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Math Lessons - Mathematics education

Teaching, learning and scientific research in mathematics

In contemporary education, the Math class the practice of teaching and learning mathematics along with related scientific research

Researchers in math class are primarily concerned with the tools, methods, and approaches that make it easier to practice or study practice. However, mathematical educational research, known on the European continent as didactics or pedagogy of mathematics, has developed into an extensive field of study with its concepts, theories, methods, national and international organizations, conferences and literature. This article describes some of the history, influences, and recent controversies.


Elementary mathematics was part of the educational system in most ancient civilizations, including ancient Greece, the Roman Empire, Vedic society, and ancient Egypt. In most cases, formal education was only available to male children of sufficiently high status, wealth, or caste.

Illustration at the beginning of the translation of Euclid Elements from the 14th century.

In Plato's division of the liberal arts into the trivium and quadrivium, the quadrivium contain the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education developed in medieval Europe. The teaching of geometry was almost everywhere based on Euclid's elements . Craft trades trainees such as bricklayers, merchants and moneylenders could expect to learn the practical math relevant to their profession.

During the Renaissance, mathematics declined in academic status as it was heavily linked to trade and commerce and was viewed as somewhat unchristian. Although it continued to be taught in European universities, it was viewed as subject to the study of natural, metaphysical, and moral philosophy. The first modern arithmetic curriculum (starting with addition, then subtraction, multiplication and division) emerged in schools in Italy in the 13th century. These methods, which spread along the trade routes, were developed for trade. They contrasted with Platonic mathematics taught at universities, which was more philosophical and concerned numbers more as concepts than as methods of computation. They were also in contrast to mathematical methods learned by apprentices who were specific to the tasks and tools involved. For example, dividing a board into thirds can be accomplished with a piece of string instead of measuring the length and using the arithmetic operation of division.

The first math textbooks in English and French were published by Robert Recorde, starting with The Grounde of Artes in 1543. However, there are many different writings on mathematics and mathematical methodology that date back to 1800 BC. BC Reaching back. These were mainly in Mesopotamia, where the Sumerians practiced multiplication and division. There are also artifacts demonstrating their methodology for solving equations such as the quadratic equation. After the Sumerians, some of the best known ancient mathematical works came from Egypt in the form of the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus. The better known Rhind Papyrus was dated around 1650 BC. Dated, but believed to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students.

The social status of mathematics studies improved in the 17th century when the University of Aberdeen created a Chair of Mathematics in 1613, followed by the Chair of Geometry at Oxford University in 1619 and the Lucasian Chair in Mathematics at Aberdeen University at Cambridge in 1662 .

In the 18th and 19th centuries, the industrial revolution resulted in an enormous increase in the urban population. Basic math skills like the ability to tell time, count money, and perform 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.

Up until the 20th century, mathematics was part of the core curriculum in all industrialized countries.

During the 20th century, mathematics teaching was established as an independent research area. Here are some of the key events in this development:

In the 20th century, the cultural effects of the "electronic age" (McLuhan) were also taken up in education and mathematics classes. While the previous approach focused on "working with specific" problems "in arithmetic," the emerging structural knowledge approach "had young children meditate on number theory and" sets. "


Boy makes buzz, Guinea-Bissau, 1974.

At different times and in different cultures and countries, math classes have tried to achieve different goals. These goals included:

  • Teaching and learning basic numeracy skills for all students.
  • The teaching of practical mathematics (arithmetic, elementary algebra, plane and fixed geometry, trigonometry) for most students to equip them to practice a trade or craft
  • Teaching abstract math concepts (such as quantity and function) at a young age
  • Teaching selected areas of mathematics (such as Euclidean geometry) as an example of an axiomatic system and as a model for deductive reasoning
  • Teaching selected areas of mathematics (such as analysis) as an example of the intellectual achievements of the modern world
  • The advanced math classes for those students who wish to pursue careers in science, technology, engineering, and math (STEM).
  • Teaching heuristics and other problem-solving strategies to solve non-routine problems.
  • Teaching mathematics in actuarial sciences, social sciences and, in some selected parts, arts under liberal arts education, in liberal arts colleges or universities.


