Mathematics education

Encyclopedia

In contemporary education,

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.

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

The first mathematics textbooks to be written in English and French were published by Robert Recorde

, beginning with

and the Moscow Mathematical Papyrus

. The more famous Rhind Papyrus has been dated to approximately 1650 BCE but it is thought to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students.

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 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

'."

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

Formative assessment

Effective instruction

Students with difficulties

Bilingualism

Learning statistics

Algebraic reasoning

What can we learn from research?

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.

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.

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

**mathematics education**is the practice of teaching and learningLearning

Learning is acquiring new or modifying existing knowledge, behaviors, skills, values, or preferences and may involve synthesizing different types of information. The ability to learn is possessed by humans, animals and some machines. Progress over time tends to follow learning curves.Human learning...

mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, along with the associated scholarly research

Research

Research can be defined as the scientific search for knowledge, or as any systematic investigation, to establish novel facts, solve new or existing problems, prove new ideas, or develop new theories, usually using a scientific method...

.

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

Didactics

A didactic method is a teaching method that follows a consistent scientific approach or educational style to engage the student’s mind.The didactic method of instruction is often contrasted with dialectics and the Socratic method; the term can also be used to refer to a specific didactic method,...

or pedagogy

Pedagogy

Pedagogy is the study of being a teacher or the process of teaching. The term generally refers to strategies of instruction, or a style of instruction....

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

Elementary mathematics was part of the education systemEducation

Education in its broadest, general sense is the means through which the aims and habits of a group of people lives on from one generation to the next. Generally, it occurs through any experience that has a formative effect on the way one thinks, feels, or acts...

in most ancient civilisations, including Ancient Greece

Ancient Greece

Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

, the Roman empire

Roman Empire

The Roman Empire was the post-Republican period of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean....

, Vedic society and ancient Egypt

Ancient Egypt

Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...

. In most cases, a formal education was only available to male

Male

Male refers to the biological sex of an organism, or part of an organism, which produces small mobile gametes, called spermatozoa. Each spermatozoon can fuse with a larger female gamete or ovum, in the process of fertilization...

children with a sufficiently high status, wealth or caste

Caste

Caste is an elaborate and complex social system that combines elements of endogamy, occupation, culture, social class, tribal affiliation and political power. It should not be confused with race or social class, e.g. members of different castes in one society may belong to the same race, as in India...

.

In Plato

Plato

Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

's division of the liberal arts

Liberal arts

The term liberal arts refers to those subjects which in classical antiquity were considered essential for a free citizen to study. Grammar, Rhetoric and Logic were the core liberal arts. In medieval times these subjects were extended to include mathematics, geometry, music and astronomy...

into the trivium and the quadrivium

Quadrivium

The quadrivium comprised the four subjects, or arts, taught in medieval universities, after teaching the trivium. The word is Latin, meaning "the four ways" , and its use for the 4 subjects has been attributed to Boethius or Cassiodorus in the 6th century...

, the quadrivium included the mathematical fields of arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

and geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

. This structure was continued in the structure of classical education

Medieval university

Medieval university is an institution of higher learning which was established during High Middle Ages period and is a corporation.The first institutions generally considered to be universities were established in Italy, France, and England in the late 11th and the 12th centuries for the study of...

that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclid

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

'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

Robert Recorde

Robert Recorde was a Welsh physician and mathematician. He introduced the "equals" sign in 1557.-Biography:A member of a respectable family of Tenby, Wales, he entered the University of Oxford about 1525, and was elected a fellow of All Souls College in 1531...

, beginning with

*The Grounde of Artes*in 1540. However, there are many different writings on mathematics and math methodology that date back to 1800 BCE. These were mostly located in Mesopotamia where the Sumerians were practicing multiplication and division. There are also artifacts demonstrating their own methodology for solving equations like the quadratic equation. After the Sumerians some of the most famous ancient works on mathematics come from Egypt in the form of the Rhind Mathematical PapyrusRhind Mathematical Papyrus

The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...

and the Moscow Mathematical Papyrus

Moscow Mathematical Papyrus

The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...

