Five things to try when you're completely stuck

Most students treat being stuck as information about themselves. I don’t get this. I’m not a maths person. It isn’t. Being stuck is information about the problem — specifically, that you haven’t yet found the right way in.

Strong problem-solvers aren’t stuck less often. They’re just quicker to move through it, because they have a list of things to try. Here’s mine.

1. Try a small case

If the question is about $n$, try $n = 1$. Then $n = 2$. Then $n = 3$.

This feels like cheating. It isn’t — it’s the single most productive move in mathematics. Small cases show you the pattern, tell you what the answer should look like, and very often reveal why the general statement is true.

Suppose you’re asked to show that

\[1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\]

and you can’t see where to begin. Write out $n=4$: $1+2+3+4 = 10$, and $\frac{4 \times 5}{2} = 10$. Now pair the terms from the outside in: $1+4 = 5$ and $2+3 = 5$. Two pairs of $5$. There’s your proof, and you found it by playing with a specific case rather than staring at the general one.

2. Draw it

Even when it isn’t obviously a geometry question. Sketch the function. Draw the situation. Put the vectors on a diagram with the angle marked.

A surprising proportion of “I don’t know how to start” is really “I haven’t formed a picture of what’s going on”, and the picture usually suggests the method. In mechanics this is close to universal: almost every mechanics question that goes wrong went wrong because there was no force diagram.

3. Name what you want

Write down, in words, the thing you’re trying to find. Then write it as a symbol.

It sounds trivial. It is astonishingly effective, because a lot of stuckness comes from holding too much in your head at once. “I want the value of $k$ that makes the discriminant zero” is a sentence you can act on. “I want to do question 7” isn’t.

4. Work backwards from the answer

If the question tells you what to show — and in STEP, in proofs, and in “hence show that” questions, it usually does — start from the target and ask what would be enough to get there.

To show $x = 3$, it would be enough to show $x - 3 = 0$. It would be enough to show that $x$ is a root of some polynomial you can factorise. Now you have a smaller, more concrete goal than you started with.

5. Ask what you’ve been given that you haven’t used

This is the one I use most in exams, and it’s nearly always the answer to a “hence” question.

Exam questions are constructed. Every condition in the stem is there for a reason. If you’ve reached part (iii) and haven’t used the fact that the sequence is increasing, or that $n$ is even, or the result you proved in part (i) — that unused fact is your next step. Part (i) of a STEP question is almost never a warm-up. It’s the hint.

The habit underneath all five

Notice that none of these require you to know more content. They’re all ways of changing your relationship to the question — from “I must recall the method for this” to “let me find out what’s going on here”.

That shift is most of what separates students who find maths frustrating from students who find it interesting. It’s also entirely learnable, which is the good news.