Key takeaways
- Discrete math answers stand or fall on exact definitions and complete cases, so make definitions the first thing you revise.
- Test yourself rather than rereading: practice testing and spreading study over days are the two techniques a major review rated most useful.
- Practise proofs from a blank page, then compare with a model answer line by line.
- Mix problem types in later sessions so you practise recognising which method a question needs.
- Sit at least one full mock exam under timed conditions about a week out, and spend the remaining days on what it exposed.
A discrete math exam rarely asks you to calculate something hard. It asks you to apply a definition exactly, count without double-counting, and write an argument that holds for every case. That changes how you should study. Rereading notes feels productive, but the review described below rated it low utility; testing yourself on definitions and writing proofs from scratch is uncomfortable, and practice testing was rated high utility.
What the research says about studying
A widely cited review of study techniques, by John Dunlosky and colleagues in Psychological Science in the Public Interest (2013), evaluated ten common methods. Two received its high utility rating: practice testing and distributed practice, because they benefit learners of different ages and abilities across many kinds of test, including in real classrooms. Elaborative interrogation, self-explanation and interleaved practice were rated moderate, promising but with more limited evidence.
For discrete math, that translates into three habits:
- Retrieve, don't reread. Close the notes and write the definition, formula or proof from memory, then check.
- Spread it out. Plan several short sessions over two weeks rather than the same hours crammed into two days.
- Mix topics later on. Once you know each method, practise sets where you must decide which method applies; our guide to interleaving versus blocked practice explains why.
Week one: definitions and core methods
Days 1–2: list and learn the definitions. Go through each topic on the syllabus and write down every definition, word for word: injective, surjective, equivalence relation, partial order, congruence, tree, planar, and so on. Then test yourself: cover the definition and write it out. A one-page cheat sheet is a good starting list, but writing your own is the actual revision.
Why definitions first? Because a proof is graded on whether each step follows from them. "f is injective because each input has one output" loses the marks; that is the definition of a function. The correct statement, f(a) = f(b) implies a = b, gets them.
Days 3–5: one method per day. Work through problems topic by topic: truth tables and equivalences, proof techniques, counting, number theory, graphs. For each problem, attempt it fully before looking at any solution. If you are stuck for more than ten minutes, read only the first line of the solution and try again.
Days 6–7: proof practice from a blank page. Pick five standard proofs from the course, such as √2 irrational, a sum formula by induction or the handshake lemma. For each, write it from a blank page, then compare it line by line with the model. Note exactly where yours went wrong: a missing definition, a step that does not follow, a case left out.
Week two: mixed practice and a mock exam
Day 8: a full mock exam. Use a past paper or a practice exam, in exam conditions: timed, closed book, no phone. It is the most informative step in this plan, because it tests recall, method choice and speed together.
Days 9–12: fix what the mock exposed. Sort every lost mark into a cause:
| Cause | Fix |
|---|---|
| Did not know a definition | Rewrite it from memory daily until exact |
| Chose the wrong method | Mixed practice sets; for each, name the method before solving |
| Logic error in a proof | Rewrite the proof; check each step cites a definition or earlier result |
| Counting error | Check small cases by listing them by hand |
| Arithmetic slip | Slow down; check the final answer against a quick estimate |
The last row matters more than it seems. Our guide to accuracy before speed explains why rushing produces exactly the slips that cost easy marks.
Day 13: a second, shorter mock, focusing on the weakest topics.
Day 14: light review only. Definitions and the cheat sheet. No new material.
Proof-writing habits that save marks
- State what you are proving and how. "We prove the contrapositive" or "by induction on n" at the top tells the marker your plan.
- Start from definitions. Write out what the hypothesis and conclusion mean in symbols. The direct proof guide gives a template.
- Use arbitrary elements. "Let n be any integer", never "let n = 5".
- Check small cases first, in scratch work, to see why the claim is true before writing the proof.
- End with the conclusion in words, matching the statement you were asked to prove.
Levin's Discrete Mathematics: An Open Introduction is a good companion for this stage: its section on proofs works through direct proof, proof by contrapositive, contradiction, (counter)example and cases, followed by exercises.
In the exam room
- Read the whole paper first and do the questions you are surest of.
- For proofs, write the setup even if you cannot finish: the definitions and structure often earn partial marks.
- For counting answers, sanity-check with a tiny case you can list.
- Leave five minutes to reread for missing cases and unjustified steps.
Using the app for this plan
Discrete Math: Study & Quiz fits the plan's steps: topic quizzes for week one, the review of missed questions for days 9–12, the custom homework mode to build a set around the topics your exam covers, and full-length practice exams for the mock. Its progress screen shows mastery by unit, which helps decide where the second week goes. The app guide explains each mode. MIT's free 6.042J readings on OpenCourseWare are another source of problems at university level.
It is listed with our education and brain-training apps, and every topic guide in this series is in the discrete math hub.
Good to know
Frequently asked questions
How should I study for a discrete math exam?
How do I get better at writing proofs?
What should I memorise for a discrete math exam?
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Sources
- Improving Students' Learning With Effective Learning Techniques (Dunlosky et al., Psychol Sci Public Interest 2013;14(1):4–58) (opens in a new tab)PubMed, U.S. National Library of Medicine — accessed
- 6.042J Mathematics for Computer Science, Spring 2015: Readings (opens in a new tab)MIT OpenCourseWare — accessed
- Proofs (Discrete Mathematics: An Open Introduction, 3rd edition, Oscar Levin) (opens in a new tab)Open Math Books — accessed
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