Mathematics education
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.
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 moneylenders could expect to learn such practical mathematics as was relevant to their profession.
 Important results
 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
 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.) 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, wellimplemented teaching, or due to faulty teaching the students must struggle to make sense of.
 Formative assessment

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
 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
 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 shortterm memory. Techniques that have been found productive for helping such students include peerassisted learning, explicit teaching with visual aids, instruction informed by formative assessment and encouraging students to think aloud.
 Algebraic reasoning
 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...."
 Aspects of mathematics education
 North American issues
 Mathematical difficulties

Computerbased math an approach based around use of mathematical software as the primary tool of computation.

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.

Exercises: the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding vulgar fractions or solving quadratic equations.

Historical method: teaching the development of mathematics within an historical, social and cultural context. Provides more human interest than the conventional approach.,

Mastery: an approach in which most students are expected to achieve a high level of competence before progressing

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."

Problem solving: the cultivation of mathematical ingenuity, creativity and heuristic thinking by setting students openended, 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.

Recreational mathematics: Mathematical problems that are fun can motivate students to learn mathematics and can increase enjoyment of mathematics.

Standardsbased mathematics: a vision for precollege mathematics education in the US and Canada, focused on deepening student understanding of mathematical ideas and procedures, and formalized by the National Council of Teachers of Mathematics which created the Principles and Standards for School Mathematics.

Relational approach: Uses class topics to solve everyday problems and relates the topic to current events. This approach focuses on the many uses of mathematics and helps students understand why they need to know it as well as helping them to apply mathematics to real world situations outside of the classroom.

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.

Euclid (fl. 300 BC), Ancient Greek, author of The Elements

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

Robert Lee Moore (1882–1974), American mathematician, originator of the Moore method

George Pólya (1887–1985), Hungarian mathematician, author of How to Solve It

Georges Cuisenaire (1891–1976), Belgian primary school teacher who invented Cuisenaire rods

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

Hans Freudenthal (1905–1990), Dutch mathematician who had a profound impact on Dutch education and founded the Freudenthal Institute for Science and Mathematics Education in 1971

Caleb Gattegno (19111988), Egyptian, Founder of the Association for Teaching Aids in Mathematics in Britain (1952) and founder of the journal Mathematics Teaching.

Toru Kumon (1914–1995), Japanese, originator of the Kumon method, based on mastery through exercise
 Pierre van Hiele and Dina van HieleGeldof, Dutch educators (1930s–1950s) who proposed a theory of how children learn geometry (1957), which eventually became very influential worldwide

Robert Parris Moses (1935–), founder of the nationwide US Algebra project

Robert M. Gagné (1958–1980s), pioneer in mathematics education research.

Lewis Carroll, pen name of British author Charles Dodgson, lectured in mathematics at Christ Church, Oxford. As a mathematics educator, Dodgson defended the use of Euclid's Elements as a geometry textbook; Euclid and his Modern Rivals is a criticism of a reform movement in geometry education led by the Association for the Improvement of Geometrical Teaching.

John Dalton, British chemist and physicist, taught mathematics at 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

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

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, American athlete and Olympic gold medalist, taught mathematics at high schools in California.

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.

Auslander, Maurice; et al. (2004). "Goals for School Mathematics: The Report of the Cambridge Conference on School Mathematics 1963" (PDF). Cambridge MA: Center for the Study of Mathematics Curriculum.

Sriraman, Bharath; English, Lyn (2010). Theories of Mathematics Education. Springer. ISBN .

Strogatz, Steven Henry; Joffray, Don (2009). The Calculus of Friendship: What a Teacher and a Student Learned about Life While Corresponding about Math. Princeton University Press. ISBN .
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