STEP, TMUA and MAT preparation at maths schools
Several UK universities set additional maths-admissions tests for competitive courses, sat well after a maths school's own entrance process — usually in Year 13, as part of applying to university. Maths schools are, by design, well placed to help prepare for these, given their specialist staff and maths-heavy timetables.
STEP (Sixth Term Examination Paper)
STEP is used by the University of Cambridge (and occasionally referenced by a small number of other university maths departments) as part of conditional offers for mathematics-related courses. It goes well beyond standard A-level in difficulty and format, rewarding depth of understanding and unconventional problem-solving over speed or rote method.
TMUA (Test of Mathematics for University Admission)
The TMUA is used by a number of universities — historically including Warwick, LSE, Durham and UCL for some courses, though which universities require it changes year to year — as part of admission to competitive maths, economics and related degrees. It tests mathematical thinking and logical reasoning applied to unfamiliar problems, rather than further content beyond the A-level syllabus.
MAT (Mathematics Admissions Test)
The MAT is used by the University of Oxford (and, in the past, by Imperial College London for some courses) for admission to maths, computer science and joint courses. Like STEP and the TMUA, it's a reasoning-heavy test rather than a content-heavy one.
Which universities require which test changes from year to year, and not every maths school student will need any of them — it depends entirely on where you apply. Always check the current requirement directly on the university admissions page for the specific course, not from a general guide like this one.
What to realistically expect from a school
Specialist staff, small class sizes, and a maths-first timetable all help — but exactly how much dedicated STEP/TMUA/MAT preparation is built into the timetable varies by school and by year. If this is a priority for you, it's a reasonable and specific question to ask at open days or interview, rather than something to assume comes automatically with "maths school" on the prospectus.