The method or methods used in a given context are largely determined by the goals that the particular educational system is trying to achieve. Methods of teaching math include:

Games can motivate students to hone skills that are usually memorized. In "Number Bingo", players roll 3 dice and then perform basic math operations on those numbers to get a new number that they cover on the board and try to cover 4 spaces in a row. This game was played on a "Discovery Day" organized by Big Brother Mouse in Laos.
  • Computational Mathematics An approach based on the use of mathematical software as the primary calculation tool.
  • Computer-aided math lessons using computers to teach math. Mobile applications have also been developed to help students learn math.
  • Conventional approach : the step-by-step and systematic guidance through the hierarchy of mathematical terms, ideas and techniques. Starts with arithmetic and is followed by Euclidean geometry and elemental algebra taught at the same time. Requires the instructor to be well informed about elementary math as didactic and curriculum decisions are often driven by subject logic rather than pedagogical considerations. Other methods emerge from highlighting some aspects of this approach.
  • Discovery Mathematics : A constructivist method of teaching (discovery learning) math that focuses on problem-based or research-based learning using open-ended questions and manipulative tools. This type of math teaching was introduced in different parts of Canada from 2005. Discovery-based math is at the forefront of the Canadian Math Wars debate. Many criticize their effectiveness for falling math grades compared to traditional teaching models that have direct value teaching, memorization, and memorization.
  • Exercises : Strengthening math skills by performing a large number of exercises of a similar type, e.g. B. Add vulgar fractions or solve quadratic equations.
  • Historical method : Communicating the development of mathematics in a historical, social and cultural context. Offers more human interest than the traditional approach.
  • Mastery : An approach where most students are expected to achieve a high level of competence before making any progress.
  • New math : A method of teaching math that focuses on abstract concepts such as set theory, functions, and fundamentals other than ten. Adopted in the US in response to the challenge of Soviet early technical superiority in space, it began to be questioned in the late 1960s. One of the most influential reviews of New Math was Morris Kline's 1973 book Why Johnny Can't Add . The New Math Method was the subject of one of Tom Lehrer's favorite parody songs, with his introductory remarks on the song: "... As you know, with the new approach it is important to understand what you are doing, rather than to get the correct answer. "
  • Troubleshooting : Promoting math ingenuity, creativity, and heuristic thinking by asking students open, unusual, and sometimes unsolved problems. Problems can range from simple word problems to problems from international math competitions such as the International Mathematical Olympiad. Problem solving is used as a means of building new math knowledge, usually by building on the students' previous understanding.
  • Leisure math : Math problems that are fun can motivate students to learn math and increase their enjoyment of math.
  • Standards-based math : A vision for pre-college math teaching in the United States and Canada focused on deepening students' understanding of math ideas and procedures, formalized by the National Council of Math Teachers, which established the principles and standards for school math .
  • Relational approach : Uses class topics to solve everyday problems and relates the topic to current events. This approach focuses on the many uses of math, helping students understand why they need to know, and helping them apply math to real-world situations outside of the classroom.
  • Red learning : teaching mathematical results, definitions and concepts through repetition and memorization, typically without meaning or aided by mathematical reasoning. A derisive term is Drill and kill . In traditional education, memorization is used to teach multiplication tables, definitions, formulas, and other aspects of math.

Content and ages

Different levels of math are taught in different age groups and in slightly different sequences in different countries. Sometimes a class may be taught at an earlier age than is usually the case as a special or honorary class.

Elementary mathematics is taught similarly in most countries, although there are differences. In most countries, fewer topics are covered in greater detail than in the United States. During the elementary school years, children learn whole numbers and their arithmetic, including addition, subtraction, multiplication, and division. Comparisons and measurements are conveyed in both numerical and figurative form, as well as in fractions and proportions, patterns and various topics related to geometry.

At the high school level in most of the United States, algebra, geometry, and analysis (precalculus and calculus) are taught as separate courses in different years. Mathematics is built into most other countries (and some US states), with subjects from all areas of mathematics being studied every year. Students in many countries choose an option or pre-defined course rather than courses as in the US à la carte too choose . Students in science-oriented curricula usually study differential calculus and trigonometry in their final year of school between the ages of 16 and 17, as well as integral calculus, complex numbers, analytical geometry, exponential and logarithmic functions, and infinite series. Probability and statistics can be taught in secondary education. In some countries these subjects are available as "advanced" or "additional" math.

