View source for Mathematics education

{{Short description|Teaching, learning, and scholarly research in mathematics}}
[[File:Konkretismus (DSC04014).jpg|thumb|260x260px|A child calculating with his fingers (2006)]]
{{Educational research}}
{{Math topics TOC}}

In contemporary [[education]], '''mathematics education'''—known in Europe as the [[didactics]] or [[pedagogy]] of [[mathematics]]—is the practice of [[teaching]], [[learning]], and carrying out [[Scholarly method|scholarly]] [[research]] 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. [[List of mathematical societies|National and international organisations]] regularly hold conferences and [[List of mathematics education journals|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]], [[Babylonia|ancient Babylonia]], [[ancient Greece]], [[ancient Rome]], and [[Vedic civilization|Vedic]] [[Ancient India|India]].{{citation needed|date=July 2023}} In most cases, formal education was only available to [[male]] children with sufficiently high status, wealth, or [[caste]].{{Citation needed|date=November 2022}} The oldest known mathematics textbook is the [[Rhind papyrus]], dated from circa 1650 BCE.<ref>{{cite journal
 | title=The World's First Mathematics Textbook
 | first=Underwood | last=Dudley
 | journal=Math Horizons
 | volume=9 | issue=4 | date=April 2002 | pages=8–11
 | publisher=Taylor & Francis, Ltd.
 | doi=10.1080/10724117.2002.11975154 | jstor=25678363
| s2cid=126067145 }}</ref>

==== Pythagorean theorem ====
Historians of [[Mesopotamia]] have confirmed that use of the [[Pythagorean theorem|Pythagorean rule]] dates back to the [[Old Babylonian Empire]] (20th–16th centuries BC) and that it was being taught in scribal schools over one thousand years before the birth of [[Pythagoras]].<ref>{{Cite book |last=Neugebauer |first=Otto |author-link=Otto E. Neugebauer |title=The exact sciences in antiquity |year=1969 |location=New York |publisher=Dover Publications |isbn=978-0-486-22332-2 |url=https://archive.org/details/exactsciencesina0000neug_c0r3 |url-access=registration |page=36 |quote=In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse.}}</ref><ref>{{cite journal |author=Friberg, Jöran |year=1981 |title=Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations |url=https://www.researchgate.net/publication/222892801 |journal=Historia Mathematica |volume=8 |issue=3 |pages=277&ndash;318 |doi=10.1016/0315-0860(81)90069-0 |doi-access=free}}: p. 306 "Although Plimpton 322 is a unique text of its kind, there are several other known texts testifying that the Pythagorean theorem was well known to the mathematicians of the Old Babylonian period."</ref><ref>{{cite conference |last=Høyrup |first=Jens |author-link=Jens Høyrup |editor-last=Renger |editor-first=Johannes |title=Babylon: Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne. 2. Internationales Colloquium der Deutschen Orient-Gesellschaft 24.–26. März 1998 in Berlin |url=http://akira.ruc.dk/~jensh/Publications/Pythrule.pdf |publisher=Berlin: Deutsche Orient-Gesellschaft / Saarbrücken: SDV Saarbrücker Druckerei und Verlag |pages=393–407 |archive-url=https://web.archive.org/web/20210225094632/http://akira.ruc.dk/~jensh/Publications/Pythrule.pdf |archive-date=2021-02-25 |access-date=2022-11-15 |contribution=Pythagorean 'Rule' and 'Theorem' – Mirror of the Relation Between Babylonian and Greek Mathematics |url-status=live}}, p. 406, "''To judge from this evidence alone'' it is therefore likely that the Pythagorean rule was discovered within the lay surveyors' environment, possibly as a spin-off from the problem treated in Db<sub>2</sub>-146, somewhere between 2300 and 1825 BC." ([[Db2-146|Db<sub>2</sub>-146]] is an Old Babylonian clay tablet from [[Eshnunna]] concerning the computation of the sides of a rectangle given its area and diagonal.)</ref><ref>{{cite book |last=Robson |first=E. |title=Mathematics in Ancient Iraq: A Social History |publisher=Princeton University Press |year=2008 |author-link=Eleanor Robson}}: p. 109 "Many Old Babylonian mathematical practitioners … knew that the square on the diagonal of a right triangle had the same area as the sum of the squares on the length and width: that relationship is used in the worked solutions to word problems on cut-and-paste 'algebra' on seven different tablets, from Ešnuna, Sippar, Susa, and an unknown location in southern Babylonia."</ref><ref>{{Cite book |last=Ferguson |first=Kitty |title=Pythagoras : His Lives and the Legacy of a Rational Universe |publisher=Icon |year=2010 |isbn=978-184831-231-9 |location=London |pages=78–84 |language=en}}</ref>

In [[Plato]]'s division of the [[liberal arts]] into the [[trivium (education)|trivium]] and the [[quadrivium]], the quadrivium included the mathematical fields of [[arithmetic]] and [[geometry]]. This structure was continued in the structure of [[Medieval university|classical education]] that was developed in medieval Europe. The teaching of geometry was almost universally based on [[Euclid|Euclid's]] ''[[Euclid's Elements|Elements]]''. Apprentices to trades such as masons, merchants, and moneylenders could expect to learn such practical mathematics as was relevant to their profession.

=== Medieval and early modern ===
[[File:Woman teaching geometry.jpg|left|thumb|Illustration at the beginning of a 14th-century translation of [[Euclid's Elements|Euclid's ''Elements'']]]]

