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In contemporary
education Education is a purposeful activity directed at achieving certain aims, such as transmitting knowledge or fostering skills and character traits. These aims may include the development of understanding, rationality, kindness, and honesty ...
, mathematics education, known in Europe as the didactics or
pedagogy Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political and psychological development of learners. Pedagogy, taken ...
of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
– is the practice of
teaching Teaching is the practice implemented by a ''teacher'' aimed at transmitting skills (knowledge, know-how, and interpersonal skills) to a learner, a student, or any other audience in the context of an educational institution. Teaching is closely ...
,
learning Learning is the process of acquiring new understanding, knowledge, behaviors, skills, values, attitudes, and preferences. The ability to learn is possessed by humans, animals, and some machines; there is also evidence for some kind of lea ...
and carrying out scholarly
research Research is "creative and systematic work undertaken to increase the stock of knowledge". It involves the collection, organization and analysis of evidence to increase understanding of a topic, characterized by a particular attentiveness ...
into the transfer of mathematical knowledge. Although research into mathematics education is primarily concerned with the tools, methods and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods. National and international organisations regularly hold conferences and publish literature in order to improve mathematics education.


History


Ancient

Elementary mathematics were a core part of education in many ancient civilisations, including ancient Egypt,
ancient Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...
,
ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cu ...
,
ancient Rome In modern historiography, ancient Rome refers to Roman people, Roman civilisation from the founding of the city of Rome in the 8th century BC to the collapse of the Western Roman Empire in the 5th century AD. It encompasses the Roman Kingdom ...
and
Vedic upright=1.2, The Vedas are ancient Sanskrit texts of Hinduism. Above: A page from the '' Atharvaveda''. The Vedas (, , ) are a large body of religious texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute th ...
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...
. In most cases, formal education was only available to
male Male (symbol: ♂) is the sex of an organism that produces the gamete (sex cell) known as sperm, which fuses with the larger female gamete, or ovum, in the process of fertilization. A male organism cannot reproduce sexually without access to ...
children with sufficiently high status, wealth or
caste Caste is a form of social stratification characterised by endogamy, hereditary transmission of a style of life which often includes an occupation, ritual status in a hierarchy, and customary social interaction and exclusion based on cultur ...
. The oldest known mathematics textbook is the Rhind papyrus, dated from circa 1650 BCE.


Pythagorean theorem

Historians of
Mesopotamia Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the ...
have confirmed that use of the Pythagorean rule dates back to the
Old Babylonian Empire The Old Babylonian Empire, or First Babylonian Empire, is dated to BC – BC, and comes after the end of Sumerian power with the destruction of the Third Dynasty of Ur, and the subsequent Isin-Larsa period. The chronology of the first dynasty ...
(20th to 16th centuries BC) and that it was being taught in scribal schools over one thousand years before the birth of
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His poli ...
. In
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's division of the
liberal arts Liberal arts education (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as La ...
into the
trivium The trivium is the lower division of the seven liberal arts and comprises grammar, logic, and rhetoric. The trivium is implicit in ''De nuptiis Philologiae et Mercurii'' ("On the Marriage of Philology and Mercury") by Martianus Capella, but t ...
and the
quadrivium From the time of Plato through the Middle Ages, the ''quadrivium'' (plural: quadrivia) was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the ...
, the quadrivium included the mathematical fields of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. This structure was continued in the structure of classical education that was developed in medieval Europe. The 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.


Medieval and early modern

In the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
, 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 This is a list of lists of universities and colleges by country, sorted by continent and region. The lists represent educational institutions throughout the world which provide higher education in tertiary, quaternary, and post-secondary educatio ...
, it was seen as subservient to the study of
Natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
,
Metaphysical Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
and
Moral Philosophy Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong behavior".''Internet Encyclopedia of Philosophy'' The field of ethics, along with aesthetics, concerns ...
. The first modern arithmetic curriculum (starting with
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
, then
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
, 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 An artisan (from french: artisan, it, artigiano) is a skilled craft worker who makes or creates material objects partly or entirely by hand. These objects may be functional or strictly decorative, for example furniture, decorative art ...
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 Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and las ...
, beginning with ''The Grounde of Artes'' in 1543. 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 methodology for solving equations like the
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
. After the Sumerians, some of the most famous
ancient Ancient history is a time period from the beginning of writing and recorded human history to as far as late antiquity. The span of recorded history is roughly 5,000 years, beginning with the Sumerian cuneiform script. Ancient history cov ...
works on mathematics came from Egypt in the form of the
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scotland, Scottish antiquarian, who ...
and the Moscow Mathematical Papyrus. The more famous Rhind Papyrus has been dated back 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 , mottoeng = The fear of the Lord is the beginning of wisdom , established = , type = Public research universityAncient university , endowment = £58.4 million (2021) , budget ...
creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in
University of Oxford , mottoeng = The Lord is my light , established = , endowment = £6.1 billion (including colleges) (2019) , budget = £2.145 billion (2019–20) , chancellor ...
in 1619 and the Lucasian Chair of Mathematics being established by the
University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...
in 1662.


