A Guide to Problem Solving

The art of problem solving is how to proceed when confronted with a problem where you cannot immediately see which method to use to solve it. 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 you can use other sources of problems, such as TMUA papers, old MAT papers and UKMT Maths Challenges.  The most important aspect is that it should be a question that you cannot see how to do immediately.

For STEP questions practice the best starting points are the STEP Support Programme Foundation Modules, and Stephen Siklos' book Advanced Problems in Mathematics, and old STEP 1 papers can also be a good source.

One Cambridge University student 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". As you become a more experienced problem solver you will start to internalise some of these steps.
 

Understanding the problem

• What area of mathematics is this?
• What exactly am I being asked to do?  (Read the question more than once!)
• 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?

It might not always be immediately obvious what area of mathematics is relevant, and STEP questions often combine lots of different areas together. You might like to underline or highlight parts of the question, such as restrictions like "n is a positive integer" as these are easy to forget when working through the problem. STEP questions can often have a lot of words setting them up, and it can be useful to summarise what the question is asking as a series of bullet points. Also pay attention to the "command words", for example "Write down" implies that the answer needs no working out to find, but "determine" means that you should expect to have to do some working to reach a solution.
 

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?

As you gain more experience this part of the process will become much quicker.  If you are finding it difficult to come up with a plan, then having a bit of a play with the problem, picking some values or trying to consider a simpler case might give you insights on how to deal with the original problem. A carefully devised plan is usually very helpful, but sometimes "Brute force and ignorance" can take you a surprisingly long way.  

Carrying out the plan ... expect to get STUCK!

• Re-read the question!
• 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.

Often algebraic manipulation mistakes can cause the plan to fail. It helps if you set out your working logically and legibly, and space it out so that you have room to correct mistakes (although sometimes the best thing to do it just start again and go a little bit more slowly second time around).

Looking back

• Have I answered the question?
• Sanity check for sense and consistency
• Check the problem has been fully solved (and re-read the question!)
• Read through the solution and check the flow of the logic.

Re-read the question to make sure you have answered the question that was actually set and not the one in your head! Sometimes substituting some numbers to check that your answer working for a simple case is a good idea.  

Throughout the process

It's a good idea to periodically assess your progress and state of mind whilst solving problems.  

• Is my plan working?
• Am I spending too long on this?
• Am I focussing on the problem?
• Is my work becoming chaotic, do I need to slow down, go back and tidy up?
• Am I getting stressed?
• Do I need to STOP, PEN DOWN, THINK?

If you feel like you are making no progress, then it is often a good idea to take a break from the question. This could be 10 mins to go and make a hot drink, a quick walk or even leaving the question for another day.
 

Final thoughts

Finally, don't forget that STEP questions are designed to take well prepared students about 30 minutes to solve, and to start with they will take you much longer than that. If after trying all the advice here, including taking a break away from the problem, you are making no further progress with the problem then you can use published answers to help you complete a solution, but not until you have spent a long time just thinking about the problem, making notes and trying things out. 

Even if you have found a complete solution by yourself, it is still very useful to look at other solutions and examiner reports to see what other methods could be used, what other students struggled with or perhaps parts of your solution which could have been explained better.