Mathematics education is the practice of teaching and learning mathematics, as well as the field of scholarly research on this practice. Researchers in mathematics education are primarily 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 mathematics education as a practice.

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 being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics being 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 organisation
• A new interest in mathematics education emerged in the 1960s, and the commission was revitalised
• 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]

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, Technology, Engineering, and Mathematics (STEM) fields.
• 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.

Research

An increasing amount of research has been done in the area of mathematics education in the last few decades. The National Council of Teachers of Mathematics has summarized the state of current research in mathematics education in nine areas of current interest, as follows.^[2] (Though the NCTM has special interest in American education, the research summarized is international in scope.)

What can we learn from research?

Instead of just looking at whether a particular program works, we must also look at why and under what conditions it works. Teachers can adapt tasks used in studies for their own classrooms. Individual studies are often inconclusive, so it is important to look at a consensus of many studies to draw conclusions. Theory can put practice in a new perspective. For example, research shows that when students invent their own algorithms first, and then learn the standard algorithm, they understand better and make fewer errors. Such findings can have an impact on classroom practice.

Homework

Homework which leads students to practice past lessons or prepare future lessons are more effective than those going over today's lesson. Assignments should be a mix of easy and hard problems and ideally based on the student's learning style. Students must receive feedback. Students with learning disabilities or low motivation may profit from rewards. Shorter homework is better than long homework and group homework is sometimes effective, though these findings depend on grade level. Homework helps simple skills, but not broader measures of achievement.

Student learning

Most bilingual adults switch languages when calculating. Such code-switching has no impact on math ability and should not be discouraged.

When studying statistics, children need time to explore, study and share reasoning about centers, shape, spread and variability. The ability to calculate averages does not mean students understand the concept of averages, which students conceptualize in a variety of ways—from a simplistic "typical value" to a deeper idea of "representative value." Learning when to use mean, median and mode is difficult.

Algebra

It is important for elementary school children to spend a long time learning to express algebraic properties without symbols before learning algebraic notation. When learning symbols, many students believe letters always represent unknowns and struggle with the concept of variable. They prefer arithmetic reasoning to algebraic equations for solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns. Students often have trouble with the minus sign and understand the equals sign to mean "the answer is...."

American Curriculum

The US National Research Council has found it difficult to evaluate any given program, but two general patterns have become clear from large-scale studies: (1) Students achieve greater conceptual understanding from standards-based curricula compared to traditional curricula. (2) Students achieve the same procedural skill level in both types of curricula as measured by traditional standardized tests.

Effective instruction

The two most important criteria for helping students gain conceptual understanding are making connections and intentionally struggling with important ideas. Skill efficiency is best attained by rapid pacing, direct traditional teaching and a smooth transition from teacher modeling to error-free practice. Students who learn skills in conceptually-oriented instruction are better able to adapt their skills to new situations.

Students with difficulties

Students with genuine difficulties (unrelated to motivation or past instruction) struggle with basic facts, answer impulsively, struggle with mental representations, have poor number sense and have poor short-term memory. Techniques that have been found productive for helping such students include peer-assisted learning, explicit teaching with visual aids, instruction informed by formative assessment and encouraging students to think aloud.

Formative assessment

Formative assessment is both the best and cheapest way to boost student achievement, student engagement and teacher professional satisfaction. Results surpass those of reducing class size or increasing teachers' content knowledge. Only short-term (within and between lessons) and medium-term (within and between units) assessment is effective. Effective assessment is based on clarifying what students should know, creating appropriate activities to obtain the evidence needed, giving good feedback, encouraging students to take control of their learning and letting students be resources for one another.

Mathematics specialists and coaches

Little research has been done so far on mathematics coaches and the studies that have been done are hard to evaluate because coaching is usually part of larger programs. What research has been done seems to show that coaches can improve teaching, but the coaching program must be well designed.

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, realistic for, and considered socially appropriate 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) summarised the research of others who found, based on nationwide data, that students with higher scores on standardised 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. ^[3]

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.

Content and age levels

Different levels of mathematics are taught at different ages and in somewhat different sequences in different countries. Sometimes a class may be taught at an earlier age than typical as a special or "honors" class.

Elementary mathematics in most countries is taught in a similar fashion, though there are differences. In the United States fractions are typically taught starting from 1st grade, whereas in other countries they are usually 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.^[4]

In most of the US, algebra, geometry and analysis (pre-calculus and calculus) are taught as separate courses in different years of high school. Mathematics in most other countries (and in a few US states) is integrated, with topics from all branches of mathematics 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 the United States. 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 secondary school.

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 this 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 memorisation typically without meaning or supported by mathematical reasoning. A derisory term is drill and kill. Parrot Maths was the title of a paper critical of rote learning. Within the conventional approach, rote learning 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. Problem solving is used as a means to build new mathematical knowledge, typically by building on students' prior understandings.
• 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 method 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 pre-college mathematics education in the US and Canada, focused on deepening student understanding of mathematical ideas and procedures, and formalized by the National Council of Teachers of Mathematics which created the Principles and Standards for School Mathematics.

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.