In the [[Middle Ages]], the academic status of mathematics declined, because it was strongly associated with trade and commerce, and considered somewhat un-Christian.<ref name="taboo">{{cite web |url=https://www.npr.org/templates/transcript/transcript.php?storyId=486172977 |title=Why We Learn Math Lessons That Date Back 500 Years |date=23 Jul 2016 |author=Gabrielle Emanuel |publisher=[[National Public Radio]] |access-date=10 April 2018 |archive-date=10 April 2018 |archive-url=https://web.archive.org/web/20180410135222/https://www.npr.org/templates/transcript/transcript.php?storyId=486172977 |url-status=live }}</ref> Although it continued to be taught in [[Lists of universities and colleges by country|European universities]], it was seen as subservient to the study of [[Natural Philosophy|natural]], [[Metaphysics|metaphysical]], and [[moral philosophy]]. The first modern arithmetic curriculum (starting with [[addition]], then [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]]) arose at [[reckoning school]]s in Italy in the 1300s.<ref name="taboo2">{{cite web|url=https://www.npr.org/sections/ed/2016/07/23/486172977/a-history-lesson-when-math-was-taboo|title=Why We Learn Math Lessons That Date Back 500 Years|website=NPR.org|access-date=2018-04-10|archive-date=2018-04-10|archive-url=https://web.archive.org/web/20180410135438/https://www.npr.org/sections/ed/2016/07/23/486172977/a-history-lesson-when-math-was-taboo|url-status=live}}</ref> 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.<ref name="taboo2" /> They also contrasted with mathematical methods learned by [[artisan]] apprentices, which were specific to the tasks and tools at hand. For example, the division of a board into thirds can be accomplished with a piece of string, instead of measuring the length and using the arithmetic operation of division.<ref name="taboo" />

The first mathematics textbooks to be written in English and French were published by [[Robert Recorde]], 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]]. After the Sumerians, some of the most famous [[Ancient history|ancient]] works on mathematics came from Egypt in the form of the [[Rhind Mathematical Papyrus]] 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]] creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in [[University of Oxford]] in 1619 and the [[Lucasian Professor of Mathematics|Lucasian Chair of Mathematics]] being established by the [[University of Cambridge]] in 1662.

=== Modern ===
In the 18th and 19th centuries, the [[Industrial Revolution]] led to an enormous increase in [[urban area|urban]] populations. Basic numeracy skills, such as the ability to tell the time, count money, and carry out simple [[arithmetic]], became essential in this new urban lifestyle. Within the new [[public education]] systems, mathematics became a central part of the [[curriculum]] from an early age.

By the twentieth century, mathematics was part of the core curriculum in all [[developed country|developed countries]].

During the twentieth century, mathematics education was established as an independent field of research. Main events in this development include the following:

* In 1893, a Chair in mathematics education was created at the University of Göttingen, under the administration of [[Felix Klein]].
* The [[International Commission on Mathematical Instruction]] (ICMI) was founded in 1908, and Felix Klein became the first president of the organisation.
* The professional [[periodical literature]] on mathematics education in the United States had generated more than 4,000 articles after 1920, so in 1941 William L. Schaaf published a [[subject indexing|classified index]], sorting them into their various subjects.<ref>William L. Schaaf (1941) [http://babel.hathitrust.org/cgi/pt?id=uc1.$b417103;view=1up;seq=11 A Bibliography of Mathematical Education] {{Webarchive|url=https://web.archive.org/web/20200110155051/https://babel.hathitrust.org/cgi/pt?id=uc1.$b417103;view=1up;seq=11 |date=2020-01-10 }}, Forest Hills, N.Y. : Stevinus Press, link from [[HathiTrust]]</ref>
* A renewed interest in mathematics education emerged in the 1960s, and the International Commission was revitalized.
* In 1968, the [https://www.mathshell.org/ Shell Centre for Mathematical Education] was established in [[Nottingham]].
* The first [[International Congress on Mathematical Education]] (ICME) was held in [[Lyon]] in 1969. The second congress was in [[Exeter]] in 1972, and after that, it has been held every four years.

Midway through the twentieth century, the cultural impact of the "[[Information Age|electronic age]]" (McLuhan) was also taken up by [[educational theory]] and the teaching of mathematics. While previous approach focused on "working with specialized 'problems' in [[arithmetic]]", the emerging structural approach to knowledge had "small children meditating about [[number theory]] and '[[set theory|sets]]'."<ref>[[Marshall McLuhan]] (1964) ''[[Understanding Media]]'', p.13 {{cite web|url=http://www9.georgetown.edu/faculty/irvinem/theory/McLuhan-Understanding_Media-I-1-7.html |title=McLuhan: Understanding Media |access-date=2007-09-04 |url-status=dead |archive-url=https://web.archive.org/web/20081208070250/http://georgetown.edu/faculty/irvinem/theory/McLuhan-Understanding_Media-I-1-7.html |archive-date=2008-12-08 }}</ref> Since the 1980s, there have been a number of efforts to reform the traditional curriculum, which focuses on continuous mathematics and relegates even some basic discrete concepts to advanced study, to better balance coverage of the continuous and discrete sides of the subject:<ref>{{cite book|last1=Hart|first1=Eric W.|last2=Martin|first2=W. Gary|date=2018|chapter=Discrete mathematics is essential mathematics in a 21st century school curriculum.|chapter-url=https://link.springer.com/chapter/10.1007/978-3-319-70308-4_1|title=Teaching and learning discrete mathematics worldwide: Curriculum and research|series=ICME-13 Monographs |pages=3–19|doi=10.1007/978-3-319-70308-4_1|isbn=978-3-319-70307-7 }}</ref>
* In the 1980s and early 1990s, there was a push to make discrete mathematics more available at the post-secondary level;
* From the late 1980s into the new millennium, countries like the US began to identify and standardize sets of discrete mathematics topics for primary and secondary education;
* Concurrently, academics began compiling practical advice on introducing discrete math topics into the classroom;
* Researchers continued arguing the urgency of making the transition throughout the 2000s; and
* In parallel, some textbook authors began working on materials explicitly designed to provide more balance.
Similar efforts are also underway to shift more focus to mathematical modeling as well as its relationship to discrete math.<ref>{{cite journal|last1=Greefrath|first1=Gilbert|last2=Siller|first2=Hans‑Stefan|last3=Vorhölter|first3=Katrin|last4=Kaiser|first4=Gabriele|date=2022|title=Mathematical modelling and discrete mathematics: opportunities for modern mathematics teaching|journal=ZDM–Mathematics Education|volume=54|issue=4|pages=865–879|doi=10.1007/s11858-022-01339-5|pmid=35291444 |pmc=8908952 |hdl=11250/3054903|hdl-access=free}}</ref>