. The more famous Rhind Papyrus has been dated to approximately 1650 BCE but it is thought to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students.

In the Renaissance

Renaissance

The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...

, 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

Natural philosophy

Natural philosophy or the philosophy of nature , is a term applied to the study of nature and the physical universe that was dominant before the development of modern science...

, Metaphysical

Metaphysics

Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...

and Moral Philosophy.

This trend was somewhat reversed in the seventeenth century, with the University of Aberdeen

University of Aberdeen

The University of Aberdeen, an ancient university founded in 1495, in Aberdeen, Scotland, is a British university. It is the third oldest university in Scotland, and the fifth oldest in the United Kingdom and wider English-speaking world...

creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in University of Oxford

University of Oxford

The University of Oxford is a university located in Oxford, United Kingdom. It is the second-oldest surviving university in the world and the oldest in the English-speaking world. Although its exact date of foundation is unclear, there is evidence of teaching as far back as 1096...

in 1619 and the Lucasian Chair of Mathematics being established by the University of Cambridge

University of Cambridge

The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...

in 1662. However, it was uncommon for mathematics to be taught outside of the universities. Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

, for example, received no formal mathematics teaching until he joined Trinity College, Cambridge

Trinity College, Cambridge

Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...

in 1661.

In the 18th and 19th centuries, the industrial revolution

Industrial Revolution

The Industrial Revolution was a period from the 18th to the 19th century where major changes in agriculture, manufacturing, mining, transportation, and technology had a profound effect on the social, economic and cultural conditions of the times...

led to an enormous increase in urban

Urban area

An urban area is characterized by higher population density and vast human features in comparison to areas surrounding it. Urban areas may be cities, towns or conurbations, but the term is not commonly extended to rural settlements such as villages and hamlets.Urban areas are created and further...

populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, became essential in this new urban lifestyle. Within the new public education

Public education

State schools, also known in the United States and Canada as public schools,In much of the Commonwealth, including Australia, New Zealand, South Africa, and the United Kingdom, the terms 'public education', 'public school' and 'independent school' are used for private schools, that is, schools...

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

Developed country

A developed country is a country that has a high level of development according to some criteria. Which criteria, and which countries are classified as being developed, is a contentious issue...

.

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 KleinFelix KleinChristian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
- 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 NottinghamNottinghamNottingham is a city and unitary authority in the East Midlands of England. It is located in the ceremonial county of Nottinghamshire and represents one of eight members of the English Core Cities Group...
- The first International Congress on Mathematical EducationInternational Congress on Mathematical EducationThe International Congress on Mathematical Education is held every four years under the auspices of the International Commission on Mathematical Instruction of the International Mathematical Union ....

(ICME) was held in LyonLyonLyon , is a city in east-central France in the Rhône-Alpes region, situated between Paris and Marseille. Lyon is located at from Paris, from Marseille, from Geneva, from Turin, and from Barcelona. The residents of the city are called Lyonnais....

in 1969. The second congress was in ExeterExeterExeter is a historic city in Devon, England. It lies within the ceremonial county of Devon, of which it is the county town as well as the home of Devon County Council. Currently the administrative area has the status of a non-metropolitan district, and is therefore under the administration of the...

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

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

", the emerging structural approach to knowledge had "small children meditating about number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

and 'sets

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

'."

## 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 numeracyNumeracyNumeracy is the ability to reason with numbers and other mathematical concepts. A numerically literate person can manage and respond to the mathematical demands of life...

skills to all pupils - The teaching of practical mathematics (arithmeticArithmetic

, elementary algebraElementary algebraElementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...

