Mathematics education

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Mathematics education is the study of practices and methods of both the teaching and learning of mathematics. Furthermore, mathematics educators are concerned with the development of tools that facilitate practice and/or the study of practice.

Contents 1 History 2 Objectives 3 Standards 4 Content and age levels 5 Mathematics for the 21st Century 6 Methods 7 Recent controversy over U.S. mathematics education 8 Mathematics teachers 9 Mathematics educators 10 On-line Forums 11 Notes 12 See also 13 External links 13.1 Scholarly Journals: Print 13.2 Scholarly Journals: on-line

[edit] History

Elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece, the Roman empire, 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 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 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 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 and nineteenth centuries the industrial revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic, 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 countries.

[edit] 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) 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 nonroutine problems.

Methods of teaching mathematics have varied in line with changing objectives.

[edit] 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 National Curriculum for England, while Scotland maintains its own educational system.

In the United States, there is no central ministry of curriculum. The curriculum was traditionally set by textbook publishers which followed common teaching practices found to be successful.

[edit] Content and age levels

Different levels of mathematics are taught at different ages. This motivation can sometimes be attributed to the Van Hiele's Levels which is based on the belief that children at different ages have different cognitive preparedness. 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 : age 8; more digits ages 9-10
• Pre-algebra : ages 11-13
• Algebra : ages 13+
• Geometry : ages 14+
• Algebra II : ages 15 +
• Trigonometry : ages 16+
• Calculus : ages 17+

(To convert these ages to American grade levels, age 5 becomes kindergarten; subtract 5 from all older ages to get the relevant grade.)

[edit] Mathematics for the 21st Century

Some progressions of mathematics education include relatively greater emphasis on discrete math such as number theory and combinatorics and sometimes relating to probability, and communcations, including creating the kinds of pie and bar-and-whisker charts commonly used in Powerpoint-style presentations as early as fourth grade. In the state of Washington, for example, Prof. Don Orlich of Washington State University has observed that fourth graders are expected to know the meaning and usage of mean, median and mode. Such depth in statistics are unfamiliar to most adults who would have traditionally learned such topics in high school or college if they learned it at all.

[edit] 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, which was taught as a paradigm of deductive reasoning
• Rote learning - the teaching of mathematical results, definitions and concepts by repetition and memorisation. Typically used to teach multiplication tables. Another term used to deride traditional mathematics is drill and kill, or Parrot Math, which was the title of a paper critical of rote learning.
• Exercises - the teaching of mathematical skills by completing large numbers of exercises of a similar type, such as adding vulgar fractions or solving quadratic equations. For example, Cuisenaire rods are used as a method of teaching fractions.
• 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 problems 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 rather than practical applications. 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 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" math, it was a vision for precollege mathematics education in the 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.

[edit] Recent controversy over U.S. mathematics education

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 systemic education reform took hold based on contructivist practices and the belief in success for all groups including minorities and women. Among the development of a number of controversial standards across reading, science and history, the National Council of Teachers of Mathematics [9] of the United States produced the Curriculum and Evaluation Standards for School Mathematics in 1989. Principles and Standards for School Mathematics [10] included new goals such as equity and de-emphasized the traditional idea of teaching one way to get one correct solution.

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 frameworks. 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 communities where math and science professionals who actually used higher mathematics in real life were aghast at 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 college prep math placed them into remedial math in college, while professor David Klein (California State University Northridge) strongly challenged the idea that traditional mathematics was only suitable for white males in science and engineering.

The standards set forth a democratic vision that for the first time set out to promote equity and mathematical power as a goal for all students, including women and underrepresented minorities. The use of calculator and manipulatives are encouraged, but algebra skills and rote memorization are deemphasized, and there is writing about mathematics as well as computation. Some controversial math curricula such as Investigations in Numbers, Data, and Space were based on research papers such as those by Constance Kamii which assert that teaching of traditional arithmetic methods such as borrowing "not only are not helpful in learning arithmetic, but also hinder children’s development of numerical reasoning".^[1] All students are expected to master enough mathematics to succeed in college, and rather than defining success by rank order, uniform, high standards are set for all students. Explicit goals of standards based education reform are to require all students to pass high standards of performance, to improve international competitiveness, eliminate the achievement gap and produce a productive labor force. Such beliefs, which are congruent with the democratic vision of outcome-based education and standards based education reform that all students will meet standards, refute past research which shows an achievement gap in scores between groups of different education development on every test and assessment, even those aligned with reformed mathematics standards and instruction. The U.S. Department of Education would name several standards based curricula as "exemplary", though academics would respond in protest with an ad taken out the in the Washington Post, and they would note selection was made largely on which curricula implemented the standards most extensively rather than on demonstrated improvements in test scores. The reform standards, while widely accepted as a consensus by education agencies from local to federal levels were met with intense criticism from groups such as Mathematically Correct widely characterized by newspapers such as the Wall Street Journal as "math wars".

