TMUA, ESAT, and STEP Tuition
Specialist one-to-one preparation for competitive university Maths admissions tests
Harry Williams offers individual online preparation for the Test of Mathematics for University Admission (TMUA) and Engineering and Science Admissions Test (ESAT) Mathematics 1 and Mathematics 2.
Harry is a fully qualified Maths teacher and former Head of Mathematics at Harrow School Online and Highgrove Education, with regular experience teaching students preparing for these tests.
A strong A Level or International A Level grade is an excellent foundation, but admissions-test questions can require a different approach: applying familiar ideas in unfamiliar settings, recognising efficient methods, reasoning accurately and managing severe time pressure.
Harry's tutorials develop these skills through challenging questions, discussion of alternative approaches, timed practice and targeted work between lessons. He can help students prepare for the specific Mathematics components required by their chosen university course.
How TMUA and ESAT tuition works
Harry works through the student's existing mathematical knowledge and the demands of the chosen examination. He identifies whether a mistake arose from missing knowledge, an inefficient approach, inaccurate working, difficulty interpreting the question or time pressure.
Tuition may include:
Mathematical reasoning and unfamiliar problem-solving.
Efficient approaches to multiple-choice questions.
Timed practice and decisions about when to move on.
Discussion of why an approach works and when a different method would be better.
Harry sets homework after every tutorial and personally marks work submitted by students. Model solutions may also be provided for additional self-review, with any outstanding questions discussed in the following tutorial.
All Maths tutorials last 55 minutes and are scheduled in one-hour slots. Prices are per tutorial. Lessons take place online, one-to-one with Harry. The emphasis is on developing the mathematical thinking required for the test, rather than memorising a collection of tricks.
Tutorials with Harry: Specialist teaching for high-attaining Maths students
Harry is a fully qualified Maths teacher with Qualified Teacher Status (QTS) and a PGCE in Mathematics from the University of Reading.
He has more than ten years’ experience teaching Mathematics and Further Mathematics, including at Abingdon School and as Head of Mathematics at Harrow School Online and Highgrove Education.
Harry holds a First-Class Master’s degree in Mathematics from the University of Sussex and has extensive experience teaching students working at the highest levels of school Mathematics.
He is also a current Pearson Edexcel examiner for International A Level Mathematics and Further Mathematics.
His admissions-test tuition focuses particularly on:
Recognising the mathematical structure behind unfamiliar problems
Developing flexible problem-solving approaches
Selecting efficient methods
Mathematical reasoning and proof
Working accurately under time pressure
Learning from unsuccessful approaches rather than simply being shown a solution
Developing the independence needed for university-level mathematical study
TMUA Tuition: Test of Mathematics for University Admission
Harry offers specialist one-to-one preparation for the TMUA.
The TMUA assesses mathematical knowledge, application, and reasoning rather than simply testing whether a student can reproduce standard A Level methods.
The test consists of two papers:
Paper 1 – Applications of Mathematical Knowledge
Paper 1 assesses a student’s ability to apply mathematical knowledge in unfamiliar situations.
Students answer 20 multiple-choice questions in 75 minutes.
Paper 2 – Mathematical Reasoning
Paper 2 assesses mathematical reasoning and includes ideas from elementary logic.
Students answer 20 multiple-choice questions in 75 minutes.
Calculators are not permitted.
TMUA preparation with Harry can focus on:
Applying familiar mathematics in unfamiliar contexts
Identifying efficient solution routes
Mathematical reasoning and logic
Multiple-choice strategy
Eliminating implausible answers
Working accurately without a calculator
Managing the considerable time pressure of the test
Timed practice using official and appropriate preparation materials
Harry can tailor the programme to the student’s existing mathematical strengths, target universities, and time remaining before the examination.
ESAT Mathematics Tuition: Engineering and Science Admissions Test
Harry offers tuition for the mathematical components of the ESAT, specifically:
Mathematics 1
Mathematics 2
Harry does not provide tuition for the Biology, Chemistry, or Physics modules.
ESAT Mathematics 1
Mathematics 1 is taken by all ESAT candidates.
The module contains 27 multiple-choice questions to be completed in 40 minutes.
The combination of relatively short questions and significant time pressure means that students need both secure mathematical foundations and the ability to recognise efficient methods quickly.
ESAT Mathematics 2
Mathematics 2 is required for some mathematically demanding courses.
It also consists of 27 multiple-choice questions in 40 minutes.
Students should check the current requirements for their specific university and course before choosing their ESAT modules.
