Yesterday, the British newspaper, The Telegraph, published a piece by Jo Boaler, Professor of Mathematics Education at Stanford University. In the article, Boaler celebrates the achievement of Maryam Mirzakhani, the first female winner of the Fields Medal – often described as the maths equivalent of the Nobel Prize – and she laments the gender gap in PISA maths achievement in England. Those who have followed Boaler’s arguments since the 1990s will not be surprised by her solution. It is a proposal that has echos of Herbert Spencer, John Dewey and the rest, although the focus on gender is more recent. And it is a proposal that is wrong.
Talking about a PhD viva that she recently attended, Boaler notes, “The young woman defending her PhD that day paced the room, sharing conjectures and connecting different theories. The mathematics we saw was visual, creative and alive. Sadly, however, the approach of Maryam and her students is far removed from that taught in British classrooms, and it is this chasm that stands in the way of gender equity in maths in Britain, and of better maths education in general.”
Boaler continues, “I am British and was educated in Britain. In our maths classrooms today, students do not make conjectures, or learn creatively. Instead they sit watching teachers demonstrate standard methods, which they are forced to reproduce. Of course, the mathematics at PhD level and in a school classroom are not the same. But even so, the classroom has something vital to learn from the way Maryam practises mathematics.”
In order to better emulate the mathematics practised by professional mathematicians, Boaler prescribes that, “We need a maths revolution in schools. We need to stop implying that female students are incapable; we need to stop offering a mathematics that is procedural; and we need to start teaching problem-solving mathematics better for students in general and for girls in particular.”
Setting aside for the moment the notion that we are somehow implying that female students are incapable, this argument follows a classic structure: Experts behave in a certain way and so, in order to become experts, we need novices to emulate these behaviours – the route to expertise is the emulation of experts.
This is wrong. The differences between novices and experts are profound. Experts have vast stores of knowledge in their long term memories – both knowledge of principles and concepts and episodic knowledge, the knowledge gained from the experience of solving many problems in the past. This enables experts to pace the room, sharing conjectures and connecting different theories. Novices don’t have theories to connect together and so the many and various attempts to make children learn maths this way over the past century have failed. And yet professors of maths education like Jo Boaler keep popping up to suggest them.
This fallacy is the subject of Chapter 6 of Dan Willingham’s book “Why don’t students like school?”. He asks the question, “What can be done to get students to think like scientists, mathematicians and historians?” Before answering that, “a flawed assumption underlies the logic, namely that students are cognitively capable of doing what scientists or historians do… Cognition early in training is fundamentally different from cognition late in training.”
Willingham discusses the seductive research that implies that skilled readers make fewer eye movements than novices and how this led to some to conclude that we need to teach children to recognise whole words rather than sounding them out – a flawed approach that has led to a downgrading of the teaching of phonics with a cost in more children developing reading difficulties.
The novice versus expert difference is the reason for the traditional form of maths teaching that Boaler criticises. Teachers show students how to solve problems and then ask them to practise doing this because, at this stage, students are not able to work out fundamental principles of arithmetic or algebra or whatever for themselves. Indeed, when I teach my advanced students about the long division of polynomials, I don’t teach the subtraction of fractions – a necessary part of the procedure – because they can already do this. Here we have an example of a transitional stage; students who are expert in some areas but not others. And students will pass through many such transitional stages on the path to expertise.
What conjectures, exactly, would our students be developing in the absence of sound procedural knowledge? What connections would they be making? Such an approach either doesn’t work at all or, more commonly, large amounts of time will be spent developing false conjectures about trivial problems.
The gender part of Boaler’s argument is also interesting. Apparently, “Research shows that girls need to explore subjects in depth, while boys are more prepared to accept rote learning. Girls want to understand, not just follow rules.” I am not sure what research Boaler is referring to and it would be good to see it. I have to say that this statement seems rather dismissive of boys and provokes something of an emotional response in me.
The other key issue around gender is, “the routine advice, intended to be helpful, of primary school teachers to young girls that ‘maths might not be for them’.” I didn’t realise that such advice was so routine in primary schools. There is an overwhelming majority of female primary school teachers and so it seems odd that they would set out to crush girls’ dreams in this way. Do we have the evidence?
I am not downgrading the issue of a gender disparity in PISA. There is a similar, worrying difference opening up in Australia. I am deeply sceptical of Boaler’s diagnosis and suggested solution.
However, there clearly are problems in maths teaching. The main issue is that maths graduates and those graduates of degree courses that lend themselves to the teaching of maths – such as the physical sciences and engineering – are in high demand in industry and can earn much more money there than in teaching. Many students therefore do not have a specialist maths teacher in secondary school and hardly any have one in primary school. Indeed, some professors of maths education who are teaching the next wave of maths teachers do not hold maths degrees themselves. So, the situation is endemic and self-perpetuating.
I would assert that all students want and need to understand maths, not just the girls. The reason why confusion often reigns in the maths classroom is not due to the teaching methods that are used but due to the thin grasp that many teachers have on the maths that they are teaching. This means that they struggle to explain it and resort to just listing the steps. Confusion, difficulties and anxiety ensue and this is a real problem.
I’m with Hung-Hsi Wu on this. Rather than abandon traditional approaches, we need a better educated group of maths teachers and we should start with training primary teachers on how best to explain the standard algorithms.
Perhaps we should be providing our maths teachers with professional development on how to explain maths better instead of constantly calling for a revolution in teaching methods based upon a misconceived philosophy.