Learning university-level maths on your own

For many adults, mathematics is a mixed memory, sometimes a painful one. And it is often what trips people up in a science, data or economics path: a machine learning course that assumes linear algebra, a finance course that assumes calculus, a statistics course that assumes probability.

Yet mathematics can be learned remarkably well on your own, at any age. On two conditions that are often forgotten: follow the right order, and do lots of exercises. This article offers a complete programme, from the end of secondary school to university-level mathematics, built from the Atlas courses and free textbooks.

Order matters more than anything

Mathematics is a discipline that builds up. Every concept rests on earlier ones, and it is almost impossible to understand a step if you skipped the one before. That is the most common reason self-learners get stuck: they tackle linear algebra with shaky algebra skills, and conclude they are “not a maths person”, when a single step was simply missing.

Here is the order we recommend, with, for each step, the Atlas courses, their length and a free textbook to go with them.

The first four steps form a backbone best followed in order. The last two, on the other hand, can be taken in whichever order you like, depending on your goal. Discrete mathematics is essential for computing; probability and statistics are essential for data, economics and the experimental sciences.

If your horizon is machine learning, two free textbooks bridge the gap between this mathematics and its applications very well: Stephen Boyd and Lieven Vandenberghe’s Introduction to Applied Linear Algebra, and Mathematics for Machine Learning.

This programme in the Atlas curricula

This programme isn’t theoretical: it is, more or less, the backbone of the General Science curriculum. The first semester revisits algebra and precalculus, pairing them with Python. The second covers calculus and linear algebra, applied to mechanics. The third is devoted to Mathematics for Engineers and differential equations.

The Data Science curriculum takes a faster route. It goes through calculus in the second semester, then moves on to discrete mathematics in the third and probability in the fourth.

If your goal is narrower, three Atlas paths focus on mathematics: pre-university science preparation (228 hours), the maths and physical science teacher path (318 hours) and the operations research and optimisation path (349 hours).

A method for progressing alone

Following the right order is not enough. You also need to work in a certain way, and this is where many self-learners go wrong.

Exercises first. Watching a proof on video gives a very convincing feeling of understanding. Yet it evaporates at the first problem set. In mathematics, you only really understand by doing. Allow at least as much time for exercises as for videos, and don’t hesitate to draw on the textbook exercises as well as the course ones. OpenStax textbooks, for example, give answers to the odd-numbered exercises, so you can check your work.

A course and a textbook, together. The course sets the pace and gives intuition; the textbook supplies the details, the full proofs and an endless supply of exercises. Get into the habit of reading, each week, the textbook section that matches that week of the course. The two complement each other remarkably well.

Redo it without looking. After reading a worked example, hide the solution and redo it entirely. If you get stuck, uncover a single line, then hide it again and carry on. It is slower, but that is how the method really sinks in.

Explain it out loud. If you can’t explain in two sentences why a method works, you haven’t quite understood it yet. Try explaining it to someone, or even to yourself: the gaps show up immediately.

Space out your reviews. Go back to earlier chapters a week later, then a month later, with a few exercises. This spacing effect is one of the most robust findings in memory research. We explain it in taking effective notes.

Code the maths. Computing a numerical derivative in Python, programming matrix multiplication, simulating thousands of dice rolls: turning a concept into code forces you to understand it with a precision you don’t always reach on paper. That is in fact why the Atlas curricula pair Python with mathematics from the first semester.

When you get stuck

It will happen, and that is normal. A few reflexes help you get going again.

The first is to go back one step. Being stuck in calculus very often comes from weak algebra; being stuck in linear algebra, from discomfort with systems of equations. Consolidating the base often unlocks what comes next.

The second is to change source. Another explanation, in another textbook or video, often sheds light on what the first one left in the dark. Every teacher has their own way of presenting things, and one of them will eventually click.

The third is to ask. Course forums, or sites like Mathematics Stack Exchange, are happy to answer well-asked questions. The key is to explain what you tried and exactly where you are stuck.

Finally, accept slowness. A single page of mathematics can take an hour of work, sometimes more. It is not a sign that maths isn’t for you: it is simply the normal pace of mathematics, for everyone.