Preparing for ESAT Mathematics
Tuition can focus on:
Identifying what a question is really testing
Applying school mathematics in less familiar contexts
Finding efficient approaches under time pressure
Mental and written mathematical accuracy
Multiple-choice strategy
Knowing when to move on from a difficult question
Timed practice
Reviewing errors and developing more reliable approaches
Because ESAT module requirements differ by university and course, families should confirm exactly which modules the student is required to take.
STEP Tuition: STEP 2 and STEP 3
Harry has experience supporting students with STEP 2 and STEP 3. These papers require extended mathematical reasoning and clearly constructed solutions, and the best approach depends on a student's existing Further Maths knowledge, timescale and goals.
STEP is substantially different in style from TMUA and ESAT. Rather than short multiple-choice questions, STEP requires students to engage with long, demanding mathematical problems and construct complete mathematical solutions.
It is designed to assess mathematical reasoning, problem-solving, and the ability to work with unfamiliar material at a level closer to undergraduate Mathematics.
STEP 2
Harry can support students preparing for STEP 2, including work on advanced problem-solving, selecting questions strategically, constructing rigorous solutions, and developing the persistence needed for longer mathematical problems.
STEP 3
Harry also offers preparation for STEP 3, which assumes a broader mathematical background and is particularly relevant to students studying Further Mathematics.
Preparing for STEP
STEP tuition can focus on:
Tackling unfamiliar and extended mathematical problems
Breaking difficult problems into manageable stages
Choosing productive starting points
Mathematical proof and reasoning
Writing complete and coherent solutions
Learning when an approach should be abandoned or adapted
Selecting questions strategically in the examination
Developing stamina for extended mathematical problem-solving
Reviewing past solutions in detail rather than simply checking final answers
STEP preparation is normally most effective when students begin sufficiently early to develop their problem-solving skills gradually rather than treating the examination as a short revision exercise.
Which Maths Admissions Test Do I Need?
The required admissions test depends on the university, course, and year of application.
For example, TMUA is currently used for a number of Mathematics, Computer Science, Economics, and related courses, while ESAT is used for selected Engineering, Physics, Natural Sciences, and other science-based courses.
STEP continues to be used for some highly mathematical university courses and offers.
Admissions-test requirements can change from year to year.
Students should therefore always check the current requirements directly on the official university course page before booking preparation.
Harry can help students understand the mathematical demands of the relevant test, but responsibility for confirming the correct admissions test, registration requirements, and deadlines remains with the student and family.
Oxford Maths Admissions: TMUA Has Replaced MAT
Students applying for Oxford Mathematics and related courses should be aware that Oxford now uses the TMUA rather than the former Mathematics Admissions Test (MAT).
The MAT is no longer used for the current admissions cycle.
Students who have previously worked through MAT material may still find some of the mathematical problem-solving useful, but preparation should now be aligned to the current TMUA specification and format.
Preparing Alongside A Level or International A Level Maths
Admissions-test preparation is separate from normal A Level or International A Level tuition.
However, the two can complement each other.
Students may need to strengthen particular areas of their school Mathematics course while simultaneously developing the much less familiar problem-solving and reasoning skills required for an admissions test.
Harry also offers separate tuition for:
Where a student is receiving both curriculum and admissions-test tuition, the programme can be planned around their school workload, examination timetable, university application, and admissions-test date.
Testimonials and Google Reviews
“I wanted to improve the way I approached ESAT problems. Harry gave incredibly clear explanations and really helpful advice for dealing with the challenging time pressure of the test. Working with him changed the way I approach difficult problems and helped me spot the most time-efficient way of solving them. Harry’s help is 100% worth it. Even after my first session, I could already see ESAT questions differently and felt much more confident.”
These are a selection of our most recent Google reviews. Read them all on our Google profile.
Frequently Asked Questions
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Yes. Harry offers specialist one-to-one preparation for both papers of the Test of Mathematics for University Admission (TMUA):
Paper 1 – Applications of Mathematical Knowledge: 20 multiple-choice questions in 75 minutes.
Paper 2 – Mathematical Reasoning: 20 multiple-choice questions in 75 minutes, including elementary logic and mathematical reasoning.
Calculators are not permitted. The test rewards students who can apply familiar mathematics to unfamiliar problems, recognise efficient solutions, and reason accurately under time pressure.
Harry's tuition develops problem-solving strategies, mathematical reasoning, multiple-choice techniques, and examination time management, supported by timed practice and personally marked work.
The programme is tailored to the student's current mathematical level, intended university course, and examination date.
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Yes. Harry provides specialist preparation for the two mathematical components of the Engineering and Science Admissions Test (ESAT):
Mathematics 1: 27 multiple-choice questions in 40 minutes. This module is compulsory for all ESAT candidates.
Mathematics 2: 27 multiple-choice questions in 40 minutes. This module is required for certain courses.
