Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods. National and international organisations regularly hold conferences and publish literature in order to improve mathematics education.
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. The teaching of geometry was almost universally based on Euclid'sElements. Apprentices to trades such as masons, merchants, and moneylenders could expect to learn such practical mathematics as was relevant to their profession.
Medieval and early modern
Illustration at the beginning of a 14th-century translation of Euclid's Elements
In the Middle Ages, the academic status of mathematics declined, because it was strongly associated with trade and commerce, and considered somewhat un-Christian.[7] Although it continued to be taught in European universities, it was seen as subservient to the study of natural, metaphysical, and moral philosophy. The first modern arithmetic curriculum (starting with addition, then subtraction, multiplication, and division) arose at reckoning schools in Italy in the 1300s.[8] Spreading along trade routes, these methods were designed to be used in commerce. They contrasted with Platonic math taught at universities, which was more philosophical and concerned numbers as concepts rather than calculating methods.[8] They also contrasted with mathematical methods learned by artisan apprentices, which were specific to the tasks and tools at hand. For example, the division of a board into thirds can be accomplished with a piece of string, instead of measuring the length and using the arithmetic operation of division.[7]
The first mathematics textbooks to be written in English and French were published by Robert Recorde, beginning with The Grounde of Artes in 1543. However, there are many different writings on mathematics and mathematics 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 methodology for solving equations like the quadratic equation. After the Sumerians, some of the most famous ancient works on mathematics came from Egypt in the form of the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus. The more famous Rhind Papyrus has been dated back 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 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. Main events in this development include the following:
In 1893, a Chair in mathematics education was created at the University of Göttingen, under the administration of Felix Klein.
The professional periodical literature on mathematics education in the United States had generated more than 4,000 articles after 1920, so in 1941 William L. Schaaf published a classified index, sorting them into their various subjects.[9]
A renewed interest in mathematics education emerged in the 1960s, and the International Commission was revitalized.
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'."[10]
Objectives
Boy doing sums, Guinea-Bissau, 1974
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 students[11]
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:
Games can motivate students to improve skills that are usually learned by rote. In "Number Bingo," players roll 3 dice, then perform basic mathematical 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.
Computer-based math: an approach based on the 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.
Relational approach: uses class topics to solve everyday problems and relates the topic to current events.[19] This approach focuses on the many uses of mathematics and helps students understand why they need to know it as well as helps them to apply mathematics to real-world situations outside of the classroom.
Historical method: teaching the development of mathematics within a historical, social, and cultural context. Proponents argue it provides more human interest than the conventional approach.[20]
Discovery math: a constructivist method of teaching (discovery learning) mathematics which centres around problem-based or inquiry-based learning, with the use of open-ended questions and manipulative tools.[21] This type of mathematics education was implemented in various parts of Canada beginning in 2005.[22] Discovery-based mathematics is at the forefront of the Canadian "math wars" debate with many criticizing it for declining math scores.
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."
Recreational mathematics: mathematical problems that are fun can motivate students to learn mathematics and can increase their enjoyment of mathematics.[23]
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.
Math walk: a walk where experience of perceived objects and scenes is translated into mathematical language.
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 similarly, though there are differences. Most countries tend to cover fewer topics in greater depth than in the United States.[24] During the primary school years, children learn about whole numbers and arithmetic, including addition, subtraction, multiplication, and division.[25] Comparisons and measurement are taught, in both numeric and pictorial form, as well as fractions and proportionality, patterns, and various topics related to geometry.[26]
At high school level in most of the US, algebra, geometry, and analysis (pre-calculus and calculus) are taught as separate courses in different years.
On the other hand, in most other countries (and in a few US states), mathematics is taught as an integrated subject, with topics from all branches of mathematics studied every year;
students thus undertake a pre-defined course - entailing several topics - rather than choosing courses à la carte as in the United States.
Even in these cases, however, several "mathematics" options may be offered, selected based on the student's intended studies post high school.
(In South Africa, for example, the options are Mathematics, Mathematical Literacy and Technical Mathematics.)
Thus, a science-oriented curriculum typically overlaps the first year of university mathematics, and includes 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 are similarly often taught.
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,[29] while Scotland maintains its own educational system. Many other countries have centralized ministries which set national standards or curricula, and sometimes even textbooks.
