Mathematics Degrees

Further Mathematics

Studying A level Further Mathematics is excellent preparation for a degree in Mathematics. Many university mathematics departments encourage students to take Further Mathematics at A level as it introduces a wider range of pure and applied content, such as matrices and complex numbers. Students who have studied Further Mathematics often find the transition to university far more straightforward. Some leading universities now specify Further Mathematics as an entry requirement for their mathematics degrees.

33% of Mathematics BSc degree courses mention Further Mathematics in their entry requirements, including it in their A level offers or encouraging students to take it if possible. For those universities in the Russell Group, this proportion rises to approximately 60% (August 2016).

STEP and AEA Mathematics

In addition to A level grades some universities also require students to pass Sixth Term Examination Papers (STEP) or an Advanced Extension Award (AEA) in Mathematics. Currently the universities requiring STEP/AEA qualifications for their mathematics degrees are:

Some other universities encourage students to take these papers and may include STEP or AEA in their offers, for example:

Oxford University and Imperial College require students to sit the Mathematics Admissions Test (MAT). This is taken in November prior to selecting candidates to invite for interviews.

For further details on how to prepare for these examinations go to STEP, AEA and MAT courses.

Entry Requirements for Mathematics Degrees

Some universities explicitly encourage students to take A or AS Further Mathematics by:

  • making it a requirement for entry;
  • differentiating the grades in their offer for students with Further Mathematics qualifications;
  • offering financial support to students with Further Mathematics qualifications;
  • including encouraging statements about the benefits of studying A or AS level Further Mathematics.

Below are some examples of the sorts of statements you will find when looking at university websites and prospectuses.(Whilst we try to maintain up-to-date information about the entry requirements for mathematics degrees, we strongly recommend visiting the university's own website for the most recent information.)

Courses requiring Further Mathematics

Warwick University

The standard offers for applicants taking A levels are one of:

  • A* (Maths), A* (Further Maths), A (third subject), grade 2 STEP
  • A* (Maths), A (Further Maths), A* (third subject)
  • A* (maths), A* (Further Maths), A* (third subject)
  • A* (Maths), A* (Further Maths), A (third subject), A (fourth subject)
  • Distinction in AEA is accepted instead of grade 2 in a STEP paper. We do not mind in which of the three STEP papers you achieve the grade 1 or 2. Distinction in AEA is accepted instead of grade 2 in a STEP paper.

    We believe that prospective students can best prepare to study Mathematics at university by taking STEP or AEA papers in addition to their other school-leaving examinations. The questions on these papers are closer to the style of mathematical thinking you will meet at university, and will help you develop mathematically. We therefore recommend, but do not insist, that applicants should take AEA, or at least one STEP paper.

    If your school does not offer Further Maths or cannot help with STEP, the Further Maths Support Programme (FMSP) can help.

    Durham University

    Entry requirements are:

  • Suitable performance on the forthcoming University’s Admission Test* in addition to grades A* and A in Mathematics and Further Mathematics at A-level or equivalent (A* for either), together with grade A in a third A-level or equivalent.
  • Alternatively, grades A*A* in Mathematics and Further Mathematics at A-level or equivalent, together with grade A in a third A-level or equivalent
  • Alternatively, 1 in any STEP in addition to grades A* and A in Mathematics and Further Mathematics at A-level or equivalent (A* for either) together with grade A in a third A-level or equivalent
  • Alternatively, suitable performance on the University’s Admissions Test* in addition to grade A* in Mathematics and grade A in Further Mathematics at AS-level or equivalent, and two further A-levels at grade A.
  • *We strongly encourage applicants to sit the forthcoming University’s Admissions Test if it is available to them. For more information about this test visit the admissions page.

    University College, London (UCL)

    A*A*A, or A*AA and a 1 in any STEP paper or distinction in Mathematics AEA. Mathematics and Further Mathematics required at A*, or Mathematics at A* and Further Mathematics at A if STEP or AEA offered.

    Courses making differentiated offers

    University of York

  • AAB in three A levels, including A in Mathematics and A in Further Mathematics, or
  • AAA in three A levels, including Mathematics plus A in Further Mathematics AS level.
  • AAA in three A levels, including Mathematics
  • We will offer you an interview if you present with a strong school performance and application form. Although the interview is not part of your offer and you do not need to attend, if you do, your offer could be reduced by one A Level grade or equivalent.

