Further Mathematics is the title given to a number of advanced secondary

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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 The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Northwestern Europe, off the coast of European mainland, the continental mainland. It comprises England, Scotlan ...

, 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 ...

courses. In the state of

Victoria Victoria most commonly refers to: * Queen Victoria (1819–1901), Queen of the United Kingdom and Empress of India * Victoria (state), a state of Australia * Victoria, British Columbia, Canada, a provincial capital * Victoria, Seychelles, the capi ...

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

Victorian Certificate of Education The Victorian Certificate of Education (VCE) is the credential available to secondary school students who successfully complete year 10, 11 and 12 in the Australian state of Victoria (state), Victoria as well as in some international schools i ...

(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 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 int ...

. In other words, more mathematics can also be referred to as part of advanced mathematics, or advanced level math.

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, where 4 exists for the

AQA AQA Education, trading as AQA (formerly the Assessment and Qualifications Alliance), is an awarding body in England, Wales and Northern Ireland. It compiles specifications and holds Test (assessment), examinations in various subjects at Genera ...

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 and Warwickshire, England. The university was founded in 1965 as part of ...

, the

University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

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.

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 int ...

, 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 matrix (mathemat ...

- studies on

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

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

s, linear and geometric transformations *Topic 2 -

Geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

- closer look on

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

s and

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

s *Topic 3 -

Statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

and

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

- the

geometric Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

and

negative binomial In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Berno ...

distributions, unbiased estimators,

statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. T ...

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

relations Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...

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 iden ...

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 re ...

ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...

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, a binary operation ...

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) whe ...

*Topic 5 -

Calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

- 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 cal ...

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 i ...

, limits,

improper integral In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integral ...

s and various first-order ordinary differential equations *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 mathematical proof, 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), \dots all hold. This is done by first proving a ...

, linear

Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...

Fermat's little theorem In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as a^p \equiv a \pmod p. For example, if and , t ...

route inspection problem In graph theory and combinatorial optimization, 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) undirected graph at ...

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 into 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

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

See also

References

External links