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.
Three books that you may be interested in reading prior to applying for a Mathematics degree course are:
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.
You may also like to try the questions in Cambridge University's pre-course Mathematics workbook.
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!).
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 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
|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|
Normal distribution; Central Limit theorem
Discrete / continuous random variables
Probability generating functions
Hypothesis testing: p-values, z-test, t-test
|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.
Proof by Induction
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
Damped and undamped motion
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
Groups; Symmetry groups;
Subgroups and Lagrange'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 |
Intermediate Value Theorem
L’ Hopital’s Rule
Mean Value Theorem and Rolle's Theorem
Radius of convergence
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|
Elementary logic in programming
In subsequent years you may have the opportunity to study Operational Research (O.R.). Some sample topics are outlined below.
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.
Simulation and Queueing
Planar graphs and graph colouring
The principle of inclusion/exclusion and the matrix-tree theorem