Niall MacKay's third post in his mathematics and
problem-solving series
It is part of a declinist narrative with a long history to argue
that English education is going to hell in a handcart. The
Programme for International Student Assessment (PISA) results in mathematics
are actually rather unexceptional, especially once one has
understood that PISA's
statistical methods are on built on sand. And Further Maths A
level remains, for the mathematical elite, an outstanding
qualification.
But we had an unwonted piece of good news recently, in that PISA
found that English children
were good at solving problems. Here I want to argue that the
encouragement of problem-solving has traditionally been an
important cultural strength of English mathematical and scientific
education, helping us to grow creative achievement, but that it has
become somewhat unloved in recent years.
Toby
Young, no doubt, would wonder why I called problem-solving
'unloved,' but, as I explained in the first part of this post, in
mathematics, at least, we have had 30 years of development - in
textbooks, in assessment, perhaps in teaching which is
certainly inimical to it.
So what kind of teaching helps to grow the creative achievers of
maths and science? Obviously technical development and mastery is
crucial. A system which selects the elite early and then challenges
them constantly may well help those selected, as in Russia's
Kolomogorov schools and the French Grandes Ecoles - but
actually England has a partial elite system, with selection at 18
for a small number of elite universities.
I'm not going to talk here about how best to teach technical
mastery. Rather I want to begin with an observation of another
feature of the really great researchers: that they pursue -
doggedly, tenaciously, without any sense of deference or proportion
- the questions which bother them. Many a great research result has
begun with a refusal to set aside a niggling question, something
which everyone (else) thinks is of no import, but which, like a
loose tooth, someone just cannot leave alone.
I think we are, or used to be, rather good at encouraging this.
A culture which encourages eccentrics and contrarians is a good
start. In the education system, however, there's a subtle tension.
Typically the teacher knows rather more than the student, and a
student whose misplaced self-confidence leads incorrigibly to the
wrong answer needs to learn the limits of his own expertise -
and then extend them. On the other hand, I think that in English
education there's a general disinclination to appeal to mere
authority - whether of the teacher or of a text - so that a
questioning attitude is typically used by a good teacher to arrive
at a higher truth. We're quite good at the Socratic method, and I
hope we're committed to the Enlightenment, too - to 'daring to
know'.
But, in universities, we've certainly noticed a move, especially
among weaker students, in the wrong direction over the last 20
years. Too many students come to university having been
taught-to-the-test, wanting to know the answer. One
colleague commented to me that students have a particular notion of
how we mark their exams, as if we peer at their solutions through a
template, looking for scraps of fact and 'correctness' to which we
can award marks. I explained to him that this is exactly
how school exams have come to be marked - and that it's going to be
a long if not impossible job to convince the students that we do
otherwise.
So my position, in the end, is that it's not our curriculum but
our exam system that needs reform. It's pretty easy to write a
curriculum that's about right. We also have many teachers who, left
to themselves, will encourage the kind of disciplined enquiry that
typifies the finest minds. I think that in our culture most
teachers, whether at school or university, love to teach students
who are brighter than themselves, who can transcend them.
But we have a crazy system, in which schools choose from among
commercially competing examining organizations which often produce
fragmented, trivial exams, and which sell shallow textbooks
tailored to those exams to accompany them. There's no sense any
more of a great textbook empowering and inspiring students, let
alone teachers. If the Education Secretary is feeling radical,
reform of the awarding bodies, and their replacement by a single
organization, is where he should begin.
Dr Niall MacKay is an ACME member and Reader in the Department of
Mathematics, University of York.