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.