, plane and solid geometryGeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, trigonometryTrigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

) to most pupils, to equip them to follow a trade or craft - The teaching of abstract mathematical concepts (such as set and functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

) at an early age - The teaching of selected areas of mathematics (such as Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

) as an example of an axiomatic systemAxiomatic systemIn mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...

and a model of deductive reasoningDeductive reasoningDeductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis... - The teaching of selected areas of mathematics (such as calculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

) as an example of the intellectual achievements of the modern worldModern WorldModern World or The Modern World may refer to:*Modernity, a popular academic term.*The modern era, the age in which people today now live.*Modern World, a song by Wolf Parade from their 2005 album Apologies to the Queen Mary.... - The teaching of advanced mathematics to those pupils who wish to follow a career in Science, Technology, Engineering, and Mathematics (STEM)STEM fieldsSTEM fields is a US Government acronym for the fields of study in the categories of science, technology, engineering, and mathematics. The acronym is in use regarding access to work visas for immigrants who are skilled in these fields. Maintaining a citizenry that is well versed in the STEM fields...

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 assessmentFormative assessmentFormative assessment is a range of formal and informal assessment procedures employed by teachers during the learning process in order to modify teaching and learning activities to improve student attainment. It typically involves qualitative feedback for both student and teacher that focuses on...

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 instructionDirect instructionDirect Instruction is an instructional method that is focused on systematic curriculum design and skillful implementation of a prescribed behavioral script....

, 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 factsRecall of factsRecall of facts is aWork Research InstituteFrom Wikipedia, the free encyclopediaThe Work Research Institute is a social science research institute based in Oslo, Norway. It is state owned, but independent in its research. Its purpose is to "produce systematic knowledge on working life"...

, answer impulsively, struggle with mental representations, have poor number senseNumber senseIn mathematics education, number sense can refer to "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations." Many other definitions exist, but are similar to the one given...

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 assessmentFormative assessmentFormative assessment is a range of formal and informal assessment procedures employed by teachers during the learning process in order to modify teaching and learning activities to improve student attainment. It typically involves qualitative feedback for both student and teacher that focuses on...

and encouraging students to think aloud.

Bilingualism

- Most bilingual adults switch languages when calculating. Such code-switchingCode-switchingIn linguistics, code-switching is the concurrent use of more than one language, or language variety, in conversation. Multilinguals—people who speak more than one language—sometimes use elements of multiple languages in conversing with each other...

has no impact on math ability and should not be discouraged.

Learning statistics

- When studying statisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, children benefit from exploring, studying, and sharing reasoning about centers, shape, spread, and variability. The ability to calculate averageAverageIn mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

s does not signify an understanding of population and sample mean. Students have difficulty remembering to use the sample median as a robustRobust statisticsRobust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

estimatorEstimatorIn statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....

of locationLocation parameterIn statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...

, which is less sensitive than the sample mean or the sample modeMode (statistics)In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....

when the data-set has outlierOutlierIn statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....

s or when the population is skewedSkewnessIn probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...

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 variableVariable (mathematics)In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

. 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 signEquals signThe equality sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. The equals sign is placed between the things stated to have the same value, as in an equation...

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

Epidemiology

Epidemiology is the study of health-event, health-characteristic, or health-determinant patterns in a population. It is the cornerstone method of public health research, and helps inform policy decisions and evidence-based medicine by identifying risk factors for disease and targets for preventive...

, political science, economics, sociology

Sociology

Sociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...

, anthropology

Anthropology

Anthropology is the study of humanity. It has origins in the humanities, the natural sciences, and the social sciences. The term "anthropology" is from the Greek anthrōpos , "man", understood to mean mankind or humanity, and -logia , "discourse" or "study", and was first used in 1501 by German...

, 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

United States Department of Education

The United States Department of Education, also referred to as ED or the ED for Education Department, is a Cabinet-level department of the United States government...

) 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

England

England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

, 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

Common core state standards initiative

The Common Core State Standards Initiative is a U.S. education initiative that seeks to bring diverse state curricula into alignment with each other by following the principles of standards-based education reform. The initiative is sponsored by the National Governors Association and the Council...

.

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

Principles and Standards for School Mathematics

Principles and Standards for School Mathematics are guidelines produced by the National Council of Teachers of Mathematics in 2000, setting forth recommendations for mathematics educators. They form a national vision for preschool through twelfth grade mathematics education in the US and Canada...