Modern

In the 18th and 19th centuries, the
Industrial Revolution The Industrial Revolution was the transition to new manufacturing processes in Great Britain, continental Europe, and the United States, that occurred during the period from around 1760 to about 1820–1840. This transition included going f ...
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 Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, became essential in this new urban lifestyle. Within the new
public education State schools (in England, Wales, Australia and New Zealand) or public schools (Scottish English and North American English) are generally primary or secondary educational institution, schools that educate all students without charge. They are ...
systems, mathematics became a central part of the
curriculum In education, a curriculum (; plural, : curricula or curriculums) is broadly defined as the totality of student experiences that occur in the educational process. The term often refers specifically to a planned sequence of instruction, or to ...
from an early age. By the twentieth century, mathematics was part of the core curriculum in all
developed countries A developed country (or industrialized country, high-income country, more economically developed country (MEDC), advanced country) is a sovereign state that has a high quality of life, developed economy and advanced technological infrastruct ...
. 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 A periodical literature (also called a periodical publication or simply a periodical) is a published work that appears in a new edition on a regular schedule. The most familiar example is a newspaper, but a magazine or a journal are also example ...
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, th
Shell Centre for Mathematical Education
was established in
Nottingham Nottingham ( , locally ) is a city and unitary authority area in Nottinghamshire, East Midlands, England. It is located north-west of London, south-east of Sheffield and north-east of Birmingham. Nottingham has links to the legend of Robi ...
. * The first
International Congress on Mathematical Education The International Commission on Mathematical Instruction (ICMI) is a commission of the International Mathematical Union and is an internationally acting organization focussing on mathematics education. ICMI was founded in 1908 at the Internationa ...
(ICME) was held in
Lyon Lyon,, ; Occitan: ''Lion'', hist. ''Lionés'' also spelled in English as Lyons, is the third-largest city and second-largest metropolitan area of France. It is located at the confluence of the rivers Rhône and Saône, to the northwest of ...
in 1969. The second congress was in
Exeter Exeter () is a city in Devon, South West England. It is situated on the River Exe, approximately northeast of Plymouth and southwest of Bristol. In Roman Britain, Exeter was established as the base of Legio II Augusta under the personal comm ...
in 1972, and after that, it has been held every four years In the 20th century, the cultural impact of the "
electronic age The Information Age (also known as the Computer Age, Digital Age, Silicon Age, or New Media Age) is a historical period that began in the mid-20th century. It is characterized by a rapid shift from traditional industries, as established during t ...
" (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 () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
", the emerging structural approach to knowledge had "small children meditating about
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
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 Numeracy is the ability to understand, reason with, and to apply simple numerical concepts. The charity National Numeracy states: "Numeracy means understanding how mathematics is used in the real world and being able to apply it to make the bes ...
skills to all students * The teaching of practical mathematics (
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
,
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entail ...
, plane and solid
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
,
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
,
statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
) to most students, to equip them to follow a trade or craft and to understand mathematics commonly used in news and
internet The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, p ...
(
percentage In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also use ...
s,
chart A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent ...
s,
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
,
statistic A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
, etc.) * The teaching of abstract mathematical concepts (such as
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
and function) at an early age * The teaching of selected areas of mathematics (such as
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
) as an example of an
axiomatic system In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...
and a model of
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
* The teaching of selected areas of mathematics (such as
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
) as an example of the intellectual achievements of the
modern world The term modern period or modern era (sometimes also called modern history or modern times) is the period of history that succeeds the Middle Ages (which ended approximately 1500 AD). This terminology is a historical periodization that is appli ...
* The teaching of advanced mathematics to those students who wish to follow a career in science, technology, engineering, and mathematics (STEM) fields * The teaching of
heuristics A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
and other problem-solving strategies to solve non-routine problems *The teaching of mathematics in
social science Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of s ...
s and actuarial sciences, as well as in some selected
arts The arts are a very wide range of human practices of creative expression, storytelling and cultural participation. They encompass multiple diverse and plural modes of thinking, doing and being, in an extremely broad range of media. Both ...
under
liberal arts Liberal arts education (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as La ...
education in
liberal arts college A liberal arts college or liberal arts institution of higher education is a college with an emphasis on undergraduate study in liberal arts and sciences. Such colleges aim to impart a broad general knowledge and develop general intellectual ca ...
s or universities


