Mathematics education is the practice of teaching and learning mathematics, as well as the field of scholarly research on this practice. Researchers in math education are in the first instance concerned with the tools, methods and approaches that facilitate practice or the study of practice. However mathematics education research, known on the continent of Europe as the didactics of mathematics, has developed into a fully fledged field of study, with its own characteristic concepts, theories, methods, national and international organisations, conferences and literature. This article describes some of the history, influences and recent controversies concerning math education as a practice.
A mathematics lecture at Helsinki University of Technology.
Illustration at the beginning of a 14th century translation of Euclid's Elements.
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 and French were published byRobert 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 theUniversity 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.
During the twentieth century mathematics education was established as an independent field of research. Here are some of the main events in this development:
- In 1893 a Chair in mathematics education was created at the University of Göttingen, under the administration of Felix Klein
- The International Commission on Mathematical Instruction (ICMI) was founded in 1908, and Felix Klein became the first president of the organization
- A new interest in mathematics education emerged in the 1960s, and the commission was revitalized
- In 1968, the Shell Centre for Mathematical Education was established in Nottingham
- The first International Congress on Mathematical Education (ICME) was held in Lyon in 1969. The second congress was in Exeter in 1972, and after that it has been held every four years
In the 20th century, the cultural impact of the "electric age" (McLuhan) was also taken up by educational theory and the teaching of mathematics. While previous approach focused on "working with specialized 'problems' in arithmetic", the emerging structural approach to knowledge had "small children meditating about number theory and 'sets'."
At different times and in different cultures and countries, mathematics 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, while Scotland maintains its own educational system.
Ma (2000) summarized the research of others who found, based on nationwide data, that students with higher scores on standardized math tests had taken more mathematics courses in high school. This led some states to require three years of math instead of two. But because this requirement was often met by taking another lower level math course, the additional courses had a “diluted” effect in raising achievement levels. 
In North America, the National Council of Teachers of Mathematics (NCTM) has published the Principles and Standards for School Mathematics. In 2006, they released theCurriculum Focal Points, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards are not nationally enforced in US schools.
Content and age levels
Different levels of mathematics are taught at different ages. 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 certain topics of arithmetic are taught in the United States is as follows:
The ages at which other math subjects (rational numbers, geometry, measurement, problem solving, logic, algebraic thinking, probability, statistics, reasoning skills and so on) are taught vary considerably from state to state.
Elementary mathematics in other countries is similar, though fractions (typically taught from 1st grade in the United States) are often taught later, since the metric system does not require young children to be familiar with them. Most countries tend to cover fewer topics in greater depth than in the United States.
A typical pre-college sequence of mathematics courses in the United States would include some of the following, especially Geometry and Algebra I and II:
- Pre-algebra: ages 11-13 (Pre-Algebra taught in schools as early as 6th grade as an honor course. Algebraic reasoning can be taught in elementary school, though)
- Algebra I (basic algebra): ages 12+ (Algebra I is taught at 9th grade on average, or as early as 7th or 8th grade for an honors course)
- Geometry: ages 13+ (Geometry taught at 10th grade on average, or as early as 8th grade as an honors course)
- Algebra II: ages 14+; usually includes powers and roots, polynomials, quadratic functions, coordinate geometry, exponential and logarithmic functions, probability, matrices, and basic trigonometry
- Trigonometry or Algebra 3 or Pre-Calculus: ages 15+
- Statistics: ages 15+ (Probability and statistics topics are taught throughout the curriculum from early elementary grades, but may form a special course in high school.)
- Calculus: ages 16+ (usually seen in 12th grade, if at all; some honors students may see it earlier)
Mathematics in most other countries and in a few US states is integrated, with topics of algebra, geometry and analysis (pre-calculus and calculus) studied every year. Students in many countries choose an option or pre-defined course of study rather than choosing courses à la carte as in North America. Students in science-oriented curricula typically study differential calculus and trigonometry at age 16-17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions and infinite series their final year of high school.
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:
- Conventional approach - the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with arithmetic and is followed byEuclidean geometry and elementary algebra taught concurrently. Requires the instructor to be well informed about elementary mathematics, since didactic and curriculum decisions are often dictated by the logic of the subject rather than pedagogical considerations. Other methods emerge by emphasizing some aspects of the conventional approach.
- Classical education - the teaching of mathematics within the classical education syllabus of the Middle Ages, which was typically based on Euclid's Elements taught as a paradigm ofdeductive reasoning.
- Rote learning - the teaching of mathematical results, definitions and concepts by repetition and memorization. A derisory term is drill and kill. Parrot Maths was the title of a paper critical of rote learning. Within the conventional approach, is used to teach multiplication tables.
- Exercises - the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding vulgar fractions or solving quadratic equations.
- Problem solving - the cultivation of mathematical ingenuity, creativity and heuristic thinking by setting students open-ended, unusual, and sometimes unsolved problems. The problems can range from simple word problems to problems from internationalmathematics competitions such as the International Mathematical Olympiad.
- New Math - a method of teaching mathematics which focuses on abstract concepts such as set theory, functions and bases other than ten. Adopted in the US as a response to the challenge of early Soviet technical superiority in space, it began to be challenged in the late 1960s. One of the most influential critiques of the New Math was Morris Kline's 1973 bookWhy Johnny Can't Add. The New Math was the topic of one of Tom Lehrer's most popular parody songs, with his introductory remarks to the song: "...in the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer."
- Historical method - teaching the development of mathematics within an historical, social and cultural context. Provides more human interest than the conventional approach.
- Standards-based mathematics - a vision for precollege mathematics education in the USand Canada, based on constructivist ideas, and formalized by the National Council of Teachers of Mathematics which created the Principles and Standards for School Mathematics.