Mathematics education
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In contemporary
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.
History
Ancient
Elementary mathematics were a core part of education in many ancient civilisations, including
Pythagorean theorem
Historians of Mesopotamia have confirmed that use of the Pythagorean rule dates back to the Old Babylonian Empire (20th–16th centuries BC) and that it was being taught in scribal schools over one thousand years before the birth of Pythagoras.[2][3][4][5][6]
In
Medieval and early modern
In the
The first mathematics textbooks to be written in English and French were published by
The social status of mathematical study was improving by 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.
Modern
In the 18th and 19th centuries, the
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 International Commission on Mathematical Instruction (ICMI) was founded in 1908, and Felix Klein became the first president of the organisation.
- 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 1968, the Shell Centre for Mathematical Education was established in Nottingham.
- The first in 1972, and after that, it has been held every four years.
In the 20th century, the cultural impact of the "
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 and learning of basic numeracy skills to all students[11]
- The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid geometry, trigonometry, probability, statistics) to most students, to equip them to follow a trade or craft and to understand mathematics commonly used in news and Internet (such as percentages, charts, probability, and statistics)
- 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)[12] as an example of an axiomatic system[13] and a model of deductive reasoning
- The teaching of selected areas of mathematics (such as modern world
- The teaching of advanced mathematics to those students who wish to follow a career in science, technology, engineering, and mathematics (STEM)fields
- The teaching of heuristics[14]and other problem-solving strategies to solve non-routine problems
- The teaching of mathematics in social sciences and actuarial sciences, as well as in some selected arts under liberal arts education in liberal arts colleges or universities
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:
- Computer-based math: an approach based on the use of mathematical software as the primary tool of computation.
- Computer-based mathematics education: involves the use of computers to teach mathematics. Mobile applications have also been developed to help students learn mathematics.[15][16][17]
- Classical education: the teaching of mathematics within the quadrivium, part of the classical education curriculum of the Middle Ages, which was typically based on Euclid's Elements taught as a paradigm of deductive reasoning.[18]
- 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 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]
- Standards-based mathematics: a vision for pre-college mathematics education in the United States 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.
- Mastery: an approach in which most students are expected to achieve a high level of competence before progressing.
- 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.
- simple fractions or solving quadratic equations.
- 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.
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
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,
At college and university level,
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,[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
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[37]
- 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[37]
- 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[39]
- 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[40]
- 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[40]
- 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[40]
- 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
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]
In the United States, the
Organizations
- Advisory Committee on Mathematics Education
- American Mathematical Association of Two-Year Colleges
- Association of Teachers of Mathematics
- Canadian Mathematical Society
- C.D. Howe Institute
- Mathematical Association
- National Council of Teachers of Mathematics
- OECD
- International Association for the Evaluation of Educational Achievement
See also
Aspects of mathematics education
- Cognitively Guided Instruction
- Critical mathematics pedagogy
- Ethnomathematics
- Number sentence, primary level mathematics education
- Pre-math skills
- Sir Cumference, children's mathematics educational book series
- Statistics education
North American issues
Mathematical difficulties
References
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Further reading
- 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.
- 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. Archived (PDF) from the original on 2010-07-15. Retrieved 2009-08-06.
- 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 online Archived 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. ISBN 978-1934137178.
- 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.
- ISBN 978-3-642-00774-3.
- ISBN 978-0-691-13493-2.
- Strutchens, Marilyn E., et al. The mathematics education of prospective secondary teachers around the world (Springer Nature, 2017) online Archived 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.
External links
- Math Education at Curlie
- History of Mathematical Education
- A quarter century of US 'math wars' and political partisanship. David Klein. California State University, Northridge, United States