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.

My High Five

Ben Gordon on Twitter suggested this idea, and started off with his brilliant "Teach Innovate Reflect" blog. I am going to use the same format that Ben suggested, which can be seen below:

Format:

• What you learnt
• What was the source
• Implications on your practice

I really love the idea of educators sharing a few key ideas from their professional learning. Sometimes it can seem really hard just to filter all of the great practice that comes across your timeline. This seems to me to be a great way of giving others a bitesize of the things that have made the biggest impact on them. So here are my 5 main things I have learnt this year:

1) That there are three levels of curriculum planning

What I learnt: The difference between the intended, implemented and enacted curriculum.

Source: Bauersfeld 1979, via Dylan Wiliam's paper on principled curriculum design.

Implications on my practice: I guess this really helped frame many of our conversations in department, particularly around developing material for our new scheme. What was really useful is having a language to discuss the separation between what we plan, what we teach and what kids learn. I wrote more about it in my article for TES here.

2) About the use of blocked practice

What I learnt: That blocked practice is useful in early concept development

Source: I honestly cannot remember

Implications on my practice: First of all, I am sure I am not the only person that reads/hears things, and has near perfect recall of what they read/heard but cannot at all remember where? But anyway, we have heard a lot in the last year or two about the importance of interleaved practice. What emerged from somewhere recently (although like I say I cannot say where) is the idea that when pupils are first developing a new idea, or applying it in a new way, that they need time to just focus on that idea. Later on, interleaved practice is really helpful in promoting far transfer, but this does need to wait otherwise the new concept/application becomes confused. In terms of my practice, this basically means we do a lot of back and forth stuff in class around the main idea, and it is only when the more independent work starts that interleaved practice comes in.

3) The best ways to use examples

What I learnt: How much backwards fading of examples improves learning.

Source: It wasn't the first place I read/heard it (again can't remember where that was), but most recently on Craig Barton's podcast with Mark McCourt.

Implications on my practice: Quite obvious really, I plan example sets rather than two or three full examples, and within those sets I gradually reduce the support until pupils can work through a few from beginning to end.

4) Some important things about Direct Instruction programmes

What I learnt: That DI programmes need to be under 15 pupils and that over 80% of each hour revisiting rather than teaching new material.

Source: Chloe Sanders (@Chloe_jo) courtesy of the @DITrainingHub at St Martins Catholic School, Stoke Golding.

Implication on my practice: We are trialling a small group intervention using Connecting Maths Concepts with a possible view to expand this a little next year. As part of our preparation for this I was lucky enough to have the opportunity to visit St Martins and see DI in action with the amazing Chloe Sanders. It was in discussion with Chloe that I found about these important rules for DI programmes. Admittedly we were going to use the same programmes, but just getting it straight early on, I think, has helped with the implementation and will help it have more impact when we choose to use it.

5) How great Frayer models are

What I learnt: That Frayer models exist, and how great they are for capturing mathematical concepts.

Source: Jo Gledhill (@JoLocke1) at #mathsconf18.

Implication on my practice: I have been openly critical of the use of knowledge organisers for mathematics teaching. Kris Boulton summed up a lot of the problems with them in his blog. However, when Jo showed a Frayer model (like the one below), I couldn't believe I had never seen them before. I immediately saw how useful they could be for summarising ideas in maths. Over this summer I am going to write them into our new scheme, although I do need to give a little more thought as to how.

Maybe one of these things is a new thing for you. Whether it is or it isn't, I hope you will consider sharing the 5 things you have learned this year, so others have the opportunity to learn from you (and make sure you tag me in!)