![Mathematics lecture at the Helsinki University of Technology](https://upload.wikimedia.org/wikipedia/commons/8/86/Mathematics_lecture_at_the_Helsinki_University_of_Technology.jpg)
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
A didactic method ( el, διδάσκειν ''didáskein'', "to teach") is a teaching method that follows a consistent scientific approach or educational style to present information to students. The didactic method of instruction is often contr ...
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 – 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 and carrying out
scholarly
The scholarly method or scholarship is the body of principles and practices used by scholars and academics to make their claims about the subject as valid and trustworthy as possible, and to make them known to the scholarly public. It is the me ...
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,
ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...
,
ancient Rome
In modern historiography, ancient Rome refers to 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 (753–509 BC ...
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 the ...
India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
. 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. The oldest known mathematics textbook is the
Rhind 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 Scottish antiquarian, who purchased ...
, 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 politi ...
.
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 "free" and "art or principled practice") is the traditional academic course in Western higher education. ''Liberal arts'' takes the term '' art'' in the sense of a learned skill rather than specifically th ...
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 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 ...
. This structure was continued in the structure of
classical education Classical education may refer to:
*''Modern'', educational practices and educational movements:
**An education in the Classics, especially in Ancient Greek and Latin
**Classical education movement, based on the trivium (grammar, logic, rhetoric) an ...
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
![Woman teaching geometry](https://upload.wikimedia.org/wikipedia/commons/d/d4/Woman_teaching_geometry.jpg)
In the
Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history
The history of Europe is traditionally divided into four time periods: prehistoric Europe (prior to about 800 BC), classical antiquity (800 BC to AD ...
, 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
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 are ...
,
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, then
subtraction,
multiplication, and
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
) 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 q ...
. 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 and the
Moscow Mathematical Papyrus
The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...
. The more famous
Rhind 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 Scottish antiquarian, who purchased ...
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
The Lucasian Chair of Mathematics () is a mathematics professorship in the University of Cambridge, England; its holder is known as the Lucasian Professor. The post was founded in 1663 by Henry Lucas, who was Cambridge University's Member of Pa ...
being established by the
University of Cambridge
The University of Cambridge is a public collegiate research university in Cambridge, England. Founded in 1209 and granted a royal charter by Henry III in 1231, Cambridge is the world's third oldest surviving university and one of its most pr ...
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
Urban means "related to a city". In that sense, the term may refer to:
* Urban area, geographical area distinct from rural areas
* Urban culture, the culture of towns and cities
Urban may also refer to:
General
* Urban (name), a list of people ...
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
In education, a curriculum (; : 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 a view ...
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
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
.
* The
International Commission on Mathematical Instruction 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 International ...
(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, th
Shell Centre for Mathematical Educationwas 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 (ICME) was held in
Lyon
Lyon,, ; Occitan language, Occitan: ''Lion'', hist. ''Lionés'' also spelled in English as Lyons, is the List of communes in France with over 20,000 inhabitants, third-largest city and Urban area (France), second-largest metropolitan area of F ...
in 1969. The second congress was in
Exeter in 1972, and after that, it has been held every four years
In the 20th century, the cultural impact of the "
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
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, "Mat ...
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 students
* The teaching of practical mathematics (
arithmetic,
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 ...
,
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 speakin ...
,
statistic) 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, pub ...
(
percentages,
charts,
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 speakin ...
,
statistic, etc.)
* The teaching of abstract mathematical concepts (such as
set and
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
) 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
* The teaching of selected areas of mathematics (such as
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
) as an example of the intellectual achievements of the
modern world
* 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 so ...
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 "free" and "art or principled practice") is the traditional academic course in Western higher education. ''Liberal arts'' takes the term '' art'' in the sense of a learned skill rather than specifically th ...
education in
liberal arts colleges 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:
![Number bingo improves math skills LPB Laos](https://upload.wikimedia.org/wikipedia/commons/c/c9/Number_bingo_improves_math_skills_LPB_Laos.jpg)
*
Computer-based math: an approach based on the use of mathematical software as the primary tool of computation.
*
Computer-based mathematics education: 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 of
deductive reasoning.
* 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
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
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels.
In the Canadian curriculum, there are six basic strands in Elementary Mathematics: Number, Algebra, Data, Spatial Sense, Finan ...
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 reader ...
than the conventional approach.
*Discovery math: a constructivist method of teaching (
discovery learning
Discovery learning is a technique of inquiry-based learning and is considered a constructivist based approach to education. It is also referred to as problem-based learning, experiential learning and 21st century learning. It is supported by the ...
) 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 conce ...
,
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
Morris Kline (May 1, 1908 – June 10, 1992) was a professor of mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.
Education and career
Kline was born to a Jewish fami ...
's 1973 book ''
Why Johnny Can't Add''. The New Math method was the topic of one of
Tom Lehrer
Thomas Andrew Lehrer (; born April 9, 1928) is an American former musician, singer-songwriter, satirist, and mathematician, having lectured on mathematics and musical theater. He is best known for the pithy and humorous songs that he recorded in ...
'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
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
: 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 tot ...
