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. It thought to be a course of high-calibre as compared to the standard AS-Level mathematics course.
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) - build on knowledge from the core mathematics modules, the applied modules may start from first principles.
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 FP1, at least one of FP2 or FP3, and 4 other modules. With regard to Mathematics degrees, most universities do not require Further Mathematics, and may incorporate foundation math modules or offer "catch-up" classes covering any additional content. Exceptions are the University of Warwick, the University of Cambridge which requires Further Mathematics to at least AS level; University College London requires or recommends an A2 in Further Maths for its maths courses; Imperial College requires an A in A level Further Maths while other universities may recommend it or may promise lower offers in return. Some schools and colleges may not offer Further mathematics but online resources are available  Although the subject has about 60% of its cohort obtaining "A" grades, students choosing the subject are assumed to be more able in mathematics.
Some medicine courses do not count maths and further maths as separate subjects for the purposes of making offers due to the perceived overlap in content and the potentially narrow education a candidate with maths, further maths and just one other subject may have.
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 two modules chosen by the student (or usually by the school or teacher) from a list of four. The core modules are Univariate Data, Bivariate Data, Time Series, Number Patterns and Business-Related Mathematics. The optional modules are Geometry and Trigonometry, Graphs and Relations, Networks and Decision Mathematics, or Matrices. See also: Mickey Mouse
Further Mathematics is available as a second and higher mathematics course at A Level (now H2), in addition to the Mathematics course at A Level. Students can offer this subject if they have A2 and better in 'O' Level Mathematics and Additional Mathematics, depending on the school. 
International Baccalaureate Diploma
Further Mathematics, as studied within the International Baccalaureate Diploma Programme, is a Higher Level (HL) course that can be taken in conjunction with Mathematics HL or on its own. It consists of studying all four of the options in Mathematics HL, plus two additional topics.
Topics studied in Further Mathematics include:
- Topic 1 - Linear algebra - studies on matrices, vector spaces, linear and geometric transformations
- Topic 2 - Geometry - closer look on triangles, circles and conic sections
- Topic 3 - Statistics and probability - the geometric and negative binomial distributions, unbiased estimators, statistical hypothesis testing and an introduction to bivariate distributions
- Topic 4 - Sets, relations and groups - algebra of sets, ordered pairs, binary operations and group homomorphism
- Topic 5 - Calculus - infinite sequences and series, limits, improper integrals and various first-order ordinary differential equations
- Topic 6 - Discrete mathematics - complete mathematical induction, linear Diophantine equations, Fermat's little theorem, route inspection problem and recurrence relations
- "Medical school a-level requirements". The Student Room.
- IB DP Further mathematics HL guide (first examinations 2014) (Electronic (PDF)). Cardiff, Wales, United Kingdom: International Baccalaureate Organization. June 2012.