Three teachers from the Cambridge Maths Hub are currently in Shanghai.  Each day they will send some thoughts, which we will post here.

Collaborative learning across the key stages and the world!

Some Primary Observations

Reflection 1: Relationship between 3,6 and 9 times tables.

By Grade 2 (Year 3), the children in Shanghai are fluent in their multiplication facts up to 9 x 9.

We found this interesting considering that we are teaching up to 12 x 12 by the end of Year 4. Their thinking is, like our thinking has been previously, that if you learn up to the ten times table you can use this knowledge to calculate any other multiplication fact (as long as you have a deep understanding of the relationship).

After children are fluent in recalling multiplication facts, teaching moves on to expose the relationship between fact families with the ultimate goal of being able to use this knowledge to solve problems e.g. “My house number is a multiple of 3,6 and 9. The number is between 30 and 40. What is my house number?”

On discussion with the teachers, after the lesson (in a TRG), we explained that while we do look at the relationship between the multiples in England, this is often to support children to learn seemingly harder multiplication facts e.g. double a multiple of 3 to find a multiple of 6.

Reflection 2: Classifying triangles

On Wednesday, we observed a lesson in Grade 3 (Year 4) that focused on classifying triangles. After recapping the previous learning, that a triangle had 3 angles and 3 sides, and having already classified triangles by their angles, the children were given plenty of sticks (4 colours/lengths – AND plastic joiners!) to explore and make different triangles. After a few minutes of exploration, the teacher began to collect triangles in groups. She displayed in on the board, in a very specific arrangement (equilateral followed by isosceles with a space left for scalene triangles).  The isosceles triangles were collected in such a way that the children were able to identify the missing triangles (based on the stick pattern that was missing from the display).

After collecting all triangles, the teacher asked the pupils to circle the isosceles triangles – this included the equilateral triangle as isosceles triangles (as these are classified as having AT LEAST 2 equal sides). This generated an interesting discussion between the English teachers at the back of the room, as many teachers in England are still classifying triangles into 3 different groups. This lesson exposed the equilateral triangle as a special type of isosceles triangles (like we see a square as a special type of rectangle). We referred back to the NCETM website and confirmed that English thinking is moving in this direction but not all of the English teachers were necessarily aware of this.

The layout of this collection of triangles allowed the pupils to come out and circle the triangles to create, what we would see as, a Venn diagram.

During the TRG, after the lesson, we were able to share the NCETM resource (that we had earlier referred to) with the Chinese teachers which they found interesting and were keen to use in future to develop challenge activities. Go NCETM! It was great to be involved in such a collaborative discussion with both sides learning from each other.

Day 6: Forwards and Backwards - the power of the inverse

In Chinese lessons we saw expressions being explored both ways.  So in fractions students know that 

But in England we do not focus on the reverse:  (something that Alevel students struggle with when simplifying).

Factorisation is the inverse of multiplying or expanding.

In a reflection lesson students were reflecting shapes in a line,

At the end of the lesson they had to find the mirror line.

The first picture is a nearly finished version where he reversed the method in  the lesson.  The second student chose to construct the perpendicular bisector.

Day 5: Relationship between fractions and ratio

In Shanghai mathematics is about generalising.  The image shows the key vocabulary used in the first few ratio lessons, the basic properties of fractions and three algebraic representations,  ratio, fractions and equivalent division, and how they are  connected.

This language and representation is used from the start of year 7.

Day 4: Representations and when they need to change /amend

Have you ever used the crocodile/pacman image for inequalities?

Talking to primary colleagues I admit that I use it, however they were adamant it is no longer used.

Instead they gave me a better representation.  1 is less than 4 can be shown using cubes.

This works well until you want to consider negative numbers.  At this point considering the position on the numberline is more useful.  We need to be sharing the representations we use and how and when they work.

Day 3: Planning of a starter example/practice to feed into later part of lesson

In a  lesson on circles, student had to find the circumference of 2 circles.  The answers to these questions was written on the board (here is a reconstruction):

Later in the lessons students had to find the radius of a circle when the circumference was 9.42m.  They could look at the earlier calculation and realise that it was the inverse, so the radius = 3

The teacher then looked at how you could efficiently work out the radius as 

Students could see that 6.28 came from 2π

The teacher had planned the examples to support work later in the lesson – students can see the calculations being used in reverse (another idea that is common in Shanghai lessons).

Below is the board at the end of the lesson showing most of the work the class has covered and the final question:

Day 2: Planning for a misconception and how to address it

In a Year 4 lesson students were asked to find how many lines of symmetry a rectangle, square, equilateral triangle and circle have.  Students were given pre-cut shapes (manipulatives) and an answer sheet. Many students chose to not use the shapes. Some students made mistakes but these were not addressed until the teacher went over the answers.


When discussing how many lines in a rectangle, the teacher identified the lines of symmetry one at a time.  When the diagonal was chosen as a line of symmetry she made the students fold a rectangle to see that it did not work. Students were self-correcting their work.

I have heard discussions about when it is appropriate to bring in misconceptions as students might learn the wrong answer. In this case they already had the misconception and they were given a concrete example to correct it.

(Translation - The circle has many lines of symmetry )

Day 1: The teacher photographs errors that have been made throughout the topic and then uses them at the end of the topic to do an overview - with students spotting the mistake and someone coming to the board to do it correctly.

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The Maths Hubs programme brings together Mathematics education professionals in a collaborative national network of 35 hubs, each locally led by an outstanding school or college.

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