Mathematics education
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2008-06-23T19:12:03Z
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/* Content and age levels */
'''Mathematics education''' is a term that refers both to the practice of [[teaching]] and [[learning]] [[mathematics]], as well as to a field of scholarly [[research]] on this practice. Researchers in math education are in the first instance concerned with the tools, methods and approaches that facilitate practice and/or the study of practice. However [[mathematics education research]], known on the continent of Europe as the [[didactics]] of mathematics has developed into a fully fledged field of study, with its own characteristic concepts, theories, methods, national and international organizations, conferences and literature. This article describes some of the history, influences and recent controversies concerning math education as a practice.
[[Image:Math lecture at TKK.JPG|300px|right|thumb|A mathematics lecture at Helsinki University of Technology.]]
==History==
[[Image:Woman teaching geometry.jpg|250px|left|thumb|Illustration at the beginning of a 14th century translation of Euclid's ''Elements''.]]
'''Elementary mathematics''' was part of the [[Education|education system]] in most ancient civilizations, including [[Ancient Greece]], the [[Roman empire]], [[Vedic civilization|Vedic society]] and [[ancient Egypt]]. In most cases, a formal education was only available to [[male]] children with a sufficiently high status, wealth or [[caste]].
In [[Plato]]'s division of the [[liberal arts]] into the [[trivium (education)|trivium]] and the [[quadrivium]], the quadrivium included the mathematical fields of [[arithmetic]] and [[geometry]]. This structure was continued in the structure of [[classical education]] that was developed in medieval Europe. Teaching of geometry was almost universally based on [[Euclid]]'s ''Elements''. Apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession.
The first mathematics textbooks to be written in English and French were published by [[Robert Recorde]], beginning with ''The Grounde of Artes'' in 1540.
In the [[Renaissance]] the academic status of mathematics declined, because it was strongly associated with trade and commerce. Although it continued to be taught in European universities, it was seen as subservient to the study of Natural, Metaphysical and Moral Philosophy.
This trend was somewhat reversed in the [[seventeenth century]], with the [[University of Aberdeen]] creating a Mathematics Chair in 1613, followed by the Chair in Geometry set up in [[University of Oxford]] in 1619 and the [[Lucasian Professor of Mathematics|Lucasian Chair of Mathematics]], established by the [[University of Cambridge]] in 1662. However, it was uncommon for mathematics to be taught outside of the universities. [[Isaac Newton]], for example, received no formal mathematics teaching until he joined [[Trinity College, Cambridge]] in 1661.
In the [[eighteenth century|eighteenth]] and [[nineteenth century|nineteenth]] 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. Here are some of the main events in this development:
* In 1893 a Chair in mathematics education was created at the University of Göttingen, under the administration of [[Felix Klein]]
* The International Commission on Mathematical Instruction ([[ICMI]]) was founded in 1908, and Felix Klein became the first president of the organization
* A new interest in mathematics education emerged in the 1960s, and the commission was revitalized
* In 1968, the [[Shell Centre for Mathematical Education]] was established in [[Nottingham]]
* The first International Congress on Mathematical Education ([[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
In the 20th century, the cultural impact of the "[[electric 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 [http://www9.georgetown.edu/faculty/irvinem/theory/McLuhan-Understanding_Media-I-1-7.html]</ref>
==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 of basic [[numeracy]] skills to all pupils
* The teaching of practical mathematics ([[arithmetic]], [[elementary algebra]], plane and solid [[geometry]], [[trigonometry]]) to most pupils, to equip them to follow a trade or craft
* The teaching of abstract mathematical concepts (such as [[set]] and [[function (mathematics)|function]]) at an early age
* The teaching of selected areas of mathematics (such as [[Euclidean geometry]]) as an example of an [[axiomatic system]] and a model of [[deductive reasoning]]
* The teaching of selected areas of mathematics (such as [[calculus]]) as an example of the intellectual achievements of the [[modern world]]
* The teaching of advanced mathematics to those pupils who wish to follow a career in [[science]]
* The teaching of [[heuristics]] and other problem-solving strategies to solve non routine problems.
Methods of teaching mathematics have varied in line with changing objectives.
==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 and realistic 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 [http://www.nc.uk.net/webdav/servlet/XRM?Page/@id=6004&Subject/@id=22 National Curriculum for England], while Scotland maintains its own educational system.
Ma (2000) summarized the research of others who found, based on nationwide data, that students with higher scores on standardized math tests had taken more mathematics courses in high school. This led some states to require three years of math instead of two. But because this requirement was often met by taking another lower level math course, the additional courses had a “diluted” effect in raising achievement levels.
