Further Mathematics

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.

United Kingdom

Background

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 FP1, at least one of FP2 or FP3, and 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 as it is the only subject to further the study of a particular subject. Although the subject has about 60% of its cohort obtaining "A" grades, students taking the subject tend to be more able.

Some medical schools 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 potential narrow education a candidate with maths, further maths and just one other subject may have.

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
  • Complex Numbers
  • Use of iterative methods to solve equations including Newton-Raphson method
  • Parabolas and Rectangular Hyperbola
  • Matrices
  • Summation of series using standard results
  • Proof by induction
• Further Pure 2
  • Inequalities
  • Summation of series by the method of differences
  • Further Complex Numbers
  • Linear, Ordinary Differential Equations, of the first and second order
  • Taylor series
  • Polar Coordinates
• Further Pure 3
  • Hyperbolic functions, including their differentiation and integration
  • Conic Sections
  • Calculus, including reduction formulae, surfaces of revolution and the inverse trigonometric functions.
  • Vectors, including the cross product and the triple scalar product
  • Further Matrices

Australia (Victoria)

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.

Singapore

Further Mathematics was formerly available as a second and higher mathematics course at A level, in addition to the Mathematics course at A level. It was discontinued with the launch of the new 'A' level curriculum in 2006 for examinations from 2007, leaving a single mathematics course at H2 level (equivalent to the former A level). Further Mathematics at H2 level is being reintroduced to the 'A' level curriculum for teaching beginning in 2016 for examinations from 2017.

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

See also

• Additional Mathematics

External links

• The Further Mathematics Support Programme
• Mechanics M1 Material