    University of Bristol

    The usual entry requirements are:

  • A*A*A including A* in Mathematics, plus at least one of Physics, Chemistry, Biology, Economics or Computer Science, or
  • A*AA including A* in Mathematics and A in Further Mathematics
  • STEP achievement may be included as part of an alternative offer.

    Loughborough University

    AAA - AAB, including Mathematics at grade A. Applicants with any of the following will usually be given the lower offer (AAB): Further Mathematics, Physics, Chemistry, Computing or Economics; Further Mathematics AS-Level at grade A; AEA or STEP in Maths.
    If you want to study Further Maths but your school or college is unable to provide tuition, you can find tuition through the Further Mathematics Support Programme.

    University of Bath

    Typical Offers: A*AA

    Double Mathematics:
    For applicants taking both A-level Mathematics and Further Mathematics and STEP/AEA a typical offer is:

  • obtain grades A* in Mathematics, A in Further Mathematics and B in one further A-level subject plus at least 2/Merit in one STEP/AEA Mathematics paper.

  • If STEP/AEA is not taken a typical offer is:

  • obtain grades A* in A-level Mathematics, A in A-level Further Mathematics and A in one further A-level subject.

  • AS Further Mathematics:
    For applicants taking A-level Mathematics and AS Further Mathematics and STEP/AEA a typical offer is:

  • obtain grade A* in A-level Mathematics, grades A A in two further A-level subjects and grade A in AS Further Mathematics plus at least 2/Merit in one STEP/AEA Mathematics paper.

  • Single Mathematics:

  • obtain A* in A-level Mathematics with A's in each module, grades A* and A in two further A- level subjects plus at least 2/Merit in one STEP/AEA Mathematics paper. There should be a second scientific or quantitative subject preferably physics.

  • Further Information:
    Bath wishes to allow access to its courses in the mathematical sciences to applicants who are not able to take a full A level in Further Mathematics and will continue to make offers to exceptional candidates in this category. Accordingly we shall require such applicants to achieve at least Merit in Advanced Extension Mathematics or 2 in one STEP paper.

    Nevertheless we would like to encourage applicants to take Further Mathematics at A levels if possible, and to similarly encourage the taking of Advanced Extension papers or STEP papers for all applicants. There are two reasons for this. Firstly, having studied a wider mathematics curriculum and having attempted more demanding questions is an undoubted advantage in the transition to university mathematics. Secondly, it is very hard to assess an applicant's real mathematical potential unless they take the full A level Further Mathematics. In many cases an applicant's university place depends on their grade in Biology or Physics or English etc whereas we would prefer a better measure of mathematical ability.

    King’s College, London

    AAA including Mathematics and Further Mathematics. Further Mathematics will be accepted at AS Level grade A only if you additionally achieve a 3 in any STEP paper or a Merit in AEA Mathematics.
    The following are not required for entry, but you may find these useful further preparation for this degree:

  • study of mathematical mechanics and statistics
  • King’s also strongly encourages you to extend your mathematical interests by studying for the Advanced Extension Award or STEP
  • Performance in STEP or AEA will be taken into account should you narrowly fail to meet your A-level offer.

    Lancaster University

    Our minimum requirement for A Level applicants is that you should be studying at least three A Levels including A Level Mathematics. We encourage, but do not require, you to study A Level Further Mathematics as well - our standard offer is slightly lower for those taking A Level Further Mathematics.
    If it is not possible for you to study A Level Further Mathematics, think about studying AS Level Further Mathematics - the style of mathematics in Further Mathematics, especially Further Pure Mathematics, is similar to university mathematics and will be excellent preparation for further study.
    For the majority of our degrees, our offer will usually be as follows. Firstly, we require grade A or higher in either A Level Mathematics or A Level Further Mathematics. Secondly, we require that your grades from your best 3 A Levels satisfy one of the following conditions:

  • Your best 3 A Levels not including Further Mathematics, without Grade 3 or higher in any STEP, are AAA;
  • Your best 3 A Levels including Further Mathematics, without Grade 3 or higher in any STEP, are AAB;
  • Your best 3 A Levels not including Further Mathematics, with Grade 3 or higher in any STEP, are AAB;
  • Your best 3 A Levels including Further Mathematics, with Grade 3 or higher in any STEP, are ABB.