In North America, the [http://www.nctm.org National Council of Teachers of Mathematics (NCTM)] has published the [http://standards.nctm.org/ Principles and Standards for School Mathematics]. In 2006, they released the [http://www.nctm.org/focalpoints/ Curriculum Focal Points], which recommend the most important mathematical topics for each grade level.
Ma, X. (2000). A longitudinal assessment of antecedent course work in mathematics and subsequent mathematical attainment. ''Journal of Educational Research, 94'', 16-29.
==Content and age levels==
Different levels of mathematics are taught at different ages. Sometimes a class may be taught at an earlier age as a special or "honors" class. A rough guide to the ages at which the sub-topics of arithmetics and algebra are taught in the [[United States]] is as follows:
* [[Addition]]: ages 5-7; more digits ages 8-9
* [[Subtraction]]: ages 5-7; more digits ages 8-9
* [[Multiplication]]: ages 7-8; more digits ages 9-10
* [[Division (mathematics)|Division]]: age 8; more digits ages 9-10
* [[Pre-algebra]]: ages 11-13
* [[Algebra I]]: ages 13+
* [[Geometry]]: ages 14+
* Algebra II: ages 15+
* [[Trigonometry]] (or [[Statistics]] or Algebra 3): ages 16+
* [[Calculus]] (or [[Pre-Calculus]]): ages 17+
For comparison to American grade levels, most Americans begin [[kindergarten]], the year before first grade in the American schooling system, between the ages of 4 and 6.
==Methods==
The method or methods used in any particular context are largely determined by the objectives that the relevant educational system is trying to achieve. Methods of teaching mathematics include the following:
*'''[[Classical education]]''' - the teaching of mathematics within the classical education syllabus of the [[Middle Ages]] was typically based on [[Euclid's Elements|Euclid's ''Elements'']], which was taught as a [[paradigm]] of [[deductive reasoning]]
*'''[[Rote learning]]''' - the teaching of mathematical results, definitions and concepts by repetition and memorization. Typically used to teach [[multiplication table]]s. A derisory term is [[drill and kill]]. [[Parrot Maths]] was the title of a paper critical of rote learning.
*'''Exercises''' - the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding [[vulgar fraction]]s or solving [[quadratic equation]]s.
*'''[[Problem solving]]''' - the cultivation of mathematical ingenuity, creativity and [[heuristic]] thinking by setting students open-ended, unusual, and sometimes insoluble problems. The problems can range from simple [[word problem]]s to problems from international [[mathematics competitions]] such as the [[International Mathematical Olympiad]].
*'''[[New math]]''' - a method of teaching mathematics which focuses on abstract concepts such as [[set theory]], functions and bases other than ten. Adopted as a response to the challenge of early Soviet technical superiority in space, it would be largely abandoned and discredited by the late 1960s. The new math 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."
*'''Historical method''' - teaching the development of mathematics within an historical, social and cultural context. Provides more [[human interest]] than a purely abstract approach.
*'''Reform''' or [[Standards-based mathematics]]. Often called the "new new" maths, it was a vision for precollege mathematics education in the [[United States|US]] and [[Canada]], based on the [[Constructivism (learning theory)]], and formalized by the [[National Council of Teachers of Mathematics]] which created the [[Principles and Standards for School Mathematics]].
==Recent controversy over U.S. mathematics education==
{{main | Math wars | National Council of Teachers of Mathematics }}
Near the end of the 20th century diverse and changing ideas about the purpose of mathematical education would lead to wide adoption of reform-based standards and curricula funded by the US federal government, and also adopted by other national curriculum standards These were based on [[student-centered learning]] methods and equity in mathematics as a centerpiece of the [[standards based education reform]] movement. This movement in turn was met with opposition which called for a return to traditional direction instruction of time-tested arithmetic methods by the start of the 21st century as some schools and districts supplemented or replaced standards-based curricula.
With the adoption of substantially different teaching reform standards and the development and widespread adoption of federally funded curricula during the 1990s, mathematics education has become the most hotly debated subject since the original 1960s "[[New Math]]" in mainstream news journals such as the Wall Street Journal and New York Times. There is a significant difference in perspective between the relative few who practice mathematics in their careers, and those who have been tasked with teaching mathematics to children. The goals for educators since the 1990s have been expanded in the context of systemic [[standards based education reform]] in the United States and other nations to promote increased learning for all students. It is a goal to achieve equity and success for all groups in society. It is no longer acceptable to many in the education community that some were historically excluded from the full range of opportunities open to those who learned the most advanced mathematics.
