Mathematics education is the study of the practices and methods of teaching mathematics. It has been a hotly debated subject in modern society. There is an ambiguity in the term for it refers both to these practices in classrooms around the world, but also to an emergent discipline with its own journals, conferences, etc. The main international body involved is the International Commission on Mathematical Instruction.  (http://www.kurims.kyoto-u.ac.jp/IMU/ICMI/)
Elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most cases, a formal education was only available to male children with a sufficently high status, wealth or caste.
In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession.
The first mathematics textbooks to be written in English were published by Robert Recorde, beginning with The Grounde of Artes in 1540.
In the Renaissance the academic status of mathematics declined, because it was strongly associated with trade and commerce. Although it continued to be taught in European universities, it was seen as subservient to the study of Natural, Metaphysical and Moral Philosophy.
This trend was somewhat reversed in the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics, established by the University of Cambridge in 1662. However, it was uncommon for mathematics to be taught outside of the universities. Isaac Newton, for example, received no formal mathematics teaching until he joined Trinity College, Cambridge in 1661.
In the eighteenth and nineteenth centuries the industrial revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic, became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age.
By the twentieth century mathematics was part of the core curriculum in all developed countries. However, diverse and changing ideas about the purpose of mathemtical education led to little overall consistency in the content or methods that were adopted.
At different times and in different cultures and countries, mathematical education has attempted to achieve a variety of different objectives. These objectives have included :-
Methods of teaching mathematics have varied in line with changing objectives.
Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to and realistic for their pupils.
In modern times there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In England, for example, standards for mathematics education are set as part of the National Curriculum for England  (http://www.nc.uk.net/webdav/servlet/XRM?Page/@id=6004&Subject/@id=22). In the USA education standards are usually set at a state level - in California, for example, the California State Board of Education sets standards for mathematics education  (http://www.cde.ca.gov/be/st/ss/mthmain.asp).
Different levels of mathematics are taught at different ages. A rough guide to the ages at which the sub-topics of arithmetics and algebra are taught is as follows :
- Addition : ages 5-7; more digits ages 8-9
- Subtraction : ages 5-7; more digits ages 8-9
- Multiplication : ages 7-8; more digits ages 9-10
- Division : age 8; more digits ages 9-10
- Pre-Algebra : ages 11-12
- Algebra : ages 13+
The method or methods used in any particular context are largely determined by the obectives that the relevant educational system is trying to achieve. Methods of teaching mathematics include the following:
- Classical education - the teaching of mathematics within the classical education syllabus of the middle ages was typically based on Euclid's Elements, which was taught as a paradigm of deductive reasoning
- Rote learning - the teaching of mathematical results, definitions and concepts by repetition and memorisation. Typically used to teach multiplication tables.
- Exercises - the teaching of mathematical skills by completing large numbers of exercises of a similar type, such as adding fractions or solving quadratic equations.
- Problem solving - the cultivation of mathematical ingenuity, creativity and heuristic thinking by setting students open-ended, unusual, and sometimes insoluble problems. The problems can range from simple word problems to problems from international mathematics competitions such as the International Mathematical Olympiad.
- New math - a method of teaching mathematics which focusses on abstract concepts such as set theory rather than practical applications.
- Historical method - teaching the development of mathematics within an historical, social and cultural context. Provides more human interest than a purely abstract approach.
These methods are not exclusive, and any given system of mathematical education will probably combine several different methods.
The following people all taught mathematics at some stage in their lives, although they are better known for other things :-
- Lewis Carroll, pen name of British author Charles Dodgson, lectured in mathematics at Christ Church, Oxford
- John Dalton, British chemist and physicist, taught mathematics in schools and colleges in Manchester, Oxford and York
- Tom Lehrer, American songwriter and satirist, taught mathematics at Harvard and MIT
- Georg Joachim Rheticus, Austrian cartographer and disciple of Copernicus, taught mathematics at the University of Wittenberg
- Edmund Rich, Archbishop of Canterbury in the 13th century, lectured on mathematics at the universities of Oxford and Paris
- Archie Williams, American athlete and Olympic gold medallist, taught mathematics at high schools in California
- History of Mathematical Education (http://www-gap.dcs.st-and.ac.uk/~history/Education/index.html)
- The Philosophy of Mathematics Education Journal homepage (http://www.ex.ac.uk/~PErnest/)