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: *
Computer-based math Computer-Based Math is an educational project started by Conrad Wolfram in 2010 to promote the idea that routine mathematical calculations should be done with a computer. Conrad Wolfram believes that mathematics education should make the greatest ...
: an approach based on the use of mathematical software as the primary tool of computation. *
Computer-based mathematics education Computer-based mathematics education (CBME) is an approach to teaching mathematics that emphasizes the use of computers. Computers in math education Computers are used in education in a number of ways, such as interactive tutorials, hyperme ...
: involves the use of computers to teach mathematics. Mobile applications have also been developed to help students learn mathematics. * Classical education: the teaching of mathematics within the
quadrivium From the time of Plato through the Middle Ages, the ''quadrivium'' (plural: quadrivia) was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the ...
, part of the classical education curriculum of the
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
, which was typically based on Euclid's ''Elements'' taught as a
paradigm In science and philosophy, a paradigm () is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitute legitimate contributions to a field. Etymology ''Paradigm'' comes f ...
of
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
. * Conventional approach: the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
and is followed by
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
and
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entail ...
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. *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 helps them to apply mathematics to real-world situations outside of the classroom. *Historical method: teaching the development of mathematics within a historical, social and cultural context. Proponents argue it provides more
human interest In journalism, a human-interest story is a feature story that discusses people or pets in an emotional way. It presents people and their problems, concerns, or achievements in a way that brings about interest, sympathy or motivation in the reade ...
than the conventional approach. *Discovery math: a constructivist method of teaching ( discovery learning) mathematics which centres around problem-based or inquiry-based learning, with the use of open-ended questions and manipulative tools. This type of mathematics education was implemented in various parts of Canada beginning in 2005. Discovery-based mathematics is at the forefront of the Canadian Math Wars debate with many criticizing it for declining math scores. * New Math: a method of teaching mathematics which focuses on abstract concepts such as
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, 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." * Recreational mathematics: Mathematical problems that are fun can motivate students to learn mathematics and can increase their enjoyment of mathematics. * Standards-based mathematics: a vision for pre-college mathematics education in the US and
Canada Canada is a country in North America. Its ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, covering over , making it the world's second-largest country by to ...
, 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. *
Mastery A skill is the learned ability to act with determined results with good execution often within a given amount of time, energy, or both. Skills can often be divided into domain-general and domain-specific skills. For example, in the domain of w ...
: an approach in which most students are expected to achieve a high level of competence before progressing. *
Problem solving Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
: the cultivation of mathematical ingenuity,
creativity Creativity is a phenomenon whereby something new and valuable is formed. The created item may be intangible (such as an idea, a scientific theory, a musical composition, or a joke) or a physical object (such as an invention, a printed lit ...
and
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
thinking by setting students open-ended, unusual, and sometimes
unsolved problems List of unsolved problems may refer to several notable conjectures or open problems in various academic fields: Natural sciences, engineering and medicine * Unsolved problems in astronomy * Unsolved problems in biology * Unsolved problems in chem ...
. The problems can range from simple word problems to problems from international
mathematics competitions Mathematics competitions or mathematical olympiads are competitive events where participants complete a math test. These tests may require multiple choice or numeric answers, or a detailed written solution or proof. International mathematics comp ...
such as the
International Mathematical Olympiad The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre- university 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, excep ...
. Problem-solving is used as a means to build new mathematical knowledge, typically by building on students' prior understandings. *
Exercise Exercise is a body activity that enhances or maintains physical fitness and overall health and wellness. It is performed for various reasons, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic ...
s: the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding vulgar fractions or solving
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
s. *
Rote learning Rote learning is a memorization technique based on repetition. The method rests on the premise that the recall of repeated material becomes faster the more one repeats it. Some of the alternatives to rote learning include meaningful 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 Traditional education, also known as back-to-basics, conventional education or customary education, refers to long-established customs that society has traditionally used in schools. Some forms of education reform promote the adoption of progressiv ...
, 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 similarly, though there are differences. Most countries tend to cover fewer topics in greater depth than in the United States. During the primary school years, children learn about whole numbers and arithmetic, including addition, subtraction, multiplication, and division. Comparisons and
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
are taught, in both numeric and pictorial form, as well as fractions and proportionality, patterns, and various topics related to geometry. At high school level, in most of the U.S.,
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and analysis ( pre-calculus and
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
) are taught as separate courses in different years. 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 In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...
and
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
at age 16–17 and
integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
, complex numbers,
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
, exponential and
logarithmic function In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s, and infinite series in their final year of secondary school.
Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
may be taught in secondary education classes. In some systems, such as South Africa, the subject may be offered re functionality (Mathematics, Mathematical Literacy and Technical Mathematics). At college and university,
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
- and engineering students will be required to take
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with Function of several real variables, functions of several variables: the Differential calculus, di ...
,
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
; at several US colleges, the minor or AS in mathematics substantively comprises these courses. Mathematics majors continue, to study various other areas within
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
- and often in applied mathematics - with the requirement of specified advanced courses in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
and
modern algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
.
Applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
may be taken as a
major Major ( commandant in certain jurisdictions) is a military rank of commissioned officer status, with corresponding ranks existing in many military forces throughout the world. When used unhyphenated and in conjunction with no other indicato ...
subject in its own right, while specific topics are taught within other courses: for example,
civil engineers This list of civil engineers is a list of notable people who have been trained in or have practiced civil engineering. A B C D E F G H I J K L M N O P Q R S T U ...
may be required to study
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, and "math for computer science" might include
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
,
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
, probability, and formal
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every pr ...
s. Pure and applied math degrees often include modules in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
/
mathematical statistics Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...
; while a course in numerical methods is a common requirement for applied math. (Theoretical)
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
is mathematics intensive, often overlapping substantively with the pure or applied math degree. ( "Business mathematics" is usually limited to introductory calculus and, sometimes,
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
calculations. Economics programs additionally cover
optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
, often
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, sometimes analysis.)


