How to Take Notes in Math Class: What the Board Leaves Out

To take notes in math class, write down what the lecturer says as well as the board. Beside each step, add a short phrase saying why it happens, and keep every worked example whole. After class, explain each step to yourself and redo the examples on a blank page instead of rereading your notes.

That advice draws on studies of university mathematics, several of them in proof-based lectures, plus one online calculus course taught by video. The findings describe those settings, so in a school math class, read the steps below as our advice.

What gets lost when you copy only the board?

In one study of proof-based lectures, the part that went missing from students’ notes was the reasoning the lecturer spoke aloud. Fukawa-Connelly and colleagues (2017) looked for what they called informal content: “ways of thinking and reasoning about advanced mathematics that are not captured by formal symbolic statements”. Think of a lecturer explaining why a step is taken, what an idea means, or how to approach a proof.

They describe their method in one line: “We recorded 11 80-minute mathematics lectures and photographed the notes of 96 students.” They found that “informal content was common (with, on average, 32 instances per lecture)”, that “most informal content was presented orally”, and that “typically students recorded written content while not recording oral content in their notes.” The abstract reports no measure of grades or learning, so it shows a gap in the notes and leaves open what that gap costs.

Feudel and Panse (2022) start from the same observation: “In traditional mathematics lectures the instructor normally writes the definitions, theorems, and proofs covered on the board, and gives informal oral explanations that help to make sense of them.” They name the cost that shows up afterwards: “Making sense of the content later is also difficult because many students do not include the lecturer’s oral explanations in their notes.”

Why is it hard to listen and write at the same time?

Because note-taking asks for three jobs at once. Krapf and Pfefferkorn (2022) put it plainly: “Note-taking in tertiary education is a challenging activity which requires listening, processing and recording new information at the same time.” They go on: “However, in lectures at a high presentation rate, students often have to choose between these activities.” Feudel and Panse describe what gives way: “Students often cannot think about the information presented during the lecture as they are busy writing.”

Our suggestion is to abbreviate your own words freely and copy carefully only what must be exact: definitions, theorem statements and worked examples. That leaves some attention for thinking. Pen or keyboard is a separate question: our comparison of handwriting vs typing notes gives handwriting a modest edge, and suggests typing when a fast technical lecture full of equations would outrun a pen.

How do you take notes in math class?

Give the spoken reasoning its own place on the page. These habits are our suggestions for any math class, in a lecture hall or a classroom.

1. Split the page. Copy the board down the left two-thirds and keep a narrow right-hand column for what is said. Later, one glance shows which steps have a reason beside them and which are bare.
2. Write the reason in a few words. A margin note such as why: x = 0 ruled out earlier or idea: squeeze it between two functions with the same limit is enough. You are catching a hook to expand after class.
3. Treat certain phrases as a cue. When the lecturer says something like the idea here is, why would we do this, intuitively or this is the step people get wrong, move your pen to the right-hand column.
4. Keep each worked example whole. Copy every line, including steps the lecturer does aloud and skips on the board, and put a reason beside each one. An example with a gap in the middle is hard to redo later.
5. Mark what you missed. A question mark and a line of white space where you fell behind tell you what to fill in after class.

In an online calculus course, van de Sande, Abramson and Judson-Garcia (2017) asked whether an example’s position in a video and its completeness related to whether it reached notes. They found that “leading worked examples within a video were salient, but that example completeness was not a factor in inclusion”. In other words, the worked examples that came first in a video stood out, and whether an example was complete made no difference to whether it was noted. We read that as a reason to give the later examples in a lecture the same care as the first.

Our guide to taking lecture notes still applies: capture the structure in your own words, and note what the lecturer says around the slide instead of the slide itself. Math changes one part of that advice: definitions, theorem statements and worked examples need to be exact, so copy those in full and put your own words in the column beside them.

What if your instructor hands out guided notes?

