Difficulties in Maths: gaps, method or lack of practice?

Each of these three mechanisms calls for a different teaching response. Giving fifty equations to a student who has not yet mastered negative numbers adds repetition on a fragile foundation. Re-explaining a chapter at length to a student who understands it but lacks fluency addresses a different problem. And giving a “standard method” to someone who can apply a procedure but does not recognise when to use it can reinforce dependence on the model.

After six years of private tutoring, I therefore prefer to ask a simple question: at exactly what point does the reasoning break down? The aim is not to put a label on the student, but to choose the next task that will provide useful information.

1. The three main causes to distinguish

“Gaps”, “lack of method” and “lack of practice” are not three medical diagnoses, nor three categories that explain every difficulty at school. Anxiety, attention, the relationship with mistakes, understanding of school language or a learning disorder can also play a part. But these three categories provide a practical framework for analysing many of the blockages encountered in a Maths tutoring session.

What you observe	Main hypothesis	Question to test
The student gets stuck on a basic step in the middle of a more advanced exercise	Missing prerequisite	Do they succeed at this same step when it is isolated?
They can do the calculations but do not know how to start	Method problem	Can they explain what the problem is asking and compare two approaches?
They understand the method, then make many mistakes or remain very slow	Lack of practice	Does success become more consistent across several similar exercises?
They succeed with the tutor but fail alone the next day	Dependence on guidance or fragile memory	What can they redo without hints and without the correction in front of them?

This logic of placement assessment also exists in the school system. The placement tests at the start of Seconde (the first year of lycée, upper secondary school) published by Éduscol are designed to identify what students have acquired and what they need, in order to target support. In Maths, the results distinguish several areas and a specific test of automatic skills, as in the early Maths exam in Première (the second year of lycée). The idea to take away for one-to-one support is the same: an overall difficulty must be broken down before choosing the remedy.

Why the mark is not enough

Two students who get 8/20 can have almost opposite profiles. The first loses a lot of marks because they have not yet mastered fractions and algebra; the second has mastered calculations but misreads instructions and does not know how to select a strategy. Giving them the same revision programme because they have the same mark would not make much sense.

A marked paper gives more information if you look at the nature of the mistakes: are they concentrated on the same technique? do they only appear when several steps follow one another? does the student leave questions blank or start with an unsuitable strategy? does a mistake corrected in the first part reappear later?

You also need to distinguish a one-off mistake from a reproducible one. An isolated calculation slip does not necessarily justify going back over everything. On the other hand, if the same confusion appears in several contexts, it becomes a diagnostic lead. The tutor can then build a very short exercise that targets only this confusion, and then check whether the correction holds when you go back to the original problem.

2. Spotting a gap in prerequisites

Maths is cumulative: a new concept often reuses older knowledge without saying so. In Seconde, an exercise on functions can be blocked by handling a fraction, calculating with negative numbers or substituting into an expression. In geometry, the problem may come from a property that is not yet secure rather than from the chapter of the day.

The Seconde curriculum applicable from the 2026-2027 school year replaces the old curriculum. The official Éduscol page on Maths at lycée brings together, in particular, the new curricula, resources on automatic skills, reasoning and proof, and the placement tests designed to identify needs at the transition to Seconde.

Isolate the suspect step

Suppose a student cannot solve 3(x - 2) = 2x + 5. Saying “they don’t understand equations” is too hasty. You can isolate, one after the other:

• 3(x - 2): can they expand correctly?
• 3x - 6 = 2x + 5: do they understand what they can do to both sides?
• -6 - 5: have they mastered negative numbers?
• at the end: do they know how to check the value obtained?

A mistake on the first line calls for a different response from a strategy mistake on the third. The visible chapter is “equations”; the real prerequisite may be distributivity or calculating with signs.