==Objectives==
[[File:ASC Leiden - Coutinho Collection - C 05 - School in Sara, Guinea-Bissau - Boy in front of blackboard - 1974.tif|thumb|Boy doing sums, Guinea-Bissau, 1974]]
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<ref>{{Cite web|url=https://medium.com/inspired-ideas-prek-12/5-approaches-to-teaching-prek-12-numeracy-f756f41bbb2c|title=5 Approaches to Teaching PreK-12 Numeracy|last=Education|first=McGraw-Hill|date=2017-10-20|website=Inspired Ideas|access-date=2019-02-12|archive-date=2021-12-26|archive-url=https://web.archive.org/web/20211226164138/https://medium.com/inspired-ideas-prek-12/5-approaches-to-teaching-prek-12-numeracy-f756f41bbb2c|url-status=live}}</ref>
* The teaching of practical mathematics ([[arithmetic]], [[elementary algebra]], plane and solid [[geometry]], [[trigonometry]], [[probability]], [[statistics]]) to most students, to equip them to follow a trade or craft and to understand mathematics commonly used in news and [[Internet]] (such as [[percentage]]s, [[chart]]s, [[probability]], and [[statistics]])
* The teaching of abstract mathematical concepts (such as [[Set (mathematics)|set]] and [[function (mathematics)|function]]) at an early age
* The teaching of selected areas of mathematics (such as [[Euclidean geometry]])<ref>{{Cite web|url=https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid_fifth_postulate/index.html|title=Euclidean Geometry|website=www.pitt.edu|access-date=2019-02-12|archive-date=2019-01-30|archive-url=https://web.archive.org/web/20190130192629/http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid_fifth_postulate/index.html|url-status=live}}</ref> as an example of an [[axiomatic system]]<ref>{{Cite web|url=http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxiomaticSystems.htm|title=Axiomatic Systems|website=web.mnstate.edu|access-date=2019-02-12|archive-date=2019-07-17|archive-url=https://web.archive.org/web/20190717132454/http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxiomaticSystems.htm|url-status=dead}}</ref> and a model of [[deductive reasoning]]
* The teaching of selected areas of mathematics (such as [[calculus]]) as an example of the intellectual achievements of the [[modern world]]
* The teaching of advanced mathematics to those students who wish to follow a career in [[STEM fields|science, technology, engineering, and mathematics (STEM)]] fields
* The teaching of [[heuristics]]<ref>{{Cite web|url=http://theory.stanford.edu/~amitp/GameProgramming/Heuristics.html|title=Heuristics|website=theory.stanford.edu|access-date=2019-02-12|archive-date=2019-04-06|archive-url=https://web.archive.org/web/20190406233524/http://theory.stanford.edu/~amitp/GameProgramming/Heuristics.html|url-status=live}}</ref> and other problem-solving strategies to solve non-routine problems
*The teaching of mathematics in [[social science]]s and [[actuarial science]]s, as well as in some selected [[Mathematics and art|arts]] under [[Liberal arts education|liberal arts]] education in [[liberal arts college]]s or universities

==Methods==
{{See also|Financial literacy curriculum|List of open-source software for mathematics|List of mathematical art software}}
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:

