Mathematics education

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In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research.

Researchers in mathematics education are primarily concerned with the tools, methods and approaches that facilitate practice or the study of practice. However mathematics education research, known on the continent of Europe as the didactics or pedagogy of mathematics, has developed into an extensive field of study, with its own concepts, theories, methods, national and international organisations, conferences and literature. This article describes some of the history, influences and recent controversies.

Contents

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 and French were published by Robert Recorde, beginning with The Grounde of Artes in 1540. However, there are many different writings on mathematics and math 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 own methodology for solving equations like the quadratic equation. After the Sumerians some of the most famous ancient works on mathematics come from Egypt in the form of the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus. The more famous Rhind Papyrus has been dated 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.

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 being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics being 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 18th and 19th 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 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 '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:

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

Research

The following results are examples of some of the current findings in the field of mathematics education:[2]

Formative assessment[2]
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. Only short-term (within and between lessons) and medium-term (within and between units) assessment is effective. 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.
Effective instruction[2]
The two most important criteria for helping students gain conceptual understanding are making connections and intentionally struggling with important ideas. Research in the 70s and 80s concluded that skill efficiency is best attained by rapid pacing, direct instruction, and students practicing at least until mastery level. According to summaries of recent research, students who learn skills in conceptually-oriented instruction are better able to adapt their skills to new situations.
Students with difficulties[2]
Students with genuine difficulties (unrelated to motivation or past instruction) struggle with 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.
Bilingualism[2]
Most bilingual adults switch languages when calculating. Such code-switching has no impact on math ability and should not be discouraged.
Learning statistics[2]
When studying statistics, children benefit from exploring, studying, and sharing reasoning about centers, shape, spread, and variability. The ability to calculate averages does not signify an understanding of population and sample mean. Students have difficulty remembering to use the sample median as a robust estimator of location, which is less sensitive than the sample mean or the sample mode when the data-set has outliers or when the population is skewed or heavy-tailed.
Algebraic reasoning[2]
It is important for elementary school children 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. They prefer arithmetic reasoning to algebraic equations for solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns. Students often have trouble with the minus sign and understand the equals sign to mean "the answer is...."
What can we learn from research?
Studies can examine the interaction of treatments with covariates and with blocking factors, instead of just estimating the main effects of treatments. Such examinations of interaction give insight into the contexts in which treatments work. Even well-done randomized experiments can yield different results, so it is important to read reviews of many well-done randomized experiments studies when deciding on practical implications.

Methodology controversy

In the scholarly literature in mathematics education, relatively few articles report random experiments in which teaching methods were randomly assigned to classes.[3][4] Randomized experiments have been done to evaluate non-pedagogical programs in schools, such as programs to reduce teenage-pregnancy and to reduce drug use.[5] Randomized experiments were commonly done in educational psychology in the 1880s at American universities.[6][7] However, randomized experiments have been relatively rare in education in recent decades. In other disciplines concerned with human subjects, like biomedicine, psychology, and policy evaluation, controlled, randomized experiments remain the preferred method of evaluating treatments.[8][9] Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods.[4] On the other hand, many scholars in educational schools have argued against increasing the number of randomized experiments, often because of philosophical objections.[3]

Both randomized experiments and observational studies are useful for suggesting new hypotheses, which can eventually be tested by new randomized experiments. The scientific community recognizes that observational studies remain valuable in education—just as observational studies remain valuable in epidemiology, political science, economics, sociology, anthropology, etc.[10]

In the United States, the National Mathematics Advisory Panel (NMAP) published a report in 2008 based on studies using randomized and nonrandom assignments that involved various statistically controlled experiments; the NMAP report's preference for randomized experiments received criticism from some scholars.[11] In 2010, the What Works Clearinghouse (essentially the research arm for the Department of Education) responded to ongoing controversy by extending its research base to include non-experimental studies, including regression-discontinuity designs and single-case studies. [12]

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, while Scotland maintains its own educational system. In the USA, the National Governors Association Center for Best Practices and the Council of Chief State School Officers have published the national mathematics Common Core State Standards Initiative.

Ma (2000) summarised the research of others who found, based on nationwide data, that students with higher scores on standardised 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.[13]

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 through grade 8. However, these standards are not nationally enforced in US schools.