Go back just far enough

When a gap appears, you do not necessarily need to “redo the whole of collège”. Go back to the first skill that is not secure, check it on two or three tasks, then work back up to the original exercise, as set out in our method for making progress in Maths in 3e (the final year of collège) and preparing for the Brevet (the end-of-collège exam).

1. Rework the isolated prerequisite.
2. Reuse it in a simple exercise from the chapter.
3. Go back to the problem that was originally blocking.
4. Check a few days later that the difficulty does not reappear.

This working back up is important. A student can get ten isolated calculations right but keep losing their signs in a more complex problem when their attention is taken up elsewhere.

For collège (lower secondary school) students who have built up several weak points, our guide to method and frequency for private Maths tutoring at collège complements this diagnosis. On entering lycée, the article on the most common Maths difficulties in Seconde helps to target the transition points more precisely.

3. Recognising a method problem

A method problem often shows up in a student who can do the necessary operations but does not know how to organise them. They can solve an equation when the tutor says “set up an equation”, then get stuck in a problem where they have to decide alone that an equation is relevant.

In this case, doing more identical exercises can create an illusion of progress: the student recognises the format of the worksheet, not necessarily the mathematical structure of the problem.

Three signs of a method problem

• The student asks “which formula do I need to use?” before rephrasing the question.
• They know several techniques but choose between them almost at random.
• They cannot explain why a step is valid, even when it is correct.

An effective method should not become a recipe to recite. It helps the student make decisions: what are we looking for? which information is useful? which representation simplifies the problem? which property links the data to the unknown? how can the result be checked?

The What Works Clearinghouse of the Institute of Education Sciences recommends, for algebra in middle and high school, having students analyse solved problems, teaching the structure of algebraic representations and teaching students to choose deliberately between several strategies. The level of evidence varies between recommendations; the guide gives, in particular, a moderate level of evidence to work on choosing between alternative strategies.

Have them compare two solutions

A very simple method activity is to present two solutions to the same problem and ask: where do the approaches diverge? which one is valid? which is the most efficient here? in what other situation might the other be preferable?

You can also show a deliberately wrong solution and ask the student to identify the first incorrect line. They then have to judge the reasoning rather than mechanically reproduce a procedure.

4. Identifying a lack of practice

A lack of practice sometimes looks like a gap, but the profile is different. The student understands the explanation, can justify the method and succeeds when they take their time, but carrying it out remains costly: sign errors, hesitant mental arithmetic, slow expansion, forgotten checks.

Automatic skills have a practical function: they reduce the number of basic operations that take up attention during a more complex problem. If every fraction or algebraic transformation requires considerable effort, the overall reasoning becomes more fragile.

How can you test this hypothesis?

Set four or five similar tasks, with no new concept. If the method is understood but the execution remains inconsistent, practice is probably part of the answer.

The aim is not to impose fifty identical exercises. The Institute of Education Sciences summarises several learning principles that apply to Maths: spacing work out over time, alternating worked examples and problem-solving, using quizzes to re-expose students to content and asking explanatory questions. This summary covers several school levels, so the quantity and difficulty must be adapted to the student.

In practice, ten regular minutes on a specific skill are often better than two hours of repetition the night before a test. Then the form of the questions needs to be varied gradually to check that the student recognises the skill when it is no longer announced.

Do not confuse speed with mastery

A slow student is not necessarily struggling conceptually. If they choose the right strategy, justify their steps and get reliable results, the need may be to automate certain procedures, not to redo the whole chapter. Conversely, a very fast student may be applying a recipe without understanding why it works. Speed is therefore only one indicator among others.

5. The role of written presentation and mathematical language

Some “blockages in Maths” are partly language problems. A student understands the situation intuitively but misuses symbols, confuses an implication with an equivalence, skips steps or turns the “=” sign into simple punctuation between several calculations.

Writing up is not just a school requirement added after the reasoning. It often makes the reasoning visible.