. 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

Mathematics education in the United States

From kindergarten through high-school, the mathematics education in public schools in the United States varies widely from state to state, and often even varies considerably within individual 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 arithmeticArithmetic

and is followed by Euclidean geometryEuclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

and elementary algebraElementary algebraElementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...

taught concurrently. Requires the instructor to be well informed about elementary mathematicsElementary mathematicsElementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry...

, 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 quadriviumQuadriviumThe quadrivium comprised the four subjects, or arts, taught in medieval universities, after teaching the trivium. The word is Latin, meaning "the four ways" , and its use for the 4 subjects has been attributed to Boethius or Cassiodorus in the 6th century...

, part of the classical education curriculum of the Middle AgesMiddle AgesThe Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...

, which was typically based on Euclid's*Elements*Euclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

taught as a paradigmParadigmThe word paradigm has been used in science to describe distinct concepts. It comes from Greek "παράδειγμα" , "pattern, example, sample" from the verb "παραδείκνυμι" , "exhibit, represent, expose" and that from "παρά" , "beside, beyond" + "δείκνυμι" , "to show, to point out".The original Greek...

of deductive reasoningDeductive reasoningDeductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

.**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 isRote learningRote learning is a learning technique which focuses on memorization. The major practice involved in rote learning is learning by repetition by which students commit information to memory in a highly structured way. The idea is that one will be able to quickly recall the meaning of the material the...*drill and kill*. In traditional educationTraditional educationTraditional education or back-to-basics refers to long-established customs found in schools that society has traditionally deemed appropriate. Some forms of education reform promote the adoption of progressive education practices, a more holistic approach which focuses on individual students'...

, rote learning is used to teach multiplication tableMultiplication tableIn mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....

s, 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 equationQuadratic equationIn mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

s.**Problem solving**- the cultivation of mathematical ingenuity, creativity and heuristicProblem solvingProblem solving is a mental process and is part of the larger problem process that includes problem finding and problem shaping. Consideredthe most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of...HeuristicHeuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

thinking by setting students open-ended, unusual, and sometimes unsolved problems. The problems can range from simple word problemWord problem (mathematics education)In mathematics education, the term word problem is often used to refer to any math exercise where significant background information on the problem is presented as text rather than in mathematical notation...

s to problems from international mathematics competitions such as the International Mathematical OlympiadInternational Mathematical OlympiadThe International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...

. 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 theoryNew mathNew Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S...Set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, 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'sMorris KlineMorris Kline was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.Kline grew up in Brooklyn and in Jamaica, Queens...

1973 book*Why Johnny Can't Add*. The New Math method was the topic of one of Tom LehrerTom LehrerThomas Andrew "Tom" Lehrer is an American singer-songwriter, satirist, pianist, mathematician and polymath. He has lectured on mathematics and musical theater...

'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 USReform mathematicsReform mathematics is an approach to mathematics education, particularly in North America. It is based on principles explained in 1989 by the National Council of Teachers of Mathematics . The NCTM document, Curriculum and Evaluation Standards for School Mathematics, attempted to set forth a vision...United StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...

and CanadaCanadaCanada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...

, focused on deepening student understanding of mathematical ideas and procedures, and formalized by the National Council of Teachers of MathematicsNational Council of Teachers of MathematicsThe National Council of Teachers of Mathematics was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally....

which created the Principles and Standards for School MathematicsPrinciples and Standards for School MathematicsPrinciples and Standards for School Mathematics are guidelines produced by the National Council of Teachers of Mathematics in 2000, setting forth recommendations for mathematics educators. They form a national vision for preschool through twelfth grade mathematics education in the US and Canada...

.**Relational approach**- Uses class topics to solve everyday problems and relates the to topic to current events. This approach focuses on the many uses of math and helps students understand why they need to know it as well as helping them to apply math to real world situations outside of the classroom.

## Mathematics teachers

The following people all taught mathematics at some stage in their lives, although they are better known for other things:- Lewis CarrollLewis CarrollCharles Lutwidge Dodgson , better known by the pseudonym Lewis Carroll , was an English author, mathematician, logician, Anglican deacon and photographer. His most famous writings are Alice's Adventures in Wonderland and its sequel Through the Looking-Glass, as well as the poems "The Hunting of the...

