# Mathematics teaching in China: reflections from an Ofsted HMI

In late February I was a member of a delegation representing HM Government that visited the three Chinese provinces of Shanghai, Beijing and Hubei with a specific focus on mathematics education.

I have waited until now to reflect on my visit to China because I wanted to go back into some English schools to test out the thinking I developed while there. The differences in maths outcomes for our young people between the two countries are stark and worrying for us, unless we act now to catch up – and I do not mean just in terms of PISA test scores. I am coming at this not only from an inspector’s point of view, but also from my background of being a physics teacher and so frequent user of maths, reliant on pupils being able to handle and manipulate numbers confidently. In this respect, Chinese children are streets ahead of ours, so the benefits of their high standards in mathematics go way beyond just this core subject.

As everyone knows, Her Majesty’s Inspectors are not concerned about the ‘how’ but ‘how effective’ with teaching. This approach requires a clear focus on the outcomes for the pupils and their response to the teaching, including crucially the evidence of learning and progress over time in their work books and folders. These were impressive in the classes we observed in China, and told a story of a consistency of approach and expectations that has led to the pupils being confident mathematicians, willing to have a go and able to tackle problems in different contexts.

For example, given this problem…:

X = 2√ (7/14 x 28/7 x 3/9 x 24/8 x 18/9)

… none of the 12-year-old pupils reached for the calculator; they couldn’t because they have been banned from their classrooms. They calmly looked for the potential to cancel and reduce the fractions, and spotted that this expression is really just the square root of 4. Not a job for the calculator; not for them at least. This was clearly not about them learning ‘tricks’ either. This problem was one of just 4 or 5 set by the teacher in a 5 minute burst of practice, to help the pupils master the concepts covered by her in the latest part of the lesson before they moved on confidently together to the next stage of increasingly challenging maths. The key was not the teacher’s ‘performance’ in this lesson, but the demonstration of the depth of the pupils’ mathematical learning over time and the impressive armoury of knowledge and skills they had built up to deploy as and when needed. Evidence of solidly knowing their times tables was absolutely apparent across the pupils, as was the ability to use efficient methods of calculation without having to really think. Their mathematical toolkit was there to be used as surely as a mechanic’s spanners, or a surgeon’s scalpel.

In another lesson in a primary school, the 6-year-old pupils worked with the teacher on the link between addition and subtraction for 2 digit numbers; they demonstrated their clear knowledge of the inverse relationship here and could use that to check their own answers. The work in their books over time showed consistency and accuracy in setting out calculations, and the discipline of drawing diagrams carefully with a pencil and ruler had been instilled from an early age. However, as with most children across the world, these pupils were full of fun; and when their teacher stopped to talk to us near the end of the lesson, the level of chatter rose significantly and it wasn’t about maths!

Compare these observations with a fairly typical Year 11 maths lesson I observed in a school once I returned to England. This top set, all predicted at least B grades at GCSE, the majority on for As, were learning how to use the sine rule:

a/sin A = b/sin B = c/sin C

During the lesson, and while the pupils worked their way through the 20 or so examples they had been set by the teacher, I asked about a third of the class individually how they would rearrange ‘a’ to be the subject of the formula in the context of one of the examples which asked them to find one side length of a triangle. Some mentioned using ‘formula triangles’ – a trick often used by science teachers when pupils’ basic algebra is weak. It is depressing if these are used in maths where algebra should be taught rigorously without such tricks; it was even more depressing in this case because the necessary formula had 4 variables, so doesn’t fit in a 3-sided shape. Just one pupil could explain how they would do this in a way that demonstrated they understood the basic algebra underlying this task. Much of what they said demonstrated weak understanding that multiplication and division are inverse operations; a fact that I had observed 6-year-olds in China using successfully a couple of weeks earlier.

In another lesson I recently visited in England, most but by no means all of the 7 and 8-year-olds I asked to take away 98 from 103 could give the correct answer. However, when asked to check their answers, few could explain a simple, efficient method of doing so. Most tried using variants of their different subtraction methods and got confused; a few realised (correctly) that counting on was an option when the difference was so small. None got close to demonstrating an understanding that subtraction and addition are inverse operations and so using that fact to gain confidence in their answer. This was second nature to the same 6-year-old Chinese children I mentioned above.

My point is not that pupils in English schools are terrible at maths. It is that the basic mathematical facts (like number bonds and times tables), operations, appreciation of size and handling of number are not second nature to all our pupils – and they need to be if we are to develop confident, young mathematicians.

In the coming months, I will be urging my inspectors to ask pupils similar questions to assess the real level of knowledge and understanding of basic mathematical concepts in the schools we inspect. I want them to get under the skin of national test scores and levels to really assess what pupils know, understand and can do in this vital subject.

My diagnosis of the difference in our systems is that mathematics in the Chinese curriculum is not seen as an elite subject. It is viewed as an essential of life, and one in which everyone can be highly competent if you work at it. ‘Maths gets you everywhere’ is a common phrase used in China and far from turning pupils off the subject, this focus and respect for it gives their pupils confidence and purpose. The strong leadership and management of the subject, and curriculum focus are demonstrated in a number of ways. Mathematics is compulsory to age 15, but any applicant to university must have a qualification in maths equivalent to our ‘A’ level, regardless of the subject they wish to read. While maths is mandated to get a minimum of 3 hours a week on the secondary timetable in China, the schools we visited used their curriculum freedoms to provide 5 periods of maths for each class per week. Maths teachers in China typically teach just two periods per day; although balance that with class sizes of 48-50 pupils and you can see how this can be done within budget.

Assessment of attainment is rigorous, frequent and regular through ‘traditional’ tests; and it is diagnostic for the class. Maths homework is set, marked and handed back by the teachers every day. Teachers use their non-contact time in the day to do this and handing in homework is rigorously enforced by teachers and adhered to by pupils. This discipline is instilled in the children from a young age. The maths teachers in a school meet each other most days, in what they call Teacher Research Groups. This time is used to analyse what is going well with their classes, what needs to be tackled differently and what the frequent tests are telling them they need to adjust about their teaching to address any misunderstandings arising among the pupils in their classes. This commitment to teacher collaboration is impressive and seems to pay off.

The Chinese education colleagues we met were very open to our views throughout the trip and keen to learn about the ways we educate our young people in a variety of subjects. I have every confidence that a strong relationship is developing between our countries and we have much to learn from each other.

I would like to thank the DfE for inviting me to represent Her Majesty’s Chief Inspector and Ofsted on the visit.

**Sean Harford HMI**

**National Director, Initial Teacher Education**

**Ofsted**