Ask the student to read their own line

Take 2x + 3 = 11 = 2x = 8 = x = 4. The student may reach the right value, but the chain of equalities is wrong: the expressions written are not all equal. Asking them to read each equality aloud often makes the problem visible.

Other useful signals:

• a letter used without knowing what it represents;
• a property quoted without checking its conditions;
• a graph described as “it goes up” without specifying the quantity or the interval;
• a numerical answer without a unit when the unit is meaningful;
• a conclusion that does not answer the question asked.

Having the student rephrase aloud helps to distinguish clumsy writing from a lack of understanding. If the student can correctly explain what they meant, the priority is formalisation. If they cannot justify the step, you need to go back to the concept or the method.

6. Building a diagnosis in twenty minutes

A useful diagnosis does not need to look like a full test. Above all, it should offer tasks that separate the hypotheses. Here is a twenty-minute format that serves as a working grid, not a standardised test.

Time	Task	What you observe
0 to 3 min	Have the student explain the blockage and show a recent mistake	Vocabulary, how they picture the problem, the type of mistake they perceive
3 to 7 min	A simpler prerequisite task	The technical foundation actually available
7 to 12 min	A typical exercise at the current level	Choice of strategy, execution, written presentation
12 to 16 min	A variant with the same idea but a different form	Transfer, or mere recognition of a standard exercise
16 to 20 min	Have them explain the mistake, then redo a similar task without hints	Understanding of the correction and immediate autonomy

What to note down

• the first step where the student hesitates;
• the smallest hint needed to get going again;
• the type of mistake that keeps coming back;
• the ability to explain their choice;
• what they correct alone after a question;
• what they can redo without the model.

This diagnosis does not replace a full school assessment nor, when lasting and unusual difficulties are suspected, the opinion of qualified professionals.

7. Choosing the right teaching response

Once the dominant cause has been identified, the response must change.

Likely diagnosis	Priority response	Mistake to avoid
Fragile prerequisite	Go back to the first unstable skill, then work back up to the current chapter	Piling up exercises that are too difficult
Fragile method	Have the student put things into words, compare strategies and analyse examples	Giving yet another recipe to memorise
Lack of practice	Short, regular and gradually varied practice	Re-explaining the whole lesson at every mistake
Fragile language or written presentation	Have each symbol, property and conclusion made explicit	Correcting only the final presentation

Example: three students stuck on functions

Student A: they cannot correctly substitute a value into f(x) = 2x² - 3x + 1. Priority: algebra and writing conventions.

Student B: they can calculate an image but confuse “image” and “preimage” when the question is set within a problem. Priority: language, representation and choice of approach.

Student C: they understand all the questions but lose a lot of time on calculations and make execution errors. Priority: automation and checking.

The chapter is the same. The remedy should not be.

How can you tell whether the student is really making progress?

A rise in marks can be encouraging but is not enough to identify what has changed. To track progress without promising an automatic result, look for signs closer to the reasoning:

• does the student start an exercise without waiting for the first hint?
• does the same mistake come back less often?
• do they spot some mistakes when rereading?
• can they explain why a method is appropriate?
• do they succeed at a variant that does not look exactly like the model exercise?
• does a skill they have worked on remain available a few days later?

The Education Endowment Foundation’s guidance on teaching Maths, focused on the levels roughly corresponding to the end of primary school and collège in the United Kingdom, also stresses assessing students’ knowledge and understanding in order to choose the next steps. Its recommendations are not a protocol for private tutoring in France, but they support the idea that an intervention must start from what the student can really do.

On NOEO, tutors are listed by subject and level, and the guidance request lets you specify the student’s level, goals and difficulties. Once the support has started, the session report can record strengths, areas for improvement, the work to prepare and the next priority. The point is not to label the student, but to check from one session to the next whether the type of help needed is decreasing.

Useful follow-up also consists in periodically redoing a task comparable to the one that caused problems, without telling the student which technique is expected. If success still depends on a reminder from the tutor, the skill is not yet fully independent.