Then use them, and write the spoken reasoning into their margins. Feudel and Panse define guided notes as “a modified version of the instructor’s notes with certain blanks the students have to fill in during the lecture”, so only your instructor can offer them. Between them, three studies report what students perceived, preferred and wrote down.

Krapf and Pfefferkorn surveyed students in an undergraduate mathematics course: “A self-report survey reveals that students value guided notes as a tool to remain focussed during the lecture, to process and store new information and to foster active engagement.” And: “Moreover, they prefer guided notes both compared to lectures without instructor-provided notes and with full notes.” That is self-report about how the handouts felt to use.

Feudel and Panse’s quantitative data suggest that “guided notes are perceived as beneficial by many students for several aspects of their note-taking”. The use of guided notes can address some note-taking problems, they conclude, “while it can also lead to new problems that one needs to be aware of.”

Iannone and Miller’s case study starts from earlier research in which students recorded only what was written on the board, and reports a difference from it and a limit. With guided notes, “some students in our study recorded the non-written comments as well as some of their own links between sections of the lecture.” They add: “We did not, however, find students’ attitude towards those comments to be different from what previous research found.” They conclude that “guided notes can be an appropriate way of teaching university mathematics but on their own cannot make the pedagogical intentions of the lecturer clearer to the students.”

What should you do with math notes after class?

Work with them. Within about a day, fill the gaps while you still remember what was said, then close the notes and test yourself. Our guide to reviewing notes after class sets out that repair pass and the retrieval after it. Math notes then support two more jobs.

Explain each step to yourself

Hodds, Alcock and Inglis (2014) tested self-explanation training for reading proofs. They report that “a simple booklet containing self-explanation training, designed to focus students’ attention on logical relationships within a mathematical proof, can significantly improve their proof comprehension”. Their first experiment gives a number: “Experiment 1 demonstrated that students who received the training generated higher quality explanations and performed better (effect size d = 0.950) on a comprehension test.” The authors argue that transition-to-proof courses should include the training.

That finding is about reading proofs. Our own extension, beyond what the abstract reports, is to ask of each line of a worked example in your notes which earlier line, rule or condition allows it. Answer in a few words, then check against the reasoning you caught in the lecture. Our guide to the self-explanation study method covers the habit, including why the benefit comes from generating the explanation yourself.

Redo each example from a blank page

Then cover each worked example, redo it on a blank page, and uncover it only to check. Where you get stuck, the margin note for that step shows what you missed. Our guide on how to study for a math test puts working problems from memory, plus an error log, at the centre of math revision. Your lecture notes feed that, and once the examples come easily, move on to new problems.

Do better notes lead to better math grades?

In one online calculus course, better notes did not go with better exam results. Van de Sande and colleagues describe the setting: “The study presented here explored note-taking from video lectures by students who were genuinely invested in doing well in an online course.” Their abstract reports that “note-taking participation and quality was not positively correlated with exam performance”.

They are clear about its status: “This observational study sets the stage for future experimental research”. So it supports a narrow reading: in that course, notes alone did not predict exam results. That fits our view of notes as raw material for problem practice.

Where GeniusPal fits

GeniusPal suits the recall layer of a math course: definitions, theorem statements, the conditions a result needs, and when a method applies. Upload a PDF, Word, PowerPoint, plain text, Markdown or CSV file of 10 MB at most, or paste a link, and it writes a study set of questions from that material: 10 per set on the Free plan, and up to 30 on paid plans. Free sets run as a multiple-choice quiz. The paid Student and Genius plans add flashcard mode, recall mode and written quiz answers.

It cannot read photos, or scans that lack a text layer, so photographing or scanning handwritten math notes will not work. Typed notes, or a PDF with real text in it, will. Nor does it schedule reviews. Its daily review, free on every plan, is a round of up to 20 questions drawn from all your sets, putting first the ones you got wrong or have not tried yet. Problem practice stays on paper.