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.
Illustration at the beginning of a 14th-century translation of Euclid's Elements.
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.
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 1893, a Chair in mathematics education was created at the University of Göttingen, under the administration of Felix Klein
In the 20th century, the cultural impact of the "electronic 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[edit]
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 and learning of basic numeracy skills to all pupils
The teaching of heuristics and other problem-solving strategies to solve non-routine problems.
Methods[edit]
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:
Conventional approach: the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with arithmetic and is followed by Euclidean geometry and elementary algebra taught concurrently. Requires the instructor to be well informed about elementary mathematics, since didactic and curriculum decisions are often dictated by the logic of the subject rather than pedagogical considerations. Other methods emerge by emphasizing some aspects of this approach.
Games can motivate students to improve skills that are usually learned by rote. In "Number Bingo," players roll 3 dice, then perform basic math operations on those numbers to get a new number, which they cover on the board trying to cover 4 squares in a row. This game was played at a "Discovery Day" organized by Big Brother Mouse in Laos.
Rote learning: the teaching of mathematical results, definitions and concepts by repetition and memorisation typically without meaning or supported by mathematical reasoning. A derisory term is drill and kill. In traditional education, rote learning is used to teach multiplication tables, definitions, formulas, and other aspects of mathematics.
Problem solving: the cultivation of mathematical ingenuity, creativity and heuristic thinking by setting students open-ended, unusual, and sometimes unsolved problems. The problems can range from simple word problems to problems from international mathematics competitions such as the International Mathematical Olympiad. Problem solving is used as a means to build new mathematical knowledge, typically by building on students' prior understandings.
New Math: a method of teaching mathematics which focuses on abstract concepts such as set theory, functions and bases other than ten. Adopted in the US as a response to the challenge of early Soviet technical superiority in space, it began to be challenged in the late 1960s. One of the most influential critiques of the New Math was Morris Kline's 1973 book Why Johnny Can't Add. The New Math method 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 the conventional approach.,^{[2]}
Relational approach: Uses class topics to solve everyday problems and relates the topic to current events.^{[3]} This approach focuses on the many uses of math and helps students understand why they need to know it as well as helping them to apply math to real world situations outside of the classroom.
Recreational mathematics: Mathematical problems that are fun can motivate students to learn mathematics and can increase enjoyment of mathematics.^{[4]}
Computer-based math an approach based around use of mathematical software as the primary tool of computation.
Content and age levels[edit]
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.^{[citation needed]} Most countries tend to cover fewer topics in greater depth than in the United States.^{[5]} K-12 topics include elementary arithmetic (addition, subtraction, multiplication, and division), and pre-algebra.
In most of the U.S., 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 U.S. 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. Probability and statistics may be taught in secondary education classes.
Science and engineering students in colleges and universities may be required to take multivariable calculus, differential equations, linear algebra. Applied mathematics is also used in specific majors; for example, civil engineers may be required to study fluid mechanics,^{[6]} while "math for computer science" might include graph theory, permutation, probability, and proofs.^{[7]} (Mathematics students obviously would continue to study potentially any area.)
Standards[edit]
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,^{[8]} 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.^{[9]}
"Robust, useful theories of classroom teaching do not yet exist".^{[10]} However, there are useful theories on how children learn mathematics and much research has been conducted in recent decades to explore how these theories can be applied to teaching. The following results are examples of some of the current findings in the field of mathematics education:
Important results^{[10]}
One of the strongest results in recent research is that the most important feature in effective teaching is giving students "opportunity to learn". Teachers can set expectations, time, kinds of tasks, questions, acceptable answers, and type of discussions that will influence students' opportunity to learn. This must involve both skill efficiency and conceptual understanding.
Conceptual understanding^{[10]}
Two of the most important features of teaching in the promotion of conceptual understanding are attending explicitly to concepts and allowing students to struggle with important mathematics. Both of these features have been confirmed through a wide variety of studies. Explicit attention to concepts involves making connections between facts, procedures and ideas. (This is often seen as one of the strong points in mathematics teaching in East Asian countries, where teachers typically devote about half of their time to making connections. At the other extreme is the U.S.A., where essentially no connections are made in school classrooms.^{[11]}) These connections can be made through explanation of the meaning of a procedure, questions comparing strategies and solutions of problems, noticing how one problem is a special case of another, reminding students of the main point, discussing how lessons connect, and so on.
Deliberate, productive struggle with mathematical ideas refers to the fact that when students exert effort with important mathematical ideas, even if this struggle initially involves confusion and errors, the end result is greater learning. This has been shown to be true whether the struggle is due to challenging, well-implemented teaching, or due to faulty teaching the students must struggle to make sense of.
Formative assessment^{[12]}
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. 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.
