I was asked recently to deliver a training session for two maths departments on the topic of problem solving. After internally balking (problem solving as a discrete entity is something that gets on my nerves, “problem solving lessons” even more so) I decided it was the perfect opportunity to talk about cognitive load and relate it to the requested topic.

If you don’t already know about cognitive load, you really need to get hold of Daniel Willingham’s *Why Don’t Students Like School?* In the meantime, it goes a little bit like this. For learning purposes our minds have two main parts: the long-term memory and the working memory. Long-term memory consists of the things we know, like our address, or the answer to 5 x 5, or the colours in Joseph’s Amazing Technicolour Dreamcoat (What? You mean that’s just me?) When we are thinking about something we use our working memory. You can bring things out of long-term memory into working memory to help you think about something. The standard example is multiplication. If I ask you to calculate 56 x 3 in your head you’ll probably do it quite quickly (have a go now). If you’re like me you did 50 x 3, then added on 6 x 3. If, however, I ask you to calculate 47865 x 3 you’re going to struggle, but not because the component parts are any harder. You can do 40000 x 3 and 7000 x 3 and 800 x 3 and 60 x 3 and 5 x 3, but to hold the answer to so many constituent parts in your head while you add on more is extremely difficult. The second problem contains so many more chunks for your brain to think about in one go that you experience cognitive overload.

This is what happens to our students in lessons. When they first learn something it has not been committed to long-term memory yet, they are still very much novices on the topic. To expect them to use this new learning to any depth is naive, they need much more practice and familiarisation before they can be adept at applying the knowledge.

So, onto problem solving. For me, a problem is *anything that isn’t straightforward to solve*. So for a five year-old a problem may look like this:

21 + 3

This is a problem because the child is still getting to grips with the concept of place value. They need to identify the tens and units in the first number, then know what to do when they see the addition symbol (including not counting up from 1 to 21 before adding on 3!), then add the second number to the units of the first. There are a lot of chunks of information for the novice to deal with there. You or I know, of course, that the answer is 24 without thinking, but that’s because everything required to solve this problem is in our long-term memory, so it’s no longer a problem.

What about for the secondary school student? Perhaps a problem like this (from LaSalle’s free problem solving booklet) is a pretty tricky one:

Well “tricky” all depends on how confident you are with percentage change and reverse percentages. If your skills in that area pretty solid, your working memory only need focus on how to solve the problem. If you are still needing to practise the maths as well as figure out how to solve the problem then there are going to be problems (“I can’t do it” and myriad forms of attempt-dodging). In short, you are setting your students up for a fall if the maths is not committed to long-term memory.

To illustrate the point further, take a look at this problem:

If you are a maths teacher, chances are you’ve studied the maths required to answer this at some point, although perhaps not for quite a while. Indeed, you might find it hard as there is a lot to process there. Give it a try if you like, see how you feel. If you struggled to remember the mathematical processes involved, as well as figuring out how to solve the problem, then you have just had a glimpse into what many problem solving lessons are like for our students. (If you want the answers, it was taken from the Oxford/Imperial MAT 2014)

So what can we do about this? I propose a rather simple suggestion: if you want students to practise solving problems, give them problems on maths they are solid at. This means if you’ve just done three lessons on Pythagoras’ Theorem, don’t give them Pythagoras problems yet. Distribute their practice over the next few weeks/months, and when you know they’ve totally nailed Pythagoras and really don’t need to think about the process any more, *then *give them some problems. That way their working memory only has to work on how to solve the problem and they won’t be overloaded. Fewer chunks, more success!

This means the standard teach-the-topic-this-week-and-finish-the-sequence-of-lessons-with-some-problems approach needs to change. The teachers I was working with certainly agreed in principle. What do you think? If you’ve got anything to add to the discussion, please do leave a comment.

June 29, 2016 at 6:04 am

Great clarity in what you’ve written. I’m intending to give some of my students “problems” where the maths is straight forward for them but the problem has to be broken down, in an attempt to release part of the thinking load and concentrate on interpreting longer written problems. As you say, I’m concerned the maths they’ve not fully nailed puts them off approaching these longer problems.

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June 29, 2016 at 7:01 am

I think this is why so many attempts at problem solving go badly, they haven’t sorted out the maths enough to cope with the whole scenario.

