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. Imperial College London also have a recommended reading list which includes some technical reading suitable to support their first year undergraduate degree courses.

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
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
Eigenvalues/vectors
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
Covariance
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
Conics
Polar form; Implicit / Parametric form
Mechanics
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.
Kinematics
Newtonian Mechanics; Momentum
Simple Harmonic Motion
Variable acceleration
Projectiles
Damped and undamped motion
Circular Motion
Relative velocity
Kepler’s Laws
Algebra
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.
Sets
Equivalence relations; Permutations
Cycle notation
Groups; Symmetry groups;
Cayley Tables
Subgroups and Lagrange's Theorem
Isomorphism
Rings; Fields
Cayley’s theorem
Analysis
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
Algorithms
Pseudocode
Selection statements
Repetition statements
Elementary logic in programming
MATLAB
Python

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
Forecasting
Network flow
Linear Programming
Integer programming
The principle of inclusion/exclusion and the matrix-tree theorem
Inventory control
Quality control