What is the commutative law of multiplication? Quickly now, spit it out. Sorry, too slow. Several of the six and seven-year-olds in Farhana Nazu’s year 2 class had their hands up first and they all know the answer: the commutative law means that you can swap the numbers around and the answer will still be the same.
Not hard, but perhaps, like me, you panicked when the question was thrown out there. Let us take a seat at the back of the class and observe as the children do simple multiplication. They are asked to work out how many biscuits there are if there are five jars and each jar contains four biscuits.
Some of the children physically count off groups of four on a bead string. Others take a pictorial approach, drawing jumps along a number line. Those who are able write the equation: 5 × 4 = 20.
The lesson proceeds remarkably smoothly. When the children move from their desks to the carpet and back they count out loud together in multiples. They work quietly and diligently. “Maths is my favourite subject,” one girl tells me. “Honestly,” Nazu says, “they love maths. It’s a natural thing just to enjoy maths.”
Enjoy maths? Now there is an intriguing concept. I managed to spend weeks at a time at primary school staring at the same page of the maths textbook, lesson after lesson. Without the groundwork in the early years I was tormented by the subject all the way through the desperate tutoring that enabled me to scrape the crucial grade C at O level (required for university) without actually understanding much of what I was doing.
My recurring nightmare takes place in the corridor of the maths department of my secondary school where my teacher informs me that the O-level exam is the next day, then asks grimly where I have been for the past two months. The dream is almost worth it for the blissful relief of waking to find that I am an adult who no longer has to worry about this stuff.
It is possible that maths today is generally taught better than it was in the 1970s. Certainly my two children have got a better grip on the subject than I had. Yet Britain still fails its children when it comes to teaching them maths. The latest Pisa (Programme for International Student Assessment) results from the OECD (from 2012) show that the UK’s 15-year-olds were in 26th place for their maths skills. The top of the table was dominated by Asian countries and cities, with Shanghai in first place followed by Singapore, Hong Kong, Taiwan and South Korea.
There may be cause for hope in lessons like this one at Grazebrook primary school in Hackney where children are taught using mathematics mastery, a system based on the successful approach known as Singapore maths (or sometimes “shangapore maths” because it also takes in elements of the way maths is taught in Shanghai). Under the maths mastery system fewer subjects are studied, but in greater depth. The whole class studies the same subject at the same time, with the abler pupils exploring in greater detail, before everyone together moves on to the next subject. There is an emphasis on problem solving and thorough understanding and less focus on memorising methods and procedures.
A recent study by the Institute of Education at University of London and the University of Cambridge looked at the progress made by pupils being taught using the maths mastery scheme at 90 primary and 50 secondary schools. The researchers found that after one year of maths mastery there were “small but positive” effects. They concluded that while the system “cannot be seen as a ‘silver bullet’ that will guarantee a country success in mathematics”, it “shows signs of promise, and should now be tested over a longer time horizon and a greater number of schools.” The report suggested that the findings, which are the first evidence that British students could benefit from the Asian approach to maths, had potentially important implications for education policy.
The government has been watching closely. Nick Gibb, the schools minister, has lamented the “stagnation” in the maths performance of Britain’s children and praised maths mastery because it “embodies the idea that every pupil can do well and achieve high standards in maths”. An £11 million Department for Education scheme has seen 30 Shanghai maths teachers brought to English primary schools.
Maths mastery was developed by the education charity Ark, which runs a string of academies in the UK. The maths programme, which began in 2012, will be running in more than 200 primary schools in September and 67 secondary schools.
Dr Helen Drury, the director of mathematics mastery, says in her book, Mastering Mathematics, that Britain needs to change it’s thinking. “It’s time to stop acting as if mathematics is for ‘clever’ people. It is not ‘OK’ to be ‘bad at mathematics’,” she writes. “Every child can succeed in mathematics, whatever their socio-economic background or prior attainment, as long as they are given the appropriate learning experiences.” She says that if you go into a maths classroom in Singapore it is hard to work out which ability set you are observing. The teachers act as if they expect the children to succeed.
I am one of the “bad at mathematics” people. My kids, aged 11 and 9, gave up expecting help with maths homework when they discovered that I could no longer remember how to do long division. When I talk to Drury she certainly makes her scheme sound sensible. The problem with the teaching of maths in recent history has not been that schools are teaching too little maths, but they are trying to cover too much, with the result that topics are skimmed over.
“Maths is such a connected subject but it gets sort of popped into tiny buckets, so you might spend a day or two days or three days learning about multiplication. A lot of the class will only be beginning to make sense of the idea but you move on and look at area.” Some pupils will work out the area of a shape by counting squares because they haven’t mastered multiplication. “The teacher would feel it would confuse some of the children and they’d be right because they hadn’t spent long enough with multiplication in the first place so that they can then make the connections.”
In Singapore, she says, students in the first year of school spend several weeks looking at the number 10, even though many of them can already count above 100. This gives them deeper understanding, which is lacking in many older pupils here. “If you talk to people here about their experience of mathematics there was a point where it stopped making sense. They resort to memorising methods and procedures.”
Drury was on the drafting team for the maths national curriculum. Schools using maths mastery still need to cover the national curriculum, which Drury says they do over the course of the year. However, they focus on teaching fewer topics in greater depth each term, rather than trying to cover the whole year’s topics in the first term then returning to them — an approach that she believes is driven by schools wanting to test pupils before Christmas. Drury also has concerns that the maths national curriculum becomes too advanced at the top of key stage 2 (end of primary school). One example: she doesn’t think that pupils need to be doing algebra in primary school.
