How to Study for a Math Test (What Actually Works)

In 2006 Roediger and Karpicke ran an experiment that has since been replicated in dozens of variations. One group of students studied a prose passage four times over, reading it an average of 14.2 times. Another studied it once and then tried to recall it from memory three times, with no feedback and no second look, reading it only 3.4 times in total. Tested five minutes later, the re-readers were ahead: 83% against 71%. This is the result that keeps re-reading alive, and it is real. Tested a week later, the ordering flipped hard: 61% for the recall group, 40% for the re-readers.

Then the finding that should change your behaviour. Asked to predict how well they would remember the passage, the re-readers were the more confident group. They had read four times as much material, they were substantially worse at producing it, and they did not know.

The mechanism is fluency. A page you have read before goes down smoothly, and your brain reads that smoothness as knowledge. It is evidence of familiarity, which is a different thing, and in mathematics the gap is enormous: you can read a worked solution to a related-rates problem, feel every line click into place, and be unable to produce the first line an hour later. Every technique below denies you that comfortable feeling.

Dunlosky and colleagues reviewed ten study techniques in 2013 and graded each on how reliably its benefits generalise across ages, abilities, materials, and outcome measures. Two came out high utility: practice testing and distributed practice. Three were moderate: elaborative interrogation, self-explanation, and interleaved practice. Five were rated low, and that list is worth memorising because it is most of what students actually do — summarisation, highlighting, the keyword mnemonic, imagery for text, and re-reading.

The protocol:

• Close everything. Notes, textbook, phone, the worked solution you were about to peek at.
• Set a timer for the length the question would get in the real test. A five-mark question gets about six minutes.
• Write the full solution as if it were being marked, including the lines you would normally do in your head. Those lines are where marks live, which is covered in how markers award method marks.
• Only then check, and note the first line where yours diverged — not the final answer, the first divergence.

Retrieval pays twice, and the second payment is the one people miss. The first is the memory effect: pulling something out of your head strengthens the route back to it in a way that putting it in again does not. The second is diagnostic — a failed retrieval tells you immediately that you did not know something you believed you knew. Re-reading cannot deliver that message, because everything on the page looks familiar by construction.

A second form most students never try: reconstruct the topic rather than a problem. Take a blank sheet, write "integration by parts" at the top, and produce from memory the formula, the rule for choosing $u$, one example that works in a single pass, and one that needs two. Then check against the guide. The holes are a precise revision list, generated in four minutes.

One caution, because "test yourself" is often done badly. Looking at a problem, thinking "yes, I know how to do that", and moving on is not retrieval. Neither is checking the solution after ninety seconds of feeling stuck. Retrieval requires that you write the answer out to the end and be wrong if you are wrong. The version that counts is the one where failure is possible.

Distributed practice was the other high-utility technique in the Dunlosky review. Each spaced session begins with partial forgetting, and reloading that material is the work that makes it durable. Cramming eliminates the forgetting, which feels efficient and removes the mechanism.

Be honest about the exception. Cramming produces a real short-term gain — that is the 83% in the five-minute test above. If your test is in eleven hours, cram. The problem is that the gain has a half-life measured in days, and the material will be back in front of you in the final exam.

A spacing rule that does not require software: revisit a topic one day after you learn it, then three days later, then a week later, then before the test. Four touches, each shorter than the last.

Open any textbook exercise set. Section 4.3 introduces the Pythagorean theorem and then gives you twelve problems, all solved with the Pythagorean theorem. You know the method before you read the question. This is blocked practice, and it quietly removes the hardest step in the subject.

Rohrer, Dedrick and Stershic tested this directly with 126 seventh-grade students over three months. Both groups received exactly the same practice problems; only the arrangement differed. One day after a review session, the interleaved group scored 80% against 64% for the blocked group. Thirty days after, the gap widened: 74% against 42%. Same problems, same total practice, a thirty-two point difference a month later, purely from the order they appeared in.

The explanation is not mysterious. In a test, nobody tells you which method a question needs, and choosing correctly is most of the difficulty. Blocked practice never rehearses that choice; interleaved practice rehearses nothing else. It also feels worse — accuracy during practice drops, and students rate the blocked version as more effective while performing worse on it later.

