Maths flashcards: what to put on the cards and what to keep for exercises

1. Why knowing your maths lesson is not always enough

You can recite Pythagoras’ theorem and still hesitate as soon as a triangle is drawn differently from the one in your exercise book. You can also know an algebraic identity but not recognise that it would help you factorise an expression. These two situations do not call for quite the same work: in the first, what may be missing is recognising the hypotenuse; in the second, choosing a strategy.

A session with a private tutor helps tell these blocks apart. After the session, it would be a shame to turn the whole correction into twenty cards: you would be revising steps that were already guided, without knowing how to find them again in front of a different exercise. Conversely, immediately redoing ten long problems when you systematically forget a condition of application can be needlessly frustrating.

The solution is to separate the knowledge you need to be able to recall from the skills you need to practise. Cards make useful elements available for solving problems; exercises teach you how to combine those elements. The two complement each other, but they do not measure the same thing.

2. What should go on a maths flashcard, and what should you keep for an exercise?

A good card asks a question you can answer without seeing the back, then check unambiguously whether you are right. “Everything about right-angled triangles” is too broad. “In a triangle with a right angle at A, which side is the hypotenuse?” can be checked: it is the side opposite the right angle, BC.

Lesson content	Preferred format	Example
Definition or vocabulary	Flashcard	“What is the hypotenuse?”
Property and condition of application	Flashcard	“In what type of triangle do you use Pythagoras’ equation?”
Small automatic skill or common mistake	Short flashcard	“What is 7²?” or “Can you add the squares to find any side?”
Choosing the method for a given problem	Exercise	Decide whether to use Pythagoras’ equation, its converse or another property.
Multi-step calculation, diagram, justification	Written exercise	Find a missing length and explain why the equation used is valid.

The dividing line is not “what is easy” versus “what is hard”. A card can test a demanding idea if its answer remains precise. And a small calculation can deserve an exercise if the aim is to learn to choose the right operation. Ask yourself this question: do I want to check that I can retrieve a piece of information, or that I know how to use it in a situation?

This choice reflects the place given to both automatic skills and reasoning in the official maths resources for cycle 4 (the last three years of collège, lower secondary school). Learning properties does not exempt you from searching or proving.

To go further in designing cards without confusing memorisation and problem-solving, also read the method for creating and using effective flashcards.

3. Why this distinction helps you learn

When you reread a formula, it can feel familiar without you being able to write it on a blank page. A well-used card creates a small moment of active retrieval: you try to answer, and only then do you turn the card over. So what counts is not owning a deck of flashcards, but what you do before reading the answers.

The review by Dunlosky and colleagues distinguishes in particular practice testing and spaced revision, whose general usefulness is well supported, from simple rereading, which is less reliable when it makes up most of the work. An experimental study by Karpicke and Blunt also shows the value of retrieval from memory for learning science texts. This research supports the principle of recall, but does not prove that learning cards is enough to solve maths problems.

In maths, an extra difficulty arises: you have to recognise which method to use. Experimental work by Rohrer and colleagues on interleaved practice points to the value, in some contexts, of mixing types of problems rather than only working through exercises of the same form. Mixing forces you more to identify the right strategy. It does, however, assume that the methods have first been sufficiently explained and practised.

To go deeper into the mechanisms, you can read how active recall works, so that this article does not turn into a general guide to memorisation.

4. The 25-minute routine after a real maths session

Here is a routine you can use after a session on a concept that has already been explained. The timings are practical benchmarks, not guaranteed learning times. If you do not yet understand the principle of the lesson, start by asking for an explanation or working through a corrected example again.

1. 0 to 3 minutes — Identify the difficulty. Write down what you need to be able to do alone, then note your most frequent mistake. Example: “I can state Pythagoras, but I mix up the sides when the hypotenuse is given.”
2. 3 to 8 minutes — Select three to five cards. Choose a definition, a condition, a property and, if necessary, a card on the mistake you spotted. Do not turn a whole correction into questions and answers.
3. 8 to 12 minutes — Answer without looking. Hide each back. Say the answer or write down the essential symbols, then compare with the correction. Set aside the cards you know and pick out those that reveal genuine hesitation.
4. 12 to 22 minutes — Solve a new exercise. Start by understanding the figure or the question, decide on the method and write out the calculation. If you get stuck, work out whether the problem comes from a forgotten rule, a choice of strategy or an operation.
5. 22 to 25 minutes — Correct and plan another attempt. Note the mistake that deserves another card or an extra exercise. Come back to these elements later, ideally with a slightly different question.

With 15 minutes, keep two or three cards and a short exercise. With 30 minutes, add a second situation where the method is not given. The extra time should go on searching, not on decorating the cards. A flashcard never replaces written practice when the goal is a written-up solution.

