This is a trap I have fallen into more than once while teaching maths to my own children. You introduce a new idea. They understand it almost immediately. You demonstrate an example; they can copy the method, then solve one independently. You make the questions progressively harder and they continue getting them right. Something that might occupy several weeks of classroom lessons appears to have been mastered in an afternoon.

Excellent. As a proud home-educating parent, you let them go and play, feeling that you have done your job for the day, the week and possibly the rest of their mathematical education.

Except that two months later you put the same idea in front of them — and they make mistakes, get confused, or seem to have forgotten how to start.

I confused understanding with retaining

I have seen this with long multiplication and division, BODMAS, adding and multiplying fractions, and other topics that seemed completely secure at the time.

That led me to realise that I had confused being able to do something very well today with having learnt it permanently.

A child who learns quickly may need very little initial repetition. But that does not necessarily mean they need less long-term practice. If we never come back to an idea, a skill that once looked remarkably strong can fade.

So what do we actually do?

In theory, I could maintain a carefully calculated revision schedule for every topic we cover. In reality, that could become another full-time job.

I try to build maths revision into our normal week instead. Mondays are for warm-ups. We bring back a handful of things we have learned quite recently. Fridays are for a random revision topic: something we haven’t touched for a while.

There is room to make this more systematic. Careful scheduling — and increasingly AI — could help keep track of what a child has studied and what hasn’t appeared recently. But I don’t think we should let the perfect system get in the way of a very simple question:

What haven’t we done in a while?

Pick something and bring it back. That is already better than allowing a topic that looked completely secure in March to disappear until September.

This is also the idea behind our Practice Lab — Times Tables Recall. Rather than treating times tables as something to learn once and then leave behind, it keeps bringing them back into view, helping knowledge become durable rather than merely familiar for a few days.

We’re also working on a growing collection of random and unusual revision questions for “same old” topics. The idea is not to give children another page of near-identical exercises, but to bring familiar mathematics back in slightly unexpected contexts. A topic may be old; the question doesn’t have to be.

Maths revision doesn’t have to mean another worksheet

There is another problem. If the child understood the topic easily the first time, another page of almost identical calculations is boring.

Instead, I increasingly prefer one wordy, catchy problem that gives us a reason to use the old mathematics.

How many people could actually fit inside our house?

Immediately there are decisions to make. Do we use only the available floor area? How much space does one person occupy? Do we include furniture? Or do we interpret “inside” literally and pack the entire volume? In that case, what is the volume of our house — and what is the volume of an average person?

Suddenly we are estimating, measuring, multiplying, dividing, working with area or volume, converting units and arguing about whether our assumptions are sensible.

Then there are lifts. A lift might specify a maximum number of people and a maximum weight. How are those numbers calculated? What average weight is being assumed? Could eight people exceed the weight limit?

One slightly ridiculous question can resurrect a surprising amount of old mathematics without announcing: Today we are revising multiplication.

And because the problem doesn’t tell the child which operation to perform, they have to decide what mathematics might actually help.

A warning from home education

Of course, home education has a remarkable tendency to turn one interesting question into an entirely different day’s education.

Before long, “How many people fit in our house?” can become a discussion about architecture, lift engineering, crowd density, human anatomy or whether it would be possible to fill the house with guinea pigs instead.

Sometimes those diversions are exactly what we want. But if today’s purpose is to retrieve some mathematics that hasn’t been used for a while, it is worth keeping an eye on the ball. Write down the fascinating side question for another day and come back to:

Right. So how are we actually going to calculate this?

Move forward — but leave paths back

I don’t want old maths revision to slow down a child who is ready for new mathematics. Quite the opposite.

Let them move forward. Stop the repetitive practice when it has clearly stopped being useful. Explore harder and more interesting ideas.

But leave little paths leading back to the mathematics they have already learned.

Not twenty questions at once. Perhaps just one interesting question on Monday, an old topic on Friday, and another unexpected encounter a few weeks later.

The goal isn’t for a child to complete twenty fraction calculations on the afternoon we teach fractions. The goal is for fractions to become part of the mathematical equipment they can still reach for a year later.

Further reading

The ideas in this post connect with research on retrieval practice and distributed (spaced) practice: bringing previously learned material back to mind and revisiting learning over time.

For a detailed review of the evidence, see Dunlosky et al. (2013), Improving Students’ Learning With Effective Learning Techniques: Promising Directions From Cognitive and Educational Psychology. The review identifies practice testing and distributed practice as particularly well-supported techniques for improving learning and retention.

For a shorter, practical example in mathematics, the Education Endowment Foundation’s Retrieval practice: a game of ‘hide and seek’ looks at how mathematical games and varied activities can provide retrieval practice without simply giving children another quiz.

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. I wrote more about this problem — and how we try to handle it in our own home education — in Where Did I Go Wrong? When Understanding Maths Isn’t Enough.

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.