You can see the difference on one pair of equations. Given $x^2-5x+6=0$ in a factoring exercise, you factor, because the heading said so. Given the same equation next to $x^2-5x+3=0$ in a mixed set, you have to notice that the first has an integer factor pair and the second does not, and pick a different route for each. The second situation is the exam.

How to interleave without rebuilding your textbook:

• Note each problem you want to practise on a slip of paper. Shuffle before each session.
• Take one problem from each of the last four topics rather than four from the current one — including the topic before last, which is the one everyone stops practising exactly when it starts to fade.
• Write the method you are going to use before you start solving. If you cannot name it, that is the gap, and solving the problem will not close it.

Keep a running record of every mistake, with four columns: the problem, what you wrote, what was correct, and — the only column that matters — the name of the underlying error.

"Got question 6 wrong" is not actionable. "I multiply out a squared binomial as if the exponent distributes" is a nameable defect that takes ten minutes to fix and was probably costing you marks on four question types. Most students discover, after two weeks of logging, that their entire error profile reduces to five or six recurring faults. Ours is a list of twelve algebra errors; your personal list will be shorter.

Then use it. Every session, start by re-solving one problem from the log at random, from a blank page. If you get it right twice on separate days, retire it. That is retrieval, spacing, and interleaving at once, driven by your own weaknesses rather than a textbook's chapter order. Here is a re-solve when the log entry says cleared denominators wrongly on a rational equation; forgot to check excluded values.

Note what step 1 does. Writing the excluded values before the algebra starts costs four seconds and makes the final check automatic rather than something you remember to do. Most durable fixes look like this: not more effort, but a step moved earlier.

Assume a test covering four topics and roughly seven hours available. Adjust the durations, not the structure.

1. Day 1 — diagnose (60 min). Take the past paper or the end-of-chapter review under exam timing, before revising anything. It will go badly. That is data, and it tells you where the other six hours should go. Log every error by name.
2. Day 2 — attack the weakest topic (90 min). Work the topic with the most log entries. Read the method once, then close it and produce three problems from a blank page. Finish by writing the method out from memory in your own words.
3. Day 3 — second-weakest topic, plus five minutes on day 2 (90 min). Open by re-solving one problem from yesterday cold. This is the first spaced repetition, and skipping it is the most common way this plan fails.
4. Day 4 — rest, or fifteen minutes of formula recall. Write the formula sheet from memory, check it, mark the gaps. Nothing else.
5. Day 5 — interleaved set (90 min). Twelve problems, shuffled, three from each topic, method named before each one is started. Expect this to feel harder than day 2 and day 3. It is doing more.
6. Day 6 — second full paper under timing (75 min). Compare against day 1 and log only the errors that are new. The ones that reappear are the real problem and get tomorrow.
7. Day 7 — targeted repair and stop early (60 min). Work only the errors that survived day 6. Then stop. An extra hour on the last evening buys very little and costs sleep, which measurably matters more.

The two days people cut are day 1 and day 4. Day 1 feels pointless, and it is the step that makes the other six days targeted rather than generic. Day 4 feels lazy, and it is the spacing interval that makes day 5 work.

Do not learn anything new. A topic met for the first time at 10pm will not be available at 9am, and the anxiety it generates degrades your access to what you do know. Accept the marks you will lose and protect the rest.

Do a short, easy retrieval set. Six problems you can already do, from a blank page, twenty minutes. This is not learning; it is confirming that retrieval works.

Sleep the normal amount. The trade is between an extra hour of low-quality revision and an hour of sleep that consolidates the previous six days. It is not close.

In the room, read the whole paper before writing anything, and start with the question you are most confident about rather than question 1. Write down the lines you would normally skip — the substitution, the rearrangement, the formula before the numbers go into it — because that is where partial credit lives when something goes wrong later in the question.

Six problems from six different topics, in no useful order. Name the method before you start each one. If naming the method is the hard part, that is the finding.

When one goes wrong, compare your working against a full solution line by line and find the first divergence. That is what the algebra solver is for, and our position on using a solver without crossing into copying is worth reading before you make it part of your routine.