Further Mathematics is the title given to a number of advanced

secondary Secondary may refer to: Science and nature * Secondary emission, of particles ** Secondary electrons, electrons generated as ionization products * The secondary winding, or the electrical or electronic circuit connected to the secondary winding i ...

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

courses. The term "Higher and Further Mathematics", and the term "Advanced Level 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 The A-Level (Advanced Level) is a subject-based qualification conferred as part of the General Certificate of Education, as well as a school leaving qualification offered by the educational bodies in the United Kingdom and the educational aut ...

courses. In the state of

Victoria Victoria most commonly refers to: * Victoria (Australia), a state of the Commonwealth of Australia * Victoria, British Columbia, provincial capital of British Columbia, Canada * Victoria (mythology), Roman goddess of Victory * Victoria, Seychelle ...

in Australia, it describes a course delivered as part of the

Victorian Certificate of Education The Victorian Certificate of Education (often abbreviated VCE) is one credential available to secondary school students who successfully complete year 11 and 12 in the Australian state of Victoria. The VCE is the predominant choice for students ...

(see § Australia (Victoria) for a more detailed explanation). Globally, it describes a course studied in addition to GCE AS-Level and A-Level Mathematics, or one which is delivered as part of the

International Baccalaureate Diploma The International Baccalaureate Diploma Programme (IBDP) is a two-year Curriculum, educational programme primarily aimed at 16-to-19-year-olds in 140 countries around the world. The programme provides an internationally accepted qualification fo ...

United Kingdom

Background

A qualification in Further Mathematics involves studying both

pure Pure may refer to: Computing * A pure function * A pure virtual function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool * Pure-FTPd, F ...

and applied modules. Whilst the pure modules (formerly known as Pure 4–6 or Core 4–6, now known as Further Pure 1–3, where 4 exists for the AQA board) build on knowledge from the core mathematics modules, the applied modules may start from first principles. The structure of the qualification varies between exam boards. 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 Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands (county), West Midlands and Warwickshire, England. The university was founded i ...

, the

University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...

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 proficient in mathematics, and there is much more overlap of topics compared to base mathematics courses at A level. Some medicine courses do not count maths and further maths as separate subjects for the purposes of making offers. This is due to the overlap in content, and the potentially narrow education a candidate with maths, further maths and just one other subject may have. The IGCSE equivalent of GCSE Further Maths is IGCSE Additional Mathematics.

Support

There are numerous sources of support for both teachers and students. The AMSP (formerly FMSP) is a government-funded organisation that offers professional development, enrichment activities and is a source of additional materials via its website. Registering with AMSP gives access to Integral, another source of both teaching and learning materials hosted by Mathematics Education Innovation (MEI). Underground Mathematics is another resource in active development which reflects the emphasis on problem solving and reasoning in the UK curriculum. A collection of tasks for post-16 mathematics can be also found on the

NRICH The Millennium Mathematics Project (MMP) was set up within the University of Cambridge in England as a joint project between the Faculties of Mathematics and Education in 1999. The MMP aims to support maths education for pupils of all abilities fr ...

site.

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 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 pursue this subject if they have A2 and better in 'O' Level Mathematics and Additional Mathematics, depending on the school. Some topics covered in this course include mathematical induction, complex number, polar curve and conic sections, differential equations, recurrence relations, matrices and linear spaces, numerical methods, random variables and hypothesis testing and confidence intervals.

International Baccalaureate Diploma

Further Mathematics, as studied within the

International Baccalaureate Diploma Programme The International Baccalaureate Diploma Programme (IBDP) is a two-year educational programme primarily aimed at 16-to-19-year-olds in 140 countries around the world. The programme provides an internationally accepted qualification for entry into ...

, was a Higher Level (HL) course that could be taken in conjunction with Mathematics HL or on its own. It consisted of studying all four of the options in Mathematics HL, plus two additional topics. Topics studied in Further Mathematics included: *Topic 1 -

Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

- studies on

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

s, linear and geometric transformations *Topic 2 -

Geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

- closer look on

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

s and

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

s *Topic 3 -

Statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

and

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

- the

geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

and

negative binomial In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-r ...

distributions, unbiased estimators,

statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...

and an introduction to bivariate distributions *Topic 4 - Sets, relations and

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

algebra of sets In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the ...

ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

s and

group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...

*Topic 5 -

Calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

- infinite

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

s and

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

improper integral In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...

s and various first-order ordinary

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s *Topic 6 -

Discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

- complete

mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...

, linear

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...

Fermat's little theorem Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...

route inspection problem In graph theory, a branch of mathematics and computer science, Guan's route problem, the Chinese postman problem, postman tour or route inspection problem is to find a shortest closed path or circuit that visits every edge of an (connected) undire ...

and

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s From 2019, the course has been discontinued and transited info the followings modules:Mathematics
/ref> * Mathematics: analysis and approaches SL * Mathematics: analysis and approaches HL * Mathematics: applications and interpretation SL * Mathematics: applications and interpretation HL

India

CBSE does not include any 'further' mathematics courses. While the

JEE Advanced Jee may refer to: People * Joe Jee (1883–1919), English football player * Joseph Jee (1819–1899), English surgeon * Martha Jee, also known as Martha Wong (1939–), Texas politician * Rupert Jee (born 1956), American entrepreneur and televi ...

syllabus is a slightly extended variant of the CBSE Class 11 and 12 syllabus, that is not a school-leaving examination.

References

{{reflist

External links

The Further Mathematics Support ProgrammeMechanics M1 MaterialAMSP
(Advanced Math Support Program)
Integral
(High level support for AS/A level Maths & Further Maths)
Underground Mathematics
(Resources on A level mathematics) Mathematics education

United Kingdom

Background

Support

Australia (Victoria)

Singapore

International Baccalaureate Diploma

India

See also

References

External links