## Maths Memos

## Mathematics and Problem-solving (II)

**Niall MacKay shares some further thoughts on
problem-solving...**

In a previous post I talked about process versus facts in the learning of mathematics, and about problem-solving. What I had in mind there was principally problem-solving in pure mathematics, which together with theorem-proving form the systole and diastole of the subject (see Tim Gowers' thoughts). Here I want to say a little about another of Tim's interests, namely how mathematics should be taught to non-mathematicians - by which he means those who choose not to study Mathematics at Advanced level, and who will therefore not study calculus at school.

I've been on various groups involved in thinking-through such 'Core Maths' qualifications. Sometimes I have the impression that our political masters would prefer to see more attempts to test to destruction a student's capacity to absorb formal mathematical facts and algorithms. Our thinking, in contrast, has focussed on 'problem solving'. You can find the kind of problems we mean on Tim's blog - typically, they are what engineers might call 'back of the envelope' calculations and physicists ' Fermi estimates' (after Enrico Fermi's love of order-of-magnitude estimation). They also include problems in finance, medicine, and a variety of other areas of modern life and society.

What they all have in common is the need to work with very small
or very large numbers. Indeed, this is largely *all* you
need, for whilst there are some nice problems which begin with
algebra or geometry, they are rarer and have a greater tendency to
be *sui generis*. I'm personally very concerned that even
highly intelligent people in this country are often unable to deal
with large numbers - so that if your method of working out by
hand one thousand four hundred billion divided by sixty million
involves writing down many zeros, I think you have a problem,
although perhaps not one as large as the national debt.

Part of the problem arose when we threw out the teaching of logarithms before the age of 16. Grim memories of printed 'log tables' have been replaced by a syllabus which introduces logarithms algebraically, at A level, alongside the exponential function. What has been lost is the development of students' ability to turn multiplication into addition - that is, to use standard form, so that instead of writing 10000 x 1000000 one writes 10^4 x 10^6 = 10^10, and multiplication has become addition. To turn those integer powers of 10 into a continuous, logarithmic scale is then only a small step further. If you have understood this idea - and it takes time to absorb - then you can deal with astronomical sums and remote risks, debunk false statistics in the news, and turn growth, decay and scaling processes into straight lines. You'll also understand that the ability to work intelligently and comfortably with large and small numbers is a necessity if one is to engage competently with modern life - and a failure to do so can even be deadly. In the first instalment of this post I began with Toby Young's recent paean to facts. I wonder what his attitude will be to a course which contains so few of them?

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