Mathematics education

A child calculating with his fingers (2006)

In contemporaryeducation,mathematics education—known in Europe as thedidacticsorpedagogyofmathematics—is the practice ofteaching,learning,and carrying outscholarlyresearchinto 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 organisationsregularly hold conferences andpublish literaturein order to improve mathematics education.

History

Ancient

Elementary mathematics were a core part of education in many ancient civilisations, includingancient Egypt,ancient Babylonia,ancient Greece,ancient Rome,andVedicIndia.In most cases, formal education was only available tomalechildren with sufficiently high status, wealth, orcaste.The oldest known mathematics textbook is theRhind papyrus,dated from circa 1650 BCE.

Pythagorean theorem

Historians ofMesopotamiahave confirmed that use of thePythagorean ruledates back to theOld Babylonian Empire(20th–16th centuries BC) and that it was being taught in scribal schools over one thousand years before the birth ofPythagoras.

InPlato's division of theliberal artsinto thetriviumand thequadrivium,the quadrivium included the mathematical fields ofarithmeticandgeometry.This structure was continued in the structure ofclassical educationthat was developed in medieval Europe. The teaching of geometry was almost universally based onEuclid'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 ofEuclid'sElements

In theMiddle Ages,the academic status of mathematics declined, because it was strongly associated with trade and commerce, and considered somewhat un-Christian.Although it continued to be taught inEuropean universities,it was seen as subservient to the study ofnatural,metaphysical,andmoral philosophy.The first modern arithmetic curriculum (starting withaddition,thensubtraction,multiplication,anddivision) arose atreckoning schoolsin Italy in the 1300s.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.They also contrasted with mathematical methods learned byartisanapprentices, 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.

The first mathematics textbooks to be written in English and French were published byRobert Recorde,beginning withThe Grounde of Artesin 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 thequadratic equation.After the Sumerians, some of the most famousancientworks on mathematics came from Egypt in the form of theRhind Mathematical Papyrusand theMoscow Mathematical Papyrus.The more famousRhind Papyrushas 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.

The social status of mathematical study was improving by the seventeenth century, with theUniversity of Aberdeencreating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up inUniversity of Oxfordin 1619 and theLucasian Chair of Mathematicsbeing established by theUniversity of Cambridgein 1662.

Modern

In the 18th and 19th centuries, theIndustrial Revolutionled to an enormous increase inurbanpopulations. Basic numeracy skills, such as the ability to tell the time, count money, and carry out simplearithmetic,became essential in this new urban lifestyle. Within the newpublic educationsystems, mathematics became a central part of thecurriculumfrom an early age.

By the twentieth century, mathematics was part of the core curriculum in alldeveloped 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 ofFelix Klein.
• TheInternational Commission on Mathematical Instruction(ICMI) was founded in 1908, and Felix Klein became the first president of the organisation.
• The professionalperiodical literatureon mathematics education in the United States had generated more than 4,000 articles after 1920, so in 1941 William L. Schaaf published aclassified index,sorting them into their various subjects.
• A renewed interest in mathematics education emerged in the 1960s, and the International Commission was revitalized.
• In 1968, theShell Centre for Mathematical Educationwas established inNottingham.
• The firstInternational Congress on Mathematical Education(ICME) was held inLyonin 1969. The second congress was inExeterin 1972, and after that, it has been held every four years.

Midway through the twentieth century, the cultural impact of the "electronic age"(McLuhan) was also taken up byeducational theoryand the teaching of mathematics. While previous approach focused on "working with specialized 'problems' inarithmetic",the emerging structural approach to knowledge had" small children meditating aboutnumber theoryand 'sets'. "Since the 1980s, there have been a number of efforts to reform the traditional curriculum, which focuses on continuous mathematics and relegates even some basic discrete concepts to advanced study, to better balance coverage of the continuous and discrete sides of the subject:

• In the 1980s and early 1990s, there was a push to make discrete mathematics more available at the post-secondary level;
• From the late 1980s into the new millennium, countries like the US began to identify and standardize sets of discrete mathematics topics for primary and secondary education;
• Concurrently, academics began compiling practical advice on introducing discrete math topics into the classroom;
• Researchers continued arguing the urgency of making the transition throughout the 2000s; and
• In parallel, some textbook authors began working on materials explicitly designed to provide more balance.

