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==

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. In the Renaissance, the academic status of

mathematics declined, because it was strongly associated with trade and commerce, and considered

somewhat un-Christian. Although it continued to be taught in European

universities, it was seen as subservient to the study of Natural, Metaphysical and Moral

Philosophy. The first modern arithmetic curriculum (starting

with addition, then subtraction, multiplication, and division) arose at reckoning schools in

Italy in the 1300s. Spreading along trade routes, these methods

were designed to be used in commerce. They contrasted with Platonic math taught

at universities, which was more philosophical and concerned numbers as concepts rather than

calculating methods. They also contrasted with mathematical methods

learned by artisan apprentices, which were specific to the tasks and tools at hand. For example, the division of a board into

thirds can be accomplished with a piece of string, instead of measuring the length and

using the arithmetic operation of division.The first mathematics textbooks to be written

in English and French were published by Robert Recorde, beginning with The Grounde of Artes

in 1540. However, there are many different writings

on mathematics and mathematics 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 Papyrus

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. The social status of mathematical study was

improving by 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. 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

The professional periodical literature on mathematics education in the U.S.A. had generated

more than 4000 articles after 1920, so in 1941 William L. Schaaf published a classified

index, sorting them into their various subjects. A renewed interest in mathematics education

emerged in the 1960s, and the International 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 yearsIn the 20th century, the cultural impact

of the “electronic 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 and learning 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==

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: 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. Computer-based math an approach based around

use of mathematical software as the primary tool of computation. Computer-based mathematics education involving

the use of computers to teach mathematics. Mobile applications have also been developed

to help students learn mathematics. 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. 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. Historical method: teaching the development

of mathematics within an historical, social and cultural context. Provides more human interest than the conventional

approach. Mastery: an approach in which most students

are expected to achieve a high level of competence before progressing

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.” 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. Recreational mathematics: Mathematical problems

that are fun can motivate students to learn mathematics and can increase enjoyment of

mathematics. 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. Relational approach: Uses class topics to

solve everyday problems and relates the topic to current events. This approach focuses on the many uses of

mathematics and helps students understand why they need to know it as well as helping

them to apply mathematics to real world situations outside of the classroom. 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.==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. Most countries tend to cover fewer topics

in greater depth than in the United States.In most of the U.S., 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 U.S. 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. Probability and statistics may be taught in

secondary education classes. Science and engineering students in colleges

and universities may be required to take multivariable calculus, differential equations, and linear

algebra. Applied mathematics is also used in specific

majors; for example, civil engineers may be required to study fluid mechanics, while “math

for computer science” might include graph theory, permutation, probability, and proofs. Mathematics students would continue to study

potentially any area.==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. Many other countries have centralized ministries

which set national standards or curricula, and sometimes even textbooks. Ma (2000) summarised the research of others

who found, based on nationwide data, that students with higher scores on standardised

mathematics tests had taken more mathematics courses in high school. This led some states to require three years

of mathematics instead of two. But because this requirement was often met

by taking another lower level mathematics course, the additional courses had a “diluted”

effect in raising achievement levels.In North America, the National Council of Teachers

of Mathematics (NCTM) published the Principles and Standards for School Mathematics in 2000

for the US and Canada, which boosted the trend towards reform mathematics. In 2006, the NCTM released Curriculum Focal

Points, which recommend the most important mathematical topics for each grade level through

grade 8. However, these standards were guidelines to

implement as American states and Canadian provinces chose. In 2010, the National Governors Association

Center for Best Practices and the Council of Chief State School Officers published the

Common Core State Standards for US states, which were subsequently adopted by most states. Adoption of the Common Core State Standards

in mathematics is at the discretion of each state, and is not mandated by the federal

government. “States routinely review their academic standards

and may choose to change or add onto the standards to best meet the needs of their students.” The NCTM has state affiliates that have different

education standards at the state level. For example, Missouri has the Missouri Council

of Teachers of Mathematics (MCTM) which has its own pillars and standards of education

listed on its website. The MCTM also offers membership opportunities

to teachers and future teachers so they can stay up to date on the changes in math educational

standards.==Research==

“Robust, useful theories of classroom teaching do not yet exist”. However, there are useful theories on how

children learn mathematics and much research has been conducted in recent decades to explore

how these theories can be applied to teaching. The following results are examples of some

of the current findings in the field of mathematics education: Important results

One of the strongest results in recent research is that the most important feature in effective

teaching is giving students “opportunity to learn”. Teachers can set expectations, time, kinds

of tasks, questions, acceptable answers, and type of discussions that will influence students’

opportunity to learn. This must involve both skill efficiency and

conceptual understanding.Conceptual understanding Two of the most important features of teaching

in the promotion of conceptual understanding are attending explicitly to concepts and allowing

students to struggle with important mathematics. Both of these features have been confirmed

