Mathematics

== Training and practice ==

=== Education ===
{{main|Mathematics education}}
Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a [[human activity]], the practice of mathematics has a social side, which includes [[Mathematics education|education]], [[Mathematician|careers]], [[List of mathematics awards|recognition]], [[Popular mathematics|popularization]], and so on. In education, mathematics is a core part of the curriculum and forms an important element of the [[STEM]] academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, [[statistician]], [[actuary]], [[financial analyst]], [[economist]], [[accountant]], [[commodity trader]], or [[Information technology consulting|computer consultant]].<ref>{{cite book
 | title=Mathematicians and Statisticians: A Practical Career Guide
 | first=Kezia
 | last=Endsley
 | year=2021
 | series=Practical Career Guides
 | isbn=978-1-5381-4517-3
 | publisher=Rowman & Littlefield
 | pages=1–3
 | url={{GBurl|id=1cEYEAAAQBAJ|p=3}}
 | access-date=November 29, 2022
}}</ref>

Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.<ref>{{cite book
 | title=The Oxford Handbook of the History of Mathematics
 | first=Eleanor | last=Robson | author-link=Eleanor Robson
 | year=2009
 | chapter=Mathematics education in an Old Babylonian scribal school
 | editor1-first=Eleanor | editor1-last=Robson
 | editor2-first=Jacqueline | editor2-last=Stedall | editor2-link=Jackie Stedall
 | publisher=OUP Oxford
 | isbn=978-0-19-921312-2
 | chapter-url={{GBurl|id=xZMSDAAAQBAJ|p=199}}
 | access-date=November 24, 2022
}}</ref> Comparable evidence has been unearthed for scribal mathematics training in the [[ancient Near East]] and then for the [[Greco-Roman world]] starting around 300 BCE.<ref>{{cite book
 | chapter=Mathematics Education in Antiquity
 | first1=Alain | last1=Bernard
 | first2=Christine | last2=Proust | author2-link=Christine Proust
 | first3=Micah | last3=Ross
 | title=Handbook on the History of Mathematics Education
 | editor1-last=Karp | editor1-first=A.
 | editor2-last=Schubring | editor2-first=G.
 | year=2014 | pages=27–53 | isbn=978-1-4614-9154-5
 | publisher=Springer | publication-place=New York
 | doi=10.1007/978-1-4614-9155-2_3
}}</ref> The oldest known mathematics textbook is the [[Rhind papyrus]], dated from {{Circa|1650 BCE}} in Egypt.<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> Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized [[oral tradition]] since the [[Vedic period]] ({{c.|1500|500 BCE}}).<ref>{{cite conference
 | title=Indian pedagogy and problem solving in ancient Thamizhakam
 | last=Subramarian
 | first=F.
 | conference=History and Pedagogy of Mathematics conference, July 16–20, 2012
 | url=http://hpm2012.onpcs.com/Proceeding/OT2/T2-10.pdf
 | access-date=November 29, 2022
 | archive-date=November 28, 2022
| archive-url=https://web.archive.org/web/20221128082654/http://hpm2012.onpcs.com/Proceeding/OT2/T2-10.pdf
 | url-status=live
 }}</ref> In [[Imperial China]] during the [[Tang dynasty]] (618–907 CE), a mathematics curriculum was adopted for the [[Imperial examination|civil service exam]] to join the state bureaucracy.<ref>{{cite book
 | chapter=Official Curriculum in Mathematics in Ancient China: How did Candidates Study for the Examination?
 | first=Man Keung | last=Siu
 | series=Series on Mathematics Education
 | title=How Chinese Learn Mathematics
 | pages=157–185 | year=2004 | volume=1 | isbn=978-981-256-014-8
 | doi=10.1142/9789812562241_0006
 | url=https://scholar.archive.org/work/3fb5lb2qsfg35gf2cv6viaydny/access/wayback/http://hkumath.hku.hk:80/~mks/Chapter%206-Siu.pdf
 | access-date=November 26, 2022 }}</ref>

