Maths Memos

Mathematics and Problem-solving (III)


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. 


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