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June 29, 2016 at 11:43 am

Fascinating & important stuff, and becoming moreso with new GCSEs.

It’s definitely important to make the most of working memory, and a good way to do this is to focus the load on applying some knowledge they already are secure in. If they don’t have the requisite maths, working memory can just spend a lot of time ‘searching’ and guessing, which doesn’t result in much learning.

However, there’s also a bit of a reversal effect here too. Applying the maths makes them more secure on it, and so problem solving can be a useful exercise in building depth/security in understanding. Which means that it’s all about timing, gradual phasing/fading etc.

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June 29, 2016 at 12:34 pm

Thanks for your comment, Peps, I see what you mean. I didn’t talk about it here but in the training we went on to look at scaffolding and the expertise reversal effect, which I guess ties into what you’re saying. With a good scaffold, the problem can be used to reinforce the maths so that thinking becomes focused on the maths alone.

So then the question becomes, why are we doing this problem? If it’s to build understanding of the maths, design a scaffold that helps students to focus on what you want and be very aware of when to fade out elements of the scaffold depending on their proficiency. If it’s to just practise “problem solving”, make sure the (lack of) security in the maths doesn’t create an obstacle to that aim.

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June 29, 2016 at 5:05 pm

Excellent insights on problem solving. In the algebra books I had in high school, every chapter had a few sections of word problems, so the spacing/distribution effect was effectively implemented. And the word problems employed whatever new concepts had been explored. So in the chapter on rational expressions/algebraic fractions, the word problems yielded equations with algebraic fractions. In the chapter on quadratics, the word problems yielded quadratic equations and so on. But in all such chapters, basic problem solving skills in distance/rate, work, number, and mixture problems remained. I’m structuring my pre-algebra and algebra courses to do this. Thus, in pre-algebra, mastery of solving two step and multi-step equations is necessary before embarking on word problems. Then, word problems require certain tools: e.g., given a length of wood that is 15 in, cut into two pieces of unknown length, we can express those lengths as x and 15-x. This is a structure used in many word problems. Basic structures carry through to many word problem types.

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June 29, 2016 at 5:28 pm

See also: http://www.ams.org/notices/201310/rnoti-p1340.pdf

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June 29, 2016 at 7:04 pm

Thanks for the link. I like this bit at the end:

“It is more the case that, as students increase their expertise, more nonroutine problems appear to them as routine.”

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July 1, 2016 at 12:01 pm

Hi Jem, an interesting read, thank you.

There seems to be a blanket acceptance of Daniel Willingham’s views, I think it’s possible that the interplay of memory and learning are a lot more complex than the way his ideas are often expressed.

I agree that “if you want students to practise solving problems, give them problems on maths they are solid at”. I think there is also a place for attempting problems that build on their solid maths and invite them to think about new concepts. A famous example of this is the young Andrew Wiles encountering Fermat’s Last Theorem. Had he only ever attempted problems he knew the maths of, then he would never have solved it.

Staying with Andrew Wiles, at the beginning of the famous documentary, he describes his experience of doing maths using the metaphor of entering a dark mansion and only gradually coming to understand what is in the rooms: https://www.youtube.com/watch?v=Sj8TbbIuOL4

I can contrast this with him having been given a plan of the mansion and a torch beforehand. Obviously Andrew Wiles is a special case, however I think the mathematical experience for everyone, from a novice to a professor, does involve a ‘journey into the dark’. As a teacher, we can plan this journey so that illumination arrives regularly; I wonder though that if we preclude the possibility of not understanding something at first, then our students miss out on the full vista of mathematical knowledge.

To the problem involving the graph of the polynomial above, I relish the opportunity to see if I can solve it, and I think that it is possible for all students to have a similiar mindset when encountering a new maths problem.

Best, Rufus

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July 1, 2016 at 12:59 pm

Thanks for such a thoughtful comment, Rufus. I completely agree that learning is complex, and of course we would be silly to oversimplify it and not expose our students to challenges. This post was more addressed to the idea that we can teach “problem solving” by giving students more problems to solve, which (believe it or not) is a position I encounter a lot in schools, where teachers don’t seem to have thought any more than that about it. The ideas from cognitive psych (Willingham et al) are not widely known in my experience, so the idea that working memory and lack of understanding (highly structured long-term memory, to use Peps McCrae’s words) limits our ability to solve a problem is often met with keen surprise, I find. I don’t see a blanket acceptance of Willingham’s views simply because I don’t see many people who even know his views. Perhaps ideas filter through quicker in the capital?!

