tower remains standing.

So by ensuring that the foundations are solid before moving on, children develop a more stable platform from which to build.


Curriculum: Sprial vs Linear

Many primary schools are still following a spiral curriculum where topics get revisited each term. The nature of this is that you have to move on quite quickly from one topic to the next, often spending only one week on something and not really having time to ensure that the children have properly and deeply undersood the concepts being taught. This suited the old national curriculum as it was content heavy and didn’t require any depth of understanding.

An alternative to this approach is following a linear curriculum in which topics are blocked together and covered only once in the year. Doing so allows more time for children to gain a deeper understanding of the key topics as there is less pressure to move on before they are ready. Evidence shows that children who have a deeper understanding of number and an enhanced ability to reason mathematically can progress through other topics more quickly.

TC 3

This is an example of how a linear curriculum might be structured. Note the emphasis that is placed on number in the first two terms. It is important to note that just because the curriculum is linear, it does not mean that the skills learnt in one block are not revisited again throughout the year as research clearly shows that interleaving ideas and revisting them aids long term retention. For example in the scheme above, during the geometry in the spring term you might look at perimeter and in doing so revisit addition and subtraction.

It is important at this stage in the year that schools consider whether the scheme they are following really provides their children with the most solid foundations.

There are many schemes availble on the market to suppot teaching for mastery (two of which we review in Textbooks: Compare and contrast). In addition to these, there is a FREE scheme that has been created by the White Rose Maths Hub, which provides an excellent starting point for any primary school wishing to adopt a new approach. We are currently working with the White Rose Maths Hub to further enhance these resources, details of what we are adding and expected timelines can be found here.

Depth is achieved through variation

There are three elements that are critical on the journey towards mastery in maths and that we need to develop in our children. Fluency, reasoning and problem solving. Without one, the next cannot follow and it is only by developing these three skills that children can move towards mastery.

TC 7

This flow of learning was first seen in Shanghai by the teachers who went on the exchange trip. Children must first gain fluency in whatever topic they are studying before they move on to reason mathematically and then begin to solve a variety of different problems which probe and challenge their depth of understanding.

Here’s a few examples to help illustrate how this development might look.

Example 1:


TC 8


TC 9

Problem Solving:

TC 10

Example 2:


TC 4


TC 6

Problem Solving:

TC 5

Can you see how each task builds on the previous one? The skills required are the same but the depth of understanding required is very different. When these activities are put in the context of whole-class, mixed-ability teaching where the aim is for all children to develop the skills of fluency, reasoning and problem solving at the same time, the slow progression through these three elements is even more critical. More will be written on this blog shortly regarding whole-class, mixed-ability teaching and how it can and does work.

Intelligent Practice

Variation is a phrase which has been banded about all too often since the first teachers returned from the Shanghai exchange visit. It think it has taken us a very long time to fully appreciate what is meany by variation…

Variation is NOT:

  • more of the same thing but a bit harder.
  • the same as variety!

When using intelligent practice, all tasks are selected and sequenced carefully with purpose, offering appropriate variation so that when viewed together they reveal something about the underlying mathematical structure, concept or process. Put simply, variation reveals concepts.

Variation is an approach to teaching. It is the art of sequencing similar but increasingly complex problems to “generate disturbance of some sort for the learner” Festinger (1957)

Consider the following example:

23 + 10 = [   ]

23 + 11 = [   ]

23 + 12 = [   ]


23 + 9 = [   ]

23 + 8 = [   ]

23 + 7 = [   ]

By changing just one small element at a time children are given the chance to develop their understanding of a concept rather than just learning a process to get the right answers. However, it is not as simple as just finding patterns. Care must be taken to ensure the structural concept is understood; not just a superficial procedure found, as it is this deep conceptual understanding that leads to mastery.

If you just rearrange the questions a little you can quickly see how big a difference the order the questions are in can make to the development of a child’s understanding.

23 + 9 = [   ]

23 + 12 = [   ]

10 + 23 = [   ]

23 + 7 = [   ]

23 + 11 = [   ]

8 + 23 = [   ]

It is this intelligent practice, the appropriate selection and sequencing of questions, that can make a fundamental difference to a child’s conceptual understanding.

