How to Solve Problems

Alison Kiddle

How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya, suggesting the following steps when solving a mathematical problem:

First, you have to understand the problemAfter understanding, then make a planCarry out the planLook back on your work.

Understanding the problem

Restate the problem in your own words

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?

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?

Carrying out the plan

I’m STUCK! How do I get UNSTUCK?Work backwardsSubgoalsSimpler problem, special casesMore general caseAct it outGuess and checkHas the plan failed? When is it time to abandon the plan and try something else?

Looking back

Have I answered the question?

Does it make sense?

Have I fully solved the problem?

Is my solution consistent?

Control

Am I getting stressed?

Am I spending too long on this?

Is my plan working?

Could I move on to something else and come back to this later?

Am I focussing on the problem?

Solving an unseen problem

On the next slide is a problem that I have only seen very briefly. I will attempt to solve it, and talk through my mental process.

Try to note down any problem solving techniques you notice me using.

The picture shows four squares on the sides of a quadrilateral. Prove that the if the diagonals of the quadrilateral intersect at right-angles then the sum of the areas of one pair of opposite squares equals the sum of the areas of the other two squares.