[[File:Digital carrel classroom.webp|thumb|3D sketch of [[Carrel desk|desk cubicles]] to get [[computers in the classroom]] for [[Computer-based mathematics education|computer-based mathematics]], [[Computer-aided design|CAD]], [[Computer-aided manufacturing|CAM]], [[Building information modeling|BIM]], [[List of computer-aided engineering software|computer-aided engineering]], [[Computer programming|computer programming]], [[animation software]], [[List of free and open-source software packages#Science|science software applications]], and more.]]
[[File:Number bingo improves math skills LPB Laos.jpg|thumb|Games can motivate students to improve skills that are usually learned by rote. In "Number Bingo," players roll 3 dice, then perform basic mathematical operations on those numbers to get a new number, which they cover on the board trying to cover 4 squares in a row. This game was played at a "Discovery Day" organized by [[Big Brother Mouse]] in Laos.]]
* '''[[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.<ref>{{Cite news|url=https://www.techzim.co.zw/2018/06/passing-mathematics-just-got-easier-for-students-with-this-new-platform-mathematica/|title=Passing Mathematics Just Got Easier For Students With This New Platform: Mathematica - Techzim|date=2018-06-16|work=Techzim|access-date=2018-06-19|language=en-US|archive-date=2018-06-19|archive-url=https://web.archive.org/web/20180619140241/https://www.techzim.co.zw/2018/06/passing-mathematics-just-got-easier-for-students-with-this-new-platform-mathematica/|url-status=live}}</ref><ref>{{Cite news|url=https://edtechmagazine.com/k12/article/2017/10/5-apps-help-all-students-math|title=5 Apps to Help All Students with Math|date=2017-10-13|work=Technology Solutions That Drive Education|access-date=2018-06-19|language=en|archive-date=2021-12-26|archive-url=https://web.archive.org/web/20211226164138/https://edtechmagazine.com/k12/article/2017/10/5-apps-help-all-students-math|url-status=live}}</ref><ref>{{Cite news|url=https://www.huffingtonpost.com/2014/10/22/photomath-app-solve-math-problems_n_6025948.html|title=This Free App Will Solve Math Problems For You|last=Mosbergen|first=Dominique|date=2014-10-22|work=Huffington Post|access-date=2018-06-21|language=en-US|archive-date=2017-09-22|archive-url=https://web.archive.org/web/20170922061544/http://www.huffingtonpost.com/2014/10/22/photomath-app-solve-math-problems_n_6025948.html|url-status=live}}</ref>
* '''Classical education''': the teaching of mathematics within the [[quadrivium]], part of the classical education curriculum of the [[Middle Ages]], which was typically based on [[Euclid's Elements|Euclid's ''Elements'']] taught as a [[paradigm]] of [[deductive reasoning]].<ref>{{Cite web |date=2018-01-25 |title=Classical Education and STEM: a Common Misconception |url=http://claphamschool.org/classical-education-and-stem-a-common-misconception/ |access-date=2019-02-12 |website=Clapham School |language=en-US |archive-date=2019-02-12 |archive-url=https://web.archive.org/web/20190212131018/http://claphamschool.org/classical-education-and-stem-a-common-misconception/ |url-status=live }}</ref>
* '''Conventional approach''': the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with [[arithmetic]] and is followed by [[Euclidean geometry]] and [[elementary algebra]] taught concurrently. Requires the instructor to be well informed about [[elementary mathematics]] since didactic and curriculum decisions are often dictated by the logic of the subject rather than pedagogical considerations. Other methods emerge by emphasizing some aspects of this approach.
*'''Relational approach''': uses class topics to solve everyday problems and relates the topic to current events.<ref>{{cite web |title=Mathematical Current Events |url=http://www.math.unl.edu/~s-kfield1/203currentevent.htm%20 |url-status=dead |archive-url=https://web.archive.org/web/20111120085344/http://www.math.unl.edu/~s-kfield1/203currentevent.htm |archive-date=2011-11-20 |access-date=2011-11-29}}</ref> 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 [[history of mathematics|development of mathematics]] within a historical, social, and cultural context. Proponents argue it provides more [[human interest]] than the conventional approach.<ref>{{cite book |last=Sriraman |first=Bharath |url=https://books.google.com/books?id=P38a50_NT8oC |title=Crossroads in the History of Mathematics and Mathematics Education |publisher=IAP |year=2012 |isbn=978-1-61735-704-6 |series=Monograph Series in Mathematics Education |volume=12}}</ref>
*'''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 (mathematics education)|manipulative]] tools.<ref name=":0">{{cite journal |last1=Ansari |first1=Daniel |date=March 2016 |title=No More Math Wars |journal=The Education Digest |volume=81 |issue=7 |pages=4–9 |id={{ProQuest|1761255371}}}}</ref> This type of mathematics education was implemented in various parts of Canada beginning in 2005.<ref>{{Cite book |last=Stokke |first=Anna |title=What to Do About Canada's Declining Math Scores |publisher=C.D. Howe Institute |year=2015 |isbn=9780888069498 |location=Toronto, Ontario |pages=4–5}}</ref> 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]], [[Function (mathematics)|functions]], and bases other than ten. Adopted in the US as a response to the challenge of early Soviet technical superiority in space, it began to be challenged in the late 1960s. One of the most influential critiques of the New Math was [[Morris Kline]]'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.<ref>{{cite web |author-link=David Singmaster |first=David |last=Singmaster |title=The Unreasonable Utility of Recreational Mathematics |date=7 September 1993 |work=For First European Congress of Mathematics, Paris, July, 1992 |url=http://anduin.eldar.org/%7Eproblemi/singmast/ecmutil.html |access-date=17 September 2012 |archive-url=https://web.archive.org/web/20020207051215/http://anduin.eldar.org/~problemi/singmast/ecmutil.html |archive-date=7 February 2002 |url-status=dead }}</ref>
* '''[[Reform mathematics|Standards-based mathematics]]''': a vision for pre-college mathematics education in the [[United States]] and [[Canada]], focused on deepening student understanding of mathematical ideas and procedures, and formalized by the [[National Council of Teachers of Mathematics]] which created the [[Principles and Standards for School Mathematics]].
* '''[[Mathematics mastery|Mastery]]''': 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]], and [[heuristic]] thinking by setting students open-ended, unusual, and sometimes [[Lists of unsolved problems|unsolved problems]]. The problems can range from simple [[word problem (mathematics education)|word problem]]s to problems from international [[mathematics competitions]] such as the [[International Mathematical Olympiad]]. Problem-solving is used as a means to build new mathematical knowledge, typically by building on students' prior understandings.
* '''[[Exercise (mathematics)|Exercise]]s''': the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding [[simple fraction]]s or solving [[quadratic equation]]s.
* '''[[Rote learning]]''': the teaching of mathematical results, definitions and concepts by repetition and memorisation typically without meaning or supported by [[Logical reasoning|mathematical reasoning]]. A derisory term is ''drill and kill''. In [[traditional education]], rote learning is used to teach [[multiplication table]]s, definitions, formulas, and other aspects of mathematics.
* '''[[Math walk]]''': a walk where experience of perceived objects and scenes is translated into mathematical language.

==Content and age levels==
[[File:Mathematics lecture at the Helsinki University of Technology.jpg|300x300px|thumb|A mathematics lecture at [[Aalto University School of Science and Technology]]]]
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.<ref>{{cite web |title=Foundations for Success: The Final Report of the National Mathematics Advisory Panel |year=2008 |publisher=U.S. Department of Education |url=http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf |page=20 |archive-url=https://web.archive.org/web/20150317200016/http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf |archive-date=March 17, 2015}}</ref> During the primary school years, children learn about whole numbers and arithmetic, including addition, subtraction, multiplication, and division.<ref>{{Citation|last1=Nunes|first1=Terezinha|title=Teaching and Learning About Whole Numbers in Primary School|date=2016|work=ICME-13 Topical Surveys|pages=1–50|place=Cham|publisher=Springer International Publishing|isbn=978-3-319-45112-1|last2=Dorneles|first2=Beatriz Vargas|last3=Lin|first3=Pi-Jen|last4=Rathgeb-Schnierer|first4=Elisabeth|doi=10.1007/978-3-319-45113-8_1|doi-access=free|hdl=10183/164060|hdl-access=free}}</ref> Comparisons and [[measurement]] are taught, in both numeric and pictorial form, as well as [[fraction]]s and [[Proportionality (mathematics)|proportionality]], patterns, and various topics related to geometry.<ref>{{Cite journal|last1=Mullis|first1=Ina V. S.|last2=Others|display-authors=1|date=June 1997|title=Mathematics Achievement in the Primary School Years. IEA's Third International Mathematics and Science Study (TIMSS).|url=|journal=Third International Mathematics and Science Study|publisher=[[International Association for the Evaluation of Educational Achievement]]; [[Boston College]] Center for the Study of Testing, Evaluation, and Educational Policy.|volume=|issue=|pages=|isbn=1-889938-04-1}}</ref>

At high school level in most of the US, [[algebra]], [[geometry]], and analysis ([[pre-calculus]] and [[calculus]]) are taught as separate courses in different years. 
On the other hand, in most other countries (and in a few US states), mathematics is taught as an integrated subject, with topics from all branches of mathematics studied every year; 
students thus undertake a pre-defined course - entailing several topics - rather than choosing courses ''à la carte'' as in the United States. 
Even in these cases, however, several "mathematics" options may be offered, selected based on the student's intended studies post high school.
(In South Africa, for example, [[Matriculation in South Africa#Group A: Fundamentals (Compulsory)|the options are]] Mathematics, Mathematical Literacy and Technical Mathematics.)
Thus, a science-oriented curriculum typically overlaps the first year of university mathematics, and includes [[differential calculus]] and [[trigonometry]] at age 16–17 and [[integral calculus]], [[complex numbers]], [[analytic geometry]], [[exponential function|exponential]] and [[logarithmic function]]s, and [[infinite series]] in their final year of secondary school; [[Probability]] and [[statistics]] are similarly often taught.