Conceptual understanding leads to mastery

It is critical that children are taught maths in a way the develops a deep conceptual understanding as this is the only way of securing solid foundations. One way in which we can help children to do this is by using the concrete-pictorial-abstract (CPA) approach. This is based on research by psychologist Jerome Bruner, which suggests that there are three steps (or representations) necessary for pupils to develop understanding of a concept. Reinforcement is achieved by going back and forth between these representations.


In this stage a student is first introduced to an idea or a skill by acting it out with real objects. In division, for example, this might be done by separating balls into groups of red ones and green ones or by sharing 10 biscuits among 5 children. This is a ‘hands on’ component using real objects and it is the foundation for conceptual understanding.


In this stage a student has sufficiently understood the hands-on experiences performed and can now relate them to representations, such as a diagram or picture of the problem. In the case of a division exercise this could be the action of circling objects.


In this symbolic stage a student is now capable of representing problems by using mathematical notation, for example: 10 ÷ 2 = 5 Students only use abstract numbers and figures when they have enough context to understand what they mean This is the ‘final’ and most challenging of the three stages.


The following example demonstrates the principles of CPA. It starts with a pictorial representation of the problem using a clear diagram (the use of counters is important for any child still at the concrete stage), but it then builds the abstract alongside, so that the child can make that leap between the pictorial and the abstract as and when they are comfortable in doing so.

TC 11


In this second example the concrete is used as a starting point for the question. Two oranges are divided up into quarters, three of them are circled from each which leaves six quarters which are then put back together to make one and a half oranges. The pictorial stage sees the introduction of circles instead of oranges and then the final stage is the introduction of the abstract, the numerical calculation. By putting all three alongside each other students can see the progression and when they are ready, they will naturally make the transition from concrete to pictorial to abstract.

TC 12

In Summary…

  1. Teach fewer concepts in greater depth
  2. Intelligent practice gives children a rich appreciation of the concepts through variation
  3. If you help children unlock this depth of learning then, over time, whole class, mixed ability teaching will become even more successful.


Many of the ideas in this post have been adapted from ideas seen in Tom Collin’s session at the spring conference. Click here to download the powerpoint Session 1 – Teaching for Mastery

Textbooks 2

This project is being conducted by Goldsworth Park Primary, Cranborne Primary and Uplands Primary.


Cranborne and Uplands

  • Started using the OUP Inspire Maths books with year 1 in Jan 2015 as part of the Maths Hub national project
  • Staffing changes/issues as well as some challenges associated with a mid-year start meant that it wasn’t adopted overly successfully last year, however both schools have been using it in year 1 since September.

Goldsworth Park

  • Opted to use the Maths No Problem with year 1 as part of the second phased of the Maths Hub national project

We are fortunate to be trialling both textbooks in the region and are therefore in a unique position to comment on both.

You can read the individual summaies of the two textbooks here:

Maths No Problem

OUP Inspire Maths


Compare and contrast

  • Both use a Singapore approach, however Maths No Problem! covers everything in the NC whereas Inspire doesn’t.
  • Inspire’s package included assessment materials and a teacher guide. Maths No Problem! has no teacher guide so more thinking is required of staff.
  • Inspire teacher book is overloaded whereas Maths No Problem! lessons only go over 2-3 pages.
  • The lesson approach is different: Maths No Problem! have an exploration task to start every lesson where Inspire follows a more traditional approach using teacher input.
  • Both have good variation within the tasks and the questions.
  • Thought needs to be put into the children with barriers to Maths for both schemes.
  • Inspire does challenge the fast graspers through acceleration. Thought needs to be put into challenge activities for more able in Maths No Problem! but it sticks within the end of year expectations.
  • Students love both schemes.
  • Inspire assessment materials are not great – sometimes not linked to what they been doing. Maths No Problem! assessments are in a similar format to the textbooks and follow them directly
  • Training experiences were mixed with Inspire – some key aspects of the programme were held back. Maths No Problem! training was excellent.




Textbooks 6

This project is being carried out by Cranborne Primary and Uplands.


  • The books is based on the Singapore approach ‘My Pals are Here!’.
  • Used by over 80% of primary schools in Singapore.
  • Highly scaffolded learning framework with problem solving at the core.
  • Concrete-Pictorial-Abstract (CPA) methods are throughout.

What does it consist of?