Similar efforts are also underway to shift more focus to mathematical modeling as well as its relationship to discrete math.

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 basicnumeracyskills to all students
• The teaching of practical mathematics (arithmetic,elementary algebra,plane and solidgeometry,trigonometry,probability,statistics) to most students, to equip them to follow a trade or craft and to understand mathematics commonly used in news andInternet(such aspercentages,charts,probability,andstatistics)
• The teaching of abstract mathematical concepts (such assetandfunction) at an early age
• The teaching of selected areas of mathematics (such asEuclidean geometry)as an example of anaxiomatic systemand a model ofdeductive reasoning
• The teaching of selected areas of mathematics (such ascalculus) as an example of the intellectual achievements of themodern world
• The teaching of advanced mathematics to those students who wish to follow a career inscience, technology, engineering, and mathematics (STEM)fields
• The teaching ofheuristicsand other problem-solving strategies to solve non-routine problems
• The teaching of mathematics insocial sciencesandactuarial sciences,as well as in some selectedartsunderliberal artseducation inliberal arts collegesor 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:

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 byBig Brother Mousein Laos.

• 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 thequadrivium,part of the classical education curriculum of theMiddle Ages,which was typically based onEuclid'sElementstaught as aparadigmofdeductive reasoning.
• Conventional approach:the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts witharithmeticand is followed byEuclidean geometryandelementary algebrataught concurrently. Requires the instructor to be well informed aboutelementary mathematicssince 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 thedevelopment of mathematicswithin a historical, social, and cultural context. Proponents argue it provides morehuman interestthan 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 andmanipulativetools.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 asset 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 wasMorris Kline's 1973 bookWhy Johnny Can't Add.The New Math method was the topic of one ofTom 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 theUnited StatesandCanada,focused on deepening student understanding of mathematical ideas and procedures, and formalized by theNational Council of Teachers of Mathematicswhich created thePrinciples 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,andheuristicthinking by setting students open-ended, unusual, and sometimesunsolved problems.The problems can range from simpleword problemsto problems from internationalmathematics competitionssuch as theInternational 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 addingsimple fractionsor solvingquadratic equations.
• Rote learning:the teaching of mathematical results, definitions and concepts by repetition and memorisation typically without meaning or supported bymathematical reasoning.A derisory term isdrill and kill.Intraditional education,rote learning is used to teachmultiplication 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

A mathematics lecture atAalto University School of Science and Technology

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 orhonors 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 andmeasurementare taught, in both numeric and pictorial form, as well asfractionsandproportionality,patterns, and various topics related to geometry.

At high school level in most of the US,algebra,geometry,and analysis (pre-calculusandcalculus) 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 coursesà la carteas 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 areMathematics, Mathematical Literacy and Technical Mathematics.) Thus, a science-oriented curriculum typically overlaps the first year of university mathematics, and includesdifferential calculusandtrigonometryat age 16–17 andintegral calculus,complex numbers,analytic geometry,exponentialandlogarithmic functions,andinfinite seriesin their final year of secondary school;Probabilityandstatisticsare similarly often taught.

At college and university level,scienceandengineering studentswill be required to takemultivariable calculus,differential equations,andlinear algebra;at several US colleges, theminororASin mathematics substantively comprises these courses.Mathematics majorsstudy additional other areas withinpure mathematics—and often in applied mathematics—with the requirement of specified advanced courses inanalysisandmodern algebra.Other topics in pure mathematics includedifferential geometry,set theory,andtopology.Applied mathematicsmay be taken as amajorsubject in its own right, such aspartial differential equations,optimization,andnumerical analysis.Specific topics are taught within other courses: for example,civil engineersmay be required to studyfluid mechanics,and "math for computer science" might includegraph theory,permutation,probability, andformalmathematical proofs.Pure and applied math degrees often include modules inprobability theoryormathematical statistics,as well asstochastic processes.(Theoretical)physicsis mathematics-intensive, often overlapping substantively with the pure or applied math degree.Business mathematicsis usually limited to introductory calculus and (sometimes)matrixcalculations;economics programsadditionally coveroptimization,oftendifferential equationsandlinear algebra,and sometimes analysis.