Content and age levels

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 in a similar fashion, though there are differences. In the United States fractions are typically taught starting from 1st grade, whereas in other countries they are usually taught later, since the metric system does not require young children to be familiar with them. Most countries tend to cover fewer topics in greater depth than in the United States.[14]

In most of the US, algebra, geometry and analysis (pre-calculus and calculus) are taught as separate courses in different years of high school. Mathematics in most other countries (and in a few US states) is integrated, with topics from all branches of mathematics studied every year. Students in many countries choose an option or pre-defined course of study rather than choosing courses à la carte as in the United States. Students in science-oriented curricula typically study differential calculus and trigonometry at age 16-17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions, and infinite series in their final year of secondary school.

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:

Mathematics teachers

The following people all taught mathematics at some stage in their lives, although they are better known for other things:

Mathematics educators

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

Organizations

See also

Aspects of mathematics education
General education
North American issues
Mathematical difficulties
Other

References

  1. ^ Marshall McLuhan (1964) Understanding Media, p.13 [1]
  2. ^ a b c d e f g Research clips and briefs
  3. ^ a b Thomas D. Cook, Randomized Experiments in Educational Policy Research: A Critical Examination of the Reasons the Educational Evaluation Community has Offered for Not Doing Them, Educational Evaluation and Policy Analysis 24 (2002), no. 3, 175-199.
  4. ^ a b Richard Scheaffer, ed. Working Group on Statistics in Mathematics Education Research (Richard Scheaffer, Martha Aliaga, Marie Diener-West, Joan Garfield, Traci Higgins, Sterling Hilton, Gerunda Hughes, Brian Junker, Henry Kepner, Jeremy Kilpatrick, Richard Lehrer, Frank K. Lester, Ingram Olkin, Dennis Pearl, Alan Schoenfeld, Juliet Shaffer, Edward Silver, William Smith, F. Michael Speed, and Patrick Thompson, 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. The American Statistical Association, 2007.
  5. ^ Cook.
  6. ^ Stephen M. Stigler (November 1992). "A Historical View of Statistical Concepts in Psychology and Educational Research". American Journal of Education 101 (1): 60–70. doi:10.1086/444032. 
  7. ^ Trudy Dehue (December 1997). "Deception, Efficiency, and Random Groups: Psychology and the Gradual Origination of the Random Group Design". Isis 88 (4): 653–673. doi:10.1086/383850. PMID 9519574. 
  8. ^ Shadish, William R., Cook, Thomas D., & Campbell, Donald T. (2002). Experimental and quasi-experimental designs for generalized causal inference. Boston: Houghton Mifflin Company. ISBN 0-395-61556-9. 
  9. ^ See articles on NCLB, National Mathematics Advisory Panel, Scientifically-based research and What Works Clearinghouse
  10. ^ Raudenbush, Stephen (2005). %7C . "Learning from Attempts to Improve Schooling: The Contribution of Methodological Diversity". Educational Researcher 34 (5): 25–31. doi:10.3102/0013189X034005025. http://edr.sagepub.com/cgi/content/abstract/34/5/25 %7C .. 
  11. ^ Kelly, Anthony (2008). "Reflections on the National Mathematics Advisory Panel Final Report". Educational Researcher 37 (9): 561–564. doi:10.3102/0013189X08329353. http://edr.sagepub.com/cgi/content/abstract/37/9/561.  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.
  12. ^ Sparks, Sarah (October 20, 2010). "Federal Criteria For Studies Grow". Education Week: p. 1. 
  13. ^ Ma, X. (2000). A longitudinal assessment of antecedent course work in mathematics and subsequent mathematical attainment. Journal of Educational Research, 94, 16-29.
  14. ^ (E.g., p. 20)
  15. ^ "Crossroads in the History of Mathematics and Mathematics Education". The Mathematics Enthusiast: Monograph Series in Mathematics Education. 2012. http://www.math.umt.edu/TMME/Monograph12/Mono12_TOC.pdf. 
  16. ^ "Crossroads in the History of Mathematics and Mathematics Education". The Mathematics Enthusiast: Monograph Series in Mathematics Education. 2012. http://www.infoagepub.com/products/Crossroads-in-the-History-of-Mathematics. 
  17. ^ Freddie Mercury Interview, Melody Maker, May 2, 1981
  18. ^ http://www.pims.math.ca/~hoek/teageo/TEA.pdf

Further reading

External links