How to Study Maths: A Guide for Parents

In Brief

A study method for maths rests on three habits: solving exercises with the book closed instead of rereading the theory, redoing wrong exercises from scratch, and keeping an error log of your own recurring mistakes. In this guide I explain how to recognise passive studying and what routine to suggest at home, test after test.

Since we opened Up to Ten in 2021, our tutors have built up more than 35,000 hours of lessons with over 630 students. The sentence we hear most often from parents at first contact is always the same: "they study so hard, but then they get everything wrong in the test". It is almost never a question of effort. It is a question of how they study.

Why does the test go badly when "they studied everything"?

Because rereading is not studying. Maths is the only subject in which you can spend an entire afternoon over the textbook without learning anything you can actually spend in a test: you read the worked exercise, you follow every step, it all adds up, "yes, I've got it". But understanding a solution you have read and being able to produce it yourself are two different skills — and the test measures only the second.

In our courses we see it constantly: a student reads the working of a problem on the parabola and nods at every line. Ten minutes later, with the book closed, we ask them to do it again. They freeze at the first step — the set-up — because following someone else's reasoning never requires you to choose where to start. It is the choosing, not the calculating, that rereading never trains.

There is also a basic confusion between "studying the theory" and "being able to solve the exercises". Knowing the definition of the discriminant by heart is no use at all if, faced with $3x^2 = 5x$, the student reaches for the quadratic formula instead of factorising out the $x$. The test to run at home is brutal but infallible: book closed, an exercise they have already seen, a blank sheet. If they cannot get going on their own within a couple of minutes, they had not studied — they had read.

Where are the marks really lost?

After thousands of tests marked alongside our students, the lost marks are almost always concentrated in the same four places. Not one of the four is "not knowing the theory".

1. The minus sign distributed badly. By far the most frequent mistake between middle school and the first two years of high school:

The minus is distributed over the first term and forgotten on the second. A half-second mistake that invalidates the whole exercise — and one the student does not spot on rereading, because they reread what they thought they had written.

2. Domain restrictions ignored. Take $\sqrt{x-1} = x-3$. Squaring both sides gives $x^2 - 7x + 10 = 0$, that is $x=2$ and $x=5$. Anyone who hands in both solutions loses marks: for $x=2$ you would have $\sqrt{1} = -1$, which is impossible. The final check against the original equation — thirty seconds — is the step that almost none of the students who come to us has ever done as a matter of habit.

3. The formula applied where it does not hold. The classic is the square of a binomial with no cross term: $(x+3)^2$ written as $x^2 + 9$ instead of $x^2 + 6x + 9$. This is not carelessness: it is having memorised a formula as a sequence of symbols, without knowing when and why it applies.

4. The answer never compared against the question. A geometry problem asks for a length, the calculation gives $-4$, and $-4$ ends up in the answer box. Students who have got into the habit of rereading the question before handing in pick up these marks for free.

The important point for a parent: these mistakes are not corrected by "studying more". They are corrected by changing the way the exercises are done.

How do you study maths actively?

The routine we build with our students has four elements, in order of importance.

The worked example: read it once, then redo it with the book closed. Reading the working is fine as a first contact. Straight afterwards, though, the book closes and the exercise is redone on a blank sheet, then compared line by line at the end. The difference between the two versions says exactly what had not been understood.

Redo wrong exercises from scratch; do not "look at the correction". Looking at the correction produces the same illusion as rereading: it all adds up, nothing stays. The rule we give is this: the wrong exercise goes back on the list two or three days later and has to be redone from the top, without the correction in front of you. One exercise got wrong and then genuinely solved is worth more than three new ones, because it works on exactly the weak point.

The error log. Not a list of exercises, but a list of patterns: "I get the signs wrong when I distribute a minus", "I forget the domain restrictions with square roots", "in the square of a binomial I skip the cross term". After a month the log holds 5-8 entries, and pre-test revision becomes ten targeted minutes on your own weak points instead of two generic hours on the chapter. On our paths it is the tutor who builds it with the student in the first lessons, until it turns into an automatic habit.

Spaced revision by question type, not a final marathon. Question types have to be picked up again a few days later and then a week later, not crammed into the night before. The day before the test, the only useful thing is a timed mock with mixed exercises — because in a test the exercises come shuffled, whereas in the textbook they come ordered by section, with the method to use quietly suggested by the heading at the top of the page.

What can a parent do, even without remembering any maths?

More than you might think, and without solving a single equation.

• Change the question. Not "have you done your homework?" but "show me an exercise you got wrong: can you redo it from scratch?". You do not need to be able to check the answer: you need to watch whether they get going on their own or reach straight for the correction. What you are observing is the method, not the maths.
• Look at the exercise book, not just the mark. A maths exercise book that has been studied properly is full of traces: attempts, crossings-out, checks on the answer, exercises done twice. An exercise book with nothing but neat copied-out working is the sign of passive studying, however many hours were spent at the desk.
• Protect the calendar, do not lengthen the sessions. Three 45-minute sessions spread across the week are worth more than a three-hour marathon the night before. If the test is on Friday and the book opens on Thursday, no method can work.
• Remove the alibi, gently. "I was hopeless at maths too" is an affectionate sentence that nonetheless gives permission to stop trying. It works better to move the conversation from talent to method: not "you're not a maths person", but "you're studying in a way that leaves you with nothing".

When does outside help make sense?

When the problem is not this week's chapter but something further upstream. The typical signs: mistakes on linear equations in the fourth year of high school, real effort that has produced no results for more than a term, or a concrete deadline — a subject to retake, or an admission test on the horizon, where method counts as much as syllabus.

In these cases a properly run maths tutoring path does not "explain the lesson again": it identifies the missing link — often two or three years earlier than it appears to be — and rebuilds from there the routine you have just read about, until the student carries it forward on their own. On when tutoring is genuinely needed and what makes a good tutor, we have written a dedicated guide for parents.

If you recognise your own situation in what I have described, tell us about your situation: describe how things have gone so far and we will work out together where the gap is and who on our team is the right person. Then you book the first lesson and we get to work straight away on real exercises, error log in hand. Let's get started.

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