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tables' have been replaced by a syllabus which introduces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.