  • 2 Teacher guidesTextbooks 6
  • 4 Student practice books (A, B, C, D)
  • 2 Textbooks (A, B)
  • Assessment book
  • Online support


Links to the UK curriculum

  • Based on the Singapore curriculum and so does not directly match the UK NC.
  • A correlation chart is available to map missing objectives, however this can cause difficulties with both resourcing and standardised testing.
  • Aspirational program as Year 1 covers both Year 1 and some Year 2 NC objectives.

Ease of use for teachers?

  • 5 days of teacher training provided as part of the program are essential.
  • Teachers planning guides are easy to follow but trying to fit each ‘session’ into a typical 1 hour lesson is challenging.
  • Resources are provided, saving lots of time!
  • Good variety of activities in books.
  • It is tricky to map the objectives to the UK curriculum and it definitely requires professional judgement and experience as well as flexibility in approach.

Ease of use for pupils?

  • Children need to be trained to use the books, including switching between textbooks and practise books (this is possible!)
  • Children love the books; they are bright and the layout is clear
  • Challenging for less able readers – particularly at the start of the year
  • Large volume of written work means that sometimes the less able do not reach the investigative activities (4 pages per lesson) which means they aren’t all learning the same thing.

Is there sufficient challenge?

  • Good for more able children who quickly grasp concepts due to the CPA activities.
  • Fast graspers have up to 4 pages of activities to complete per lesson, but sometimes need additional challenge.
  • Lower attainers need to be supported, particularly for reading the tasks.
  • Need to build in lesson time for lower attainers to access the high expectations.

Classroom practice

  • Strong focus on verbalising thinking into full sentences and giving clear explanations has had a hugely positive impact on the children’s learning.
  • Children need to be provided with a range of resources that can be accessed throughout the lesson which has led to greater independence.


  • The assessment book included contained multiple choice questions which the younger children struggled with.
  • The format is very different from any other part of the program and doesn’t always link to the textbooks.
  • After the first assessment, we decided to discontinue using these.

Are we going to continue?

  • Yes – we have bought into the scheme for Year 1 again next year, but not yet rolling out across the whole school.
  • Would tweak a few aspects- timings, practice books, move some to Reception.
  • Cost is a barrier that we need to try and overcome if we are going to move it into Year 2 and beyond.



Textbooks 1

This is being trialled in Goldsworth Park school with year 1.

What does the programme consist of?

  • Two textbooks per year group which cover the English National Curriculum.
  • There are two workbooks which align with the textbooks.
  • Online element with training videos and parent support videos.

How are lessons compiled?

The ‘In Focus’ section is always a very open task that leads them to explore ideas and start to think about what they already know and how it can be used (it very much links in with the idea of letting the kite fly and then reeling it in!)

Textbooks 1

The ‘Let’s Learn’ section focuses on the key mothods.

Textbooks 2

There is then a ‘Guided Practice’ question and a ‘Mind Workout’ which are designed to be some variation activities.


Why are the lessons compiled like this?

During the training course we received insight into the research and thinking of 5 people… The Maths No Problem series is underpinned by their theories:

  • Jean Piaget – ample processing time (exploration)
  • Zoltan Dienes – ideas looked at informally before formal teaching
  • Lev Vygotsky – cooperative learning
  • Richard Skemp – relational understanding (links and relationships between concepts) not just instrumental (learning rules or by rote)
  • Jerome Bruner – CPA approach



We were fortunate enough to go on a 6 day training course provided as part of this research project. This was highly valuable training and gave us the opportunity to talk to other teachers across the country doing the same.

If you buy into Maths No Problem you can buy training blocks from them depending on your needs.



Below are some examples of the work being produced by our children.

Textbooks 3Textbooks 4

Textbooks 5



  • Amazing methodology; children’s understanding is at a much higher level than in previous years.
  • The lessons are planned carefully in small steps allowing most children to keep up.
  • The textbooks have great visual representations which the children find easy to use.
  • The variation of tasks lead to a certain level of differentiation.
  • Online resource is useful for planning and the training videos are great.
  • Access to parent videos which can be placed on your website.
  • Training was excellent with Professor Yeap Ban Har.


  • Expensive!
  • Still need to differentiate for the more able, especially at the start of the year, so suitable depth was being reached.
  • Textbooks mean that levels of literacy can hold children back.
  • Workbooks need to be bought to purchase the scheme and we have not been able to build them into the work we’ve done.
  • Planning can take a little more time when splitting the lessons up – needs professional dialogue, but this is actually a huge positive too!
  • You still need to provide for the children that have barriers to their learning as you would with any scheme.