, focused on deepening student understanding of mathematical ideas and procedures, and formalized by the
National Council of Teachers of Mathematics
Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...
which created the
Principles and Standards for School Mathematics
''Principles and Standards for School Mathematics'' (''PSSM'') are guidelines produced by the National Council of Teachers of Mathematics (NCTM) in 2000, setting forth recommendations for mathematics educators. They form a national vision for pres ...
.
*
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: 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 Literature ...
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. The problems can range from simple
word problems to problems from international
mathematics competitions 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, except i ...
. Problem-solving is used as a means to build new mathematical knowledge, typically by building on students' prior understandings.
*
Exercises: the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding
vulgar fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s 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 q ...
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 table
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.
The decimal multiplication table was traditionally taught as an essenti ...
s, 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
An honors student or honor student is a student recognized for achieving high grades or high marks in their coursework at school.
United States
In the US, honors students may refer to:
# Students recognized for their academic achievement on lis ...
.
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 are taught, in both numeric and pictorial form, as well as
fractions
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
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 ...
and analysis (
pre-calculus
In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus. Schools often distinguish between algebra and trigonometr ...
and
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
) 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 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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
,
analytic geometry,
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
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 mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
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 speakin ...
and
statistics 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 Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...
- 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 to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather ...
,
differential equations, 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 matrices ...
;
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 - 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 ...
.
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 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, 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 proo ...
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
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
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 r ...
is mathematics intensive, often overlapping substantively with the pure or applied math degree.
(
"Business mathematics" is usually limited to introductory calculus and, sometimes,
matrix 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 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 matrices ...
, 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 b ...
, for example, standards for mathematics education are set as part of the National Curriculum for England, while
Scotland
Scotland (, ) is a Countries of the United Kingdom, country that is part of the United Kingdom. Covering the northern third of the island of Great Britain, mainland Scotland has a Anglo-Scottish border, border with England to the southeast ...
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
Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...
(NCTM) published the ''
Principles and Standards for School Mathematics
''Principles and Standards for School Mathematics'' (''PSSM'') are guidelines produced by the National Council of Teachers of Mathematics (NCTM) in 2000, setting forth recommendations for mathematics educators. They form a national vision for pres ...
'' in 2000 for the US and Canada, which boosted the trend towards
reform mathematics
Reform 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 (NCTM). The NCTM document ''Curriculum and Evaluation Stand ...
. 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 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
The Programme for International Student Assessment (PISA) is a worldwide study by the Organisation for Economic Co-operation and Development (OECD) in member and non-member nations intended to evaluate educational systems by measuring 15-yea ...
(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
Homework is a set of tasks assigned to students by their teachers to be completed outside the classroom. Common homework assignments may include required reading, a writing or typing project, mathematical exercises to be completed, informatio ...
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
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
. They prefer arithmetic reasoning to algebraic equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
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 includes studies that use inferential statistics
Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...
to answer specific questions, such as whether a certain teaching method
A teaching method comprises the principles and methods used by teachers to enable student learning. These strategies are determined partly on subject matter to be taught and partly by the nature of the learner. For a particular teaching method to ...
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, 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 refle ...
, discourse analysis
Discourse analysis (DA), or discourse studies, is an approach to the analysis of written, vocal, or sign language use, or any significant semiotic event.
The objects of discourse Analysis (discourse, writing, conversation, communicative event) ...
, 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”, 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 scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries between ...
, 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 unit
In statistics, a unit is one member of a set of entities being studied. It is the main source for the mathematical abstraction of a " random variable". Common examples of a unit would be a single person, animal, plant, manufactured item, or countr ...
s, 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
An education ministry is a national or subnational government agency politically responsible for education. Various other names are commonly used to identify such agencies, such as Ministry of Education, Department of Education, and Ministry of Pub ...
) responded to ongoing controversy by extending its research base to include non-experimental studies, including regression discontinuity design
In statistics, econometrics, political science, epidemiology, and related disciplines, a regression discontinuity design (RDD) is a quasi-experimental pretest-posttest design that aims to determine the causal effects of interventions by assigning a ...
s and single-case studies.
Organizations
* Advisory Committee on Mathematics Education The Advisory Committee on Mathematics Education (ACME) is a British policy council for the Royal Society based in London, England. Founded in 2002 by the Royal Society and the Joint Mathematical Council, ACME analyzes mathematics education practices ...
* American Mathematical Association of Two-Year Colleges
* Association of Teachers of Mathematics
The Association of Teachers of Mathematics (ATM) was established by Caleb Gattegno in 1950 to encourage the development of mathematics education to be more closely related to the needs of the learner. ATM is a membership organisation representing ...
* Canadian Mathematical Society
*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
The 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 ...
* National Council of Teachers of Mathematics
Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...
*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 e ...
See also
;Aspects of mathematics education
* Cognitively Guided Instruction
* Critical mathematics pedagogy
* Ethnomathematics In mathematics education, ethnomathematics is the study of the relationship between mathematics and culture. Often associated with "cultures without written expression", it may also be defined as "the mathematics which is practised among identifiabl ...
* Number sentence, primary level mathematics education
* Pre-math skills
;North American issues
* Mathematics education in the United States
From kindergarten through high school, mathematics education in public schools in the United States has historically varied widely from state to state, and often even varies considerably within individual states. With the adoption of the Common C ...
;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