At college and university level, [[Bachelor of Science|science]] and [[bachelor of engineering|engineering students]] will be required to take [[multivariable calculus]], [[differential equations]], and [[linear algebra]]; at several US colleges, the [[Academic minor|minor]] or [[Associate degree#United States|AS]] in mathematics substantively comprises these courses. 
[[bachelor of mathematics|Mathematics majors]] study additional areas of [[pure mathematics]]—and often applied mathematics—with the requirement of specified advanced courses in [[Mathematical analysis|analysis]] and [[modern algebra]]. Other topics in pure mathematics include [[differential geometry]], [[set theory]], and [[topology]]. 
[[Applied mathematics]] may be taken as a [[academic major|major]] subject in its own right, covering [[partial differential equations]], [[optimization]], and [[numerical analysis]] among other topics. Courses here are also taught within other programs: for example, [[civil engineers]] may be required to study [[fluid mechanics]],<ref>{{cite web |url=https://cee.mit.edu/undergraduate/1C-degreerequirements |title=MIT - S.B. In 1-C Civil Engineering Curriculum &#124; Department of Civil & Environmental Engineering, MIT |access-date=2014-06-18 |url-status=dead |archive-url=https://web.archive.org/web/20140714233214/https://cee.mit.edu/undergraduate/1C-degreerequirements |archive-date=2014-07-14 }}</ref> and "math for computer science" might include [[graph theory]], [[permutation]], probability, and [[formal proof|formal]] [[mathematical proof]]s.<ref>{{cite web|url=https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/|title=Mathematics for Computer Science|website=MIT OpenCourseWare|access-date=2019-01-02|archive-date=2019-05-10|archive-url=https://web.archive.org/web/20190510172404/https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/|url-status=live}}</ref> 
Pure and applied math degrees often include modules in [[probability theory]] or [[mathematical statistics]], as well as [[stochastic processes]]. 
([[Theoretical physics|Theoretical]]) [[physics]] is mathematics-intensive, often overlapping substantively with the pure or applied math degree. [[Business mathematics#University level|Business mathematics]] is usually limited to introductory calculus and (sometimes) [[Matrix (mathematics)|matrix]] calculations; [[bachelor of Economics|economics programs]] additionally cover [[mathematical optimization|optimization]], often [[differential equation]]s and [[linear algebra]], and sometimes analysis. Business and [[social science]] students [[business statistics|also typically take]] statistics and probability courses, which are often complemented by major-specific quantitative research courses. Students in the [[liberal arts]] may be offered a course in "contemporary mathematics" or "mathematics for the liberal arts," which may include topics such as [[set theory]], [[mathematical logic]], [[number theory]], [[elementary algebra]], [[linear programming]], money management and [[compound interest]] calculations, [[probability]],[[descriptive statistics]], [[geometry]], right angle [[trigonometry]], and [[voting]] and [[apportionment]], [[graph theory]], and applications of mathematics to other fields.<ref>{{cite book |last1=Kirk |first1=Donna |last2=Boschmans |first2=Brian |last3=Cathey |first3=Matthew |last4=Falduto |first4=Valeree |last5=Gerlofs |first5=Maureen |last6=Hearn |first6=Quin |last7=Walters |first7=Ian |title=Contemporary Mathematics |date=22 March 2023 |publisher=[[OpenStax]] |isbn=978-1-711470-55-9 |url=https://openstax.org/details/books/contemporary-mathematics}}</ref>

==Standards==
Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to, realistic for, and considered socially appropriate for their pupils.

In modern times, there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In [[England]], for example, standards for mathematics education are set as part of the National Curriculum for England,<ref>{{cite web |title=Mathematics curriculum |date=17 January 2013 |publisher=UK Department of Education |url=http://www.education.gov.uk/schools/teachingandlearning/curriculum/secondary/b00199003/mathematics |access-date=1 May 2012 |archive-date=2 May 2012 |archive-url=https://web.archive.org/web/20120502154746/http://www.education.gov.uk/schools/teachingandlearning/curriculum/secondary/b00199003/mathematics |url-status=live }}</ref> while [[Scotland]] maintains its own educational system. Many other countries have centralized ministries which set national standards or curricula, and sometimes even textbooks.

Ma (2000) summarized the research of others who found, based on nationwide data, that students with higher scores on standardized 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.<ref>{{cite journal |author=Ma, X. |title=A longitudinal assessment of antecedent course work in mathematics and subsequent mathematical attainment |journal=Journal of Educational Research |volume=94 |issue=1 |pages=16–29 |year=2000 |doi=10.1080/00220670009598739 |s2cid=144948416 }}</ref>

In North America, the [[National Council of Teachers of Mathematics]] (NCTM) published the ''[[Principles and Standards for School Mathematics]]'' in 2000 for the United States and Canada, which boosted the trend towards [[reform mathematics]]. In 2006, the NCTM released ''[[Principles and Standards for School Mathematics#Curriculum Focal Points|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 Initiative|Common Core State Standards]] for US states, which were subsequently adopted by most states. Adoption of the Common Core State Standards in mathematics is at the discretion of each state, and is not mandated by the federal government.<ref>{{cite web|url=http://www.corestandards.org/about-the-standards/myths-vs-facts/|title=Myths vs. Facts - Common Core State Standards Initiative|website=www.corestandards.org|access-date=2017-07-28|archive-date=2017-08-02|archive-url=https://web.archive.org/web/20170802182426/http://www.corestandards.org/about-the-standards/myths-vs-facts/|url-status=live}}</ref> "States routinely review their [[academic standards]] and may choose to change or add onto the standards to best meet the needs of their students."<ref>{{cite web|url=http://www.corestandards.org/standards-in-your-state/|title=Standards in Your State - Common Core State Standards Initiative|website=www.corestandards.org|access-date=2017-07-28|archive-date=2019-06-10|archive-url=https://web.archive.org/web/20190610015351/http://www.corestandards.org/standards-in-your-state/|url-status=dead}}</ref> The NCTM has state affiliates that have different education standards at the state level. For example, [[Missouri]] has the Missouri Council of Teachers of Mathematics (MCTM) which has its 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.<ref>{{cite web|url=https://www.moctm.org/Home.aspx|title=MoCTM - Home|website=www.moctm.org|access-date=2018-02-11|archive-date=2018-02-12|archive-url=https://web.archive.org/web/20180212083453/https://www.moctm.org/Home.aspx|url-status=live}}</ref>