Ma (2000) summarized the research of others who found, based on nationwide data, that students with higher scores on standardized mathematics tests had taken more mathematics courses in high school. This led some states to require three years of mathematics instead of two. But because this requirement was often met by taking another lower-level mathematics course, the additional courses had a “diluted” effect in raising achievement levels.[30]
In North America, the National Council of Teachers of Mathematics (NCTM) published the Principles and Standards for School Mathematics in 2000 for the United States and Canada, which boosted the trend towards reform mathematics. In 2006, the NCTM released Curriculum Focal Points, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards were guidelines to implement as American states and Canadian provinces chose. In 2010, the National Governors Association Center for Best Practices and the Council of Chief State School Officers published the Common Core State Standards for US states, which were subsequently adopted by most states. Adoption of the Common Core State Standards in mathematics is at the discretion of each state, and is not mandated by the federal government.[31] "States routinely review their academic standards and may choose to change or add onto the standards to best meet the needs of their students."[32] The NCTM has state affiliates that have different education standards at the state level. For example, Missouri has the Missouri Council of Teachers of Mathematics (MCTM) which has its pillars and standards of education listed on its website. The MCTM also offers membership opportunities to teachers and future teachers so that they can stay up to date on the changes in math educational standards.[33]
The Programme for International Student Assessment (PISA), created by the Organisation for the Economic Co-operation and Development (OECD), is a global program studying the reading, science, and mathematics abilities of 15-year-old students.[34] The first assessment was conducted in the year 2000 with 43 countries participating.[35] PISA has repeated this assessment every three years to provide comparable data, helping to guide global education to better prepare youth for future economies. There have been many ramifications following the results of triennial PISA assessments due to implicit and explicit responses of stakeholders, which have led to education reform and policy change.[35][36][21]
Research
According to Hiebert and Grouws, "Robust, useful theories of classroom teaching do not yet exist."[37] 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
One of the strongest results in recent research is that the most important feature of effective teaching is giving students "the opportunity to learn". Teachers can set expectations, times, kinds of tasks, questions, acceptable answers, and types of discussions that will influence students' opportunities 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 times 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 US, where essentially no connections are made in school classrooms.[38]) 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 result is greater learning. This is true whether the struggle is due to intentionally challenging, well-implemented teaching, or unintentionally confusing, faulty teaching.
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 assignments which lead students to practice past lessons or prepare for future lessons is more effective than those going over the current 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 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
Elementary school children need 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
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 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"[37] 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.[41] Many studies are “mixed”, simultaneously combining aspects of both quantitative and qualitative research, as appropriate.
Randomized trials
There has been some controversy over the relative strengths of different types of research. Because of an opinion that randomized trials provide clear, objective evidence on “what works”, policymakers often consider only those studies. Some scholars have pushed for more random experiments in which teaching methods are randomly assigned to classes.[42][43] In other disciplines concerned with human subjects—like biomedicine, psychology, and policy evaluation—controlled, randomized experiments remain the preferred method of evaluating treatments.[44][45] Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods.[43] 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 to be effective,[46] or the difficulty of assuring rigid control of the independent variable in fluid, real school settings.[47]
^Neugebauer, Otto (1969). The exact sciences in antiquity. New York: Dover Publications. p. 36. ISBN978-0-486-22332-2. In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse.
^Høyrup, Jens. "Pythagorean 'Rule' and 'Theorem' – Mirror of the Relation Between Babylonian and Greek Mathematics". In Renger, Johannes (ed.). Babylon: Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne. 2. Internationales Colloquium der Deutschen Orient-Gesellschaft 24.–26. März 1998 in Berlin(PDF). Berlin: Deutsche Orient-Gesellschaft / Saarbrücken: SDV Saarbrücker Druckerei und Verlag. pp. 393–407. Archived(PDF) from the original on 2021-02-25. Retrieved 2022-11-15., p. 406, "To judge from this evidence alone it is therefore likely that the Pythagorean rule was discovered within the lay surveyors' environment, possibly as a spin-off from the problem treated in Db2-146, somewhere between 2300 and 1825 BC." (Db2-146 is an Old Babylonian clay tablet from Eshnunna concerning the computation of the sides of a rectangle given its area and diagonal.)