5. A detailed example: a session on Pythagoras’ theorem

Imagine a maths session on right-angled triangles. This example has been constructed to explain the method; it does not correspond to a real student’s file. During the session, the student calculated a hypotenuse with their tutor, but then added the squares when they were looking for a different side. The work to continue is therefore targeted.

Three genuinely useful cards

• Front: “In a triangle ABC with a right angle at A, which side is the hypotenuse?” Back: “BC, the side opposite the right angle.”
• Front: “What equation does Pythagoras give in this triangle?” Back: “BC² = AB² + AC².”
• Front: “If the hypotenuse and one other side are known, what do you do to calculate the third side?” Back: “Subtract the square of the known side from the square of the hypotenuse, then take the positive square root.”

These questions are short, but they cover three possible causes of error: identifying, property, operation. A bad card would be “Explain all of Pythagoras with five examples”. Another bad card would be an exercise copied and pasted with its solution: the student could memorise the numbers without knowing how to choose the method.

What should remain an exercise

Triangle ABC has a right angle at A, with AB = 6 cm and BC = 10 cm. Calculate AC and justify your answer. The expected write-up is: “Triangle ABC has a right angle at A. By Pythagoras’ theorem, BC² = AB² + AC², so AC² = 10² − 6² = 100 − 36 = 64. Since AC is a positive length, AC = 8 cm.”

The “subtract” card can prevent a mistake, but it does not do the whole job: you have to work out where the right angle is, find the right equation, isolate the unknown and conclude with a unit.

To check transfer, then offer a different presentation: “A 5 m ladder leans against a vertical wall. Its foot is 3 m from the wall, on level ground. How high up the wall does it reach?” By correctly modelling the situation as a right-angled triangle, you get a height of 4 m. If the student manages this without a model, they have not simply memorised the previous calculation.

6. How NOEO can extend this work after the session

To apply this routine, you first need to find what was worked on and what is still shaky. On NOEO, the session report can set out strengths, areas for improvement and work to prepare, based on the information provided by the tutor. It therefore helps you choose which knowledge deserves a card and which exercise needs to be revisited, without replacing the tutor’s teaching judgement.

When a session is eligible, the student can choose to create a Learn pack. It contains only a revision sheet, flashcards and a quiz. It is not generated automatically after each session: it is the student’s initiative. The cards and questions offered should still be checked and worked on actively. Any exercises sent by the tutor through the session report are separate from the Learn pack.

A replay can also help you find a step explained live, but only if the required consent was given, the recording was made and it can still be viewed. Under these conditions, it remains accessible for 3 months, provided the consent remains valid. Rewatching an explanation can unblock a specific point; the decisive test is still to close the video and redo the reasoning alone.

The journey before, during and after a session on NOEO shows how these resources stay linked to the session. The value of the tools is to make the work easier to find again, not to guarantee progress in the student’s place.

7. The mistakes that make maths flashcards of little use

• Putting a whole exercise on the back. You end up recognising the correction. Instead, break the essential knowledge down into small questions and keep the full problem for an exercise sheet.
• Writing a formula without its conditions. Pythagoras’ property applies to a right-angled triangle. Forgetting the condition amounts to learning a tool without knowing when it is valid.
• Making lots of cards on what you already know. Target the mistakes observed in sessions or in homework. Three relevant cards can be more useful than a bulky, repetitive deck.
• Settling for recognising the answer. Wait a few seconds, formulate a precise answer and only then show the back. If you hesitate, mark it rather than counting the card as learnt.
• Only redoing identical exercises. Change the figure, the unknown or the wording of the problem. Knowing how to choose the method is a different skill from repeating it.
• Treating every mistake with a card. A sign error may call for automatic skills; poor modelling requires more searching and explanation. The right remedy depends on the block.

If difficulties persist despite revision, it is better to pinpoint the source of the blocks in maths than to keep making more cards. Flashcards are one working tool among others, not a universal diagnosis.

8. How to measure your progress without relying on the number of cards you got right

After a card session, note three separate results: recalling properties, choosing the method and writing up the solution. You can know Pythagoras’ formula perfectly and fail at one of the other two steps. This distinction lets you decide what to work on again.

A concrete test is to redo, two or three days later, a card you hesitated on and then a new exercise of comparable difficulty. Check whether you can recall the property without help, whether you justify using it and whether you carry the calculation through to the conclusion. Greater independence on these points is more informative than a score on cards shown in the same order.

If you get the cards right but not the exercises, shift more time to varied problems and talk through your choice of methods again with your tutor. If you understand the corrected exercises but regularly forget a condition, keep a few targeted cards and try them again later. Making progress in maths is not about choosing between memorising and reasoning: it is about using memory to free up attention for reasoning.