At colleges and universities, science and engineering students will take on multiple variable calculus, differential equations, and linear algebra; At several US colleges, the minor or AS in math essentially comprises these courses. Mathematics majors continue to study various other areas of pure mathematics - and often applied mathematics - with the requirement of specified advanced courses in analysis and modern algebra. Applied mathematics can be taken as an important topic in its own right, while specific topics may be taught in other courses: for example civil engineers may be required to study fluid mechanics and "mathematics for computer science" could include graph theory, permutation, probability and formal mathematical proofs. Pure and applied mathematics degrees often contain modules in probability theory / mathematical statistics; A course in numerical methods is often a prerequisite for applied mathematics. The (theoretical) physics is mathematics-intensive and often overlaps with the content of the pure or applied mathematics degree. ("Business mathematics" is usually limited to introductory calculations and sometimes to matrix calculations. Economic programs also cover optimization, often differential equations and linear algebra, sometimes analysis.)


For most of history, standards for math teaching have been set locally by individual schools or teachers, depending on the proficiency levels deemed relevant, realistic, and socially appropriate to their students.

There has been a trend today towards regional or national standards, usually under the umbrella of a more comprehensive standard curriculum for schools. In England, for example, standards for teaching math are set as part of the national curriculum for England, while Scotland has its own educational system. Many other countries have centralized ministries that set national standards or curricula, and sometimes even textbooks.

Ma (2000) summarized research by others who found, based on national data, that students with higher scores on standardized math tests had taken more math classes in high school. As a result, some states required three years of mathematics instead of two. However, since this requirement was often met by attending another math course at a lower level, the additional courses had a “diluted” effect on the increase in performance.

In North America, the National Council of Teachers of Mathematics (NCTM) published the Principles and standards for school mathematics for the US and Canada, which stimulated the trend towards reform mathematics. In 2006 the NCTM published Curriculum Focal Points in which the main math topics are recommended for each grade level up to 8th grade.However, these standards were guidelines that had to be implemented by the choice of the American states and Canadian provinces. In 2010, the National Governors Association Center for Best Practices and the Board of Chief State School Officers published the Common Core State Standards for U.S. states, which were later adopted by most states. Adoption of the Common Core State Standards in mathematics is at the discretion of each state and is not required by the federal government. "States routinely review their academic standards and may choose to change or add to the standards to best meet the needs of their students." The NCTM has state member organizations that have different educational standards at the state level. For example, Missouri has the Missouri Council of Teachers of Mathematics (MCTM), whose pillars and educational standards are listed on its website. The MCTM also offers membership opportunities to teachers and prospective teachers so they can stay up to date on changes in standards for math teaching.

Launched by the Organization for Economic Co-operation and Development (OECD), the International Student Assessment Program (PISA) is a global program that examines the reading, science and math skills of 15-year-old students. The first evaluation was carried out in 2000 with 43 participating countries. PISA has repeated this assessment every three years to provide comparable data that will help guide global education to better prepare youth for future economies. According to the results of the triennial PISA assessments, there have been many consequences due to implicit and explicit stakeholder responses that have led to educational reform and policy change.


"Robust, useful teaching theories do not yet exist". However, there are useful theories about how children learn math, and much research has been done over the past few decades to examine how these theories can be applied to the classroom. The following results are examples of some of the current findings in the field of math teaching:

Important results
One of the strongest findings in recent research is that the most important characteristic of effective teaching is giving students "opportunity to learn". Teachers can set expectations, time, types of assignments, questions, acceptable answers, and types of discussions that will affect students' learning opportunities. This must include both skill efficiency and conceptual understanding.
Conceptual understanding
Two of the most important features of teaching in promoting conceptual understanding are the explicit consideration of concepts and allowing students to struggle with important math. Both characteristics have been confirmed by a large number of studies. Explicit attention to concepts involves making connections between facts, procedures, and ideas. (This is often viewed as one of the strengths of teaching math in East Asian countries, where teachers typically devote about half their time to making connections. At the other extreme, the US, where essentially no connections are made in school classes.) These Connections can be made by explaining the importance of a procedure, asking questions to compare strategies and problem-solving, determining how one problem is a special case of another, reminding students of the main point, how the lessons connect, and so on further.
The conscious, productive struggle with math ideas refers to the fact that when students struggle with important math ideas, even if that struggle initially involves confusion and error, the result is better learning. This is true regardless of whether the struggle is due to challenging, well-executed teaching or faulty teaching. Students must strive to make sense of it.
Formative evaluation
Formative assessment is both the best and cheapest way to increase student performance, student engagement, and teacher job satisfaction. The results exceed 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 required, providing good feedback, encouraging students to take control of their learning, and allowing the Students are resources for each other.
Homework that gets students to practice past lessons or prepare future lessons is more effective than the one going through today's lesson. Students benefit from feedback. Students with learning difficulties or low motivation can benefit from rewards. For younger children, homework helps with simple skills, but not with broader achievement measures.
Students with difficulty
Students with real difficulty (regardless of motivation or previous class) struggle with basic facts, respond impulsively, struggle with mental representations, have poor numerical comprehension, and poor short-term memory. Techniques that have proven productive in helping such students include peer-assisted learning, explicit teaching with visual aids, instructions conveyed through formative assessment, and encouraging students to think aloud.
Algebraic thinking
Elementary school children have to learn to express algebraic properties without symbols for a long time before learning algebraic notation. When learning symbols, many students believe that letters always represent the unknown and struggle with the concept of variables. They prefer arithmetic thinking over algebraic equations for solving word problems. It takes time to move from arithmetic to algebraic generalizations and to describe patterns. Students often have problems with the minus sign and understand the equal sign as "the answer is ...".