Following the [[Dark Age]]s, mathematics education in Europe was provided by religious schools as part of the [[Quadrivium]]. Formal instruction in [[pedagogy]] began with [[Jesuit]] schools in the 16th and 17th century. Most mathematical curriculum remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was ''[[L'Enseignement Mathématique]]'', which began publication in 1899.<ref>{{cite journal
 | title=The History of Mathematical Education
 | journal=The American Mathematical Monthly
 | volume=74 | issue=1 | pages=38–55
 | publisher=Taylor & Francis, Ltd.
 | doi=10.2307/2314867 | jstor=2314867 | last1=Jones
 | first1=Phillip S.
 | year=1967
 }}</ref> The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component{{emdash}}initially for its military applications.<ref>{{cite journal
 | title=Introduction: the history of mathematics teaching. Indicators for modernization processes in societies
 | first1=Gert | last1=Schubring | first2=Fulvia | last2=Furinghetti
 | first3=Man Keung | last3=Siu
 | journal=ZDM Mathematics Education
 | volume=44 | pages=457–459 | date=August 2012
 | issue=4 | doi=10.1007/s11858-012-0445-7
| s2cid=145507519 | doi-access=free }}</ref> While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.<ref>{{Cite book | chapter=Examining eTIMSS Country Differences Between eTIMSS Data and Bridge Data: A Look at Country-Level Mode of Administration Effects | title=TIMSS 2019 International Results in Mathematics and Science | first1=Matthias | last1=von Davier | first2=Pierre | last2=Foy | first3=Michael O. | last3=Martin | first4=Ina V.S. | last4=Mullis | publisher=[[TIMSS]] & [[PIRLS]] International Study Center, [[Lynch School of Education and Human Development]] and [[International Association for the Evaluation of Educational Achievement]] | isbn=978-1-889938-54-7 | page=13.1 | language=en-US | year=2020 | url=https://files.eric.ed.gov/fulltext/ED610099.pdf | access-date=November 29, 2022 | archive-date=November 29, 2022 | archive-url=https://web.archive.org/web/20221129163908/https://files.eric.ed.gov/fulltext/ED610099.pdf | url-status=live }}</ref>

During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics.<ref>{{cite journal
 | title=Social Cognitive Factors, Support, and Engagement: Early Adolescents' Math Interests as Precursors to Choice of Career
 | first1=Heather T.
 | last1=Rowan-Kenyon
 | first2=Amy K.
 | last2=Swan
 | first3=Marie F.
 | last3=Creager
 | journal=The Career Development Quarterly
 | volume=60
 | issue=1
 | date=March 2012
 | pages=2–15
 | doi=10.1002/j.2161-0045.2012.00001.x
 | url=https://www.academia.edu/download/45974312/j.2161-0045.2012.00001.x20160526-3995-67kydl.pdf
 | access-date=November 29, 2022
 | archive-url=https://web.archive.org/web/20231122212933/https://d1wqtxts1xzle7.cloudfront.net/45974312/j.2161-0045.2012.00001.x20160526-3995-67kydl-libre.pdf?1464293840=&response-content-disposition=inline%3B+filename%3DSocial_Cognitive_Factors_Support_and_Eng.pdf&Expires=1700692172&Signature=cs9KfTPxoPh859wY~ExtJyAl9NpYb3X-2P4rDel1Z3z7DwehsHLRggoZtgi1pMsamxYobu9dVK4G7OsqfvNxcuwz3uKh1pnCMZQEz~ahVtPb4kvN-4dmwExJplzoxWu31o3SJOfuBt0GGE-0Hl8eLfPBg5agmtkjSwAWQwlqGrjp3YgYZGjbNxOEAM4t1l4qvoWXidWvSHHcEUNvlKYwCDvG0~QhGTmA6ldxmfS1ovf0adog-qqvjGxxJuSjtP6O8zCTwkPXYwi2e8giI0H6b5fNarHc-2q~-NRnVVtYKhvSBcwC22kNZoA7s8sp8ix9KIdM3uxiUIBRBRC-4aaVoQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
 | archive-date=November 22, 2023
 | url-status=live
 }}</ref> Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as [[math anxiety]] or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.<ref>{{cite journal
 | title=Spotlight on math anxiety
 | first1=Silke | last1=Luttenberger
 | first2=Sigrid | last2=Wimmer
 | first3=Manuela | last3=Paechter
 | journal=Psychology Research and Behavior Management
 | year=2018 | volume=11 | pages=311–322
 | doi=10.2147/PRBM.S141421 | pmid=30123014
 | pmc=6087017 | doi-access=free }}</ref>