  • Leeds University

    For applicants taking A-levels, we always require at least Grade A in A-level Mathematics. If you are taking Further Mathematics at A-level or AS-level, we usually make an additional alternative offer.

    Typical offers are:

    • AAA or A*AB
    • AAB or A*BB, including Further Mathematics A-level
    • AAB or A*BB or A*AC, PLUS Grade A in AS-level Further Mathematics

    In all cases, the first grade quoted is the Mathematics A-level.

    University of Reading

    At the University of Reading we require candidates to have achieved at least a grade A at A level Mathematics. Our typical offer for our BSc Mathematics course is in the range AAB-ABB.

    If you are studying Further Mathematics at either A or AS level we will make an alternative offer of ABC with a grade A in A level in Mathematics and either a grade B in A level Further Mathematics or a grade A in AS level Further Mathematics.

    Courses offering scholarships and bursaries

    Plymouth University

    There is a £500 automatic scholarship for students with a grade A* in A Level Mathematics plus £500 for an A in Further Mathematics up to a total of £1,000. To be eligible for the scholarship, students must put us as their first choice before the 1st of August. The scholarship is paid in term one of the first year. There are additional prizes and certificates to reward high marks in later years.

    University of Essex

    We give you £250 if you are taking Further Mathematics A-level, and you

  • put Essex as your firm UCAS choice, and
  • obtain an A*-B grade in Further Mathematics.
  • This award is in addition to any scholarship and is not affected by the grades you achieve in your other A-level subjects.

    Applying for a Mathematics Degree

    Mathematics can be studied as a single honours degree or as a combined/joint honours degree in conjunction with another subject. Most single honours degree courses have codes starting G1, followed by two others numbers or letters, often G100.

    Before applying for a Mathematics degree course, look at the features of the course, for example:

    • How many modules are optional? The number of optional modules often increases in the second the third years of the course - sometimes all first year modules are compulsory.
    • Does the course include several areas of applied mathematics, for example, Statistics, Mechanics and Decision Mathematics (Operational Research) or does it specialise in one of these?
    • Are any of the modules assessed via coursework?
    • Do you know what each of the modules listed will involve? The FMSP have produced a brief overview of a typical first year undergraduate mathematics course, which provides exemplar resources that illustrate the different aspects of mathematics you are likely to study.
    • Is the course delivered entirely by lectures? Most universities also provide support via seminars, tutorials and additional examples classes which expand on the material covered in the lectures.

    There are a wide range of degrees which involve mathematics and these often vary, even when they have the same title e.g. ‘G100 Mathematics'. In addition, mathematics can be combined with many other subjects, such as:

    • Mathematics and Computer Science
    • Mathematics and Spanish/German/French
    • Mathematics and Economics
    • Mathematics and Physics
    • Mathematics and Education
    • Financial Mathematics
    • Mathematics and Music
    • Mathematical Biology

    It is also possible to study for a Mathematics degree that involves studying abroad for one year.

    Therefore, researching the content and structure of the degree course you plan to apply for is very important.

    Application and Interview

    All applications for degree courses are made via the Universities and Colleges Admissions Service (UCAS) website. The deadline for most courses is mid-January each year but the deadline for courses at Oxford or Cambridge, or in Medicine, Dentistry and Veterinary Science/Medicine is in mid-October (see UCAS website for exact dates). If you are thinking of applying for a medical degree and are planning to study Further Mathematics, see the latest guidance relating to University entry requirements.

    Most offers for degree courses in Mathematics are made without an interview. Of the 24 Russell Group universities, five interview applicants, with some of these interviews being informal, as shown in the table below:

    University Interview details

    The University of Cambridge

    Applications are made to a particular College within the university. All colleges have requirements beyond A level grades and there is normally at least one mathematical interview (usually two or three) of around 20-30 minutes each, potentially leading to a conditional offer involving STEP. Sometimes, the interview is based on previously prepared material or on work done under examination conditions just before the interview.

    Each college has its own particular approach which can be found in the Cambridge University Guide to Admissions in Mathematics. Some additional general guidance can be found on the University of Cambridge Interviews page.

    For further guidance, a video recording of a typical interview is available.

    The University of Manchester

    After considering applications, many applicants are invited to a Visiting Day which includes a mathematical talk, a tour and an interview which helps determine the conditional offer made.

