Problem solving in maths – You can’t build a shed without a hammer.

The latest version of GCSE maths is my fourth.

It might not be fashionable to say it, but I’m a fan.

Why?

Because it’s actually difficult again and, yes, that is a good thing.

The emphasis on problem solving is much greater and that seems to have become a driver.

Recently I have met HOD’s from 3 schools. There was talk about making problem solving central to the curriculum. One school has given up one lesson a week (out of 5) to problem solving from Year 7 onwards.

I made the decision that my school would not do that for 2 reasons.

Firstly, it seems to me that teaching problem solving as if it is a skill that can be learnt is a falsehood. It is true that Mathematical problem solving is transferable to different areas other than mathematics once and only once the skill is acquired at a high level. This must be true in the same way that surfing and skateboarding use much of the same balancing skills and have transferable elements. However, problem solving is not in and of itself a teachable skill anymore that balancing is. It is an experience and practise based on the knowledge you already hold.

This is true of even the best mathematicians.

For example, we have a very talented (even gifted) mathematics student. Let’s call him Dave.

Dave is brilliant at number problems and logic puzzles. He solves them with amazing mental agility and can provide beautiful written explanations. But give Dave a Geometry problem and things change. He is still very good at it but the pace slows down significantly and he is less sure of himself.

Dave has been given hints and strategies to help him with this area of his mathematics and that has helped him to improve, however you cannot teach someone to “see through” a problem.

Hence we can give coaching to a student into improving their problem solving, but it is not possible to teach someone how to solve any problem. If it were possible, Fermat’s Last Theorem would have been proved a long time ago and Andrew Wiles would not be famous.

Secondly and as I have alluded to, problem solving strategies are tied to a background of knowledge and if this background is incomplete, we will by necessity, teach awkward or contrived ways of solving problems that we would not have considered had we had a full mathematical tool kit.

Problem solving in mathematics is much like winning at chess. The Dutch Chess master Adriaan de Groot studied the way in which chess players decide on moves and concluded that problem solving requires 4 elements or phases.

  • Orientation phase – What’s the problem? Let’s get a few ideas. This relies in part on familiarity with similar situations.
  • Exploration phase – What maths is there in this problem? Let’s reject the worst ideas and play with the most likely.
  • Investigation phase – Get into the problem and pursue a solution.
  • Proof phase – Does my working out follow? Does my answer look right?

The Orientation and Exploration phases require memory and knowledge for them to be successful.

Practising the Investigation phase without the pre-requisite knowledge must mean manipulating the original problem to fit into the students existing knowledge base. This is the equivalent of playing Chess without knowing all the rules.

In her wonderful book “Seven Myths about Education”, Daisy Christodoulou puts it very eloquently.

“The aim of our teaching should be to equip pupils to solve these problems, and to solve them independently. That is a legitimate and achievable aim. It is also true that real-world problems do not come neatly boxed in subject categories.”

So, am I suggesting that students should not come across any form of problem solving until Year 11?

No, not at all. All students will encounter maths problems. It is inevitable. They will experience them. They will try them. We will coach them and help them.

What I am suggesting is that the we will not try to “teach” an artificial facsimile of problem solving constructed so as to fill a gap in the scheme of work.

If that sounds very much like the old traditional approach to GCSE then that’s because it is. This approach took us from no GCSE results to best in the county in 2 years.

If you have already embarked upon a course of teaching problem solving from Year 7 on and you decide that is your preferred approach then I’m happy for you. However, I remain of the opinion;

“If you want to build a shed you need a complete tool kit, some wood and some nails. Then and only then, can you build a shed.”

I don’t know why you’d read on but . . .

I never thought I had much to say. Anyway, why would anyone want to listen?

Then we had a bit of success. We became the best school for maths in the county.

Then the visits started. Heads of maths, people I looked up to asking me how we did it.

I don’t have all the answers, but perhaps I can share just a few ideas about how we teach maths. I guess many of the ideas apply to any subject.