Next steps

  • YES!
  • We think it is having massive impact and we have found a way of making it work costs wise.
  • CPD of staff will be highly important in this and we will only be introducing it one year group at a time.


Things to consider

  • If you can’t afford to buy into the whole scheme then it is worthwhile purchasing a copy for each teacher just to help them see the progression and get ideas from the ‘In Focus’ sections.


Screen Shot 2016-04-14 at 16.42.26

It’s just over a year since we held the ‘Shanghai Primary Maths Conference’ where we first started to learn about teaching for mastery, Miss Sarah taught two live demonstration lessons and we launched four research projects (same-day intervention, daily homework, classroom design and curriculum planning)…

In that time schools and individual teachers have been hard at work trialling ideas, developing their understanding of ‘teaching for mastery’ and creating resources so it was fantastic for us to have the opportunity today for them to come together and share their experiences and learn from each other.

98 primary Heads, Deputies and Maths Leads from 62 different schools to come together to talk about how we can improve maths teaching – it’s so exciting to see that the work we have been doing of the last 18 months is really having a lasting impression on schools and individuals!


Here’s a quick summary of the day with links to posts on all the different sessions that went on.



Screen Shot 2016-04-14 at 16.42.26This is Emily… She’s 11 months old and every day she surprises me! Every day she comes back from the childminders or Granny’s being able to do something new… She’s like a sponge! Like all children, she observes and soaks up every word, every phrase, every expression…

As a parent, I want the best for best for as all parents do. Yes I want her to happy and healthy, but I also want her to be curious and inquisitive and develop a love of learning. I want her to want to understand things, to want to learn everything and to grow in confidence…

Of course, that starts at home with the family and I know that Emily is very fortunate, but from day one at primary school there is another group of adults that will influence who she becomes… the way she thinks… what she loves… what she hates… what she is confident in doing… and what she struggles with.

I want my daughter and her peers and generations to come to develop vondience in maths and to enjoy solving problems and reasoning – skills that stretch far beyond knowing her times tables and being able to add fractions…

I tried out a new hair dresser over Easter and as we started to chat she ask me, “What do you do?” to which I replied, “Teacher”. She followed on by asking me, “What do you teach?” to which I replied, “Maths”… The response was a look of horror closely followed by “I was never any good at maths!”. Although I haven’t ever kept track of the exact number, I would have to guess that at least 9 times out of 10 that is the response that I get! That makes me sad…

In a recent INSET day we ran for an alliance of primary schools we asked the teachers and teaching assistants to raise their hands if they would be one of those who reacted by saying “I was never any good at maths!”… Over three quarters of them put their hands up… Was I shocked? No. Was I sadened? Yes.

It’s time for that perception, that attitude to change… As a Maths Hub we want to work on changing the mind set of all those teachers so that when Emily and her friends go to school they are met with confident maths teachers who are able to develop her curiosity and her understanding of maths so that in 20 years time when Emily’s peers are asked that same question the response is “I loved maths at school!”

So that’s the journey we’re on… As a Maths Hub, we want to support teachers, particularly primary teachers, in improving the quality of maths teaching because if we get it right from day one in reception and slowly build each year then we will get there, we will change peoples perceptions.

GermanEurosThe German football team crashed out of the Euro 2000 finishing bottom of their group. This gave them the wake up call they needed to stop and reflect on why that was and make some significant changes. They decided that their approach to youth development needed a complete overhaul. They set about learning from the best across the world – The Netherlands football association, the New Zealand All Blacks… All the while considering, what is the difference that makes the difference? They borrowed methods and adapted ideas. They poured all their resources into producing greate players starting at grassroots level. They developed full time specialist youth coaches and began working closely with schools.

By 2014 they were back on top of the world, holding the World Cup.

PISAIn 2000, the UK was ranked 8th in the world for Maths in PISA, by 2009 this had fallen to 28th and then in 2012 we’d clawed back to 26th… I’m not a fan of PISA, it has many flaws, but one thing that you cannot argue with is the fact that we were no longer moving up.

