Maths Memos

Mathematics and Problem-solving


I had a moment of revelation reading Toby Young's pamphlet the other day. I'm sure there are people queuing up to call it simplistic, but it occurred to me that his views are especially dangerously so for mathematics.

His unashamedly Gradgrindian position is that there can be no worthwhile learning of higher thought processes-argument, analysis, abstraction - that is not embedded in a substrate of facts. Well, given the howling lack of knowledge that the press likes to report as existing among the general populace, I'm not going to argue with that, although perhaps it was ever thus. (Did Magna Carta die in vain? and was it of a surfeit of palfreys?) Young imputes to educators who advocate 'problem-solving' a position which deprecates the mere learning of facts. He considers this progressive orthodoxy inimical to real achievement grounded in rote learning, whether of the dates of the kings and queens of England or of musical scales.

But maths is rather different, because maths is process. Learning the names of six sorts of quadrilateral is not mathematics. Learning how to construct arguments in plane geometry, by contrast, is a training in all sorts of higher cognitive skills-and it requires very few initial facts (five, famously), and only a little taxonomy along the way. Indeed, the whole of mathematical analysis is built on nothing more than elementary arithmetic plus one statement about sequences and/or bounds.

And so I see a problem emerging. Young's Manichaeism pits traditionalist fact-grinding against progressive problem-solving. Yet in mathematics it's the traditionalists who like problem-solving, by which they mean a mathematics not of taxonomy but of process, in which exam questions are stated simply and the test is of the student's ability to think their way through them - to construct, test and communicate mathematical arguments. By contrast the progressive orthodoxy of the last 30 years has produced an exam system of great reliability but very little validity, in which questions are broken down into small, predictable parts, promoting a brittle, superficial learning that breaks down when given an unstructured or unfamiliar task.

This inversion of the traditional and the progressive is not perfect, of course. One could argue that students still need to know mathematical formulae-but it is striking that those who have achieved most in maths usually have very few formulae committed to memory. Rather they have done the calculations so many times that they are able to re-derive results without fuss - like being able to progress quickly through the easy stages of a video game, or a familiar rock climb, or simpler exercises on the piano. What is indisputable is that one learns maths by doing maths. So, to take an example, I've discussed with various mathematicians the value of learning long division. The common view is that its value lies not in its utility, but rather in having experienced and practised the algorithm (which is only really fully understood when one extends it to the algebraic long division of polynomials).

So to learn mathematics one certainly has to do a great deal of mathematics, acquiring a set of tools with which one is familiar - but it is this practice of process, not the memorising of facts, that is requisite for mastery. But such processes once acquired are not always very applicable - several of those mathematicians said that they had to re-learn long division whenever they wanted to remember what it is. To give algorithmic process-learning any value, one also has to learn how and why the tools work, how to apply them in unfamiliar settings, and how to decide which tool (or combination of tools) to use to solve a problem. Only thus does one progress.

I do wonder whether politicians and policy-makers, especially those whose own mathematical studies did not progress beyond age 16, and who perhaps never really felt empowered by the acquisition of their own mathematical toolkit, really understand that the traditional/progressive divide does not work the same way in maths. It would certainly explain some features of the new curriculum.

Dr Niall MacKay is an ACME member and Reader in the Department of Mathematics, University of York.

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