In the era of standards based education reform, a curriculum framework is often set at a state level. For example, the California State Board of Education [11] was one of the first to embrace the 1989 standards, and also among the first to move back towards rigorous traditional standards^[2]. In a standards based system, the curriculum is aligned with the standards. The final step in the system is that by 2006, nearly two-thirds of students in the USA would have to pass high school graduation examination set to World class standards of what every student must know and be able to do to succeed in the 21st century. However in states such as Washington, the success of mathematics reform was in question as half of sophomores and four-fifths of minorities were still struggling to pass the math standard needed to make the promise made in the 1993 education reform bill a reality that most or all would graduate two years later with a diploma. While some officials blamed this on incomplete adoption of the 1989 standards, other districts which had already embraced the 1989 standards were deciding instead to replace or supplemen standards-based curricula with more traditional instruction such as Saxon math or Singapore Math in face of poor test results.

The style of instruction can also vary from traditional direct instruction of multi-digit multiplication in books such as Singapore Math to standards-based instruction such as Investigations in Numbers, Time, and Space which may omit instruction or even discourage use of any standard calculation algorithm or method in favor of guiding students to invent their own mathematical power by using 100 charts, colored pencils, glue, writing, and singing songs in different languages. Some education officials have stated that achieving a numerically correct result is secondary to the higher order thinking process. ^[3]

In standards-based curriculum frameworks, math topics and goals may include the history and legacy of diverse multicultural groups in mathematics, mathematical communication, number sense, mathematical power, and equity. Real life examples integrate contemporary issues such as the rain forests, environment, careers, and other topics which integrate other fields of knowledge. Critics including US senators would dub one such text as "rainforest algebra" with 812 pages of seemingly anything but algebra content.^[4]

Related to issues of equity in mathematics, where some groups are under-represented in math and science fields, and others tend to dominate mathematics research, the field of Mathematical Relationships concerns how persons form relationships with mathematics, how they identify with the subject and how they disidentify with it, around social class, gender, race/ethnicity, dis/ability, nationality, and sexuality.^[5] Some critics such as David Klein of California State University Northridge believe such issues belong in social studies, not mathematics, and that mathematics should be taught in a classical method to all students without regard to a student's group affinities.

In a February 9, 1994 article in Education Week on the Web, Steven Leinwand wrote: "It's time to recognize that, for many students, real mathematical power, on the one hand, and facility with multidigit, pencil-and-paper computational algorithms, on the other, are mutually exclusive. In fact, it's time to acknowledge that continuing to teach these skills to our students is not only unnecessary, but counterproductive and downright dangerous." Leinwand was part of the expert panel that in early October of 1999 directed the United States Department of Education to endorse ten K-12 mathematics as"exemplary" or "promising." The "exemplary" programs announced by the Department of Education were:

• Cognitive Tutor Algebra
• College Preparatory Mathematics (CPM)
• Connected Mathematics Program (CMP)
• Core-Plus Mathematics Project
• Interactive Mathematics Program (IMP)

The "promising" programs were:

• Everyday Mathematics
• MathLand
• Middle-school Mathematics through Applications Project (MMAP)
• Number Power
• The University of Chicago School Mathematics Project (UCSMP)

The American Institutes for Research lauded the new U.S. standards for giving greater than nations like Singapore to developing important 21st century mathematical skills that go beyond the skill sets used to develop 20th century technologies such as computers and space flight:^[6]

• Representation
• Reasoning
• Making connections
• Communication
• Statistics, powerpoint-style charts and probability

Mathematics professionals such as David Klein of California State University Northridge challenged the primacy given to gender and race "equity" in the mathematics reform movement. ^[7] One of the themes of the mathematics reform movement is that traditional mathematics fails because women and members of ethnic minority groups learn mathematics differently than white males. Objections to mathematics curricula which introduced cutting, coloring, pasting and multicultural writing while often omitting arithmetic methods recognizable to parents came largely from professionals who used math in real life rather than educators whose "real life" applications might be to use linear algebra to compute bake sale proceeds ^[8].