, pen name of BritishUnited KingdomThe United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...

author Charles Dodgson, lectured in mathematics at Christ Church, OxfordChrist Church, OxfordChrist Church or house of Christ, and thus sometimes known as The House), is one of the largest constituent colleges of the University of Oxford in England... - John DaltonJohn DaltonJohn Dalton FRS was an English chemist, meteorologist and physicist. He is best known for his pioneering work in the development of modern atomic theory, and his research into colour blindness .-Early life:John Dalton was born into a Quaker family at Eaglesfield, near Cockermouth, Cumberland,...

, British chemist and physicist, taught mathematics at schools and colleges in ManchesterManchesterManchester is a city and metropolitan borough in Greater Manchester, England. According to the Office for National Statistics, the 2010 mid-year population estimate for Manchester was 498,800. Manchester lies within one of the UK's largest metropolitan areas, the metropolitan county of Greater...

, OxfordOxfordThe city of Oxford is the county town of Oxfordshire, England. The city, made prominent by its medieval university, has a population of just under 165,000, with 153,900 living within the district boundary. It lies about 50 miles north-west of London. The rivers Cherwell and Thames run through...

and YorkYorkYork is a walled city, situated at the confluence of the Rivers Ouse and Foss in North Yorkshire, England. The city has a rich heritage and has provided the backdrop to major political events throughout much of its two millennia of existence... - Tom LehrerTom LehrerThomas Andrew "Tom" Lehrer is an American singer-songwriter, satirist, pianist, mathematician and polymath. He has lectured on mathematics and musical theater...

, AmericanUnited StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...

songwriter and satirist, taught mathematics at Harvard, MIT and currently at University of California, Santa CruzUniversity of California, Santa CruzThe University of California, Santa Cruz, also known as UC Santa Cruz or UCSC, is a public, collegiate university; one of ten campuses in the University of California... - Brian MayBrian MayBrian Harold May, CBE is an English musician and astrophysicist most widely known as the guitarist and a songwriter of the rock band Queen...

, rock guitarist and composer, worked briefly as a mathematics teacher before joining QueenQueen (band)Queen are a British rock band formed in London in 1971, originally consisting of Freddie Mercury , Brian May , John Deacon , and Roger Taylor... - Georg Joachim RheticusGeorg Joachim RheticusGeorg Joachim von Lauchen, also known as Rheticus , was a mathematician, cartographer, navigational-instrument maker, medical practitioner, and teacher. He is perhaps best known for his trigonometric tables and as Nicolaus Copernicus's sole pupil...

, AustriaAustriaAustria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

n cartographer and disciple of Copernicus, taught mathematics at the University of Wittenberg - Edmund RichEdmund RichEdmund Rich was a 13th century Archbishop of Canterbury in England...

, Archbishop of CanterburyArchbishop of CanterburyThe Archbishop of Canterbury is the senior bishop and principal leader of the Church of England, the symbolic head of the worldwide Anglican Communion, and the diocesan bishop of the Diocese of Canterbury. In his role as head of the Anglican Communion, the archbishop leads the third largest group...

in the 13th century, lectured on mathematics at the universities of OxfordOxfordThe city of Oxford is the county town of Oxfordshire, England. The city, made prominent by its medieval university, has a population of just under 165,000, with 153,900 living within the district boundary. It lies about 50 miles north-west of London. The rivers Cherwell and Thames run through...

and ParisParisParis is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region... - Éamon de ValeraÉamon de ValeraÉamon de Valera was one of the dominant political figures in twentieth century Ireland, serving as head of government of the Irish Free State and head of government and head of state of Ireland...