Challenging our students and using problems as inspiration or as a means to secure understanding certainly has its place, but I do think a more thought-out approach is necessary in order to create more success stories.

I’m interested in this: “I wonder though that if we preclude the possibility of not understanding something at first, then our students miss out on the full vista of mathematical knowledge.” In what way do you envisage that someone would miss out? I am genuinely interested to know your thoughts. Do you mean that they might learn some maths from solving a difficult problem that they would otherwise not have learnt or am I misunderstanding you?

Thanks, Jemma.

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July 1, 2016 at 1:36 pm

Thanks Jemma,

I will try to give an example of what I meant.

Lets try the ‘Goat on the rope problem’. I’ve given this problem to a post-gcse, pre-KS5 class. It goes like this:

A shed is in the middle of a large field. It measures 2m by 3m. At one corner a goat is tethered by a rope that measures 5m long. What is area of grass that the goat can eat?

So, the kids can work this out by finding the compound area made up of parts of circles. So far, so straightforward, they use knowledge they already know to solve a problem.

Then, the problem is adjusted. Exactly the same scenario, but this time the rope is 6m long. What is the area of grass the goat can eat now?

This throws up all kinds of difficulties, and the kids can’t solve it. What is interesting mathematically is what they then do to solve the problem, and by trying to solve it what mathematical questions arise.

My view is that by attempting to solve this problem, and by thinking about the maths involved, the students are being mathematicians, and this is an important part of learning maths: getting stuck, thinking about ideas that might work, making conjectures, and articulating what they find difficult.

I may well be wrong, and it may be the wrong thing to do, but at the moment, I think it has merit.

Rufus

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July 1, 2016 at 2:59 pm

I agree with you about the usefulness of opportunities to ‘get stuck’ and think your way out. That’s a great example of a useful problem in that sense. Interestingly, though, these are students that still (in theory) have all the required knowledge to solve it, they just have to apply it in an unfamiliar way. That provides them with the experience of coping with and overcoming an unfamiliar situation by using what they already know which, absolutely, is an important experience for them to have.

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July 1, 2016 at 3:04 pm

Yes, and thanks for engaging with my feedback, it’s been a good conversation. I’m on part (c) of the graph problem.

The goat on the rope problem though, I don’t think you can solve it with gcse maths can you?

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July 4, 2016 at 2:10 pm

Sorry, Rufus, I’ve just reread your problem-yes it is more advanced than GCSE!!

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July 4, 2016 at 5:43 am

Hi Jemma,

I’ve been thinking about this some more; thanks for writing it, it’s helped to clarify some things for me.

I agree with you that ‘problem solving lessons’ are not the way to go. For me, it’s one of the things I’ve tried to do in the past, and the way that you articulate why they don’t work seems to me exactly right. I would add to this some more lessons that I’ve tried and that I now think are not the right way to teach: ‘using an ICT package such as geogebra to explore, for example graph transformations’; ‘using a great activity as the basis for learning in a lesson’.

The common thread is that problem solving, using ICT and great activities each have no agency in the learning process. They are only helpful when mediated in a thoughtful, expert way be the teacher. In my opinion, the judicious use of explanations, examples and problems from a good textbook is a much better way to teach than using a hodgepodge of ‘exciting’ ICT, activities and problems. Certainly, I’ve been guilty of this in the past.

Saying that, I think it is possible to move beyond the benefits of using a textbook, and that would be by planning a sequence of lessons based on experience, collaboration and research (as I have tried to do in my blog about Transformations), and then reflecting on that sequence to improve it.

So, I say, it all comes down to sequence planning. What do you think?

Thanks, Rufus

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July 4, 2016 at 2:33 pm

I think there will be a much bigger emphasis on sequence planning in the near future for this very reason. A more coherent, structured, consistent approach is necessary.

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July 4, 2016 at 1:17 pm

Reblogged this on The Echo Chamber.

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