By the late 1980s, a movement for [[outcome-based education|systemic education reform]] took hold based on constructivist practices and the belief in success for all groups including minorities and women. Among the development of a number of controversial standards across [[whole language|reading]], [[inquiry-based science|science]] and history, the [[National Council of Teachers of Mathematics]] [http://nctm.org] of the United States produced the Curriculum and Evaluation Standards for School Mathematics in 1989. [[Principles and Standards for School Mathematics]] [http://standards.nctm.org/] included new goals such as equity and de-emphasized the traditional idea of relying solely on standard algorithms to get solutions.
The controversial 1989 [[NCTM standards]] recommended teaching elements of algebra as early as grade 5, and elements of calculus as early as grade 9, though this was rarely adopted even as late as the 2000s. In [[standards based education reform]], all students, not only the college bound must take advanced mathematics. In some large school districts, this means requiring algebra of all students by the end of junior high school, compared to the tradition of tracking only college bound and the most advanced junior high school students to take algebra.
The standards soon became the basis for many new federally funded curricula such as the [[Core-Plus Mathematics Project]] and became the foundation of many local and state [[curriculum framework]]s. Although the standards were the [[consensus]] of those teaching mathematics in the context of real life, they also became a lightning rod of criticism as math wars erupted in some communities that were opposed to some of the more radical changes to mathematics instruction such as [[Mathland]]'s Fantasy Lunch and what some dubbed "rainforest algebra". Some students complained that their new math courses placed them into remedial math in college.
In 2000 and 2006 NCTM released another standards document and the Curriculum Focal Points which expanded on the work of the previous standards documents. Refuting reports and editorials <ref>Wall Street Journal, New York Times, Chicago Sun Times</ref> that it was largely an admission that the previous standards had mistakenly de-emphasized instruction of basic skills, NCTM spokesmen maintained that it provided more grade by grade specificity on key areas of study for a coherent and consistent development of mathematical understanding and skill.
In 2000 and 2006, the same [[NCTM]] issued new studies that criticized American math standards as a "mile wide and an inch deep" in comparison to the math of nations such as [[Singapore Math]]. Rather than backing research which had called them harmful, it called for strong instruction of basic skills. The New York Times and the Wall Street Journal called it a significant retreat back towards traditional mathematics, and some warned it might lead to a generation who could solve equations accurately, but not deeply understand mathematics, or relate it to real life issues such as the environment.<ref>[http://www.nytimes.com/2006/09/13/education/13math.html?_r=1&ref=us] Report Urges Changes in the Teaching of Math in U.S. Schools by TAMAR LEWIN New York Times September 13, 2006</ref>
==Mathematics teachers==
The following people all taught mathematics at some stage in their lives, although they are better known for other things:
* [[Lewis Carroll]], pen name of [[United Kingdom|British]] author Charles Dodgson, lectured in mathematics at [[Christ Church, Oxford]]
* [[John Dalton]], British chemist and physicist, taught mathematics at schools and colleges in [[Manchester]], [[Oxford]] and [[York]]
* [[Tom Lehrer]], [[United States|American]] songwriter and satirist, taught mathematics at [[Harvard]], [[MIT]] and currently at [[University of California, Santa Cruz]].
* [[Brian May]], rock guitarist and composer, worked briefly as a mathematics teacher before joining [[Queen (band)|Queen]]<ref>[http://mr-mercury.co.uk/freddie_1981_interview.htm Freddie Mercury Interview, Melody Maker, May 2, 1981]</ref>
* [[Georg Joachim Rheticus]], [[Austria]]n cartographer and disciple of [[Copernicus]], taught mathematics at the [[University of Wittenberg]]
* [[Edmund Rich]], [[Archbishop of Canterbury]] in the 13th century, lectured on mathematics at the universities of [[Oxford]] and [[Paris]]
* [[Éamon de Valera]], a leader of Ireland's struggle for independence in the early 20th century and founder of the [[Fianna Fáil]] party, taught mathematics at schools and colleges in [[Dublin]]
* [[Archie Williams]], [[United States|American]] athlete and Olympic gold medalist, taught mathematics at high schools in [[California]]
==Mathematics educators==
The following people had a significant influence on the teaching of mathematics at various periods in history:
