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Six aspects of problem-solving that exams don't test
04 February 2026
Exams are designed to test mathematical problem-solving, but they often inadvertently strip away the features that make real world problem-solving so valuable.
In everyday life there are countless problems that require mathematics. These problems invariably have several features:
1. Check understanding
Before going down the rabbit hole of tackling a problem, it’s always important to check you fully understand the problem first. Suppose a house buyer tells an estate agent that they want a home that is within a mile of the station. What do they mean by ‘within a mile’? Is that as the crow flies? Or do they mean driving distance? The latter makes a big difference to the locus of ‘homes within a mile of the station’, because it’s unlikely that the road will go in a direct straight line to the station.
2. Collaboration
Real world mathematical problem solving is usually collaborative. Maybe you’re working out the route to take on a Duke of Edinburgh trek, or working out the catering when planning a party. People have different skills and knowledge, and a key part of problem solving is the ability to communicate and review ideas with other people.
3. No marks
Imagine if the annual form you receive from HMRC began “Please fill out your tax return ? marks)”. Telling you how many marks a problem is worth is a huge clue as to how much effort you are going to be required to use. But real world problems don’t do this. You often don’t know if a new problem you are confronted with is going to be a 1-marker or a 50- marker. That’s part of the joy and frustration of problem-solving in the real world.
4. Modelling
It's rare that maths presents itself when you first encounter a problem. Usually it's necessary to turn the problem into mathematical form so that you can then solve it. Even something as simple as a monthly budget needs you to make assumptions about (say) your monthly food spend, travel costs and so on.
5. Empathy and context It’s unusual to find any problem in everyday life that is purely mathematical. There are usually other practical factors to take into account, such as the personalities of the people involved. Health guidelines might say that a blood pressure reading of 150/100 requires medication, but a GP knows that this single data point doesn’t tell the full story. If the patient suffers from ‘white coat’ syndrome (anxiety when talking to doctors)then a high blood pressure reading is probably misleading.
6. Taking a break
How many of us solve a tricky problem in a single sitting? It’s normal to find yourself stuck and unable to make progress. What should you do? Take a break of course. Have a cup of tea, do some gardening, go for a walk. With the mind refreshed, it’s amazing how returning to the problem a few hours later can lead to an immediate breakthrough with an ‘aha’ moment.
These five factors are extremely important, but they’re almost impossible to incorporate within a traditional.
ARTICLE
I wrote a full article on this for the Scottish Maths Council's journal and the same article appeared in AMiE's journal Mathematics In School. You can download the full article here (pdf format).
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