Image by Sofia Jone from Pixabay
By Louis Vanhove
I needed a maths tutor myself in 2020, and the thing I remember is how little teaching he actually did. He mostly listened. He let me work through a problem until I stopped, asked me what I thought the next line should be, and then we fixed whatever was wrong underneath. That is the mentality I have carried into tutoring other people ever since.
Here is the claim, and it is the whole article. The topic you cannot do is almost never the topic that is broken. Something earlier is broken, and the topic you are staring at is just where you finally noticed. So before you ask anyone for help, you can do the most useful part yourself: find the earlier thing.
First, the honest part. This does not always work, and I have been on the wrong side of it. A student once booked me for trigonometry identities. I had not touched them in five years, sat down with the exercises, and found I could not properly do them either. I could have improvised an hour of vague strategy talk and taken the money. I refunded him instead, because surface-level help is worse than none. If you walk backwards and still cannot find the floor, that is information, not failure, and it is exactly the point where another person is worth asking.
Work until you stop, then mark the line
Take the question you cannot do and write it out properly. Not read it, write it. Then work forward until you genuinely stop, and draw a line there.
Most people skip this because it feels like admitting defeat on paper. It is the opposite. “I cannot do quadratics” is not something anyone can help you with, including yourself. “I get to x squared plus 5x plus 6 equals zero and I do not know what two numbers to look for” is a question with an answer.
Ask what that line assumes you already know
Now look at the exact line where you stopped and name what it takes for granted. Be specific, and write the answer down as a sentence.
Solving (x plus 2) divided by (x minus 1) equals 3 assumes you can multiply both sides by a bracket without losing the minus sign. Factoring assumes you can expand brackets fast enough to check yourself. The chain rule assumes you can look at a function and say which part is inside the other. And the thing that stops most people on trigonometry identities is not trigonometry at all, it is algebra with fractions, which is why my student’s problem was never really the topic he booked me for.
That is the move. Every topic sits on two or three earlier ones, and your gap is in the earlier ones far more often than in the new material.
Test the earlier thing on purpose, with easy numbers
Do not test it inside the hard question. Pull it out and give it the smallest, ugliest version of itself you can write.
If you suspect fractions, do six of them on their own, with a pencil, no calculator. If you suspect negative numbers, do twenty of those. Use numbers small enough that you can tell the answer is right without checking, because the whole purpose here is to find out whether you can do the step, not to get the question finished.
You are looking for one of two results. Either it is fine, and you walk back up to the next assumption on your list, or it wobbles, and you have found something real.
Keep walking back until you hit something fluent
Fluent means you can do it without narrating it to yourself. If you have to think, keep going down.
This part takes nerve, because tracing a problem backwards often lands you in something from three years ago, and it is tempting to decide that cannot be it. It usually is. Fractions, negatives, and what the equals sign actually means hold up an enormous amount of later maths, quietly.
Here is where you have an advantage a classroom student does not. You can actually stop and go back. Nobody is moving you on to the next chapter on Thursday because the timetable says so. If the real gap is two years back, you are allowed to spend a week there, and that week will do more than a month of re-reading the chapter you are stuck on.
Fix one step up from the floor, then climb
Once you find the lowest thing that wobbles, do that one until it is boring. Then take the next thing up and do that until it is boring. Then go back to the original question.
The question is often easy now, and that feeling is slightly annoying. It was not a hard question. You were missing a step.
This holds outside maths too. I also teach 3D, and a student came to me four days before a retake because his UV unwrap would not work: wrong seams, a mess of a texture. He was right that it would not work, and wrong about why. The model underneath had been built without the unwrap in mind, with hidden ngons, backward faces, duplicated vertices and floating edges, all the bad stuff you could imagine, and none of it visible from where he was looking. He remodelled it and the step he had been fighting took minutes.
What to do today
Pick the one topic you have been avoiding. Write the question out, work until you stop, draw the line, and write down the three things that line assumes. Test the easiest of them with small numbers.
That is twenty minutes, and it will tell you more than another evening of staring at the chapter. If you walk all the way down and everything is fluent, then the new topic really is the new topic, and you need explaining rather than repairing. Ask someone then. You will know exactly what to ask for, which is most of the battle.
Author Bio
Louis Vanhove is the founder of iTutorOnline. He came to tutoring from the other side of the table: he needed help with maths himself in 2020, started tutoring other people in 2022, and has taught around a hundred students since, mostly maths, programming and 3D.