Homework^{[13]}
Homework which leads students to practice past lessons or prepare future lessons are more effective than those going over today's lesson. Students benefit from feedback. Students with learning disabilities or low motivation may profit from rewards. For younger children, homework helps simple skills, but not broader measures of achievement.
Students with difficulties^{[13]}
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.
Algebraic reasoning^{[13]}
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...."
Methodology[edit]
As with other educational research (and the social sciences in general), mathematics education research depends on both quantitative and qualitative studies. Quantitative research includes studies that use inferential statistics to answer specific questions, such as whether a certain teaching method gives significantly better results than the status quo. The best quantitative studies involve randomized trials where students or classes are randomly assigned different methods in order to test their effects. They depend on large samples to obtain statistically significant results.
Qualitative research, such as case studies, action research, discourse analysis, and clinical interviews, depend on small but focused samples in an attempt to understand student learning and to look at how and why a given method gives the results it does. Such studies cannot conclusively establish that one method is better than another, as randomized trials can, but unless it is understood why treatment X is better than treatment Y, application of results of quantitative studies will often lead to "lethal mutations"^{[10]} of the finding in actual classrooms. Exploratory qualitative research is also useful for suggesting new hypotheses, which can eventually be tested by randomized experiments. Both qualitative and quantitative studies therefore are considered essential in education—just as in the other social sciences.^{[14]} Many studies are “mixed”, simultaneously combining aspects of both quantitative and qualitative research, as appropriate.
Randomized trials[edit]
There has been some controversy over the relative strengths of different types of research. Because randomized trials provide clear, objective evidence on “what works”, policy makers often take only those studies into consideration. Some scholars have pushed for more random experiments in which teaching methods are randomly assigned to classes.^{[15]}^{[16]} In other disciplines concerned with human subjects, like biomedicine, psychology, and policy evaluation, controlled, randomized experiments remain the preferred method of evaluating treatments.^{[17]}^{[18]} Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods.^{[16]} On the other hand, many scholars in educational schools have argued against increasing the number of randomized experiments, often because of philosophical objections, such as the ethical difficulty of randomly assigning students to various treatments when the effects of such treatments are not yet known.^{[15]} Unlike medical subjects, students have little choice over the teaching method imposed on them, so only a method with solid evidence from other studies can be ethically used as the basis for a randomized trial. Other questions concern the limited knowledge students may have of the experimental treatment they are receiving. Within the broad frame of qualitative research, certain types of research, such as action research, may fall in and out of favor among researchers. Preferences for certain types of research and policy decisions concerning research may vary from country to country.
In the United States, the National Mathematics Advisory Panel (NMAP) published a report in 2008 based on studies, some of which used randomized assignment of treatments to experimental units, such as classrooms or students. The NMAP report's preference for randomized experiments received criticism from some scholars.^{[19]} 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.^{[20]}
Mathematics educators[edit]
The following are some of the people who have had a significant influence on the teaching of mathematics at various periods in history:
Tatyana Alexeyevna Afanasyeva (1876–1964), Dutch/Russian mathematician who advocated the use of visual aids and examples for introductory courses in geometry for high school students^{[21]}
William Arthur Brownell (1895–1977), American educator who led the movement to make mathematics meaningful to children, often considered the beginning of modern mathematics education
Caleb Gattegno (1911-1988), Egyptian, Founder of the Association for Teaching Aids in Mathematics in Britain (1952) and founder of the journal Mathematics Teaching.^{[22]}
Toru Kumon (1914–1995), Japanese, originator of the Kumon method, based on mastery through exercise
Pierre van Hiele and Dina van Hiele-Geldof, Dutch educators (1930s–1950s) who proposed a theory of how children learn geometry (1957), which eventually became very influential worldwide
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
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^ ^{a}^{b}^{c}^{d}Hiebert, James; Grouws, Douglas (2007), "9", The Effects of Classroom Mathematics Teaching on Students' Learning1, Reston VA: National Council of Teachers of Mathematics, pp. 371–404
^Institute of Education Sciences, ed. (2003), "Highlights From the TIMSS 1999 Video Study of Eighth-Grade Mathematics Teaching", U.S. Department of Education http://nces.ed.gov/timssMissing or empty |title= (help)
^Shadish, William R.; Cook, Thomas D.; Campbell, Donald T. (2002). Experimental and quasi-experimental designs for generalized causal inference (2nd ed.). Boston: Houghton Mifflin. ISBN0-395-61556-9.
^Cajori, Florian (October 1910). "Attempts made during the eighteenth and nineteenth centuries to reform the teaching of geometry". American Mathematical Monthly17 (10): 181–201. JSTOR2973645.
Anderson, John R.; Reder, Lynne M.; Simon, Herbert A.; Ericsson, K. Anders; Glaser, Robert (1998). "Radical Constructivism and Cognitive Psychology" (PDF). Brookings Papers on Education Policy (1): 227–278.