Mathematics is often taught as though learning it means being shown a method, practising that method a few times, and then remembering it when it appears on a test.
That is certainly part of learning mathematics. But it is not the whole of it.
To become confident at mathematics, you need to develop something deeper: the ability to recognise patterns, choose approaches, make connections, test ideas, explain your reasoning and keep going when the answer is not immediately obvious.
Here are a few principles that I think make a real difference.
Try problems yourself
Reading a worked solution can be extremely useful. But there is a big difference between understanding someone else’s solution when you see it and being able to find a solution yourself.
Before looking at the answer, spend some time trying the problem.
You might succeed. You might only get halfway. You might discover an approach that does not work at all. All three can be useful.
The important thing is that your brain has had to engage with the problem rather than simply recognise an explanation placed in front of it.
Don’t be afraid of getting stuck
Getting stuck is not evidence that you are bad at mathematics.
It is a normal part of doing mathematics.
Of course, there is little benefit in staring at the same problem in frustration for hours. Sometimes you need a hint, a simpler example or an explanation of an idea you have not yet met.
But there is also value in resisting the temptation to look at the solution immediately.
Some of the most useful learning happens in the gap between I don’t know how to do this and now I see it.
Understand why, not only how
Algorithms and procedures matter. There is nothing wrong with becoming fluent at them.
But whenever possible, ask why they work.
Why does multiplying two negative numbers give a positive number? Why does the formula for the area of a triangle contain a factor of one half? Why can an Euler trail exist only when a graph has at most two vertices of odd degree?
Understanding the reason behind a method makes mathematics easier to remember — but more importantly, it makes the method easier to adapt when the next problem looks slightly different.
Use examples
Abstract mathematics becomes much easier to understand when you have concrete examples to think about.
Try small cases. Draw pictures. Calculate a few examples. Change one part of the problem and see what happens.
Then ask what those examples have in common.
Good mathematics often moves backwards and forwards between the concrete and the abstract: examples suggest an idea, and the general idea then explains the examples.
Explain your thinking
After solving a problem, try explaining your solution clearly.
Could somebody else follow it? Can you explain why each step is valid? Could you describe the main idea without simply repeating the calculation?
Trying to explain something is an excellent way of discovering whether you really understand it.
Return to ideas
Learning something once does not mean you have learnt it forever.
Come back to important ideas later. Solve another problem that uses them. Mix old material with new material rather than permanently abandoning one topic as soon as you move to the next.
Mathematical understanding grows through repeated encounters and connections.
Above all, do mathematics
Mathematics is not a spectator sport.
Read explanations. Watch good teachers. Study elegant solutions. Learn useful techniques.
But then pick up a pencil and try something.
Make a conjecture. Draw a diagram. Calculate an example. Solve a problem. Get something wrong. Work out why. Try again.
That is where much of the learning happens.
This is only a starting point. In future posts I’ll look more closely at some of these ideas — including worked examples, getting stuck, practice, spacing, explanation and problem-solving — and at what research on learning mathematics can tell us about them.
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