Maths SolverI spent most of undergraduate trying to learn maths by rereading. The pattern was: confused at the lecture, vaguely confused after reading the textbook section, more confused after reading it a second time, and eventually some combination of doing problems at $1$ AM the night before the assignment was due and asking a friend who had figured it out faster than me.
What changed in graduate school was that the pace got fast enough that I had to find a method that produced understanding more predictably. The five steps below are what I converged on. I still use them — for new research areas, for textbook chapters, for anything I have to learn from scratch. They are not original to me; versions of this method exist in study-skills literature under names like the Feynman technique. They work.
Step 1: read the section once, fast, no pen
The first pass is for reconnaissance, not for understanding. I read quickly, follow the prose loosely, and pay attention only to two things: what new vocabulary the section is introducing, and what kind of problem the section claims I will be able to solve by the end. I do not stop when something is confusing, because everything is going to be a bit confusing on the first pass and that is fine.
This pass exists because the rest of the method is more efficient when I know what I am reading toward. The biggest waste of time in maths study is reading a paragraph carefully when it turns out to be a detour from the actual point of the section. The first pass tells me what the actual point is.
Time: 5 to 10 minutes for a typical textbook section.
Step 2: work through one easy example by hand
Now I find the simplest example I can — either the first worked example in the textbook, or a problem from the warm-up exercises — and do it on paper, looking only at the problem, not the solution. If the section is on integration by parts, I do the simplest possible integration-by-parts problem, even if it feels insulting. The point is not to challenge myself; the point is to make sure the basic mechanics work in my hands.
If I can do the simple example, I check the textbook’s solution and confirm my method matches. If I can’t, I read the textbook’s solution carefully and identify exactly which step I got stuck on. That step is what the rest of my study session is going to focus on.
Time: 10 to 20 minutes.
Step 3: write a one-paragraph summary in my own words
This is the step almost no one does, and it is the most important one. I close the textbook and write, in normal English, what the section is about. The constraints are that I cannot use any notation from the textbook directly (I have to translate it into something I would say out loud) and I cannot look anything up while writing.
For the chain rule, my summary might be: “If you have a function inside another function, the derivative is the derivative of the outer function (with the inner function still inside it) times the derivative of the inner function. The reason it’s called a chain is that for nested functions you keep applying this, multiplying as you go.”
This is the moment that exposes whether I understood, because if I can’t write the summary I clearly haven’t understood, no matter how confidently I nodded at the worked examples. When my summary is wrong, that is the most useful piece of information I will get all session: I know exactly where my understanding has gaps.
I keep these summaries in a single notebook organised by topic. Six months later, when I have forgotten the chain rule and need to use it in a research problem, I read my own one-paragraph summary first, then go back to the textbook only if I need to. The summary serves as both diagnostic and revision aid.
Time: 5 to 10 minutes.
Step 4: do a harder problem
Now I pick something from the middle of the exercise set — not the warm-up, not the optional bonus problem, just a representative problem of the kind a homework set would actually contain — and do it from scratch, without the textbook open. If I get stuck, I open the textbook and find the specific bit I need (which my summary helps me locate quickly), then close it again and continue.
The reason this step matters is that the easy example in step 2 is designed to make the technique obvious. The middle-difficulty problem is designed to test whether I can recognise when to apply the technique, which is a different skill. Most students who say “I understand the chain rule but I can’t do these homework problems” have plenty of step-2 understanding and zero step-4 fluency. Step 4 is where step-2 understanding becomes step-4 fluency, and the only way to do it is by stalling on real problems and figuring out how to get unstuck.
Time: variable, often 30+ minutes for one problem.
Step 5: try to teach it to an imaginary student
Last step. I imagine I am explaining the topic to someone who has algebra but has never seen this specific topic before. I say it out loud (in my head if I am in a library) and listen for the places where my own explanation gets vague. The phrase “and then you just sort of...” is the sound of a gap in my understanding. Wherever my explanation requires hand-waving is exactly where I do not yet have the topic.
This step is sometimes called the Feynman technique after Richard Feynman, who used something like it. The mechanism is the same as the written summary in step 3, but verbal: forcing yourself to produce a fluent explanation surfaces gaps that just thinking about the topic does not. There is something specific about translating an internal sense of understanding into spoken-language explanation that catches things you would otherwise miss.
If I find a gap, I go back to step 3 (rewrite the summary, more carefully this time) or step 4 (do another problem of the kind that exposed the gap). The cycle is short and the diagnosis is precise.
Time: 5 minutes, usually less.
Why this works (and why other methods don’t)
The reason this method works, when generic “reread the textbook” does not, is that it forces you to produce output at several different levels of detail: a fast skim, a worked example, a prose summary, a harder problem, a verbal explanation. Each output mode catches a different kind of gap. Rereading produces nothing, so nothing in your understanding gets tested.
The other thing this method does is keep you honest about whether you have actually learned the topic. There is a real and persistent illusion of competence that comes from following someone else’s explanation and feeling like you got it. The illusion goes away the moment you have to produce something — an example, a summary, an explanation — on your own, with no scaffolding. That is why all five steps are about producing something rather than consuming something.
Time per topic
For a typical textbook section — one of those numbered subsections that ends with an exercise set — the whole method takes me between sixty and ninety minutes, end to end. This sounds like a lot, until you compare it to how long it would take me to “learn” the same section by rereading: probably twenty minutes, and the resulting understanding would be unreliable enough that I would have to repeat the process the next time the topic came up.
Spending ninety minutes once beats spending twenty minutes four times and still feeling shaky. This is the whole argument for the method, really. Most studying advice ends up being a version of “do fewer things, but do them properly,” and this is the version of that advice that is specific to maths.
What to skip
A few things I tried in undergraduate that I no longer do:
- Highlighting. I have never learned anything by highlighting it. The active version of highlighting is “close the book and write down what the highlighted passage said,” which is just step 3.
- Watching three videos before doing any problems. Videos can be a useful supplement, but if I watch more than one without doing anything in between, I end up in the illusion-of-competence state. One video, then a problem.
- Doing every problem in the exercise set. Diminishing returns hit fast. Do five problems carefully and review what each one taught; do not grind twenty for the sake of completion.
None of these are bad on their own; they are just bad as substitutes for actually producing your own work. The five-step method is, in the end, just a slightly more structured version of “sit down, do the problem, and force yourself to articulate what is happening.” That is the entire trick.