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?

Greg,

I think your critique is well made. The two authors are clearly going out of they way to avoid learning about the other point of view.

But I think you are always missing an argument with those that want children to be more engaged with the subject. Clearly the authors are appealing to those that value students being engaged and learning material out of curiosity and through making their own steps. And they see the alternative as always and ever a direct transfer of what the teacher knows to the students by explicitly telling them and rote regurgitation.

This is what I like about Jumpmath.orgs’ John Mighton labelling their form of instruction as guided discovery. He argues for breaking things down into smaller steps whenever things don’t work but wants the teacher to lead the students to see say a pattern in the nine times tables as opposed to telling them about every pattern.

You will find more research behind their approach here under the statement by them “In 2011, L. Alfieri et al. conducted a meta analysis of 164 studies of discovery-based learning and concluded that “Unassisted discovery does not benefit learners, whereas feedback, worked examples, scaffolding and explicit instruction do.”

https://jumpmath.org/jump/en/supporting_research

The claim from Mighton is you can have your cake and eat it – you can have students discovering math but to make it work you use explicit instruction. This is a better argument than you make as the term explicit instruction will always carry the implication that it doesn’t involve students doing more than recalling what the instructions were.

I’m not entirely convinced by this, just as I am not entirely convinced by the Alfieri et al paper.

Firstly, I think there are a lot of assumptions about engagement and motivation here that may not stand up. In the main, we seem to be motivated by getting better at things. So you want an effective maths teaching programme if you want motivation and engagement. The fact that motivation and success correlate does not mean the former causes the latter.

Secondly, it depends what you mean by discovery. The aim of explicit teaching is to eventually be able to apply all the tools you have been taught to solve ill-structured problems. So there is always an element of discovering which solution steps to take etc. The aim is to equip students to do this as well as possible. However, if you mean discovering the pattern in the 9 times table then I would suggest that’s trivial, there is little to be gained by kids figuring it out themselves – which will differentially be the more advanced kids anyway – and so a teacher might as well just point it out.

If the question is one of tactics and spin then that’s a game for others to play.

You seem to be saying some form of ability to discover things is a goal but then deriding it at the same time.

Something can hardly be trivial yet only achievable by the advanced students.

If the point of math is to foster some ability to find a solution without simply recalling a set of steps then there is plenty room to discuss how in a range of shades of gray.

Suggesting someone’s description that addresses this is perhaps just spin is the sort of ad hominem you usually decry.

I don’t believe that the ‘ability to discover things’ can ever be a goal because it is not a generic ability. It depends almost entirely on what you already know. More advanced kids know more. I wrote about this here: https://gregashman.wordpress.com/2019/01/03/solving-problems-and-being-creative/

No one is talking about a generic ability here except you who is using it as a strawman.

What is being discussed is how to get students acting like experts in very restricted domains of math. So in the 9 times tables being able to get to the pattern based on simple knowledge of patterns in other cases and even breaking it down to the patterns for the 9’s ones digit sequence and tens digit sequence.

(Here acting like experts means exactly what you describe in the other post you reference except the domain is very restricted.)

You seem to agree on the motivational aspect so I am not sure why you don’t like this. Mighton’s claim is with explicit instruction it is both possible and productive to aim at this for almost all students.

Yes. I think that teachers particularly of younger kids would approach this differently and still follow the principles Greg explains. Some differences between elem and high school teaching perhaps.

I taught grade 1/2. Same approach, details different.

Mighton is impressive.