    Oxford University

    Interviews are largely mathematical in nature. You may be asked to talk about an area of mathematics you have studied; look in detail at a point of technique, or curve sketching; you may be asked `puzzle' type questions; or you may be given a mathematical definition and asked to work out some of its consequences. Applicants will be interviewed at least twice by their first choice College and have at least one interview with another College. Applicants will also be invited to take the Mathematics Admissions Test (MAT).

    Interviews occur over a 3 day period in December are around 25 minutes each, mainly of a mathematical nature. The aim is for tutors to see how you think when you do Mathematics, and you may be asked to work at the board and talk through your thought processes.

    To help you prepare, there are some helpful videos from tutors. In addition, general interview advice, including eight sample portions of interviews of potential students is available, as well as some general sample questions. An Interviews Guide for Students, produced by Oxford University, also provides useful information.

    University College London (UCL)

    If your application is sufficiently strong you will be invited to visit the department for an applicant afternoon. Alternatively, some invitations are for an academic interview. You will also be able to talk to current students and staff and will be given a tour.

    The University of York

    All students who are made an offer are invited to attend a Visit Day between November and April, which includes a one-to-one chat with a member of academic staff.

    Interview Advice

    The following general links provide useful information when preparing for a Mathematics degree course interview:

    Typical Interview Questions

    A sample of some typical interview questions is given in these documents:

    Showing a general enjoyment of, and interest in, your chosen subject via wider reading would also be helpful in preparing for your interview. Some suggested texts are listed on the preparation for a mathematics degree page.

    Preparation for Mathematics

    If you are considering studying Mathematics at University the best preparation is taking A level Mathematics and Further Mathematics to at least AS level. This will introduce you to a breadth of topics that will be the foundation of your first year degree course.
    If you are unable to study AS or A level Further Mathematics at you school or college then the FMSP may be able to support you. Information on how the FMSP supports tuition in Further Mathematics can be found on our Studying Further Mathematics page.

    In addition to studying A level Mathematics and Further Mathematics you should practice and develop your problem-solving skills in mathematics. The STEP, AEA and MAT examinations are a good source of challenging problems which will help develop your mathematical thinking and are closer to the style of questions you will meet at university. Several mathematics departments encourage students to take one of these examinations as they highly value the skills students acquire from working on these papers.

    Suggested wider reading

    Three books that you may be interested in reading prior to applying for a Mathematics degree course are:

    • How to Study for a Mathematics Degree by Lara Alcock (ISBN 978-0-19-966132-9) - this book explains what to expect at university and contains a wealth of useful study advice.
    • Mathematics: A Very Short Introduction by Timothy Gowers (ISBN 978-0192853615) – this book gives an idea of the scope and spirit of mathematics but is written in an accessible style.
    • Number: A Very Short Introduction by Peter M. Higgins (ISBN 978-0199584055) - this book unravels the world of numbers, demonstrating its richness

    You might also like to read some popular mathematics books by authors such as Simon Singh and Ian Stewart.

    The University of Cambridge have a recommended reading list which includes a wide range of books including recreational mathematics and theoretical physics.

    Other sources of interesting and relevant reading include titles suggested by NRICH and the general interest articles relating to the beauty and practical uses of mathematics in Plus Magazine.

    You may also like to try the questions in Cambridge University's pre-course Mathematics workbook.

    Example first year undergraduate mathematics degree course

    The table below gives an indication of the areas of mathematics that are likely to be included in the first year of an undergraduate degree course. Click on the links in the table to find out more about these topics and try the sample tasks (answers are provided!).

