Cambridge and Warwick · STEP 2 and STEP 3 · Long-form proof
STEP Preparation
STEP is not a harder A-level paper. It is a different exercise: a small number of long questions, each building from a concrete case to a general result, where the marks are for the argument you write rather than the answer you reach.
Taught by Lucas Rey · Oxford DPhil candidate, Rolls-Royce sponsored · Cambridge MRes
Not sure where to start? 5 places to go
Start with your question
Why visitors arrive: Your Cambridge or Warwick offer depends on a STEP grade and you do not know how to prepare for a proof paper.
Your question: What does STEP actually reward, and how is preparing for it different from A-level?
You may also be asking
- Which STEP papers are actually sat?
- Are marks for the answer or the working?
- How do I choose which questions to attempt?
- How do I mark my own proof when there is no answer to check against?
- Are there free practice questions with full solutions?
Where to go next
- STEP preparation plan: What to do first, and when
- The STEP method library: Which shape of argument a question wants
- STEP 2 mock paper 1: Twelve questions with mark schemes and full solutions
- ESAT library: Hand-checked practice on overlapping pure content
- Enquire about one-to-one: Direct STEP preparation
- What it is: the Sixth Term Examination Paper, used by Cambridge and Warwick as part of a conditional offer for mathematics, and accepted by several other universities.
- Published here: no questions yet: the questions written for it are held for review before they ship.
- Also here: the ESAT library, 150 questions written and checked by hand, which is the deeper resource on this site.
- Dates, deadlines and which course needs what: these change between cycles, so check the official specification rather than any tutor's summary of it.
- One-to-one preparation: by application, limited places.
What the STEP actually tests
[01]
Few questions, long answers
A STEP question is long enough to be a session's work on its own, and you attempt only some of the paper. That makes choosing which questions to attempt a skill in itself, and one almost nobody practises.
[02]
The parts are a ladder
An early part almost always exists to make a later one reachable. If part three looks impossible, the intended route runs through what you proved in part one.
[03]
Marked on the argument
Examiners read the reasoning. An unjustified step loses marks even when it happens to lead somewhere true, and a well-argued partial attempt scores better than most candidates expect.
How to prepare for it
- Finish fewer questions, properly. A complete, well-written answer to two questions beats fragments of five. Most candidates lose more to abandoning questions than to being unable to do them.
- Read all the parts before starting. The last part usually tells you what the first part was for. Reading forwards changes what you do with the opening.
- Write as though someone will read it. Because they will. Say what you are doing and why, state where each assumption comes from, and do not leave the reader to reconstruct your reasoning.
- Learn to mark your own proof. The hardest part of preparing alone is that there is no answer to check against, and the model solution is only one correct write-up out of many. Mark against what a solution has to establish instead, which is what the mark schemes on this site are for.
Practice material on this site
The STEP papers are being rebuilt on the same machine that produces the ESAT material, so they will be measured against the official material, mapped to the specification question by question, gated before they ship and read by a person, exactly as those are. Nothing is published here until they are. The ESAT papers and worked solutions are free and complete today, and are the same writing these will be.
Alongside the papers
- The STEP method library Nineteen shapes of argument, and what the marks are actually for
- The preparation plan Dated, counting back from the sitting, with a check at the end of each phase
- Mark your own attempt One question, then your own write-up scored against the mark scheme
Found an error in one of them? Tell me and it will be fixed or withdrawn.
What the practice material covers
Counted from the questions themselves, not transcribed from a specification. For what is actually examinable in your cycle, check the official specification: it is revised between cycles and no tutor's summary of it should be trusted over the source.
| Topic | Questions |
|---|---|
| Algebra and inequalities | 2 |
| Sequences and series | 2 |
| Calculus | 1 |
| Discrete probability | 1 |
| Expectation and random variables | 1 |
| Forces and momentum | 1 |
| Functions and graphs | 1 |
| Number theory | 1 |
| Polynomials | 1 |
| Trigonometry | 1 |
12 questions across 10 topics. Thin coverage of a topic here does not mean it is unimportant in the real test; it means this practice set is small.
Common questions
- Which STEP papers are actually sat? STEP 2 and STEP 3. STEP 1 was discontinued in 2020, so a resource built around it is modelling a paper that no longer exists. Everything published here is written against STEP 2 or STEP 3 for that reason, and the papers listed above are the ones available today. Which one your offer names is in the offer, and the official specification is the place to check what each assumes.
- Is STEP just a harder A-level paper? No, and treating it as one is the common mistake. STEP asks for a small number of long questions where the marks are for the argument you write. A complete, well-written answer to two questions beats fragments of five.
- How do I choose which questions to attempt? Read all the parts of a question before starting, not just the first. The last part usually reveals what the first part was for, and a question that looks impossible at part three is often reachable through what you proved at part one.
- Are marks for the answer or the working? The working. An unjustified step loses marks even when it leads somewhere true, and a well-argued partial attempt scores better than most candidates expect.
- How do I mark my own proof when there is no answer to check against? Against what a solution has to establish, rather than against how someone else phrased it: three correct write-ups of one proposition are three different pieces of prose. Every question here publishes a written mark scheme naming the intermediate results a solution must reach, and there is a page that walks you through scoring one of your own attempts against it.
- Are there practice papers on this site for STEP? Not at the moment. They are being rebuilt on the machine that produces the ESAT papers, and nothing is published here until that is done. The official past papers remain the final word on the real paper either way.
Sitting more than one of these?
Candidates often sit two admissions tests in the same cycle, and the overlap is larger than it looks: the pure mathematics is shared, and the habits that earn marks in one paper usually earn them in another. Each page below covers what its test rewards and what free practice exists for it.
- TMUA: Test of Mathematics for University Admission Two papers and Multiple choice and Pure mathematics
- MAT: Mathematics Admissions Test Retired after 2025 and Oxford now uses the TMUA
- PAT: Physics Aptitude Test Retired after October 2025 and Oxford now uses the ESAT
- ESAT: Engineering and Science Admissions Test The deepest library on this site, written and checked by hand
Part of
This page sits within the broader knowledge structure on lucasrey.com:
- Teaching - Engineering-informed teaching, mentoring, and quantitative reasoning.
Connected work
Related content from the same research and engineering work:
Where to go from here
Every paper, practice set and guide on this site is free to read. These are the things worth doing next.
- Find out where you actually stand - Fifteen questions, twenty minutes, scored instantly, with a worked solution for every one you miss
- Work a practice set at full paper depth - Ten sets of five questions, two in every module, each one worked in full
- See what can actually be checked about this tutor - What is published, counted from the pages themselves, and the credentials behind it
- Enquire about one-to-one preparation - For candidates who want the gap closed rather than mapped. Places are limited and taken by application
- Books for admissions test preparation - The ESAT Move Register, a book of shortcuts for the ESAT, with a preview of its pages
The diagnostic and the practice sets linked above are free and stay free. One-to-one places are limited and taken by application, not by the hour.