The [[Programme for International Student Assessment]] (PISA), created by the [[OECD|Organisation for the Economic Co-operation and Development]] (OECD), is a global program studying the reading, science, and mathematics abilities of 15-year-old students.<ref>{{Cite web|url=http://www.oecd.org/pisa/|title=What is PISA?|date=2018|website=OECD|access-date=2019-10-14|archive-date=2018-03-04|archive-url=https://web.archive.org/web/20180304160008/http://www.pisa.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html|url-status=live}}</ref> The first assessment was conducted in the year 2000 with 43 countries participating.<ref name=":1">{{Cite book|title=The Experience of Middle-Income Countries Participating in PISA 2000. PISA|last=Lockheed|first=Marlaine|publisher=OECD Publishing|year=2015|isbn=978-92-64-24618-8|location=France|pages=30}}</ref> 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.<ref name=":1" /><ref>{{Cite journal|last1=Sellar, S., & Lingard, B.|first1=Sam|last2=Lingard|first2=Bob|date=April 2018|title=International large-scale assessments, affective worlds and policy impacts in education|journal=International Journal of Qualitative Studies in Education|volume=31|issue=5|pages=367–381|doi=10.1080/09518398.2018.1449982|s2cid=149999527|url=https://e-space.mmu.ac.uk/623371/3/IJQSE_blinded%20manuscript_revision%20clean%20copy%20details.pdf|access-date=2019-11-30|archive-date=2020-03-07|archive-url=https://web.archive.org/web/20200307175850/https://e-space.mmu.ac.uk/623371/3/IJQSE_blinded%20manuscript_revision%20clean%20copy%20details.pdf|url-status=live}}</ref><ref name=":0" />
[[File:PISA_Math_Scores_(2018).png|center]]

==Research==
According to Hiebert and Grouws, "Robust, useful theories of classroom teaching do not yet exist."<ref name=HiebertGrouws>{{Citation | last1 = Hiebert | first1 = James | last2 = Grouws | first2 = Douglas | title = The Effects of Classroom Mathematics Teaching on Students' Learning | place = Reston VA | publisher = National Council of Teachers of Mathematics | volume = 1 | year = 2007 | chapter = 9 | pages = 371–404}}</ref> 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.<ref name="HiebertGrouws" />

=== Conceptual understanding ===
Source:<ref name="HiebertGrouws" />

: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 US, where essentially no connections are made in school classrooms.<ref>{{Citation | editor-last = Institute of Education Sciences | contribution = Highlights From the TIMSS 1999 Video Study of Eighth-Grade Mathematics Teaching | year = 2003 | publisher = U.S. Department of Education | url = http://nces.ed.gov/timss | title = Trends in International Mathematics and Science Study (TIMSS) - Overview | access-date = 2012-05-08 | archive-date = 2012-05-08 | archive-url = https://web.archive.org/web/20120508063844/http://nces.ed.gov/timss/ | url-status = live }}</ref>) 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 intentionally challenging, well-implemented teaching, or unintentionally confusing, faulty teaching.

=== 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.<ref name="BlackWiliam">{{cite journal |last1=Black |first1=P. |last2=Wiliam |first2=Dylan |title=Assessment and Classroom Learning |journal=Assessment in Education |year=1998 |volume=5 |issue=1 |pages=7–74 |doi=10.1080/0969595980050102 |s2cid=143347721 |url=https://www.gla.ac.uk/t4/learningandteaching/files/PGCTHE/BlackandWiliam1998.pdf |access-date=2018-07-25 |archive-date=2018-07-26 |archive-url=https://web.archive.org/web/20180726005608/https://www.gla.ac.uk/t4/learningandteaching/files/PGCTHE/BlackandWiliam1998.pdf |url-status=live }}</ref>

=== Homework ===
:[[Homework]] assignments which lead students to practice past lessons or prepare for future lessons are more effective than those going over the current 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.<ref name="ResearchClips">{{cite web|url=http://www.nctm.org/news/content.aspx?id=8468|title=Research clips and briefs|access-date=2009-11-15|archive-date=2014-10-02|archive-url=https://web.archive.org/web/20141002062812/http://www.nctm.org/news/content.aspx?id=8468|url-status=live}}</ref>

=== Students with difficulties ===
Source:<ref name="ResearchClips" />
:Students with genuine difficulties (unrelated to motivation or past instruction) struggle with [[Recall of facts|basic facts]], answer impulsively, struggle with mental representations, have poor [[number sense]], and have poor short-term memory. Techniques that have been found productive for helping such students include peer-assisted learning, explicit teaching with visual aids, instruction informed by [[formative assessment]], and encouraging students to think aloud.
:In particular, research surrounding students with disabilities in a mathematics classroom is mostly done by special education researchers. Some mathematics education researchers have called for more collaboration across disciplines to better understand supports that could be helpful to mathematics students with disabilities.<ref>{{Cite journal |last=Tan |first=Paulo |date=2017-12-29 |title=Calling for Research Collaborations and the Use of Dis/ability Studies in Mathematics Education |url=https://jume-ojs-tamu.tdl.org/JUME/article/download/321/221 |journal=Journal of Urban Mathematics Education |volume=10 |issue=2 |pages=25–38 |doi=10.21423/jume-v10i2a321 |via=Texas Digital Library|doi-access=free }}</ref>

=== 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 (mathematics)|variable]]. They prefer arithmetic reasoning to [[algebraic equation]]s for solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns. Students often have trouble with the [[Plus and minus signs|minus sign]] and understand the [[equals sign]] to mean "the answer is...".<ref name="ResearchClips" />