=== Psychology (aesthetic, creativity and intuition) ===
The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a [[computer program]]. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process.<ref>{{cite journal
 | title=The Outlook of the Mathematicians' Creative Processes
 | first=Narges | last=Yaftian
 | journal=Procedia - Social and Behavioral Sciences
 | volume=191 | date=June 2, 2015 | pages=2519–2525
 | doi=10.1016/j.sbspro.2015.04.617
| doi-access=free}}</ref><ref>{{cite journal
 | title=The Frontage of Creativity and Mathematical Creativity
 | first1=Mehdi | last1=Nadjafikhah | first2=Narges | last2=Yaftian
 | journal=Procedia - Social and Behavioral Sciences
 | volume=90 | date=October 10, 2013 | pages=344–350
 | doi=10.1016/j.sbspro.2013.07.101
| doi-access=free}}</ref> An extreme example is [[Apery's theorem]]: [[Roger Apery]] provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.<ref>{{cite journal
 | title=A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)
 | first=A.
 | last=van der Poorten
 | journal=[[The Mathematical Intelligencer]]
 | volume=1
 | issue=4
 | year=1979
 | pages=195–203
 | doi=10.1007/BF03028234
 | s2cid=121589323
 | url=http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf
 | access-date=November 22, 2022
 | archive-date=September 6, 2015
| archive-url=https://web.archive.org/web/20150906015716/http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf
 | url-status=live
 }}</ref>

Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving [[puzzle]]s.<ref>{{cite book
 | title=Famous Puzzles of Great Mathematicians
 | first=Miodrag
 | last=Petkovi
 | date=September 2, 2009
| publisher=American Mathematical Society
 | pages=xiii–xiv
 | isbn=978-0-8218-4814-2
 | url={{GBurl|id=AZlwAAAAQBAJ|pg=PR13}}
 | access-date=November 25, 2022
}}</ref> This aspect of mathematical activity is emphasized in [[recreational mathematics]].

Mathematicians can find an [[aesthetic]] value to mathematics. Like [[beauty]], it is hard to define, it is commonly related to ''elegance'', which involves qualities like [[simplicity]], [[symmetry]], completeness, and generality. G. H. Hardy in ''[[A Mathematician's Apology]]'' expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.<ref>{{cite book
 | title=A Mathematician's Apology
 | last=Hardy | first=G. H. | author-link=G. H. Hardy
 | publisher=Cambridge University Press | year=1940
 | url=https://archive.org/details/hardy_annotated/
 | isbn=978-0-521-42706-7 | access-date=November 22, 2022
}} See also ''[[A Mathematician's Apology]]''.</ref> [[Paul Erdős]] expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book ''[[Proofs from THE BOOK]]'', inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the [[fast Fourier transform]] for [[harmonic analysis]].<ref>{{cite journal
 | title=Reflections on Paul Erdős on His Birth Centenary, Part II
 | first1=Noga | last1=Alon | first2=Dan | last2=Goldston
 | first3=András | last3=Sárközy | first4=József | last4=Szabados
 | first5=Gérald | last5=Tenenbaum | first6=Stephan Ramon | last6=Garcia
 | first7=Amy L. | last7=Shoemaker
 | journal=Notices of the American Mathematical Society
 | date=March 2015 | volume=62 | issue=3 | pages=226–247
 | editor1-first=Krishnaswami | editor1-last=Alladi
 | editor2-first=Steven G. | editor2-last=Krantz
 | doi=10.1090/noti1223
| doi-access=free }}</ref>

Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional [[liberal arts]].<ref>See, for example [[Bertrand Russell]]'s statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his {{cite book | title=History of Western Philosophy | year=1919 | page=60 }}</ref> One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are ''created'' (as in art) or ''discovered'' (as in science).<ref name=borel>{{Cite journal
 | last=Borel | first=Armand | author-link=Armand Borel
 | title=Mathematics: Art and Science
 | journal=The Mathematical Intelligencer
 | volume=5 | issue=4 | pages=9–17 | year=1983
 | publisher=Springer | issn=1027-488X
 | doi=10.4171/news/103/8| doi-access=free
}}</ref> The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.