Further Mathematics is the title given to a number of advanced secondary mathematics courses. Higher and Further Mathematics may also refer to any of several advanced mathematics courses at many institutions.
In the United Kingdom, Further Mathematics describes a course studied in addition to the standard mathematics AS-Level and A-Level courses. In Victoria, Australia it describes a course delivered as part of the Victorian Certificate of Education. See the section on Victoria for a more detailed explanation. Globally, it describes a course studied in addition to GCE AS-Level and A-Level Mathematics, or one delivered as part of the International Baccalaureate Diploma.
A qualification in Further Mathematics involves studying both pure and applied modules. Whilst the pure modules - formerly known as Pure 4-6 (or Core 4-6), now known as Further Pure 1-3 (4 exists for the AQA board) - are of a higher standard than those in the standard course, the applied modules need not be. The topics covered by Further Mathematics are more sophisticated and conceptually advanced compared to the single A-level Mathematics.
To achieve an A level in Further Maths, candidates must study six modules which have not already been used for their Maths A level. These six modules must consist of FP2 or FP3, and FP1 along with 4 other modules.
Some schools and colleges in places such as Pakistan, Hong Kong and India take examinations set by British boards and consequently the subject is offered internationally.
Because smaller schools and colleges may not be able to offer Further Mathematics (as it is a very low-intake course requiring well-trained teachers), most universities do not require the course, and may offer "catch-up" classes covering the additional content. Exceptions are the University of Warwick, the University of Cambridge where you must have Further Mathematics to at least AS level to study for a degree in mathematics; also University College London requires an A2 in Further Maths for its maths courses and is recommended for the Statistics courses; Imperial College also requires an A* in A2 Further Maths while other top end universities recommend it or promise lower offers in return.
Further Maths is currently the fastest-growing subject at A level, with the number of students increasing by 23% in 2006, and a network has been set up to offer the subject to pupils at schools that cannot provide it. Further Maths is commonly expressed as the most difficult A-level currently offered in the UK, this is mainly because it is the only subject to further the study (as an extra AS or full A-level) of one particular subject. Although the subject has about 60% of its cohort obtaining "A" grades, students taking the subject tend to be more able, those less likely to achieve top grades are usually discouraged from the course as it can disrupt their performance in other studies. Although over twenty years ago, it was only taken by the absolute elite pupils (usually from a college/grammar sixth form). Even then they usually only achieved a 50% pass rate, probably due to the rigour of the examination (Remember the most serious Further Maths papers were set by Oxford or Cambridge)
List of the areas of study on the syllabus
Study areas vary with the examination board and the specification they set, with Edexcel's syllabus being summarised below.
- Further Pure 1
- Further Pure 2
- Further Pure 3
In contrast with other Further Mathematics courses, Further Maths as part of the VCE is the easiest level of mathematics. Any student wishing to undertake tertiary studies in areas such as Science, Engineering, Commerce, Economics, and some Information Technology courses, must undertake one or both of the other two VCE maths subjects- Mathematical Methods or Specialist Mathematics. The Further Mathematics syllabus in VCE consists of three core modules, which all students undertake, plus three modules chosen by the student (or usually by the school or teacher) from a list of six. The core modules are Univariate Data, Bivariate Data and Time Series. The optional modules are Number Patterns, Geometry and Trigonometry, Graphs and Relations, Business-Related Mathematics, Networks and Decision Mathematics, or Matrices.
Further Mathematics, as studied within the International Baccalaureate Diploma is a Standard Level course that can only be taken in conjunction with Higher Level Higher Level (HL) Mathematics. Further Mathematics studied within the IB is considered considerably harder than that of HL Mathematics, and HL Mathematics is considered harder than typical A-Level Further mathematics. It assumes knowledge of the core syllabus of the HL course, and consists of studying all four of the options studied at Higher Level, plus an extra geometry unit.
The Higher Level syllabus consists of:
- Topic 1 - Functions
- Topic 2 - Sequences and series
- Topic 3 - Exponents
- Topic 4 - Logarithms
- Topic 5 - Natural logarithms
- Topic 6 - Graphing and transforming functions
- Topic 7 - Quadratic equations and functions
- Topic 8 - Complex numbers and polynomials
- Topic 9 - Binomial theorem
- Topic 10 - Mathematical induction
- Topic 11 - The unit circle and radian measure
- Topic 12 - Non right angled triangle trigonometry
- Topic 13 - Periodic phenomena
- Topic 14 - Matrices
- Topic 15 - Vectors in 2 and 3 dimensions
- Topic 16 - Complex numbers
- Topic 17 - Lines and planes in space
- Topic 18 - Descriptive statistics
- Topic 19 - Probability
- Topic 20 - Calculus
- Topic 21 - Differential calculus
- Topic 22 - Applications of differential calculus
- Topic 23 - Derivatives of exponential and logarithmic functions
- Topic 24 - Derivatives of circular functions and related rates
- Topic 25 - Integration
- Topic 26 - Integration (areas and other applications)
- Topic 27 - Circular function integration
- Topic 28 - Volumes of revolution
- Topic 29 - Further integration and differential equations
- Topic 30 - Statistical distributions
The Further Mathematics syllabus includes the topics Further Statistics and Probability, Series and Differential Equations, Sets, Relations and Groups, Euclidean Geometry, and Discrete Mathematics (including both Number Theory and Graph Theory). However, for first examinations in 2014 Further Mathematics will become a Higher Level only course. This new programme will include the same topics as before, with Series and Differential Equations changing its name to Calculus and a new topic, Linear Algebra, being added.