In contemporary education, mathematics education (known in Europe as the didactics or pedagogy of mathematics) is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge.
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.
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 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 the 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) as an example of an axiomatic system and a model of deductive reasoning The teaching of selected areas of mathematics (such as calculus) as an example of the intellectual achievements of the 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 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. 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. 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. 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. 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. This type of mathematics education was implemented in various parts of Canada beginning in 2005. 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. 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. 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. Exercises: the reinforcement of mathematical skills by completing large numbers of exercises of a similar type, such as adding 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 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. During the primary school years, children learn about whole numbers and arithmetic, including addition, subtraction, multiplication, and division. Comparisons and measurement are taught, in both numeric and pictorial form, as well as fractions and proportionality, patterns, and various topics related to geometry.
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 follow a pre-defined course sequence – encompassing different 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.
At college and university level, science and engineering students will be required to take multivariable calculus, differential equations, and linear algebra; at several US colleges, the minor or AS in mathematics substantively comprises these courses. Mathematics majors study additional areas of pure mathematics—and often applied mathematics—with the requirement of specified advanced courses in analysis and modern algebra. Other topics in pure mathematics include differential geometry, set theory, and topology. Applied mathematics may be taken as a major subject in its own right, covering partial differential equations, optimization, and numerical analysis among other topics. Courses here are also taught within other programs: for example, civil engineers may be required to study fluid mechanics, and "math for computer science" might include graph theory, permutation, probability, and formal mathematical proofs. Pure and applied math degrees often include modules in probability theory or mathematical statistics, as well as stochastic processes. Physics, especially theoretical, is mathematics-intensive, substantively overlapping both pure and applied math.
Coverage in other programs will vary. Business mathematics is usually limited to introductory calculus and (sometimes) matrix calculations; business majors often also take a parallel course in statistics and probability. Additional to these topics, economics programs typically cover optimization, often differential equations and linear algebra, and sometimes analysis; statistics is similarly extended, underpinning econometrics. The discipline of Management Science formalizes the use of mathematics and statistics in business. Social science curricula require statistics also, often complemented by major-specific quantitative research courses. Students in the humanities and liberal arts may be offered a course in "finite math", "contemporary mathematics," "mathematics for the liberal arts," or "quantitative reasoning," which may include topics such as set theory and mathematical logic, and applications of mathematics to other fields. Mathematics courses have low rates of course success relative to other fields of study, and students at US colleges and universities may be required to retake high school mathematics courses through remedial education programs.
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. 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.