As I go through problems with my students and show them possible solutions, it might be that from time to time I solve a problem that someone isn’t familiar with, though it uses theory that they do know; at this point a common response is “How am I supposed to know to do that if I get a question like that in the exam?” This question really shows the misunderstanding that people have about what goes into producing a solution to a problem.

Unfortunately, maths education within the UK school system has become very sterile over the years – though steps have and are being taken to change this with the introduction of the new GCSE maths specification and the proposed new A Level maths specification to begin in the not too distant future. I’m speaking, here, from a point of view of having observed many GCSE and A Level maths students over the last half a decade or so and a big problem for them has been that they don’t understand that maths is about problem solving; many of them have never really had the opportunity in school to solve a problem.

But what about all of the textbooks that they have the and worksheets that they get? What about all the homework that they do? Well when you look at the homework sheets that are given out, or if you look at the homework ‘problems’ that students are set they often are very routine and uninspiring. What happens (in many cases) is this: the teacher introduces a topic and gives a few worked examples; the teacher then sets a bunch of questions that effectively follow the exact pattern of the wroked examples; then move on to the next topic Sadly, this is not problem solving by any stretch of the imagination. This is what I call ‘pub-quiz maths’ – either you know the answer and can parrot it out or you don’t know the answer and you hit a brick wall.

The thing is with maths is that, even though you might not know the final answer to a problem or the method of solution, you can figure it out. You can use what you DO know to find out answers to what you DON’T know. You can’t do this in a pub-quiz: either you know what the highest mountain in Europe is or you don’t. With maths you can use what you know to come up with original solutions to problems that are not necessarily anything like you’ve come across before; and many schools will stop short of this. Many teachers will try to analyse past exam papers and distill out what the most common ‘types’ of problem are and try to show their students model solutions for each ‘type’ of problem. But then when these students are faced with an unfamiliar situation (and it might only be very slightly different to what they are familiar with) they haven’t got the problem solving experience to modify and apply their knowledge and they say, “I’ve never been shown how to do this.” Well that’s the point – you have to find a way of solving it yourself.

My tutoring style concentrates more on this problem-solving aspect of mathematics. It’s hard work for the students and for me but I can’t, and don’t, shy away from putting my students in situations where they have to apply their knowledge and solve problems themselves. This is what maths is about and this is what makes maths interesting and exciting. For me to take away that opportunity from my students would be disgraceful and a complete disservice to my students.

So the answer to the question, “How am I supposed to know to do that if I get a question like that in the exam?” is this; get the problem-solving experience that you need to solve problems confidently and independently. I can teach the knowledge and encourage problem-solving skill development but I can’t teach experience – you have to get that yourself and you’ll get it through practice. You have to solve problems to learn to solve problems and then eventually you become flexible enough with your thinking to apply knowledge very naturally in quite creative ways. When you learn to do this THEN you are doing maths.