A mathematics lecture at Helsinki University of Technology.

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[edit] History

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 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.

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‘.”^[1]

[edit] 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 non routine problems.

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

[edit] 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.

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. ^[2]

In North America, the National Council of Teachers of Mathematics (NCTM) has published the Principles and Standards for School Mathematics. In 2006, they released the Curriculum 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.

[edit] 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:

• 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

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.^[3]

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.

[edit] 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:

• Conventional approach – the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with arithmetic and is followed by Euclidean 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 of deductive 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 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, 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 book Why 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. Sometimes derisively called the “new new” maths, this is a vision for precollege mathematics education in the US and Canada, based on constructivist ideas, and formalized by the National Council of Teachers of Mathematics which created the Principles and Standards for School Mathematics.

[edit] Recent controversy over U.S. mathematics education

Near the end of the 20th century diverse and changing ideas about the purpose of mathematical education would lead to wide adoption of reform-based standards and curricula funded by the US federal government, and also adopted by other national curriculum standards. These were based on student-centered learning methods and equity in mathematics as a centerpiece of the standards based education reform movement. This movement in turn was met with opposition which called for a return to traditional direct instruction of standard arithmetic methods by the start of the 21st century as some schools and districts supplemented or replaced standards-based curricula.

With the adoption of substantially different teaching reform standards and the development and widespread adoption of federally funded curricula during the 1990s, mathematics education became the most hotly debated subject since the original 1960s “New Math” in mainstream news journals such as the Wall Street Journal and The New York Times. The goals for educators since the 1990s have been expanded in the context of systemic standards based education reform in the United States and other nations to promote increased learning for all students. It is a goal to achieve equity and success for all groups in society. It is no longer acceptable to many in the education community that some were historically excluded from the full range of opportunities open to those who learned the most advanced mathematics.

By the late 1980s, a movement for systemic education reform took hold based on constructivist practices and the belief in success for all groups including minorities and women. Among the development of a number of controversial standards across reading, science and history, the National Council of Teachers of Mathematics of the United States produced the Curriculum and Evaluation Standards for School Mathematics in 1989. These standards included new goals such as equity and conceptual understanding and de-emphasized the traditional direct instruction of standard algorithms.

The controversial 1989 NCTM standards recommended teaching elements of algebra as early as grade 5, and elements of calculus as early as grade 9, though this was rarely adopted even as late as the 2000s. In standards based education reform all students, not only the college bound, must take advanced mathematics. In some large school districts, this means requiring algebra of all students by the end of junior high school, compared to the tradition of tracking only college bound and the most advanced junior high school students to take algebra.

The standards soon became the basis for many new federally funded curricula such as the Core-Plus Mathematics Project and became the foundation of many local and state curriculum frameworks. Although the standards were the consensus of those teaching mathematics in the context of real life, they also became a lightning rod of criticism as math wars erupted in some communities that were opposed to some of the more radical changes to mathematics instruction such as Mathland‘s Fantasy Lunch and what some dubbed “rainforest algebra”. Some students complained that their new math courses placed them into remedial math in college.^{[citation needed]}

In 2000 and 2006 NCTM released Principles and Standards for School Mathematics and the Curriculum Focal Points which expanded on the work of the previous standards documents. Refuting reports and editorials ^[4] that it was largely an admission that the previous standards had mistakenly de-emphasized instruction of basic skills, NCTM spokesmen maintained that it provided more grade by grade specificity on key areas of study for a coherent and consistent development of mathematical understanding and skill. These documents criticized American math curricula as a “mile wide and an inch deep” in comparison to the math of nations such as Singapore.

Another issue with mathematics education has been integration with science education. This is difficult for the public schools to do because science and math are taught independently. The value of the integration is that science can provide authentic contexts for the math concepts being taught. Further, if mathematics is taught in synchrony with science, then the students benefit from this correlation.^{[citation needed]}

[edit] 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 at 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.
• Brian May, rock guitarist and composer, worked briefly as a mathematics teacher before joining Queen^[5]
• 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
• Éamon de Valera, a leader of Ireland’s struggle for independence in the early 20th century and founder of the Fianna Fáil party, taught mathematics at schools and colleges in Dublin
• Archie Williams, American athlete and Olympic gold medalist, taught mathematics at high schools in California

[edit] Mathematics educators

The following are some of the people who have 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 introductory courses in geometry for high school students.[2]
• William Brownell (1895-1977), American educator who led the movement to make math meaningful to children.
• Georges Cuisenaire, Belgian primary school teacher who invented Cuisenaire rods
• Euclid, author of The Elements
• Robert & Ellen Kaplan, international best-selling authors of Nothing That Is, Chances Are: Adventures in Probability , and The Art of the Infinite: The Pleasures of Mathematics
• Hans Freudenthal, Dutch mathematician who had a profound impact on Dutch education and founded the Freudenthal Institute for Science and Mathematics Education in 1971.
• Toru Kumon, originator of the Kumon method based on mastery through exercise
• 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
• Pierre van Hiele and Dina van Hiele-Geldof, Dutch educators who proposed a theory of how children learn geometry (1957), which eventually became very influential worldwide.