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 is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe ...
, for example, standards for mathematics education are set as part of the National Curriculum for England, while
Scotland Scotland (, ) is a country that is part of the United Kingdom. Covering the northern third of the island of Great Britain, mainland Scotland has a border with England to the southeast and is otherwise surrounded by the Atlantic Ocean to ...
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 The Common Core State Standards Initiative, also known as simply Common Core, is an educational initiative from 2010 that details what K–12 students throughout the United States should know in English language arts and mathematics at the conc ...
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 Academic standards are the benchmarks of quality and excellence in education such as the rigour of curricula and the difficulty of examinations. The creation of universal academic standards requires agreement on rubrics, criteria or other system ...
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 Missouri is a state in the Midwestern region of the United States. Ranking 21st in land area, it is bordered by eight states (tied for the most with Tennessee): Iowa to the north, Illinois, Kentucky and Tennessee to the east, Arkansas t ...
has the Missouri Council of Teachers of Mathematics (MCTM) which has its pillars and standards of education listed on its website. The MCTM also offers membership opportunities to teachers and future teachers so that they can stay up to date on the changes in math educational standards. The Programme for International Student Assessment (PISA), created by the Organisation for the Economic Co-operation and Development (OECD), is a global program studying the reading, science and mathematic abilities of 15-year-old students. The first assessment was conducted in the year 2000 with 43 countries participating. PISA has repeated this assessment every three years to provide comparable data, helping to guide global education to better prepare youth for future economies. There have been many ramifications following the results of triennial PISA assessments due to implicit and explicit responses of stakeholders, which have led to education reform and policy change.


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 of effective teaching is giving students "the opportunity to learn". Teachers can set expectations, times, kinds of tasks, questions, acceptable answers, and types of discussions that will influence students' opportunities 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 times 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 result is greater learning. This is 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 is 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. Jason Williams, secondary teacher of Maths in England, has pioneered Hegarty Maths and uses this as a way to streamline marking and assessment. ;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 In psychology, number sense is the term used for the hypothesis that some animals, particularly humans, have a biologically determined ability that allows them to represent and manipulate large numerical quantities. The term was popularized by Sta ...
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 :Elementary school children need 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 The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...
and understand the
equals sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between tw ...
to mean "the answer is....".


