Habits for Solving Maths Problems

Reading a solution is not solving

The most common illusion in maths is thinking you can solve a problem because you followed the worked solution. Following a solution is walking a path someone else has cleared; solving it yourself is choosing a direction where there is no path, and exams ask for the second. That is why people who have personally been stuck and found their way out tend to do better than those who have merely seen many problems. All the habits in this guide aim to give you more of that experience.

Think in four stages

In his 1945 book How to Solve It, the mathematician George Pólya divided problem solving into four stages: understand the problem, devise a plan, carry out the plan, and look back. Many students skip the second and fourth stages and dive straight into calculation. Start moving your pen before deciding what to do and you easily get lost; never look back after finishing and you will approach the same type next time as if for the first time.

• Understand: what is given and what is asked?
• Plan: have you seen a similar problem, and which idea applies?
• Carry out: work line by line and check each step
• Look back: is the answer right, and is there another way?

Start by organising the givens

When you read a problem, write the given conditions and what you need to find separately. Once information scattered through the sentences is in one place, you can see what is missing and which conditions you have not used yet. For problems involving shapes or situations, draw a diagram and write the given values on it. If you get stuck and a condition is still unused, that is very likely where the key lies. Starting to solve before reading to the end because the problem looks like a familiar type is another common way to lose marks.

How long to stay stuck

Opening the solution the moment you are stuck removes the chance to build your thinking, while spending an hour on one problem stalls everything else. So decide in advance how long to hold out. For example, try for 10 to 15 minutes without help, and if that fails, read only the first line of the solution and carry on alone. While stuck, try plugging in small numbers, working backwards from what you want, or dropping one condition to make an easier problem.

• Try small numbers or special cases
• Work backwards from what is asked
• Drop one condition to make an easier problem
• Re-express it as a diagram or table

Write the solution out in full

Skipping steps in your head looks fast, but most errors happen in the steps you skipped. Write one step per line with the equals signs lined up, and you will be able to find where things went wrong later. You can only record which step failed in your error log if the working is there. In written exams the method itself can earn marks. From your practice sessions on, write solutions someone else could read.

How to check an answer

When you get an answer, check it briefly before moving on. Put the value back into the original conditions, make sure the units match, and confirm that quantities which cannot be negative, such as lengths or counts, have not come out negative. Estimating the rough size is useful too; if the estimate and the calculated value are far apart, something went wrong. The check takes little time, but it is the most effective way to reduce marks lost to slips in the exam.

Mix problems and redo them

Working through problems from one chapter in a row feels smooth because you never have to decide which idea to use. But exams do not tell you which chapter a question comes from. Once you are reasonably comfortable with a chapter, mix problems from several chapters and practise choosing the method. And always mark problems you solved with help from the solution, then redo them without it a few days later. Only if you can do it alone then have you really solved it.

Common mistakes

The most common mistake is measuring progress by how many problem books you have finished. If you turned many pages with the solutions open, the count means nothing; working one book several times, focusing on the problems you got wrong, is better. Waving off calculation errors as just slips is also risky. If the same kind of slip keeps happening, it is a habit, and something to fix by deciding on a checking routine.