Harry teaches Mathematics 1 and Mathematics 2, but not the ESAT Biology, Chemistry, or Physics modules.
Tuition focuses on efficient mathematical methods, unfamiliar applications, numerical accuracy, and managing demanding time constraints.
The required ESAT modules depend on the student's university and course. Some candidates must take Mathematics 1 alongside two additional modules, which may include Mathematics 2 or science subjects.
Students should check their specific course requirements before registering.
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Yes. Harry offers one-to-one preparation for STEP 2 and STEP 3, focusing on the extended mathematical reasoning and problem-solving skills these examinations require.
Unlike the multiple-choice TMUA and ESAT, each STEP paper lasts three hours and contains twelve demanding questions, with the final mark determined by the student's best six answers.
STEP 2 draws on A Level Mathematics and AS Further Mathematics knowledge.
STEP 3 draws on A Level Mathematics and the broader A Level Further Mathematics syllabus.
Harry helps students develop rigorous mathematical arguments, complete written solutions, effective question selection, and persistence with unfamiliar problems.
Tuition can include detailed past-paper analysis, proof, strategic use of examination time, and personally marked solutions.
STEP is priced separately from TMUA and ESAT tuition, reflecting the specialist nature of the preparation.
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Yes. Harry provides preparation for the mathematical admissions tests used by Oxford and Cambridge, where they fall within the tests he teaches.
For the 2027 entry cycle:
Oxford Mathematics and Computer Science: Applicants take the TMUA, which has replaced the former MAT.
Cambridge Mathematics: Applicants take the TMUA, and STEP 2 and/or STEP 3 may subsequently form part of a conditional offer.
Cambridge Computer Science and Economics: Applicants take the TMUA.
Cambridge Engineering and certain science courses: Applicants take the ESAT, with required modules depending on the course.
Admissions requirements vary by university, degree, and entry year. Students must verify their specific requirements and registration deadlines directly with the university.
Harry can help students prepare for the mathematical demands of TMUA, ESAT Mathematics 1 and 2, and STEP 2 and 3, but does not provide preparation for the ESAT science modules.
Oxford's admissions-test requirements | Cambridge's admissions-test guidance
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Oxford has replaced the Mathematics Admissions Test (MAT) with the TMUA for Mathematics and Computer Science applicants from the 2026 admissions round.
Harry's current Oxford Mathematics admissions preparation therefore focuses on TMUA Papers 1 and 2, rather than the former MAT examination.
Older MAT questions can still be useful for developing mathematical reasoning and unfamiliar problem-solving skills. However, preparation should be aligned with the current TMUA specification, multiple-choice format, and time constraints.
Students applying to Oxford should confirm the current requirements for their chosen course.
Read Oxford's official explanation of the MAT-to-TMUA change.
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Starting several months before the examination is generally preferable to relying on a short period of intensive revision, particularly for students who have limited experience with unfamiliar mathematical problems.
The appropriate preparation period depends on the student's existing knowledge, mathematical confidence, school workload, target university, and chosen test.
For TMUA and ESAT, students need time to develop efficient problem-solving strategies and practise under demanding time constraints.
STEP generally benefits from sustained preparation, allowing students to develop deeper reasoning, longer written solutions, and confidence with challenging questions.
Harry can assess a student's current level during the first tutorial and recommend suitable preparation priorities for the time available.
Test dates and registration deadlines vary by admissions cycle.
Check current TMUA and ESAT deadlines | Cambridge STEP guidance
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Yes. Developing speed without sacrificing mathematical accuracy is a central part of Harry's TMUA and ESAT preparation.
Students may understand the mathematics but lose marks because they pursue inefficient methods, spend too long on difficult questions, or struggle to identify a productive starting point.
Harry uses timed practice and detailed analysis of the student's approaches to improve:
Recognition of efficient solution methods.
Decisions about when to persist and when to move on.
Multiple-choice strategy and elimination techniques.
Mathematical accuracy under pressure.
Confidence when facing unfamiliar problems.
For STEP, the emphasis also includes question selection, pacing across longer solutions, and developing the stamina needed for a three-hour examination.
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No. Admissions-test tuition is suitable for mathematically capable students who want to develop the reasoning and problem-solving skills required by their chosen examination, even if they are not yet consistently achieving A* grades.
A strong school Mathematics grade provides a useful foundation, but admissions tests assess skills that are not always developed through ordinary A Level or IAL examination preparation.
Some students need to strengthen particular mathematical topics, while others need help with unfamiliar questions, mathematical reasoning, efficient approaches, or time management.
Harry can assess the student's current level during the first tutorial and identify the areas requiring the most development.