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. InEngland,for example, standards for mathematics education are set as part of the National Curriculum for England,whileScotlandmaintains 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.

In North America, theNational Council of Teachers of Mathematics(NCTM) published thePrinciples and Standards for School Mathematicsin 2000 for the United States and Canada, which boosted the trend towardsreform mathematics.In 2006, the NCTM releasedCurriculum 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 theCommon Core State Standardsfor 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."States routinely review theiracademic standardsand may choose to change or add onto the standards to best meet the needs of their students. "The NCTM has state affiliates that have different education standards at the state level. For example,Missourihas 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.

TheProgramme for International Student Assessment(PISA), created by theOrganisation for the Economic Co-operation and Development(OECD), is a global program studying the reading, science, and mathematics abilities of 15-year-old students.The first assessment was conducted in the year 2000 with 43 countries participating.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.

Research

According to Hiebert and Grouws, "Robust, useful theories of classroom teaching do not yet exist."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.) 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 assessmentis 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

Homeworkassignments 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 withbasic facts,answer impulsively, struggle with mental representations, have poornumber 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 byformative 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 ofvariable.They prefer arithmetic reasoning toalgebraic equationsfor solving word problems. It takes time to move from arithmetic to algebraic generalizations to describe patterns. Students often have trouble with theminus signand understand theequals signto mean "the answer is...".

Methodology

As with other educational research (and thesocial sciencesin general), mathematics education research depends on both quantitative and qualitative studies.Quantitative researchincludes studies that useinferential statisticsto answer specific questions, such as whether a certainteaching methodgives 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 ascase studies,action research,discourse analysis,andclinical 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 understoodwhytreatment X is better than treatment Y, application of results of quantitative studies will often lead to "lethal mutations"of the finding in actual classrooms. Exploratory qualitative research is also useful for suggesting newhypotheses,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.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.In other disciplines concerned with human subjects—likebiomedicine,psychology,and policy evaluation—controlled, randomized experiments remain the preferred method of evaluating treatments.Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods.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,or the difficulty of assuring rigid control of the independent variable in fluid, real school settings.

In the United States, theNational Mathematics Advisory Panel(NMAP) published a report in 2008 based on studies, some of which used randomized assignment of treatments toexperimental units,such as classrooms or students. The NMAP report's preference for randomized experiments received criticism from some scholars.In 2010, theWhat Works Clearinghouse(essentially the research arm for theDepartment of Education) responded to ongoing controversy by extending its research base to include non-experimental studies, includingregression discontinuity designsandsingle-case studies.

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

• Mathematics education in the United States

Mathematical difficulties

• Dyscalculia
• Mathematical anxiety

References

• Voit, Rita."Accelerated Math: What Every Parent Should Know".Resources by HEROES Academy.Retrieved20 September2023.

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 fromthe original(PDF)on 2010-06-26.Retrieved2011-09-25.
• 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.Retrieved2009-08-06.
• 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-Didaktik39.2 (2018): 319-341onlineArchived2021-04-18 at theWayback 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." inMathematical 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." inThe mathematics education of prospective secondary teachers around the world(Springer, Cham, 2017) pp. 25-32.
• Sriraman, Bharath;English, Lyn(2010).Theories of Mathematics Education.Springer.ISBN978-3-642-00774-3.
• 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.ISBN978-0-691-13493-2.
• Strutchens, Marilyn E., et al.The mathematics education of prospective secondary teachers around the world(Springer Nature, 2017)onlineArchived2021-04-18 at theWayback Machine.
• Wong, Khoon Yoong. "Enriching secondary mathematics education with 21st century competencies." inDeveloping 21st Century Competencies In The Mathematics Classroom: Yearbook 2016(Association Of Mathematics Educators. 2016) pp. 33–50.

External links

• Math EducationatCurlie
• History of Mathematical Education
• A quarter century of US 'math wars' and political partisanship.David Klein. California State University, Northridge, United States