Mathematics Education

 

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 maths 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 maths education as a practice.

History

Elementary mathematics was part of the education system in most ancient civilizations, 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 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 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.

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 (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 electric age also invested 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 sets."^[1]

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 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 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 National Council of Teachers of Mathematics (NCTM) has published the Principles and Standards for School Mathematics. In 2006, they released the 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: age 8; more digits ages 9-10
• Pre-algebra: ages 11-13
• Algebra I: ages 13+
• Geometry: ages 14+
• Algebra II: ages 15+
• Trigonometry: ages 16+
• 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, 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 tables. A derisory term is drill and kill. Parrot Maths 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. 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 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

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 [10] of the United States produced the Curriculum and Evaluation Standards for School Mathematics in 1989. Principles and Standards for School Mathematics [11] 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 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 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.

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".^[2] 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; the controversy was 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 [12] was one of the first to embrace the 1989 standards, and also among the first to move back towards traditional standards^[3]. 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 supplement standards-based curricula with more traditional instruction such as Saxon math or Singapore Math in face of poor standardized 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. ^[4]

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.^[5]

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.^[6] 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:^[7]

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

Some mathematicians such as David Klein of California State University Northridge challenged the emphasis given to gender and race "equity" in the mathematics reform movement. ^[8] One of the themes of the mathematics reform movement is that traditional mathematics fails because women and members of ethnic minority groups are treated differently than white males. Objections to mathematics curricula which introduced multicultural writing while often omitting traditional arithmetic methods recognizable to parents came largely from mathematicians rather than educators whose "real life" applications might be to use linear algebra to compute bake sale proceeds ^[9].

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 traditional skills and memorization of math facts.^[10] 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 arithmetic as it had been known for generations. ^[11].

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 ^[12] 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. ^[13]

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^[14]
• 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

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.[13]
• Georges Cuisenaire, Belgian primary school teacher who invented Cuisenaire rods
• Euclid, author of The Elements
• Leonhard Euler, author of Elements of Algebra
• 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
• Robert & Ellen Kaplan, international best-selling authors of Nothing That Is, Chances Are: Adventures in Probability , and The Art of the Infinite: The Pleasures of Mathematics
• Toru Kumon, originator of the Kumon method based on mastery through exercise

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
• Dyscalculia
• Philosophy of education

References

External links

• 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.
• Mathematics Education Articles at Convergence
• History of Mathematical Education
• International Group for the Psychology of Mathematics Education
• K-8 Interactive Math Practice
• K-12 Online Math Resources
• [14] A quarter century of US 'maths wars' and political partisanship. David Klein. California State University, Northridge, USA
• Teaching Leonardo: An Integrated Approach at Convergence
• Online Math Activity Objects
• Motivational Resources Showing Valuable Applications of Math

Teacher organizations and others associated with mathematics education

• National Council of Teachers of Mathematics (NCTM (USA)
• [15] The National Centre for Excellence in Teaching Mathematics (NCETM) (England)
• [16] The Association of Teachers of Mathematics (UK)
• [17] The Mathematical Association (UK)
• [18] MatheMagic
• [19] SNM (Poland)
• [20]] APM (Portugal)
• [21] Homi Bhabha Centre for Science Education (India)

Scholarly journals: print

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

Scholarly journals: on-line

• The Philosophy of Mathematics Education Journal homepage
• The Mathematics Educator
• International Journal for Mathematics and Learning

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