In the early years, maths mastery shows pupils how to tackle a problem in three different ways: first, by using concrete objects (such as the bead string), then through pictorial representations, before moving on to abstract symbols. Some pupils will inevitably progress to the mathematical symbols quicker than others, but the idea is that a whole class will be working on the same topic. Drury says that some schools, especially at the secondary stage, will set pupils for maths but at the primary level “we strongly discourage it. Absolutely no benefit to it.”
I say that the methods don’t feel terribly new. “More than that, it was really common practice in the UK before it was in these other countries,” says Drury. Much of the practice draws on western educational research, such as the work of Jerome Bruner, the American psychologist who suggested children should learn through enactive, iconic and symbolic representation. “The idea isn’t to come up with something that is necessarily new because there is plenty that works out there,” says Drury. “In a lot of ways it is about coherence.”
Each one-hour lesson is divided into six parts. 1) Do Now: a warm-up task, done independently at desks. 2) New Learning: introduced by the teacher on the interactive whiteboard as the children sit in rows on the classroom carpet. 3) Talk Task: back at their desks the pupils do a task in pairs. 4) Develop Learning: on the carpet the teacher introduces a problem-solving element that the pupils do in pairs. 5) Independent Task: the children are given work to do by themselves. 6) Plenary: a recap of the lesson back on the carpet.
In the lesson I attend the children count in multiples of two and later five. However, I am told that maths mastery isn’t big on testing times tables. Drury says there is some confusion over times tables and their place in maths mastery. Although the scheme puts less emphasis on rote learning, she admits that it is helpful to know your tables.
“There is a tension in the UK where it is either about learning your tables or it’s about understanding the mathematics. Lots of people are pushing deep conceptual understanding of maths — that’s very important. And lots of people are pushing knowing tables. You absolutely need both. You can’t have one without the other and that feeling, as a teacher or school, that you have got to make a decision — ‘are we about knowledge or are we about understanding?’ — is ridiculous. You can’t engage in proper mathematics unless you know those times tables because it takes up too much thinking space.”
In its early stages maths mastery had some funding from the Department of Education and the Mayor of London. Schools pay for varying levels of support and the charity receives funding from Bank of America Merrill Lynch. When schools sign up to maths mastery their teachers receive training and local schools are gathered in clusters within which teachers share ideas and tips. Teachers work from lesson guides and can discuss ideas on online forums.
Professor Ruth Merttens, co-director of the Hamilton Trust, a charity that supports teachers, has reservations about prescribing how teachers should teach. She questions the logic of importing Chinese or Singaporean materials and “thinking that because it works there it will work here, without regard to cultural factors and differences”. She says that one reason why Shanghai children do well is that the teachers have much longer training.
Michelle Thomas, the executive headteacher at Grazebrook, says she had tried a variety of teaching schemes before adopting maths mastery. Less able children felt stigmatised, especially when equipment was brought out to help them, but now that equipment is fully integrated in the teaching scheme this is not an issue. The most able pupils, meanwhile, lacked a full understanding of what they were doing. “If you asked very bright children to explain how they did something they would say: ‘I don’t know, I just know how to do it.’ The conceptual understanding was weak.”
The biggest challenge may be to convince parents that maths really matters. Drury says that a cultural shift is required to make maths a subject at which adults believe it is essential for their children to be successful.
“We are in a negative cycle where parents, to reassure children, were saying: ‘I was rubbish at maths, it didn’t make any difference, don’t worry about it, relax, it didn’t do me any harm.’ If we can get children to be successful at their mathematics you take away the need for that kind of discourse and over time — this is a long game — they become parents who had positive experiences of mathematics at school and the whole thing becomes a positive cycle rather than negative one. But that isn’t going to happen overnight.”
Your starter for under-twelves (see picture gallery)
Question 1 (age 7)
There are 10 ladles of soup in a large pot. The soup is poured into three small bowls: A, B and C. Bowl A has more soup than Bowl C. Bowl B has more soup than Bowl A. Guess the number of ladles of soup in each bowl. Make sure the numbers add up to 10 ladles.
There are a number of possible answers (4, 5, 1; 2, 7, 1; 3, 6, 1; 3, 5, 2). There is a rich variety of follow-up questions: which one does the picture look most like?
Why can’t bowl B have 8 ladles? Why can’t bowl B have fewer than 5 ladles? Is there an answer where one bowl contains exactly half as much as another bowl? This last question ties in another area of the curriculum — fractions — to a question, which is originally about number bonds to ten.
For concrete support, pupils could use 10 cubes to represent the 10 ladles of soup and put them into piles representing the bowls.
Question 2 (age 11)
This shape is made up of a rectangle and two identical right-angled triangles. Its area is 48cm2. The rectangle has an area of 18cm2 and a perimeter of 18cm. What is the value of x?
If the rectangle has an area of 18cm2 and a perimeter of 18cm, what are the possible side lengths?
What must the total area of both triangles be to give an area of 48cm2?
What is the height of each triangle?
One answer: x is 26. The area of the rectangle is 18cm², so if the horizontal side of the rectangle is 6cm the vertical side must be 3cm. That makes the base of each triangle 10 cm (the area of a triangle is the base multiplied by the height divided by 2). So the length of x is 10 + 10 + 6 = 26.
What assumptions were made? The diagram appears to show the longer side of the rectangle as its base. Do we know this to be so? Is the diagram misleading? Could the orientation of the rectangle be different? Can you find a second solution?
Although the question is challenging, a range of children should enjoy and learn from it. Some pupils will use objects such as a geoboard or squared paper, many use more formal algebra. A popular saying among Shanghai teachers is: “the answer is only the beginning.”