    TopicSample Content
    Calculus is a branch of mathematics that involves differentiation and integration. Many of the concepts are based on consideration of the limit reached as points or intervals become infinitely small. It includes differential calculus (rates of change and gradients or curves) and integral calculus (area under and between curves), which are related by the Fundamental Theorem of Calculus.
    Basic calculus from A level
    Hyperbolic Functions
    Differential Equations
    Area under curves
    Fundamental Theorem of Calculus
    Volume and Surface of revolution; Arc length
    Riemann Integration
    Introduction to MAPLE
    Partial differentiation
    Multiple integration
    Linear Algebra
    Linear algebra relates to points, lines and their planes, and the algebraic analysis of their intersections. A number of new concepts are introduced, such as that of a vector space, which is essentially a collection of vectors that form a defined structure. Matrices are used to solve a range of problems, such as finding eigenvectors (vectors of fixed direction under a transformation).
    Gaussian Elimination and Row Echelon Form
    Vector space; Change of basis
    Linear maps; Subspaces; Kernel, image
    Dimension, rank, nullity
    Invertibility of matrices
    Cramers Rule
    Cayley-Hamilton Theorem
    Probability & Statistics
    Building on earlier study of the basics of probability, you are likely to meet a number of new distributions which model the probabilities likely to occur in a range of situations with defined characteristics. Study of these distributions will help to carry out hypothesis tests, which determine whether underlying assumptions about a situation are valid.
    Bernoulli trials and the Binomial distribution
    Geometric Distribution
    Poisson distributions
    Normal distribution; Central Limit theorem
    Cauchy-Schwartz inequality
    Discrete / continuous random variables
    Probability generating functions
    Hypothesis testing: p-values, z-test, t-test
    Chi-squared Distribution
    Number Systems & Geometry
    Study of number systems will enhance prior understanding of special sets of numbers, such as primes. Formal methods of proof are used to determine facts about such numbers, for example that there are an infinite number of primes.
    Geometry might include detailed analysis of sets of solid shapes, identifying shared characteristics. Alternative ways of representing points and curves in a plane might also be considered, for example using polar co-ordinates instead of the common Cartesian co-ordinates.
    Complex Numbers
    Proof by Induction
    Modular Arithmetic
    RSA Codes
    Properties of prime numbers; Mersenne primes
    Polyhedra; Euler Characteristic
    Polar form; Implicit / Parametric form
    Mechanics is the study of the physical world and the things which make objects move in the ways they do – for example, how a ball thrown vertically upwards will return to the starting point in a calculable period of time. Larger scale objects might include planets, and the laws which govern their motion in the solar system.
    Newtonian Mechanics; Momentum
    Simple Harmonic Motion
    Variable acceleration
    Damped and undamped motion
    Circular Motion
    Relative velocity
    Kepler’s Laws
    Building upon the elementary algebra which will be familiar, the focus is on abstract algebra which introduces more abstract structures such as groups – a set of objects which follow an agreed set of rules connected to how the elements in the set relate to one another. Shared characteristics of seemingly very different sets of objects can be revealed.
    Equivalence relations; Permutations
    Cycle notation
    Groups; Symmetry groups;
    Cayley Tables
    Subgroups and Lagrange's Theorem
    Rings; Fields
    Cayley’s theorem
    Mathematical analysis is the study of infinite processes, usually starting with the consideration of infinite sequences. These will be considered in a number of ways, introducing precise definitions of commonly known concepts such as convergence. There are links to calculus and the differentiability of functions, and a number of well-known theorems such as the Intermediate value Theorem will be considered, the basic idea of which you may have used at in A level maths to identify the location of a root through noting a sign change in y-values between two points.
    Sequences and convergence
    Bolzano-Weierstrass Theorem
    Cauchy Sequences
    Intermediate Value Theorem
    L’ Hopital’s Rule
    Mean Value Theorem and Rolle's Theorem
    Power Series
    Radius of convergence
    Complex sequences
    Mandelbrot set
    Computational Mathematics
    There are a wide number of ways in which computers can be used to find easier solutions to mathematical problems. These are likely to be underpinned by algorithms (precise sets of instructions) and pseudocode (a way to represent procedures using the structure of computer language but designed to be read by humans). Common computer languages used in mathematics include MATLAB and Python and you might also use statistical languages such as R or SPSS.

    The role of computing in mathematics
    Selection statements
    Repetition statements
    Elementary logic in programming

    Other areas of mathematics

    In subsequent years you may have the opportunity to study Operational Research (O.R.). Some sample topics are outlined below.

    TopicSample Content
    Operational Research
    Operational Research (O.R) involves the use of analytical techniques to make decisions and is used in a range of business contexts. Mathematical modelling, statistical analysis and mathematical optimization techniques are used to find optimal solutions to problems such as maximising profit or minimising costs.
    Network analysis
    Graph theory
    Simulation and Queueing
    Planar graphs and graph colouring
    Network flow
    Linear Programming
    Integer programming
    The principle of inclusion/exclusion and the matrix-tree theorem
    Inventory control
    Quality control