Smart but casual;Maths Year 2000
Informal methods - those methods that people devise themselves - are widespread in mathematics. Such methods are often limited in scope or efficiency and, of course, sometimes downright wrong. People who want a true appreciation of the nature and power of mathematics will need to learn formal methods, and the limitations of their informal methods will have to be made explicit. But what surely is worth establishing first, in everyone, is confidence in their ability to make sense of the maths they meet at school and in every day life.
For this to happen, we need to acknowledge and indeed celebrate informal methods that work - whether or not shortcomings are later laid bare.
Guided by this principle, I have published a pocketbook of maths questions called Maths Medicine. The book is for the general reader and is recreational (in a masochistic sort of way, you might think). It involves the kind of standard maths included in the national curriculum but the questions can all be solved using informal methods. Readers are encouraged to work through a question at their own pace and to discuss their ideas with friends. If they really get stuck they can visit an accompanying website which provides help and answers. The site also contains a selection of readers' methods to underline the point that maths can be approached in a variety of ways - even where there is only one correct answer.
When Maths Medicine was published I asked readers of The TES to send me their methods for answering this question from the book: "Is 17 nearer 16 or 18?" Of the replies received, about 60 per cent involved the standard school approach of converting the fractions to decimals or to fractions with the same denominator (usually 336 or 168). This approach is efficient and effective. Interestingly though, the other 40 per cent of replies involved methods that were just as effective, but contained features probably not taught in school. As far as the people submitting them were concerned, these methods were almost certainly original. They were also insightful. Thus, for example, several people used a qualitative method of this sort: "The fractions in this series are getting smaller, and closer together: 16, 17, 18, 19, 110. So any given fraction will be nearer its neighbour on the right than its neighbour on the left." Others reasoned by analogy: "It is clear that 12 is nearer 13 than 11: so, ..." Or a more concrete approach was used, perhaps based on dividing up a circle:
"16 of 360 degrees is 60x, 17 is about 51x and 18 is 45x." Others used school-based knowledge, but in an original way, as in "16 x 17 is 142, 17 x 18 is only 156," or: "The fraction midway between 16 and 18 is 748, but 17 is only 749."
There is great satisfaction to be had from devising and sharing methods of this kind, even when they are regarded primarily as a springboard to more formal methods. Many people see maths as impenetrable, demoralising and closed. By giving more credence to successful informal methods, school students and adults alike might (re)discover that even quite ordinary maths can be challenging, creative and fun.
Maths Medicine is published by Dexter Graphics, PO Box 24320, London SW17 7WQ. Price pound;3.60. The help-and-answers website is at www.mathsmed.co.uk