* [[Tatyana Afanasyeva|Tatyana Alexeyevna Afanasyeva]], [[Netherlands|Dutch]]/[[Russia]]n mathematician who advocated the use of visual aids and examples for introductory courses in geometry for high school students.[http://www.pims.math.ca/~hoek/teageo/TEA.pdf]
* [[Georges Cuisenaire]], [[Belgium|Belgian]] primary school teacher who invented [[Cuisenaire rods]]
* [[Euclid]], author of [[Euclid's Elements|''The Elements'']]
* [[Leonhard Euler]], author of [http://web.mat.bham.ac.uk/C.J.Sangwin/euler/, Elements of Algebra]
* [[Robert Lee Moore]], originator of the [[Moore method]]
* [[Robert Parris Moses]], founder of the nationwide [[United States|US]] [[Algebra project]]
* [[George Pólya]], author of [[How to Solve It]]
* Robert & Ellen Kaplan, international best-selling authors of [http://www.amazon.com/dp/0195142373/ Nothing That Is], [http://www.amazon.com/dp/0143038346/ Chances Are: Adventures in Probability ], and [http://www.amazon.com/dp/0195176065/ The Art of the Infinite: The Pleasures of Mathematics ]
* [[Toru Kumon]], originator of the [[Kumon method]] based on mastery through exercise
==See also==
<div style="-moz-column-count:2; column-count:2;">
* [[National Council of Teachers of Mathematics]] which created the [[Principles and Standards for School Mathematics]]
* [[U.S. Department of Education exemplary mathematics programs]] which implement the [[NCTM standards]]
* [[Mathematically Correct]] which is critical of the [[NCTM standards]]
* [[Anti-racist mathematics]] using mathematics education to fight [[racism]]
* [[Traditional mathematics]]
* [[Traditional education]]
* [[Constructivism (learning theory)|Constructivism]]
* [[Computer Based Mathematics Education]]
* [[Dyscalculia]]
* [[Math worksheet generator]]
* [[Philosophy of education]]
</div>
== References ==
{{reflist}}
==External links==
* [http://www.clso-math.com CLSO-Math] Concepts-Language-Symbols-Operation An innovative math teaching methodology developed by a Harvard graduate and based on recent learning theories and many decades of practical experience.
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=browseNode&resultsPerPage=20&categoryId=&subjectId=18&searchKeywords=&imageField.x=115&imageField.y=17 Mathematics Education Articles] at [http://mathdl.maa.org/convergence/1/ Convergence]
* [http://www-gap.dcs.st-and.ac.uk/~history/Education/index.html History of Mathematical Education]
* [http://igpme.org/ International Group for the Psychology of Mathematics Education]
* [http://www.aaamath.com/ K-8 Interactive Math Practice]
* [http://www.aolatschool.com/students/math K-12 Online Math Resources]
* [http://www.csun.edu/~vcmth00m/bshm.html] A quarter century of US 'maths wars' and political partisanship. David Klein. California State University, Northridge, USA
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=477&bodyId=595 Teaching Leonardo: An Integrated Approach] at [http://mathdl.maa.org/convergence/1/ Convergence]
* [http://www.adaptivecurriculum.com/ Online Math Activity Objects]
* [http://www.mathmotivation.com/ Motivational Resources Showing Valuable Applications of Math]
* [http://hk.geocities.com/mathphyweb/puremath.htm How to Become a Pure Mathematician ] a List of Undergraduate and Basic Graduate Textbook.
* [http://mathsolympiad.googlepages.com/ebooks Mathematics Related E-books]
===Teacher organizations and others associated with mathematics education===
* [http://www.nctm.org National Council of Teachers of Mathematics] ([[NCTM]] (USA)
* [http://www.ncetm.org.uk] The [[National Centre for Excellence in Teaching Mathematics]] (NCETM) (England)
* [http://www.atm.org.uk] [[The Association of Teachers of Mathematics]] (UK)
* [http://www.ma.org.uk] [[The Mathematical Association]] (UK)
* [http://www.mathemagic.org] [[MatheMagic]]
* [http://www.snm.org.pl] [[SNM]] (Poland)
* [http://www.apm.org.po] [[APM]] (Portugal)
* [http://web.gnowledge.org/mathedu] [[Homi Bhabha Centre for Science Education]] (India)
===Scholarly journals: print===
* [http://springerlink.metapress.com/app/home/journal.asp?wasp=79421927ccf647fcb830fafe34a31b4a&referrer=parent&backto=linkingpublicationresults,1:102875,1 Educational Studies in Mathematics]
* [http://flm.educ.ualberta.ca/ For the Learning of Mathematics]
* [http://my.nctm.org/eresources/journal_home.asp?journal_id=1 Journal for Research in Mathematics Education]
* [http://springerlink.metapress.com/content/1573-1820/ Journal of Mathematics Teacher Education]
*[http://www.infoagepub.com/products/journals/TMME/ The Montana Mathematics Enthusiast]
===Scholarly journals: on-line===
* [http://www.people.ex.ac.uk/PErnest/ The Philosophy of Mathematics Education Journal homepage]
* [http://math.coe.uga.edu/tme/tmeonline.html The Mathematics Educator]
* [http://www.montanamath.org/TMME/index.html The Montana Mathematics Enthusiast]
* [http://www.cimt.plymouth.ac.uk/journal/default.htm International Journal for Mathematics Teaching and Learning]
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