It was time for a change. Maths Hubs were born and one of the first things we did was look to the best in the world, Shanghai and Singapore to see what made the difference for them… But like the German Bundesliga did in 2000, we need to borrow and adapt the best ideas and make them work here in England.

As a Maths Hub that’s what we’ve done. Over the last year schools have been trialling some of these ideas to see how we can make them work for us.

The question that every school leader needs to be asking is…

What CAN we do to ensure our children get the very best Maths education in the world?


Session 1

Session 2

Session 3

  • Live demonstration lesson – Year 6 St Joseph’s Primary School & Mr Collins




TC 3

This project is being undertaken by Deb Harper at South Farnham School and Tom Collins at St Joseph’s Catholic Primary School, Guildford.

Scope of project

To develop a primary mastery curriculum which will be accompanied by a variation document providing examples of how to extend students laterally.

  • Three strands:
    • Arrange the KS1 and KS2 primary curriculum objectives into a coherent and progressive mastery curriculum.
    • Identify which of these objectives can be taken out of numeracy and addressed in other topics
    • Create examples of variation to address these objectives and to allow teachers to extend students laterally (this does not need to be a full set of worksheets but key examples of different variations which can be shared and then further developed. More than one example may be needed for each variation to show how a variation can be progressed).


By November, an outline of the order of topics had been drafted for each Year group. Teachers had begun to create resources that encouraged depth of learning and developed fluency, reasoning and problem solving skills. At this stage, the White Rose Maths Hub Schemes of Learning were discovered!

Although it was nice to see that another Hub was thinking along the same lines as us, it quickly became clear that the resources that their projet was considerably further ahead than ours, and there was no point in duplicating work!

  • White Rose Maths Hub
  • Based in Halifax, Yorkshire
  • Trinity Teaching School alliance
  • Published FREE scheme of learning for KS1 and KS2
    • Mastery Curriculum Model
    • Fluency, reasoning and problem solving questions
    • Assessments

Is it important to register with the White Rose Hub to ensure you are notified when new materials are produced. This package is not yet complete and further assessments and resources will be available throughout the summer term.

The White Rose Schemes of Learning give Overviews for each Year Group which detail the number of weeks that should be spent on each topic. The example below shows the overview for Year 5.

Curriculum 2

curriculum 1There is further information within the Schemes of Learning which details the specific learning objectives that should be taught in each topic section. The example here shows the objectives that should be covered in the 5 week fraction topic that takes place in the Spring term of Year 5.

Though this is extremely useful as a starting point, many Primary teachers we have spoken to are struggling to understand how to structure lesson plans so that these fractions objectives fill 25 lessons.

Collaborative Working

In order to address the areas highlighted by teachers, we will be working in collaboration with the White Rose Hub to produce some supplementary resources. Deb and Tom are working on producing a lesson by lesson breakdown for each topic to give teachers a better indication of how much time should be spent on each objective and how the learning could be scaffolded.

curriculum 4

The White Rose Schemes of Learning also give some examples of questions and tasks which develop student’s fluency, reasoning and problem solving skills of each of the national curriculum statements.

Curriculum 3

curriculum 6The White Rose documents have been extremely well received by Primary teachers, but some have commented that they would find powerpoint resources beneficial so they can be used within lesson slides. Deb and Tom are planning on linking the powerpoint resources developed by teachers at South Farnham and St Joseph’s to selected learning objectives, providing additional lesson resources. Here are some examples of some additional fluency, reasoning and problem solving questions for one fractions objective.curriculum 7curriculum 8

Textbook Mapping

The final aspect of this collaborative project is the development of a textbook mapping document. This will enable teachers to see how their current textbooks can be used to fit with the White Rose Schemes of Learning. The aim is to link exercises in 3 main textbooks (Busy Ants, Inspire and Maths No Problem) to specific learning objectives.

curriculum 5

End Goal!

The aim is to work with the White Rose Maths Hub to produce a FREE fully resourced package that includes the following:

  • Mastery Schemes of Learning for KS1 and KS2
  • Lesson by lesson learning objectives for all topics
  • Mapping to popular text books for all topics
  • Bank of questions for each learning objective
  • PowerPoint resources with fluency, reasoning and problem solving questions for selected topics
  • Termly assessments

We hope that the supplementary resources will be available for the Autumn term by June 2016.

Click here to download the powerpoint from the spring conference Session 1 – Curriculum Design