Mathematics education

Mathematics education is the study of practices and methods of both the teaching and learning of mathematics. Furthermore, mathematics educators are concerned with the development of tools that facilitate practice and/or the study of practice. Mathematics education 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.

History
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 sufficiently 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 mathematical education led to little overall consistency in the content or methods that were adopted.

Objectives
At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included:


 * The teaching of basic numeracy skills to all pupils
 * The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid geometry, trigonometry) to most pupils, to equip them to follow a trade or craft
 * The teaching of abstract mathematical concepts (such as set and function) at an early age
 * The teaching of selected areas of mathematics (such as Euclidean geometry) as an example of an axiomatic system and a model of deductive reasoning
 * The teaching of selected areas of mathematics (such as calculus) as an example of the intellectual achievements of the modern world
 * The teaching of advanced mathematics to those pupils who wish to follow a career in science
 * The teaching of heuristics and other problem-solving strategies to solve nonroutine problems.

Methods of teaching mathematics have varied in line with changing objectives.

Standards
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, while Scotland maintains its own educational system. In the USA, the National Council of Teachers of Mathematics has produced a series of documents, most recently Principles and Standards for School Mathematics, setting forth a consensus on broad goals for school mathematics. More specific education standards are generally set at a state level - in California, for example, the California State Board of Education sets standards for mathematics education.

Levels
Different levels of mathematics are taught at different ages. This motivation can sometimes be attributed to the Van Hiele's Levels which is based on the belief that children at different ages have different cognitive preparedness. Sometimes a class may be taught at an earlier age as a special or "honors" class. 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+
 * Geometry : ages 14-15+
 * Precalculus : ages 16+
 * Calculus : ages 17+

(To convert these ages to American grade levels, age 5 becomes kindergarten; subtract 5 from all older ages to get the relevant grade.)

Some progressions of mathematics education include relatively greater emphasis on discrete math such as number theory and combinatorics and sometimes relating to probability.

Methods
The method or methods used in any particular context are largely determined by the objectives that the relevant educational system is trying to achieve. Methods of teaching mathematics include the following:

These methods are not exclusive, and any given system of mathematical education will probably combine several different methods.
 * 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 vulgar fractions or solving quadratic equations. For example, Cuisenaire rods are used as a method of teaching fractions.
 * 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 focuses 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.
 * Reform - based on the Constructivism (learning theory). Students usually work on contextual problems in small groups, with the problems themselves acting as vehicles for mathematics content. The use of calculator and manipulatives are encouraged, but algebra skills and rote memorization are deemphasized.

Mathematics teachers
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, MIT and currently at University of California, Santa Cruz.
 * 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 medalist, taught mathematics at high schools in California

Mathematics educators
The following people had a significant influence on the teaching of mathematics at various periods in history:


 * Tatyana Alexeyevna Afanasyeva, Dutch/Russian mathematician who advocated the use of visual aids and examples for introductionary courses in geometry for high school students.
 * Georges Cuisenaire, Belgian primary school teacher who invented Cuisenaire rods
 * Euclid, author of The Elements
 * Robert Lee Moore, originator of the Moore method
 * Robert Parris Moses, founder of the nationwide US Algebra project
 * George Pólya, author of How to Solve It
 * Toru Kumon, originator of the Kumon method based on mastery through exercise

On-line Forums
Since the establishment of social software tools on the internet, several mathematics forums have appeared on the internet. The use of social software for educational purposes is still very experimental. On-line forums for mathematics education and problem solving include: (Art of Problem Solving and Mathlinks have merged)
 * Art of Problem Solving -- Forums for advanced students and problem solvers
 * Math Forum -- Forums and resources for mathematics students; run by Drexel University
 * S.O.S. Mathematics
 * MathLinks -- International forum for mathematics competition students

Scholarly Journals: Print

 * Educational Studies in Mathematics
 * Journal for Research in Mathematics Education
 * For the Learning of Mathematics

Scholarly Journals: on-line

 * The Philosophy of Mathematics Education Journal homepage

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