Methodology

As with other educational research (and the
social sciences Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of so ...
in general), mathematics education research depends on both quantitative and qualitative studies.
Quantitative research Quantitative research is a research strategy that focuses on quantifying the collection and analysis of data. It is formed from a deductive approach where emphasis is placed on the testing of theory, shaped by empiricist and positivist philoso ...
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 to test their effects. They depend on large samples to obtain statistically significant results.
Qualitative research Qualitative research is a type of research that aims to gather and analyse non-numerical (descriptive) data in order to gain an understanding of individuals' social reality, including understanding their attitudes, beliefs, and motivation. This ...
, such as
case studies A case study is an in-depth, detailed examination of a particular case (or cases) within a real-world context. For example, case studies in medicine may focus on an individual patient or ailment; case studies in business might cover a particular f ...
,
action research Action research is a philosophy and methodology of research generally applied in the social sciences. It seeks transformative change through the simultaneous process of taking action and doing research, which are linked together by critical refl ...
, 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 A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...
, 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”, policymakers often consider only those studies. 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 Biomedicine (also referred to as Western medicine, mainstream medicine or conventional medicine)
,
psychology Psychology is the science, scientific study of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, including feelings and thoughts. It is an academic discipline of immens ...
, 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 What Works Clearinghouse (WWC) is a digital library of educational research which focuses on evidence-based education Evidence-based education (EBE) is the principle that education practices should be based on the best available scientific eviden ...
(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.


Organizations

* Advisory Committee on Mathematics Education *
American Mathematical Association of Two-Year Colleges The American Mathematical Association of Two-Year Colleges (AMATYC) is an organization dedicated to the improvement of education in the first two years of college mathematics in the United States and Canada. AMATYC hosts an annual conference, ...
* Association of Teachers of Mathematics *
Canadian Mathematical Society The Canadian Mathematical Society (CMS) (french: Société mathématique du Canada) is an association of professional mathematicians dedicated to the interests of mathematical research, outreach, scholarship and education in Canada. It serves the ...
*
C.D. Howe Institute The C. D. Howe Institute (french: Institut C. D. Howe) is a Canadian nonprofit policy research organization in Toronto, Ontario, Canada. It aims to be distinguished by "research that is nonpartisan, evidence-based, and subject to definitive exper ...
* Mathematical Association * National Council of Teachers of Mathematics *
OECD The Organisation for Economic Co-operation and Development (OECD; french: Organisation de coopération et de développement économiques, ''OCDE'') is an intergovernmental organisation with 38 member countries, founded in 1961 to stimulate ...


See also

;Aspects of mathematics education *
Cognitively Guided Instruction Cognitively Guided Instruction is "a professional development program based on an integrated program of research on (a) the development of students' mathematical thinking; (b) instruction that influences that development; (c) teachers' knowledge a ...
* Critical mathematics pedagogy * Ethnomathematics * Number sentence, primary level mathematics education *
Pre-math skills Pre-math skills (referred to in British English as pre-maths skills) are math skills learned by preschoolers and kindergarten students, including learning to count numbers (usually from 1 to 10 but occasionally including 0), learning the prope ...
;North American issues * Mathematics education in the United States ;Mathematical difficulties *
Dyscalculia Dyscalculia () is a disability resulting in difficulty learning or comprehending arithmetic, such as difficulty in understanding numbers, learning how to manipulate numbers, performing mathematical calculations, and learning facts in mathematics. ...
* Mathematical anxiety


References


Further reading

* * * Ball, Lynda, et al. ''Uses of Technology in Primary and Secondary Mathematics Education'' (Cham, Switzerland: Springer, 2018). * Dreher, Anika, et al. "What kind of content knowledge do secondary mathematics teachers need?." ''Journal für Mathematik-Didaktik'' 39.2 (2018): 319-34
online
* Drijvers, Paul, et al. ''Uses of technology in lower secondary mathematics education: A concise topical survey'' (Springer Nature, 2016). * Gosztonyi, Katalin. "Mathematical culture and mathematics education in Hungary in the XXth century." in ''Mathematical cultures'' (Birkhäuser, Cham, 2016) pp. 71–89
online
* * Losano, Leticia, and Márcia Cristina de Costa Trindade Cyrino. "Current research on prospective secondary mathematics teachers’ professional identity." in ''The mathematics education of prospective secondary teachers around the world'' (Springer, Cham, 2017) pp. 25-32. * * * Strutchens, Marilyn E., et al. ''The mathematics education of prospective secondary teachers around the world'' (Springer Nature, 2017
online
* Wong, Khoon Yoong. "Enriching secondary mathematics education with 21st century competencies." in ''Developing 21st Century Competencies In The Mathematics Classroom: Yearbook 2016'' (Association Of Mathematics Educators. 2016) pp. 33–50.


External links

*



David Klein. California State University, Northridge, USA {{DEFAULTSORT:Mathematics Education Mathematical science occupations