Mathematics education

A mathematics lecture at Aalto University School of Science and Technology.

In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research.

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 or pedagogy of mathematics, has developed into an extensive field of study, with its own concepts, theories, methods, national and international organisations, conferences and literature. This article describes some of the history, influences and recent controversies.

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 18th and 19th 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'."

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

The following results are examples of some of the current findings in the field of mathematics education:

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.

Effective instruction

The two most important criteria for helping students gain conceptual understanding are making connections and intentionally struggling with important ideas. Research in the 70s and 80s concluded that skill efficiency is best attained by rapid pacing, direct instruction, and students practicing at least until mastery level. According to summaries of recent research, 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.

Bilingualism

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

Learning statistics

When studying statistics, children benefit from exploring, studying, and sharing reasoning about centers, shape, spread, and variability. The ability to calculate averages does not signify an understanding of population and sample mean. Students have difficulty remembering to use the sample median as a robust estimator of location, which is less sensitive than the sample mean or the sample mode when the data-set has outliers or when the population is skewed or heavy-tailed.

Algebraic reasoning

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...."

What can we learn from research?

Studies can examine the interaction of treatments with covariates and with blocking factors, instead of just estimating the main effects of treatments. Such examinations of interaction give insight into the contexts in which treatments work. Even well-done randomized experiments can yield different results, so it is important to read reviews of many well-done randomized experiments studies when deciding on practical implications.

Methodology controversy

In the scholarly literature in mathematics education, relatively few articles report random experiments in which teaching methods were randomly assigned to classes. Randomized experiments have been done to evaluate non-pedagogical programs in schools, such as programs to reduce teenage-pregnancy and to reduce drug use. Randomized experiments were commonly done in educational psychology in the 1880s at American universities. However, randomized experiments have been relatively rare in education in recent decades. In other disciplines concerned with human subjects, like biomedicine, psychology, and policy evaluation, controlled, randomized experiments remain the preferred method of evaluating treatments. Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods. On the other hand, many scholars in educational schools have argued against increasing the number of randomized experiments, often because of philosophical objections.

Both randomized experiments and observational studies are useful for suggesting new hypotheses, which can eventually be tested by new randomized experiments. The scientific community recognizes that observational studies remain valuable in education—just as observational studies remain valuable in epidemiology, political science, economics, sociology, anthropology, etc.

In the United States, the National Mathematics Advisory Panel (NMAP) published a report in 2008 based on studies using randomized and nonrandom assignments that involved various statistically controlled experiments; the NMAP report's preference for randomized experiments received criticism from some scholars. In 2010, the What Works Clearinghouse (essentially the research arm for the Department of Education) responded to ongoing controversy by extending its research base to include non-experimental studies, including regression-discontinuity designs and single-case studies.

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. In the USA, the National Governors Association Center for Best Practices and the Council of Chief State School Officers have published the national mathematics Common Core State Standards Initiative.

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.

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.

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 in 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 quadrivium, part of the classical education curriculum 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. In traditional education, rote learning is used to teach multiplication tables, definitions, formulas, and other aspects of mathematics.
• 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
• 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.

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:

• Euclid (fl. 300 BC), Ancient Greek, author of The Elements
• Tatyana Alexeyevna Afanasyeva (1876–1964), Dutch/Russian mathematician who advocated the use of visual aids and examples for introductory courses in geometry for high school students
• Robert Lee Moore (1882–1974), American mathematician, originator of the Moore method
• George Pólya (1887–1985), Hungarian mathematician, author of How to Solve It
• Georges Cuisenaire (1891–1976), Belgian primary school teacher who invented Cuisenaire rods
• William Brownell (1895–1977), American educator who led the movement to make mathematics meaningful to children, often considered the beginning of modern mathematics education
• Hans Freudenthal (1905–1990), Dutch mathematician who had a profound impact on Dutch education and founded the Freudenthal Institute for Science and Mathematics Education in 1971
• Toru Kumon (1914–1995), Japanese, originator of the Kumon method, based on mastery through exercise
• Pierre van Hiele and Dina van Hiele-Geldof, Dutch educators (1930s - 1950s) who proposed a theory of how children learn geometry (1957), which eventually became very influential worldwide
• Robert Parris Moses (1935-), founder of the nationwide US Algebra project
• Robert & Ellen Kaplan (about 1930/40s-), authors of Nothing That Is, The Art of the Infinite: The Pleasures of Mathematics, and Chances Are: Adventures in Probability (by Michael Kaplan and Ellen Kaplan).

Organizations

• Advisory Committee on Mathematics Education
• Association of Teachers of Mathematics
• Mathematical Association
• National Council of Teachers of Mathematics

See also

Aspects of mathematics education

• Anti-racist mathematics (using mathematics education to fight racism)
• Cognitively Guided Instruction
• Computer Based Mathematics Education
• Number sense
• Pre-math skills
• Van Hiele model (a theory of how children learn geometry)

General education

• Constructivism
• Philosophy of education
• Traditional education

North American issues

• Mathematics education in the United States
• Reform mathematics
• Traditional mathematics

Mathematical difficulties

• Dyscalculia
• Mathematical anxiety
• Parity of zero - Example of a difficult concept for children

Other

• Arithmetic for Parents
• Early numeracy
• Handedness and mathematical ability
• L - A Mathemagical Adventure (educational adventure game)
• Make 10: A Journey of Numbers (a puzzle/logic style adventure game)
• Math worksheet generator
• QuickSmart
• Statistics education

Further reading

• John R. Anderson, Lynne M. Reder, Herbert A. Simon, K. Anders Ericsson, and Robert Glaser, 'Radical Constructivism and Cognitive Psychology', Brookings Papers on Education Policy (1998), no. 1, 227-278.
• Auslander, Maurice, et al., "Goals for School Mathematics: The Report of the Cambridge Conference on School Mathematics 1963" - Cambridge, Massachusetts, June 18, 1963.
• Sriraman,Bharath & English, Lyn., "Theories of Mathematics Education" Springer, Berlin/London, 2010. ISBN 9783642007415
• Strogatz, Steven Henry, "The Calculus of Friendship : what a teacher and a student learned about life while corresponding about math", Princeton : Princeton University Press, 2009. ISBN 9780691134932

External links

• CLSO-Math Concepts-Language-Symbols-Operation An innovative math teaching methodology developed by a Harvard graduate and based on recent learning theories and many decades of practical experience.
• History of Mathematical Education
• How to Become a Pure Mathematician a List of Undergraduate and Basic Graduate Textbook.
• hyper-Tutorials from Salford University
• HyperMath site at Georgia State University
• Interactive Mathematics from Plymouth University
• International Group for the Psychology of Mathematics Education
• K-8 Interactive Math Practice
• K-12 Online Math Resources
• Mathematics Education Articles at Convergence
• Mathematics Related E-books
• Mathematics Teaching Resource Wiki based in the UK
• Online Math Activity Objects
• Online Math Courses
• [2] A quarter century of US 'maths wars' and political partisanship. David Klein. California State University, Northridge, USA
• Teaching Leonardo: An Integrated Approach at Convergence
• Teaching Mathematics with The Geometer's Sketchpad