A few states such as California which were early adopters of the 1989 standards would later revise their math standards and assesements, leading a new movement to reject the assumptions of the original 1989 standards as fatally flawed in favor of rigorous traditional skills and memorization of math facts.^[9] Some public schools in the mid 2000s started to supplement or replace their standards-based mathematics curricula with texts which emphasized direct instruction of traditional mathematics such as Saxon math, popularized by homeschoolers who often rejected standards-based curricula, and Singapore Math because of poor performance on standardized tests compared to other nations and frustration over standards-based approaches which de-emphasized rather than taught arithemetic as it had been known for generations. ^[10].

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 that ^[11] 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. ^[12]

[edit] 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 British author Charles Dodgson, lectured in mathematics at Christ Church, Oxford
• John Dalton, British chemist and physicist, taught mathematics in schools and colleges in Manchester, Oxford and York
• Tom Lehrer, 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^[13]
• Georg Joachim Rheticus, Austrian 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
• Archie Williams, American athlete and Olympic gold medalist, taught mathematics at high schools in California

[edit] Mathematics educators

The following people had a significant influence on the teaching of mathematics at various periods in history:

• Tatyana Alexeyevna Afanasyeva, Dutch/Russian mathematician who advocated the use of visual aids and examples for introductory courses in geometry for high school students.[12]
• Georges Cuisenaire, Belgian primary school teacher who invented Cuisenaire rods
• Euclid, author of The Elements
• Robert Lee Moore, originator of the Moore method
• Robert Parris Moses, founder of the nationwide US Algebra project
• George Pólya, author of How to Solve It
• Toru Kumon, originator of the Kumon method based on mastery through exercise

[edit] On-line Forums

Since the establishment of social software tools on the internet, several mathematics forums have appeared on the internet. The use of social software for educational purposes is still very experimental. On-line forums for mathematics education and problem solving include:

• Art of Problem Solving -- Forums for advanced students and problem solvers
• Math Forum -- Forums and resources for mathematics students; run by Drexel University
• S.O.S. Mathematics
• MathLinks -- International forum for mathematics competition students

(Art of Problem Solving and Mathlinks have merged)

• WebGraphing.com Forum -- For algebra, precalculus, and calculus students

[edit] Notes

1. ^ [1] “The Harmful Effects of Algorithms in Grades 1-4, NCTM Yearbook by Constance and Ann Dominick
2. ^ http://www.air.org/news/documents/Singapore%20Report%20(Bookmark%20Version).pdf AIR report in pdf "The California mathematics framework is modeled on Singaporean and Japanese frameworks"
3. ^ [2] February 9, 2006 "Alpine trio defend approach to math" By Laura Hancock Deseret Morning News (Utah) "while a right answer was important, it is not our belief (it's) as important to get the right answer than to get the process."
4. ^ [3] "Addison--Wesley's Secondary Math: An Integrated Approach. Following reviews of the book from numerous mathematics professors and reading the text herself, Jennings unofficially dubbed the textbook, Rain Forest Algebra."
5. ^ [http://ioewebserver.ioe.ac.uk/ioe/cms/get.asp?cid=4381&4381_0=12442 Mathematical Relationships seminars
6. ^ AMERICAN INSTITUTES FOR RESEARCH. What the United States Can Learn From Singapore’s World-Class Mathematics System [4] February 7, 2005
7. ^ [5] Published in: How To Teach Mathematics,by Steven Krantz, American Mathematical Society, January 1999. "Big Business, Race, and Gender in Mathematics Reform" by David Klein
8. ^ McDougal Littell Integrated Mathematics textbook
9. ^ [6] "A quarter century of US 'maths wars' and political partisanship" David Klein California State University, Northridge, USA. Accepted for publication in the BSHM Bulletin, the journal of the British Society for the History of Mathematics
10. ^ [7] "Back to basics on kids’ math" Alarmed by low scores, Tacoma school officials OK added Saxon textbook. by Debby Abe; The News Tribune (Tacoma WA) August 25th, 2006
11. ^ Wall Street Journal, New York Times, Chicago Sun Times
12. ^ [8] Report Urges Changes in the Teaching of Math in U.S. Schools by TAMAR LEWIN New York Times September 13, 2006
13. ^ Freddie Mercury Interview, Melody Maker, May 2, 1981

[edit] See also

• 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
• Computer Based Mathematics Education

[edit] External links

• History of Mathematical Education
• Teach Kids Math With Model Method
• International Group for the Psychology of Mathematics Education
• National Council of Teachers of Mathematics
• [13] A quarter century of US 'maths wars' and political partisanship. David Klein. California State University, Northridge, USA

[edit] Scholarly Journals: Print

• Educational Studies in Mathematics
• Journal for Research in Mathematics Education
• For the Learning of Mathematics

[edit] Scholarly Journals: on-line

• The Philosophy of Mathematics Education Journal homepage