As with other educational research (and the social sciences in general), mathematical educational research depends on both quantitative and qualitative studies. Quantitative research includes studies that use inferential statistics to answer specific questions, such as: B. whether a certain teaching method delivers significantly better results than the status quo. The best quantitative studies include randomized trials in which students or classes are randomly assigned different methods to test their effects. They depend on large samples to get statistically significant results.

Qualitative research such as case studies, action research, discourse analysis, and clinical interviews depend on small but focused samples to understand student learning and to examine how and why a particular method produces the results it does. Such studies cannot conclusively prove that one method is better than another, as randomized studies can. However, if not understood, Why Treatment X is better than treatment Y, the application of the results of quantitative studies often leads to "fatal mutations" of the finding in actual classrooms. Exploratory qualitative research is also useful in proposing new hypotheses that can eventually be tested through randomized experimentation. Both qualitative and quantitative studies are therefore seen - as in the other social sciences - as essential for education. Many studies are “mixed” and at the same time combine aspects of quantitative and qualitative research.

Randomized Studies

There has been some controversy over the relative strengths of different types of research. Because randomized trials provide clear, objective evidence of what works, often policymakers only consider these trials. Some scientists have pushed for more random experiments, where teaching methods are randomly assigned to classes. In other disciplines dealing with human subjects, such as biomedicine, psychology, and policy assessment, controlled, randomized experiments remain the preferred method for evaluating treatments. Educational statisticians and some math educators have worked to increase the use of randomized experiments to evaluate teaching methods. On the other hand, many academics in educational schools have opposed increasing the number of randomized experiments, often due to philosophical objections such as the ethical difficulty of randomly assigning students to different treatments when the effects of such treatments are not yet effective or the difficulty of rigid control of independent variables in fluid, real school environments.

In the United States, the National Mathematics Advisory Panel (NMAP) published a report in 2008 based on studies, some of which used a randomized assignment of treatments to experimental units such as classrooms or students. The NMAP report's preference for randomized experiments has been criticized by some scientists. In 2010, What Works Clearinghouse (essentially the Department of Education's research arm) responded to ongoing controversy by adding non-experimental studies to its research base, including regression discontinuity designs and case studies.


See also

Aspects of teaching math
North American problems
Math difficulties


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 June 26, 2010. Retrieved on September 25, 2011.
  • Fremder, 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.
  • Ball, Lynda, et al. Use of technology in math classes in primary and secondary schools (Cham, Switzerland: Springer, 2018).
  • Dreher, Anika, et al. "What content-related knowledge do secondary math teachers need?" Journal for Mathematics Didactics 39.2 (2018): 319-341 online.
  • Drijvers, Paul, et al. Use of Technology in Lower Secondary Math Classes: A Brief Up-to-Date Review (Springer Nature, 2016).
  • Gosztonyi, Katalin. "Mathematical Culture and Mathematics Education in Hungary in the 20th Century." in mathematical cultures (Birkhäuser, Cham, 2016) pp. 71-89. on-line
  • Losano, Leticia and Márcia Cristina de Costa Trindade Cyrino. "Current research on the professional identity of aspiring math teachers." in The math lessons from aspiring secondary school teachers around the world (Springer, Cham, 2017) pp. 25-32.
  • Wong, Khoon Yoong. "Enriching secondary mathematics lessons with 21st century skills." in the Developing 21st Century Competencies in Mathematics Classes: Yearbook 2016 (Association of Mathematics Educators. 2016) pp. 33-50.

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