In the September issue of Mathematics Teaching, the journal of the UK’s Association of Teachers of Mathematics, there are a couple of articles that are available to non-subscribers. One of these is Models for teaching mathematics revisited, by Andrew Blair and Helen Hindle. It reflects on a dichotomy created by Alan Wigley in 1992 between two different ways of teaching maths – a dichotomy that I had not heard of before. Briefly, Wigley proposed that there are two broad models: path smoothing versus challenging. Interestingly, Blair and Hindle do not suggest the best approach is a bit of both, but come out firmly in favour of the challenging method. This apparently holds the twin needs of instruction and exploration together in a ‘creative tension’. The challenging model contains elements of ‘telling or explaining’, but the aim is student agency, with agency defined as ‘the actual ways situated persons wilfully master their own life’. That’s some profound magic.
So what are these two approaches? Path smoothing involves small steps where the teacher decides on the type of problem and teaches students a procedure to solve it. This is followed by ‘repetitive exercises’ which don’t sound great when you describe them like that. So path smoothing appears to be a form of explicit teaching, or at least what explicit teaching looks like when students are first introduced to a topic. By contrast, the challenging model involves a problem that can be tackled a number of different ways with the students being given time to explore it, come up with their own conjectures and so on. This sounds to me a lot like problem based learning.
Why do Blair and Hindle object to path smoothing? Firstly, it apparently denies the creative tension described above. Secondly, it produces ‘limited and negative results’ and they wonder why it persists ‘in the face of evidence of its ineffectiveness’. They also seem to conflate the pedagogy of maths with the epistemology of professional mathematicians.
The only research evidence Blair and Hindle produce to support these strong claims is the final evaluation of a project conducted by the UK government where schools were encouraged to adopt a mastery teaching methods inspired by maths teaching in Shanghai. This found little evidence for the impact of Shanghai mastery on primary school students’ standardised test results, although there were a number of other findings and interim reports.
The Shanghai mastery study was not a randomised controlled trial. Students in the mastery schools were compared with students matched schools but neither they nor the schools were randomly assigned to mastery. Instead, assignment seems to have been quite ad hoc. As a result of this design the researchers note the, ‘need to remember that findings from quasi-experiments are better interpreted as correlation rather than causation.’
Even if we set this caution aside, it is not at all clear, or even perhaps likely, that the comparison schools used the alternative challenging teaching model. If so, it would be hard to conclude anything from it about the relative benefits of path smoothing versus mastery. The intervention itself seemed to be implemented using a variety of resources, including those produced by NCTEM and White Rose maths. Finally, there seems to have been an odd preoccupation with mixed-ability teaching or ‘all-attainment’ teaching as the researchers described it. This may well be a feature of maths teaching in Shanghai, but it seems to bear little relationship to the path smoothing model. In fact, mixed ability teaching is described as a key feature of Hindle’s own challenging approach.
Rather than hang so much on one, admittedly large, quasi-experimental trial that didn’t test the factor in question, it may be better to look at the broader literature on explicit, interactive forms of teaching when compared with more implicit approaches. There is a wealth of evidence from process-product research in the 1960s through to more recent randomised controlled trials that demonstrates the superiority of an explicit approach. We also have correlational data from PISA which addresses the two different styles.
We even have cognitive load theory to explain why explicit teaching is superior.
Strangely, Blair and Hindle do not refer to cognitive load theory at all. When speculating on the puzzling continued existence of the path smoothing model, they suggest it may be due to teaching to exams, the influence of parents or even a response to tight school budgets in the UK. I mean, teachers surely couldn’t have reasoned arguments, supported by evidence, for maintaining such an approach, could they?