=== Cultural Equity ===
:Despite the popular belief that mathematics is race neutral, some research<ref>{{Cite journal |last=Ladson-Billings |first=Gloria |date=June 1995 |title=But that's just good teaching! The case for culturally relevant pedagogy |url=http://www.tandfonline.com/doi/abs/10.1080/00405849509543675 |journal=Theory into Practice |language=en |volume=34 |issue=3 |pages=159–165 |doi=10.1080/00405849509543675 |issn=0040-5841|url-access=subscription }}</ref> suggests that effective mathematics teaching of culturally diverse students requires a [[Culturally relevant teaching|culturally relevant pedagogy]] that considers students' cultural backgrounds and experiences. The three criteria for culturally relevant pedagogy are academic success, cultural competence, and critical consciousness. More recent research<ref>{{Cite journal |last=Paris |first=Django |date=April 2012 |title=Culturally Sustaining Pedagogy: A Needed Change in Stance, Terminology, and Practice |url=https://journals.sagepub.com/doi/10.3102/0013189X12441244 |journal=Educational Researcher |language=en |volume=41 |issue=3 |pages=93–97 |doi=10.3102/0013189X12441244 |issn=0013-189X|url-access=subscription }}</ref> proposes that culturally sustaining pedagogy explicitly aims to perpetuate and foster cultural and linguistic pluralism within the educational system, ensuring that students can thrive while retaining their cultural identities.

=== Mathematics Teacher Education ===
:[[Student teaching]] is a crucial part of a teacher candidate's path to becoming a teacher. Recommended reform in mathematics teacher education includes a focus on learning to anticipate, elicit, and use students’ mathematical thinking as the primary goal, as opposed to models with an over-emphasis on classroom management and survival.<ref>{{Cite journal |last1=Leatham |first1=Keith |last2=Peterson |first2=Blake |date=2010 |editor-last=Lott |editor-first=J. W. |editor2-last=Luebeck |editor2-first=J. |title=Purposefully Designing Student Teaching to Focus on Students' Mathematical Thinking |journal=Mathematics Teaching: Putting Research into Practice at All Levels |volume=7 |pages=225–239 |via=Association of Teachers of Mathematics Monograph (AMTE)}}</ref>

===Methodology===
As with other educational research (and the [[social science]]s in general), mathematics education research depends on both quantitative and qualitative studies. [[Quantitative research]] includes studies that use [[inferential statistics]] to answer specific questions, such as whether a certain [[teaching method]] gives significantly better results than the status quo. The best quantitative studies involve randomized trials where students or classes are randomly assigned different methods to test their effects. They depend on large samples to obtain statistically significant results.

[[Qualitative research]], such as [[case studies]], [[action research]], [[discourse analysis]], and [[Jean Piaget|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"<ref name=HiebertGrouws/> of the finding in actual classrooms. Exploratory qualitative research is also useful for suggesting new [[Hypothesis|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.<ref>{{Cite journal | doi = 10.3102/0013189X034005025 | last = Raudenbush | first = Stephen | title = Learning from Attempts to Improve Schooling: The Contribution of Methodological Diversity | journal = Educational Researcher | volume = 34 | issue = 5 | pages = 25–31 | year = 2005 | citeseerx = 10.1.1.649.7042 | s2cid = 145667765 }}</ref> 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 of an opinion that 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.<ref name=Cook>{{cite journal |first=Thomas D. |last=Cook |title=Randomized Experiments in Educational Policy Research: A Critical Examination of the Reasons the Educational Evaluation Community has Offered for Not Doing Them |journal=Educational Evaluation and Policy Analysis |volume=24 |issue=3 |pages=175–199 |year=2002 |doi=10.3102/01623737024003175 |s2cid=144583638 }}</ref><ref name=Scheaffer>{{cite web |author=Working Group on Statistics in Mathematics Education Research |title=Using Statistics Effectively in Mathematics Education Research: A report from a series of workshops organized by the American Statistical Association with funding from the National Science Foundation |year=2007 |publisher=The American Statistical Association |url=http://www.amstat.org/research_grants/pdfs/SMERReport.pdf |access-date=2013-03-25 |archive-url=https://web.archive.org/web/20070202153649/http://www.amstat.org/research_grants/pdfs/SMERReport.pdf |archive-date=2007-02-02 |url-status=dead }}</ref> In other disciplines concerned with human subjects—like [[biomedicine]], [[psychology]], and policy evaluation—controlled, randomized experiments remain the preferred method of evaluating treatments.<ref>{{cite book |author1=Shadish, William R. |author2=Cook, Thomas D. |author3=Campbell, Donald T. |year=2002 |title=Experimental and quasi-experimental designs for generalized causal inference |location=Boston |publisher=Houghton Mifflin |isbn=978-0-395-61556-0 |edition=2nd}}</ref><ref>See articles on [[No Child Left Behind|NCLB]], [[National Mathematics Advisory Panel]], [[Scientifically based research]] and [[What Works Clearinghouse]]</ref> Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods.<ref name=Scheaffer/> 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,<ref>{{Citation
 | last1 = Mosteller | first1 = Frederick|author-link2=Robert F. Boruch | last2 = Boruch | first2 = Robert | title = Evidence Matters: Randomized Trials in Education Research | publisher = Brookings Institution Press | year = 2002}}</ref> or the difficulty of assuring rigid control of the independent variable in fluid, real school settings.<ref>{{cite journal | last = Chatterji | first = Madhabi | title = Evidence on "What Works": An Argument for Extended-Term Mixed-Method (ETMM) Evaluation Designs | journal = Educational Researcher | volume = 33 | issue = 9 | pages = 3–13 | date = December 2004 | doi = 10.3102/0013189x033009003 | s2cid = 14742527 }}</ref>

