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.

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.