Symbolic Maths; a feature by Richard Dunne for Kids Don't Count. Richard Dunne is an education consultant and author.
I am sure we all agree that children should learn the times tables, use appropriate techniques for adding and subtracting numbers, calculate confidently with money and deal with the sort of measurement and calculations that are used in everyday life (including giving change, even though most of us are happy to rely on the till to work it out for us). This involves a good deal of practice and repetition and rote learning, often using dice, playing cards, dominoes and computer games as well as daily practice led by the teacher.
We know that children with good memories can master all this rather easily because they can remember how to do the calculations, but there are many children who find it enormously difficult and need to learn this over a much longer period of time. We can cope with these differences in schools, especially when this learning is supported in homes.
It concerns me when it is assumed that coping with everyday life is the only reason for learning maths.
What concerns me even more, and what I believe makes learning maths hard in schools, is when everyday life is used for teaching maths. For example, children are taught about division using sandwiches; fractions using pizzas; negative numbers using temperature or bank accounts; decimals using money. This is the very root of the problem. In other words, there is a too great an emphasis on 'what is relevant' as the way of learning maths. Too much emphasis, too soon, on maths being 'functional'.
Let me risk your disagreement and continue to tell you where I believe maths has gone wrong in schools.
First, I need to clarify what is special about mathematics. Mathematics is a logical, symbolic language. It is the clarity of its logic, the economy of the symbols and the fact that 'it talks to you' that makes it intrinsically exciting. It works like written English works (only twenty-six letters but thousands of words); or how music works (with crotchets and quavers etc for thousands of tunes).
Secondly, and this will risk a lot of disagreement, the symbols make it easy to learn school maths. Algebra is the high-point in learning maths at school because it is entirely symbolic. It is entirely abstract. Algebra should be thought of as every child's birthright – yes, algebra, so often said to be irrelevant and made the subject of jokes, can have a powerful impact on how children think.
I need to come back to that. For the moment we must worry about why so many people find maths hard. Why is there so much anguish about using percentages and fractions – and especially algebra? Why do people wince at the thought of trigonometry? The problem has to lie in the way maths is taught.
However, the preferred emphasis on 'relevance' does seem so obvious because sandwiches and pizzas are so familiar in real life. What is more, because maths is so symbolic, it seems obvious that it must be hard and therefore needs the familiarity of everyday contexts to clarify it. What has been missed is that maths is easy because it is symbolic and logical. This is what makes it straightforward. Not easy, but straightforward.
Let's now concentrate on maths. Think of maths as being a very special world in which everything is logical, tidy and unambiguous, for example 2 × 4 – 1 × 3 is definitely 5 and cannot be anything else. [For more information on this, see Richard Dunne's feature on BODMAS.] Think of the real-life world as being complex, muddled and ambiguous, for example 'Three packets of sweets and a sandwich' risks the uncertainty of whether there are three sandwiches or just one. Of course, these are only simple examples, not serious examples, for the sake of illustration. I am trying to indicate that maths is special because its language is so precise, and its precision is used to impose order on the real-life world. What I want to avoid is using the muddle of real-life to try to teach the straightforwardness of maths. We need to teach maths as a language in such a way that retains its straightforwardness, its beauty and its simplicity.
The solution to all this is to mimic maths using concrete objects and physical actions. We can use cups and sticks and cards and a limited range of other important objects to mimic this logic and so create a robust learning system which is situated between the unambiguous abstract maths world and the ambiguous real-life world. As a result, children learn maths in all its simplicity, economy and absence of ambiguity. They can learn abstract maths (the very thing which is assumed to be too hard for children), by using such a learning system.
It is tempting to think I am simply advocating using concrete materials (and all teachers use concrete materials). This is not the case. It is the combination of, and progression within, the use of concrete objects, physical actions, special vocabulary, visual images and written symbols that is at the heart of a learning system. This is combined with Ten Big Ideas in primary maths so that what is taught early on structures all future learning (including in secondary school, when two more Big Ideas are added).
Strangely, my focus on symbolic maths also assists the children who tend to take a long time to learn their times tables or giving change. Careful and early introduction to the symbols, via a learning system, structures their understanding so that there is less emphasis on memory – children can visualise the mathematics through the objects and actions they use. This is important for adults as well as children. When I ask adults, including teachers, to try a relatively straightforward maths test there is often a great deal of confusion and many mistakes. This is no reflection on them. It emphasises the way in which most people are taught maths so that, apart from those that 'got it easily', items that have not been practised recently become confusing. It is not generally recognised, and is not explicitly taught, that 'the symbols speak to you'.
It is reasonable to ask why this approach has not been adopted across the country. It is completely understandable. The assumption that maths teaching should focus on the everyday real-life world is deeply embedded in all advice on teaching maths. Any suggestion to the contrary is understandably treated with horror. Teachers who want to study how to do this need the time and freedom for really detailed analysis to be able to trial something so radical. Teachers are too busy.
Let me finish by saying that my enduring interest is to provide a way of ensuring really high standards in schools, but I do not mean just better test results within the current system. I mean that the maths we teach should be designed so that all children know mathematics in a quite different way – that they enjoy the sheer logical symbolism of algebra and revel in its intrinsic interest and beauty.
We really do need a radical change in the way we teach maths.