In the United States, the [[Math wars#National Mathematics Advisory Panel|National Mathematics Advisory Panel]] (NMAP) published a report in 2008 based on studies, some of which used randomized assignment of treatments to [[experimental unit]]s, such as classrooms or students. The NMAP report's preference for randomized experiments received criticism from some scholars.<ref>{{Cite journal | doi = 10.3102/0013189X08329353 | last = Kelly | first = Anthony | title = Reflections on the National Mathematics Advisory Panel Final Report | journal = Educational Researcher | volume = 37 | issue = 9 | pages = 561–4 | year = 2008 | s2cid = 143471869 }} This is the introductory article to an issue devoted to this debate on report of the National Mathematics Advisory Panel, particularly on its use of randomized experiments.</ref> In 2010, the [[What Works Clearinghouse]] (essentially the research arm for the [[United States Department of Education|Department of Education]]) responded to ongoing controversy by extending its research base to include non-experimental studies, including [[regression discontinuity design]]s and [[Case studies|single-case studies]].<ref>{{cite news|last=Sparks|first=Sarah|title=Federal Criteria For Studies Grow|newspaper=Education Week|date=October 20, 2010|page=1}}</ref>

==Organizations==
* [[Advisory Committee on Mathematics Education]]
* [[American Mathematical Association of Two-Year Colleges]]
* [[Association of Teachers of Mathematics]]
*[[Canadian Mathematical Society]]
*[[C.D. Howe Institute]]
* [[Mathematical Association]]
* [[National Council of Teachers of Mathematics]]
*[[OECD]]
*[[International Association for the Evaluation of Educational Achievement]]
*[https://amte.net/ Association of Mathematics Teacher Educators]

==See also==
* [[List of mathematics journals]]
{{columns-start|num=3}}
===Aspects of mathematics education===
* [[Cognitively Guided Instruction]]
* [[Critical mathematics pedagogy]]
* [[Ethnomathematics]]
* [[Number sentence]], primary level mathematics education
* [[Pre-math skills]]
* ''[[Sir Cumference]]'', children's mathematics educational book series
* [[Statistics education]]
{{column}}
===North American issues===
* [[Chicago movement]]
* [[Mathematics education in the United States]]
{{column}}
===Mathematical difficulties===
* [[Dyscalculia]]
* [[Mathematical anxiety]]
{{columns-end}}

== References ==
{{Reflist}}

*{{cite web |last1=Voit |first1=Rita |title=Accelerated Math: What Every Parent Should Know |url=https://resources.njgifted.org/accelerated-math-what-every-parent-should-know/ |website=Resources by HEROES Academy |date=14 February 2020 |access-date=20 September 2023}}

== Further reading ==
{{refbegin}}
*{{cite journal |first1=John R. |last1=Anderson |first2=Lynne M. |last2=Reder |first3=Herbert A. |last3=Simon |first4=K. Anders |last4=Ericsson |first5=Robert |last5=Glaser |title=Radical Constructivism and Cognitive Psychology |journal=Brookings Papers on Education Policy |issue=1 |pages=227–278 |year=1998 |url=http://memory.psy.cmu.edu/publications/98_jra_lmr_has.pdf |access-date=2011-09-25 |archive-url=https://web.archive.org/web/20100626180748/http://memory.psy.cmu.edu/publications/98_jra_lmr_has.pdf |archive-date=2010-06-26 |url-status=dead }}
*{{cite web |author=Auslander, Maurice |title=Goals for School Mathematics: The Report of the Cambridge Conference on School Mathematics 1963 |year=2004 |publisher=Center for the Study of Mathematics Curriculum |location=Cambridge MA |url=http://www.mathcurriculumcenter.org/PDFS/CCM/summaries/cambridge_summary.pdf |display-authors=etal |access-date=2009-08-06 |archive-date=2010-07-15 |archive-url=https://web.archive.org/web/20100715133000/http://www.mathcurriculumcenter.org/PDFS/CCM/summaries/cambridge_summary.pdf |url-status=live }}
* 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): 319–341 [https://link.springer.com/article/10.1007/s13138-018-0127-2 online] {{Webarchive|url=https://web.archive.org/web/20210418093743/https://link.springer.com/article/10.1007/s13138-018-0127-2 |date=2021-04-18 }}.
* 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.&nbsp;71–89. [https://www.researchgate.net/profile/Katalin_Gosztonyi/publication/303537690_Mathematical_Culture_and_Mathematics_Education_in_Hungary_in_the_XXth_Century/links/5f38264792851cd302fa8989/Mathematical-Culture-and-Mathematics-Education-in-Hungary-in-the-XXth-Century.pdf online]
* {{cite book |title=A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form |author=Paul Lockhart |publisher=Bellevue Literary Press |year=2009 |isbn=978-1934137178}}
* 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.&nbsp;25–32.
*{{cite book |first1=Bharath |last1=Sriraman |first2=Lyn |last2=English |author2-link=Lyn English|title=Theories of Mathematics Education |url=https://books.google.com/books?id=OLAnY1i8NPkC |year=2010 |publisher=Springer |isbn=978-3-642-00774-3 |author1-link=Bharath Sriraman }}
*{{cite book |first1=Steven Henry |last1=Strogatz |first2=Don |last2=Joffray |title=The Calculus of Friendship: What a Teacher and a Student Learned about Life While Corresponding about Math |url=https://archive.org/details/calculusoffriend00stro |url-access=registration |year=2009 |publisher=Princeton University Press |isbn=978-0-691-13493-2 |author1-link=Steven Strogatz }}
* Strutchens, Marilyn E., et al. ''The mathematics education of prospective secondary teachers around the world'' (Springer Nature, 2017) [https://library.oapen.org/bitstream/handle/20.500.12657/28029/1001967.pdf?sequence=1 online] {{Webarchive|url=https://web.archive.org/web/20210418093744/https://library.oapen.org/bitstream/handle/20.500.12657/28029/1001967.pdf?sequence=1 |date=2021-04-18 }}.
* 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.&nbsp;33–50.
{{refend}}

== External links ==
{{Wikiquote}}
* [https://web.archive.org/web/20040409131958/http://www-gap.dcs.st-and.ac.uk/~history/Education/index.html History of Mathematical Education]
* [http://www.csun.edu/~vcmth00m/bshm.html A quarter century of US 'math wars' and political partisanship]. David Klein. California State University, Northridge, United States

{{Mathematics education}}
{{Areas of mathematics |collapsed}}
{{Education}}
{{Authority control}}

{{DEFAULTSORT:Mathematics Education}}
[[Category:Mathematics education| ]]