How to Actually Learn This Material

Intuition first

There is a reliable way to spend forty hours on this site and retain almost nothing: read every lesson carefully, nod along, feel that it made sense, and move on.

Reading produces recognition — the comfortable feeling that you have seen something before. Recognition is not the same as being able to reproduce it, and the gap between them is invisible from the inside. That is what makes it dangerous: you cannot detect the problem by introspection, only by attempting to produce something.

Everything below is a way of forcing production rather than recognition. None of it is motivational advice. It is all about arranging for the failure to happen now, cheaply, at your desk, rather than later in an exam or a code review.

The single most important habit

Close the lesson and reproduce the derivation on blank paper.

The reason this works is that reading uses recognition memory, which is generous and fast. Blank-paper reproduction uses recall, which is what an exam or a real problem demands. Practising the wrong one feels productive and transfers poorly.

Apply it selectively. You do not need to reproduce every algebraic step on the site. Pick the results that matter — the bias–variance decomposition, Bayes' theorem, the backpropagation update, the LoRA factorisation — and be strict about those.

Work the solved problems before reading the solution

Every solved problem is laid out with steps visible. Cover them.

The numbers are deliberately small enough to compute by hand, which means there is no excuse for skipping the arithmetic. And the arithmetic is where the misunderstandings surface: it is only when you have to decide whether to divide by 100 or by 9,900 that you discover you had precision and recall confused.

Space the repetition

Reviewing material once a week for four weeks beats four hours in one evening, by a large margin. The effect is well established and the mechanism is straightforward: retrieving something you have partially forgotten strengthens it far more than re-reading something still fresh.

A workable schedule:

What to review is not the lesson text but a short list of prompts you wrote yourself: "derive the variance of a binomial", "state the three conditions for the CLT", "why does L1 produce exact zeros".

Run the code, then break it

Every code block on this site is runnable and most print a comparison between an empirical result and the theoretical value. Run them. Then change something and predict what will happen before you look.

Specifically productive experiments:

• Change a sample size and check the $1/\sqrt{n}$ scaling actually appears.
• Change $\lambda$ in a regularisation example until coefficients hit zero.
• Remove the continuity correction and measure how much the error grows.
• Break an assumption deliberately — make trials correlated, make a distribution heavy-tailed — and watch the theory fail.

That last one is the most valuable and the least often done. Knowing that a method breaks is useful; having seen it break is what makes you check for the condition in real work.

Explain it to someone who is not there

Write a paragraph explaining the concept to an imagined colleague, in prose, without notation. The points where you stall are the points you do not understand.

This is unusually efficient because it is fast — five minutes per concept — and because the failure mode is unmistakable. You either produce a coherent paragraph or you do not.

Follow prerequisites backwards when stuck

If a lesson is not landing after a genuine attempt, the usual cause is a missing prerequisite rather than difficulty. Every lesson lists its prerequisites at the top. Follow them back until you reach something solid, then work forwards.

A concrete weekly structure

If you want something to follow rather than assemble, this works:

• Monday–Thursday: two lessons per day. Read, attempt the solved problems covered, run the code.
• Friday: no new material. Reproduce that week's key derivations on blank paper.
• Saturday: work the exercises you skipped, and one lesson from a later module that interests you — to keep motivation attached to something you actually want.
• Sunday: nothing. Consolidation happens during breaks, and a day off is not lost time.

Eight lessons a week finishes a typical module in two to three weeks and the whole curriculum in roughly eighteen months. That sounds long until you compare it with the alternative of reading everything in three weeks and retaining a tenth of it.

Signals you are doing it right

• You can state what a lesson assumed before it started.
• You notice when a formula is a special case of something earlier.
• You can say what would break if an assumption were violated.
• You catch errors in other material, including on this site.

That last one matters. If you find a mistake here — a wrong number, a step that does not follow, a claim without justification — that is the standard working correctly, not a failure of the site. Treat unexplained assertions as defects.

Exercise 1

Pick a lesson you read recently and believed you understood. Without opening it, write down: the main result, what it assumed, one worked example with actual numbers, and one situation where it fails.

Show solutionHide solution

There is no model answer — the point is the attempt, and what it reveals.

Common outcomes:

You produce all four comfortably. You genuinely know it. Schedule one review in a week and move on.

You get the result and the assumptions but stall on the numbers. Very common, and it means you have the shape without the mechanics. Work two solved problems from that lesson with the steps covered.

You get the result but cannot say what it assumed. The most dangerous outcome, because you will apply the result where it does not hold. Re-read only the assumptions section and the pitfalls.

You cannot state the main result. You read it rather than learned it. That is normal and not a reflection of ability — but it does mean the lesson needs a second pass with active retrieval rather than re-reading.

If you found the exercise uncomfortable, that discomfort is the mechanism, not a side effect.

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You are set up. Start with Sample Spaces and the Axioms of Probability, or browse the full curriculum and pick whatever you actually want to understand.