A Guide to Problem Solving

Stage: 5
When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice.

How do I become a better problem-solver?

First and foremost, the best way to become better at problem-solving is to try solving lots of problems! If you are preparing for STEP, it makes sense that some of these problems should be STEP questions, but to start off with it's worth spending time looking at problems from other sources. This collection of NRICH problems is designed for younger students, but it's very worthwhile having a go at a few to practise the problem-solving technique in a context where the mathematics should be straightforward to you. Then as you become a more confident problem-solver you can try more past STEP questions.

One student who worked with NRICH said:

"From personal experience, I was disastrous at STEP to start with. Yet as I persisted with it for a long time it eventually started to click - ‘it’ referring to being able to solve problems much more easily. This happens because your brain starts to recognise that problems fall into various categories and you subconsciously remember successes and pitfalls of previous ‘similar’ problems."

A Problem-solving Heuristic for STEP

Below you will find some questions you can ask yourself while you are solving a problem. The questions are divided into four phases, based loosely on those found in George Pólya's 1945 book "How to Solve It".

Understanding the problem
  • What area of mathematics is this?
  • What exactly am I being asked to do?
  • What do I know?
  • What do I need to find out?
  • What am I uncertain about?
  • Can I put the problem into my own words?

Devising a plan
  • Work out the first few steps before leaping in!
  • Have I seen something like it before?
  • Is there a diagram I could draw to help?
  • Is there another way of representing?
  • Would it be useful to try some suitable numbers first?
  • Is there some notation that will help?

Carrying out the plan

STUCK!
  • Try special cases or a simpler problem
  • Work backwards
  • Guess and check
  • Be systematic
  • Work towards subgoals
  • Imagine your way through the problem
  • Has the plan failed? Know when it’s time to abandon the plan and move on.

Looking back
  • Have I answered the question?
  • Sanity check for sense and consistency
  • Check the problem has been fully solved
  • Read through the solution and check the flow of the logic.

Throughout the problem solving process it's important to keep an eye on how you're feeling and making sure you're in control:
  • Am I getting stressed?
  • Is my plan working?
  • Am I spending too long on this?
  • Could I move on to something else and come back to this later?
  • Am I focussing on the problem?
  • Is my work becoming chaotic, do I need to slow down, go back and tidy up?
  • Do I need to STOP, PEN DOWN, THINK?

Finally, don't forget that STEP questions are designed to take at least 30-45 minutes to solve, and to start with they will take you longer than that. As a last resort, read the solution, but not until you have spent a long time just thinking about the problem, making notes, trying things out and looking at resources that can help you. If you do end up reading the solution, then come back to the same problem a few days or weeks later to have another go at it.