Maths SolverThe first time I tried to teach myself a new maths topic from a textbook, I sat down on a Sunday afternoon, opened the chapter at page one, and read it the way you would read a novel: front to back, following the prose, glancing at the boxed examples, nodding along. By the end of the chapter I was sure I understood it. Then I tried the first exercise and discovered I understood almost nothing.
This took me an embarrassingly long time to figure out. It is not that I was a bad reader, or that the textbook was bad, or that the topic was too hard. It is that maths textbooks are not designed to be read like novels. Reading a maths textbook is its own skill, and almost nobody teaches it to you. This post is the version of that skill I now wish someone had given me at the start.
Why front-to-back doesn’t work
A novel is designed to be processed in linear order, with each sentence adding a small amount to the next. A maths textbook is structurally different. Each section presents:
- a definition or a setup, often dense with notation;
- a couple of worked examples;
- some commentary explaining what just happened;
- a set of exercises that test whether you actually got it.
The novel-reader strategy is to read the definition, glance at the worked examples, skim the commentary, and skip the exercises. By the end you feel you understood, because the prose flowed and the worked examples made sense. The problem is that understanding the worked example and being able to do the exercises are different cognitive activities, and only the second one is actually learning.
| Reading a novel | Reading a maths textbook |
|---|---|
| Linear, front to back | Forward, backward, then forward again |
| Comprehension comes from following the prose | Comprehension comes from doing the manipulation yourself |
| Skipping a paragraph rarely costs much | Skipping a derivation usually means missing the chapter |
| You can read it in a chair, with a coffee | You need a desk, paper, and a working pen |
| One pass | Three passes, at different speeds, for different things |
The three-pass method
The method I would recommend — the one that has saved me the most time over the years — is to read each section three times, with a different goal each time.
Pass 1: skim, fast, no pen
Read the section quickly, beginning to end. The goal is not understanding; the goal is to find out what the section is going to ask you to be able to do by the end. Look at the section title, the worked examples, the boxed definitions and theorems, and the exercises. Do not try to follow the algebra in detail. Just see the shape.
This pass should take three to five minutes for most sections, and is the one almost everyone skips. It matters because the rest of your reading is far more efficient when you know what you are reading toward. You stop spending mental energy on a paragraph because you already know whether it’s the main point or a detour.
Pass 2: read carefully with pen and paper
Now go back to the start with a pen in your hand. The rule is: you do not turn the page until you have reproduced every worked example on your own paper, looking only at the question, not the solution.
This is the pass that does the actual learning. When the textbook says “by factoring, we get $x^2 - 5x + 6 = (x - 2)(x - 3)$,” you do not read past it. You write “$x^2 - 5x + 6 = $” on your paper, and you factor it yourself. If you can’t, you go back and re-read the relevant earlier section. Then you check the textbook’s factorisation against yours.
This is slow. The first time you do it for a chapter you used to “read in twenty minutes,” it might take you ninety. That is the correct cost; what you previously thought was twenty minutes of learning was twenty minutes of looking at a page, with almost no learning happening.
Pass 3: do the exercises, then go back to the prose
Try the first three or four exercises in the section. Some will go fine; some will get stuck. For each one that gets stuck, write down where it got stuck (“I don’t know what to do after the brackets are expanded”) and then go back to the section and find the part that addresses that. Sometimes you re-read a single sentence and the entire problem becomes obvious. That single sentence is the real content of the section, and you would never have noticed it on the first read.
Worked examples are the chapter
The single biggest mistake I see students make with maths textbooks is treating the worked examples as illustrations of the prose. They are the other way around. The prose is commentary on the worked examples. The worked examples are the actual content. If you only read one thing in a section, read the worked examples carefully, and only the worked examples.
A useful rule: a worked example only counts when you have done it yourself. Reading the textbook’s solution is not doing it. Cover the solution with a piece of paper, work it out, and only uncover the solution when you are stuck or done. If your method matches the textbook’s, great. If it differs, work out which one is right (if both work, you have learned an alternative method, which is genuinely useful).
The wrong rate to read at
A maths textbook is the wrong format for late-night reading on the sofa. The right time to read it is when you are awake, at a desk, with the page on the left and a notebook on the right and nothing else open. The reading speed should feel almost embarrassingly slow — if you are turning more than one page every ten or fifteen minutes for a topic that is new to you, you are probably going too fast.
The flip side: once you have read a section properly, you should be able to come back to it later and skim it in a couple of minutes for revision. The slow first pass is what makes the fast revision pass possible.
Annotation matters more than you think
I write in my maths books. Not heavily — underlining whole paragraphs is useless — but specifically. The kinds of marks that have actually helped me, in order of usefulness:
- A small star next to a sentence I had to re-read three times before it clicked. Future revision starts with those.
- A question mark next to anything that didn’t quite make sense, with a note about what I didn’t understand. (“? Why are we allowed to swap the limit and the integral here?”)
- A short summary in the top margin of each section: one sentence about what the section taught.
- Cross-references when a later chapter uses something from earlier: “p. 47 needs the chain rule from ch. 3” written in the chapter 3 margin.
This is partly so the book becomes more useful on the second reading, and partly because the act of writing the annotation forces you to articulate what you actually understood. Annotation is a small forcing function for active reading.
What about videos?
Videos are great supplements but bad replacements. The thing a video does well that a textbook does badly is intuition: a five-minute animated explanation of why the chain rule looks the way it does will beat any textbook for the first “oh, I see” moment. The thing a textbook does well that a video does badly is depth and practice. A video tells you what to do; a textbook makes you do it.
A workflow that works for many students: watch a short video first for the intuition (3Blue1Brown is the gold standard for this), then read the textbook section for the formal version, then do the exercises. The video alone won’t make you fluent. The textbook alone will work but is harder going. Together is faster than either in isolation.
The compounding effect
The reason I keep coming back to this method — even though it is slower per session than what most students do — is that it compounds. A topic you actually understood in November is one you don’t have to relearn in March. A topic you faked understanding of in November is one you have to relearn three times before the exam, and even then the foundation is shaky.
Real reading is upfront expensive and downstream cheap. Fake reading is the other way round. Over a school year, real reading wins by a huge margin, but only if you have the patience to do it on Sunday afternoon when fake reading would feel so much easier.