Tuition does not guarantee a particular admissions-test score or university offer.
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Yes. Harry is a fully qualified Mathematics teacher with a PGCE in Mathematics from the University of Reading and Qualified Teacher Status (QTS).
He has more than ten years' teaching experience, including teaching at Abingdon School and serving as Head of Mathematics at Harrow School Online and Highgrove Education.
Harry also holds a First-Class Master's degree in Mathematics from the University of Sussex and is a current Pearson Edexcel examiner for International A Level Mathematics and Further Mathematics.
His official live-paper marking to date has been in IAL Pure Mathematics 1 (P1).
Harry's admissions-test tuition combines advanced mathematical knowledge, extensive teaching experience, and a structured approach to unfamiliar problem-solving.
His Pearson Edexcel examining appointments are separate from the administration and marking of TMUA, ESAT, and STEP.
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Yes. Harry provides live, one-to-one online TMUA, ESAT Mathematics, and STEP preparation to students in the UK and internationally.
His teaching experience includes students in Italy, Slovakia, Spain, France, Poland, the UAE, Hong Kong, and Zimbabwe. Tuition is also available to students in mainland China, including Shanghai, and other countries.
Harry works with students attending independent schools, international schools, and other educational settings, including those studying for UK university entrance from overseas.
Lessons use Google Meet and a shared Miro whiteboard, with appointment times arranged around different time zones and school commitments, subject to availability.
Students taking TMUA or ESAT internationally should consult the official test provider for registration arrangements and approved examination centres.
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Yes. Harry can provide both specialist university admissions-test preparation and tuition for A Level or International A Level Mathematics.
The two forms of tuition develop related but distinct skills.
A Level and IAL tuition focuses on mastering the examination specification, strengthening mathematical understanding, and achieving strong school qualification results.
Admissions-test preparation focuses on unfamiliar problems, mathematical reasoning, efficient methods, and performance under examination conditions.
Harry can plan both strands around the student's school workload, examination timetable, university application, and admissions-test date.
He also offers separate tuition for IAL Maths, IAL Further Maths, UK A Level Maths, and UK A Level Further Maths Core Pure.
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Yes. Harry sets homework after every tutorial and personally marks work submitted by students.
Homework is tailored to the student's chosen admissions test, current mathematical level, and areas for improvement.
This may include TMUA or ESAT multiple-choice problems, timed practice, mathematical reasoning exercises, or extended STEP solutions.
Harry's personal marking helps identify inefficient methods, weaknesses in mathematical arguments, avoidable errors, and difficulties with unfamiliar problems.
The feedback informs subsequent tutorials, allowing preparation to focus on the areas where improvement is most needed.
Model solutions may also be provided for additional self-review, but they are not a substitute for Harry's personal marking.
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Yes. Harry holds an Enhanced Disclosure and Barring Service (DBS) certificate and is registered with the DBS Update Service.
A copy of his DBS certificate is automatically included in his first communication with parents, before tuition begins. There is no need to request it.
The DBS Update Service also allows eligible checks of his certificate's status, with his consent and in accordance with DBS requirements.
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Harry generally offers one-to-one Maths tutorials seven days a week, between 9 am and 10 pm UK time, although specific availability varies according to existing bookings.
Students in the UK and internationally can check the live Maths admissions-test booking calendar for available appointments.
TMUA, ESAT Mathematics, and STEP have separate tuition options and prices, so students should select the appropriate category when booking.
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All tutorials are live, one-to-one lessons lasting 55 minutes, scheduled in one-hour appointment slots.
TMUA and ESAT Mathematics 1 or 2:
Individual tutorial: £100
Ten-tutorial package: £800 (£80 per tutorial)
STEP 2 and STEP 3:
Individual tutorial: £140
Ten-tutorial package: £1,120 (£112 per tutorial)
All ten-tutorial packages are valid for 100 days from purchase.
New Maths clients can use FIRSTCLASS for 20% off their first individual tutorial, reducing the eligible TMUA or ESAT tutorial to £80, or STEP tutorial to £112.
FIRSTCLASS is limited to one use per client and excludes tutorial packages and Biology tuition.
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You can book your first one-to-one TMUA, ESAT Mathematics, or STEP tutorial directly through our Maths admissions-test booking page.
Each booking reserves a one-hour appointment slot, including a 55-minute tutorial. Choose the appropriate tuition option, select an available time, and complete your booking.
Students should confirm which admissions test and modules their university course requires before booking tuition.
Ready to begin?
All Maths tutorials last 55 minutes and are scheduled in one-hour slots.
Book a single one-to-one TMUA, ESAT Mathematics, or STEP preparation tutorial, or purchase one of our 10 lesson packages through the online scheduling system.