Derek Haylock's Mathematics Explained to Primary teachers (Sage) is one of the most influential books on mathematics teaching, and contains many of the key ideas in current practice. The writer's knowledge of mathematics is not in question, and is without doubt greater than mine. The application of this knowledge to teaching mathematics is where the controversy starts. It is a controversy that has haunted me ever since I first began to help people with mathematical problems, in the late eighties, and that continues to haunt discussion of mathematics as part of a hidden agenda that is designed to move maths teaching away from number work, but does not dare admit that number work is neglected.
My first pupil was a man in his twenties who wanted to join the police force, but had not passed O level maths and was not confident about the maths test he needed to take instead. I looked a the kind of questions asked by the test, which usually involved calculations in the context of police work, including decimals, and taught him how to do them. He passed. My more recent pupils have been children who have had literacy problems, and who also struggled with maths. They have all had difficulties with tables - any tables - and have had to work out number facts up to 20 from scratch virtually all the time. They can all count, and 2 + 2 is usually automatic, but not much more. Their current ages range from 8 to 13, and I'm teaching them number bonds and tables before we do anything else. One practice technique for number bonds, which was given to me by a parent, is to take a pack of cards, remove aces tens and face cards, and put down two cards at a time - children call out the total. Once they can do it quickly and accurately, without having to add the numbers up from scratch, we sometimes add a stopwatch.
The reason for working on number bonds and what the National Curriculum calls "multiplication facts" is that adding and subtracting depend on the bonds, and multiplication and division on the tables. Dr Haylock says, speaking of efficient mental strategies for multiplication and division (p87) that "The first prerequisite is that you know thoroughly and can recall instantly all the results in the multiplication tables up to 10 x10." He also notes at the beginning of Chapter 5 that pupils should be taught to recall all addition and subtraction facts for each number up to 20. But then his approach to the formal approaches that apply these facts to longer calculations is not simply to unpack and explain how they work, but to concentrate, particularly on division, on "only the number knowledge they are most confident with" - in other words, to compensate for weak knowledge of the prerequisites by devising methods that get round them, even when these are more cumbersome and inefficient. Dr Haylock even refuses to explain long division at all - "In my view, it is a method that could well be laid to rest in the twenty-first century. To be honest, this is mainly because I have been singularly unsuccessful whenever I have attempted to teach the method."
Dr Haylock reaches for the calculator. But would he need to do this so quickly if children were properly taught the "prerequisites"? This is the rub. Does the teacher of a ten year old who, like my pupils, does not know any tables can cannot do much more than count, teach the pre-requisites or try to get round them? On the one occasion I've been able to research the issue, the weakest twenty pupils in a South Essex comprehensive school virtually cracked their problems with number bonds and tables in three months, using a combination of Grace Fernald's techniques - virtually unknown in mathematical circles - and computer games to speed them up. The piece was published in The TES in 1994 - too early for the website - and I got support from Ann Dowker of the Oxford Department of Experimantal Psychology, here. When the Director of the London Institute of Education tried to set up a project to take it forward, he was cold-shouldered by his own academics, who included a prominent supporter of Dr Haylock. The curse of educational research is the way it has come to be controlled by people who use it for political purposes, and prevent investigation of elements that might conflict with their political goals. In the middle ages, such activities were known as "treason by the clerks." In the twentieth century, they are the bread and butter of left-wing politics.
Dr Ann Dowker's What Works for Children with Mathematical Difficulties contains important further evidence. She says that problems with arithmetic are not unitary, but may arise because of problems with several aspects of thinking. However, almost all children with mathematical difficulties have problems with remembering number facts, and are hence reduced to counting where others can use their knowledge of number facts to save time. This is in line with Fernald's thinking, and suggests that teaching and learning simple combinations is an important first step towards helping them.
Dr Haylock does address the issue of multiplication tables in his book for younger pupils. He and his co-author are in favour of children learning them, and include the interesting tip that, if they are chanted or said, each line should be said twice, so that the weaker children have a chance to get it righ the second time. But they don't go into detail on how to help the children who have difficulty in recalling and using the tables, and this is the problem facing the teachers of most pupils who have problems with maths.