, a leader of Ireland's struggle for independence in the early 20th century and founder of the Fianna FáilFianna FáilFianna Fáil – The Republican Party , more commonly known as Fianna Fáil is a centrist political party in the Republic of Ireland, founded on 23 March 1926. Fianna Fáil's name is traditionally translated into English as Soldiers of Destiny, although a more accurate rendition would be Warriors of Fál...

party, taught mathematics at schools and colleges in Dublin - Archie WilliamsArchie WilliamsArchibald Franklin Williams was an African-American athlete and teacher, winner of 400 meter run at the 1936 Summer Olympics....

, AmericanUnited StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...

athlete and Olympic gold medalist, taught mathematics at high schools in CaliforniaCaliforniaCalifornia is a state located on the West Coast of the United States. It is by far the most populous U.S. state, and the third-largest by land area...

.

## 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:- EuclidEuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

(fl.FloruitFloruit , abbreviated fl. , is a Latin verb meaning "flourished", denoting the period of time during which something was active...

300 BC), Ancient Greek, author of*The Elements*Euclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions... - Tatyana Alexeyevna Afanasyeva (1876–1964), DutchNetherlandsThe Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...

/RussiaRussiaRussia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

n mathematician who advocated the use of visual aids and examples for introductory courses in geometry for high school students - Robert Lee MooreRobert Lee MooreRobert Lee Moore was an American mathematician, known for his work in general topology and the Moore method of teaching university mathematics.-Life:...

(1882–1974), American mathematician, originator of the Moore methodMoore methodThe Moore method is a deductive manner of instruction used in advanced mathematics courses. It is named after Robert Lee Moore, a famous topologist who first used a stronger version of the method at the University of Pennsylvania when he began teaching there in 1911.The way the course is conducted... - George PólyaGeorge PólyaGeorge Pólya was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory...

(1887–1985), Hungarian mathematician, author of How to Solve ItHow to Solve ItHow to Solve It is a small volume by mathematician George Pólya describing methods of problem solving.- Four principles :How to Solve It suggests the following steps when solving a mathematical problem:... - Georges Cuisenaire (1891–1976), BelgianBelgiumBelgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...

primary school teacher who invented Cuisenaire rodsCuisenaire rodsCuisenaire rods give students a hands-on elementary school way to learn elementary math concepts, such as the four basic arithmetic operations and working with fractions.... - 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 FreudenthalHans FreudenthalHans Freudenthal was a Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education....

(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 KumonToru Kumonwas a Japanese mathematics educator, born in Kōchi Prefecture, Japan. He graduated from the College of Science at Osaka University with a degree in mathematics and taught high school mathematics in his home town of Osaka...

(1914–1995), Japanese, originator of the Kumon methodKumon methodThe Kumon method, developed by educator Toru Kumon, is a math and reading educational method that is practiced in Kumon's learning centers.-History:...

, 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 MosesRobert Parris MosesRobert Parris Moses is an American, Harvard-trained educator who was a leader in the 1960s Civil Rights Movement and later founded the nationwide U.S. Algebra project.-Biography:...

(1935-), founder of the nationwide USUnited States

Algebra projectAlgebra ProjectThe Algebra Project is a national U.S. mathematics literacy effort aimed at helping low-income students and students of color successfully achieve mathematical skills that are a prerequisite for a college preparatory mathematics sequence in high school... - 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 EducationAdvisory Committee on Mathematics EducationThe Advisory Committee on Mathematics Education is a major British organisation established in January 2002 by the Royal Society and the Joint Mathematical Council, with explicit backing from all major mathematics organisations in the UK....
- Association of Teachers of MathematicsAssociation of Teachers of MathematicsThe Association of Teachers of Mathematics was established in 1950 to encourage the development of mathematics education to be more closely related to the needs of the learner...
- Mathematical AssociationMathematical AssociationThe Mathematical Association is a professional society concerned with mathematics education in the UK.-History:It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1897. It was the first teachers' subject organisation...
- National Council of Teachers of MathematicsNational Council of Teachers of MathematicsThe National Council of Teachers of Mathematics was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally....

## 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 HenrySteven StrogatzSteven Henry Strogatz is an American mathematician and the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University...

, "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
- http://www.csun.edu/~vcmth00m/bshm.html 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