through a wide variety of studies. Explicit attention to concepts involves making

connections between facts, procedures and ideas. (This is often seen as one of the strong points

in mathematics teaching in East Asian countries, where teachers typically devote about half

of their time to making connections. At the other extreme is the U.S.A., where

essentially no connections are made in school classrooms.) These connections can be made through explanation

of the meaning of a procedure, questions comparing strategies and solutions of problems, noticing

how one problem is a special case of another, reminding students of the main point, discussing

how lessons connect, and so on.Deliberate, productive struggle with mathematical ideas

refers to the fact that when students exert effort with important mathematical ideas,

even if this struggle initially involves confusion and errors, the end result is greater learning. This has been shown to be true whether the

struggle is due to challenging, well-implemented teaching, or due to faulty teaching the students

must struggle to make sense of.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. 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.Homework Homework which leads students to practice

past lessons or prepare future lessons are more effective than those going over today’s

lesson. Students benefit from feedback. Students with learning disabilities or low

motivation may profit from rewards. For younger children, homework helps simple

skills, but not broader measures of achievement.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.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….”===Methodology===

As with other educational research (and the social sciences in general), mathematics education

research depends on both quantitative and qualitative studies. Quantitative research includes studies that

use inferential statistics to answer specific questions, such as whether a certain teaching

method gives significantly better results than the status quo. The best quantitative studies involve randomized

trials where students or classes are randomly assigned different methods in order to test

their effects. They depend on large samples to obtain statistically

significant results. Qualitative research, such as case studies,

action research, discourse analysis, and clinical interviews, depend on small but focused samples

in an attempt to understand student learning and to look at how and why a given method

gives the results it does. Such studies cannot conclusively establish

that one method is better than another, as randomized trials can, but unless it is understood

why treatment X is better than treatment Y, application of results of quantitative studies

will often lead to “lethal mutations” of the finding in actual classrooms. Exploratory qualitative research is also useful

for suggesting new hypotheses, which can eventually be tested by randomized experiments. Both qualitative and quantitative studies

therefore are considered essential in education—just as in the other social sciences. Many studies are “mixed”, simultaneously

combining aspects of both quantitative and qualitative research, as appropriate.====Randomized trials====

There has been some controversy over the relative strengths of different types of research. Because randomized trials provide clear, objective

evidence on “what works”, policy makers often take only those studies into consideration. Some scholars have pushed for more random

experiments in which teaching methods are randomly assigned to classes. 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, such as the ethical difficulty of randomly assigning students

to various treatments when the effects of such treatments are not yet known to be effective,

or the difficulty of assuring rigid control of the independent variable in fluid, real

school settings.In the United States, the National Mathematics Advisory Panel (NMAP)

published a report in 2008 based on studies, some of which used randomized assignment of

treatments to experimental units, such as classrooms or students. 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.==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 Felix Klein (1849 – 1925), German mathematician

who had substantial influence on math education in the early 20th Century, Inaugural president

of the International Commission on Mathematical Instruction

Andrei Petrovich Kiselyov (1852 – 1940) Russian and Soviet mathematician. His textbooks on basic arithmetics, algebra

and geometry were the standard for Russian classrooms since 1892 well into the 1960s,

when Russian mathematics education got embroiled in the New Math reforms. In 2006 these textbooks were re-printed and

became popular again. David Eugene Smith (1860 – 1944) American

mathematician, educator, and editor, considered one of the founders of the field of mathematics

education 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 Arthur 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

Caleb Gattegno (1911-1988), Egyptian, Founder of the Association for Teaching Aids in Mathematics

in Britain (1952) and founder of the journal Mathematics Teaching. 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

Bob Moses (1935–), founder of the nationwide US Algebra project

Robert M. Gagné (1958–1980s), pioneer in mathematics education research. David Tall (1941 – ), most cited mathematics

education researcher in modern times . Established the “Advanced Mathematic Thinking” working

group. Has contributed to the education and learning

process of advanced 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. As a mathematics educator, Dodgson defended

the use of Euclid’s Elements as a geometry textbook; Euclid and his Modern Rivals is

a criticism of a reform movement in geometry education led by the Association for the Improvement

of Geometrical Teaching. 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.==Organizations==

Advisory Committee on Mathematics Education American Mathematical Association of Two-Year

Colleges Association of Teachers of Mathematics

Mathematical Association National Council of Teachers of Mathematics==Related==

“. . . our students of mathematics would profit much more from a study of Euler’s Introductio

in Analysin Infinitorum, rather than of the available modern textbooks.” (Andr´e Weil 1979, quoted by J.D.Blanton

1988, p. xii) “..maxim of Niels Henrik Abel, “I learned

from the masters and not from the pupils” .. ”

(Extracted from Alexander Ostermann • Gerhard Wanner, Geometry by Its History )==See also