^Robson, E. (2008). Mathematics in Ancient Iraq: A Social History. Princeton University Press.: p. 109 "Many Old Babylonian mathematical practitioners … knew that the square on the diagonal of a right triangle had the same area as the sum of the squares on the length and width: that relationship is used in the worked solutions to word problems on cut-and-paste 'algebra' on seven different tablets, from Ešnuna, Sippar, Susa, and an unknown location in southern Babylonia."
^Ferguson, Kitty (2010). Pythagoras : His Lives and the Legacy of a Rational Universe. London: Icon. pp. 78–84. ISBN978-184831-231-9.
^Nunes, Terezinha; Dorneles, Beatriz Vargas; Lin, Pi-Jen; Rathgeb-Schnierer, Elisabeth (2016), "Teaching and Learning About Whole Numbers in Primary School", ICME-13 Topical Surveys, Cham: Springer International Publishing, pp. 1–50, doi:10.1007/978-3-319-45113-8_1, ISBN978-3-319-45112-1
^"Mathematics curriculum". UK Department of Education. 17 January 2013. Archived from the original on 2 May 2012. Retrieved 1 May 2012.
^Ma, X. (2000). "A longitudinal assessment of antecedent course work in mathematics and subsequent mathematical attainment". Journal of Educational Research. 94 (1): 16–29. doi:10.1080/00220670009598739. S2CID144948416.
^ abLockheed, Marlaine (2015). The Experience of Middle-Income Countries Participating in PISA 2000. PISA. France: OECD Publishing. p. 30. ISBN978-92-64-24618-8.
^ abcdHiebert, James; Grouws, Douglas (2007), "9", The Effects of Classroom Mathematics Teaching on Students' Learning, vol. 1, Reston VA: National Council of Teachers of Mathematics, pp. 371–404
^Cook, Thomas D. (2002). "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 (3): 175–199. doi:10.3102/01623737024003175. S2CID144583638.
^Shadish, William R.; Cook, Thomas D.; Campbell, Donald T. (2002). Experimental and quasi-experimental designs for generalized causal inference (2nd ed.). Boston: Houghton Mifflin. ISBN978-0-395-61556-0.
^Mosteller, Frederick; Boruch, Robert (2002), Evidence Matters: Randomized Trials in Education Research, Brookings Institution Press
^Chatterji, Madhabi (December 2004). "Evidence on "What Works": An Argument for Extended-Term Mixed-Method (ETMM) Evaluation Designs". Educational Researcher. 33 (9): 3–13. doi:10.3102/0013189x033009003. S2CID14742527.
^Kelly, Anthony (2008). "Reflections on the National Mathematics Advisory Panel Final Report". Educational Researcher. 37 (9): 561–4. doi:10.3102/0013189X08329353. S2CID143471869. 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.
^Sparks, Sarah (October 20, 2010). "Federal Criteria For Studies Grow". Education Week. p. 1.
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. Archived from the original(PDF) on 2010-06-26. Retrieved 2011-09-25.
Ball, Lynda, et al. Uses of Technology in Primary and Secondary Mathematics Education (Cham, Switzerland: Springer, 2018).
Dreher, Anika, et al. "What kind of content knowledge do secondary mathematics teachers need?." Journal für Mathematik-Didaktik 39.2 (2018): 319-341 onlineArchived 2021-04-18 at the Wayback Machine.
Drijvers, Paul, et al. Uses of technology in lower secondary mathematics education: A concise topical survey (Springer Nature, 2016).
Gosztonyi, Katalin. "Mathematical culture and mathematics education in Hungary in the XXth century." in Mathematical cultures (Birkhäuser, Cham, 2016) pp. 71–89. online
Paul Lockhart (2009). A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. Bellevue Literary Press. ISBN978-1934137178.
Losano, Leticia, and Márcia Cristina de Costa Trindade Cyrino. "Current research on prospective secondary mathematics teachers' professional identity." in The mathematics education of prospective secondary teachers around the world (Springer, Cham, 2017) pp. 25-32.
Strutchens, Marilyn E., et al. The mathematics education of prospective secondary teachers around the world (Springer Nature, 2017) onlineArchived 2021-04-18 at the Wayback Machine.
Wong, Khoon Yoong. "Enriching secondary mathematics education with 21st century competencies." in Developing 21st Century Competencies In The Mathematics Classroom: Yearbook 2016 (Association Of Mathematics Educators. 2016) pp. 33–50.
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