Wednesday, 29 May 2019

Reasoning, Problem Solving, Interpretation and Fluency

So as part of my holiday I was listening to Craig Barton's most recent podcast with US educator Michael Pershan. In one part of the podcast Craig talks about his belief that reasoning required fluency before it could be conducted, but that this belief was challenged in a session with the eminent Mike Askew at the joint ATM/MA conference held over Easter. From Craigs description Mike posed a problem similar to this one:

45 × 36 = 45 × 35 + 35           True or False?

The point was that this sort of question is one that can be considered, and the correct result arrived at, even if pupils are not capable of carrying out the calculation 45 × 36 correctly. Clearly this is true, and inspired this quote from Michael:

"Reasoning happens in the absence of fluency"

Craig reflects on this at the end of the podcast, and talks about how if you "hit a wall" with a problem, if you don't have the required tools in your toolkit (or don't recognise you do) that is when you need to reason. Craig also suggests (although he also admits that it doesn't feel right) that a possible implication of this is that teachers may hold pupils back from achieving fluency in order to allow opportunities for reasoning. Craig goes on to describe the idea of "teaching the fluency first" and then the reasoning becomes part of strategy selection - pose problems that pupils have the toolkit to solve, but pupils need to consider which strategies are appropriate. Craig offers his own SSDD problems as an example.

It struck me whilst listening to this, that it may well be of benefit to make a careful distinction between what me mean when we talk about reasoning and problem solving. Consider this from the National Curriculum document, that says pupils will reason mathematically by:

"reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language"

There isn't really reference in this to strategy selection, or working through a problem here. This is highlighted much more in aim 3:

"can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions"

This seems much more about what Craig is talking about with not having the necessary toolkit, and therefore not learning a lot (or simply not being able to persevere) whilst seeking solutions. This I think is where I agree with Craig - pupils need to have a secure knowledge of the mathematics underpinning the problem. The aim makes that clear - pupils solve problems by applying their mathematics. If they don't have the mathematics, they can't apply it. The purpose here is as Craig describes at the end of the podcast, to identify the mathematics required, to link the problem type to other problems they have previously encountered etc. The purpose is not actually to learn new mathematics at all. Ideally, it should not be obvious what mathematics will be needed to solve the problem, either from the context of the lesson in which the problem is set, or from the content of the question at all. What I think Craig talks about as "reasoning" about the problem is actually interpretation and strategy selection. There comes a point where we want pupils to be able to look at a problem where the mathematics required isn't explicit, and interpret the problem successfully to identify the mathematics required, before applying that mathematics through an appropriate strategy. One of my favourite areas for these is speed problems. Speed problems are a great source of both routine and non-routine problems. Some problems involving speed are solved by multiplication. Others are solved by division. And then even when a pupil can identify whether they are going to use multiplication or division, they need to choose a strategy for the calculation - for the best way to carry out the division or multiplication.

So where does this leave reasoning in comparison then? Well I actually did a session on this at what must have been mathsconf10 (by the dates of the materials) and it included this slide:
For me, the top question involves reasoning because the mathematics required is clear. That question is about highest common factor and lowest common multiple, and in particular their relationship to two numbers. The lower question is a true "Problem Solving" question (if you ignore the 'real life' context) in that it isn't clear what mathematics is going to be required. One could argue is that the last place you might want this question is in a lesson about LCM (particularly if not surrounded by others that aren't). By contrast, the reasoning question doesn't try and obscure the maths required, but at the same token doesn't just require application of a method. The reasoning question is trying to prompt a deeper consideration of the knowledge that the pupil is developing. To make them think about that knowledge in a way that, perhaps, they hadn't considered before. In the podcast Michael is therefore right, reasoning does happen in the absence of fluency, but that is because reasoning is an important part of developing fluency. To become fluent, one has to be able to approach questions like the first one above, and use knowledge flexibly to develop the chain of logic required. This of course means we can't talk about teaching to fluency before we offer opportunities for reasoning. For me, opportunities to reason are an integral part of the journey to fluency.