Mathematical education | Wikipedia audio article

In contemporary education, mathematics education
is the practice of teaching and learning mathematics, along with the associated scholarly research. Researchers in mathematics education are primarily
concerned with the tools, methods and approaches that facilitate practice or the study of practice;
however, mathematics education research, known on the continent of Europe as the didactics
or pedagogy of mathematics, has developed into an extensive field of study, with its
own concepts, theories, methods, national and international organisations, conferences
and literature. This article describes some of the history,
influences and recent controversies..==History==
Elementary mathematics was part of the education system in most ancient civilisations, including
Ancient Greece, the Roman Empire, Vedic society and ancient Egypt. In most cases, a formal education was only
available to male children with a sufficiently high status, wealth or caste. 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. Teaching of geometry was almost universally
based on Euclid’s Elements. Apprentices to trades such as masons, merchants
and money-lenders could expect to learn such practical mathematics as was relevant to their
profession. In the Renaissance, 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 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. 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 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.The first mathematics textbooks to be written
in English and French were published by Robert Recorde, beginning with The Grounde of Artes
in 1540. 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
own methodology for solving equations like the quadratic equation. After the Sumerians some of the most famous
ancient works on mathematics come from Egypt in the form of the Rhind Mathematical Papyrus
and the Moscow Mathematical Papyrus. The more famous Rhind Papyrus has been dated
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 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. 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. Here are some of the main events in this development: 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 U.S.A. had generated
more than 4000 articles after 1920, so in 1941 William L. Schaaf published a classified
index, sorting them into their various subjects. A renewed interest in mathematics education
emerged in the 1960s, and the International Commission was revitalised
In 1968, the Shell Centre for Mathematical Education was established in Nottingham
The first International Congress on Mathematical Education (ICME) was held in Lyon in 1969. The second congress was in Exeter in 1972,
and after that it has been held every four yearsIn 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’.”==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 pupils The teaching of practical mathematics (arithmetic,
elementary algebra, plane and solid geometry, trigonometry) to most pupils, to equip them
to follow a trade or craft 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 pupils 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.==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: 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. Computer-based math an approach based around
use of mathematical software as the primary tool of computation. Computer-based mathematics education involving
the use of computers to teach mathematics. Mobile applications have also been developed
to help students learn mathematics. 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. Exercises: the reinforcement of mathematical
skills by completing large numbers of exercises of a similar type, such as adding vulgar fractions
or solving quadratic equations. Historical method: teaching the development
of mathematics within an historical, social and cultural context. Provides more human interest than the conventional
approach. Mastery: an approach in which most students
are expected to achieve a high level of competence before progressing
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.” 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. Recreational mathematics: Mathematical problems
that are fun can motivate students to learn mathematics and can increase enjoyment of
mathematics. Standards-based mathematics: a vision for
pre-college mathematics education in the US 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. 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 helping
them to apply mathematics to real world situations outside of the classroom. 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.==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 in a similar fashion, though there are differences. Most countries tend to cover fewer topics
in greater depth than in the United States.In most of the U.S., algebra, geometry and analysis
(pre-calculus and calculus) are taught as separate courses in different years of high
school. Mathematics in most other countries (and in
a few U.S. states) is integrated, with topics from all branches of mathematics studied every
year. Students in many countries choose an option
or pre-defined course of study rather than choosing courses à la carte as in the United
States. Students in science-oriented curricula typically
study 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 may be taught in
secondary education classes. Science and engineering students in colleges
and universities may be required to take multivariable calculus, differential equations, and linear
algebra. Applied mathematics is also used in specific
majors; for example, civil engineers may be required to study fluid mechanics, while “math
for computer science” might include graph theory, permutation, probability, and proofs. Mathematics students would continue to study
potentially any area.==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) summarised the research of others
who found, based on nationwide data, that students with higher scores on standardised
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, the National Council of Teachers
of Mathematics (NCTM) published the Principles and Standards for School Mathematics in 2000
for the US 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. “States routinely review their academic standards
and 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, Missouri has the Missouri Council
of Teachers of Mathematics (MCTM) which has its own pillars and standards of education
listed on its website. The MCTM also offers membership opportunities
to teachers and future teachers so they can stay up to date on the changes in math educational
“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 in effective
teaching is giving students “opportunity to learn”. Teachers can set expectations, time, kinds
of tasks, questions, acceptable answers, and type of discussions that will influence students’
opportunity 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 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 U.S.A., 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 end result is greater learning. This has been shown to be true whether the
struggle is due to challenging, well-implemented teaching, or due to faulty teaching the students
must struggle to make sense of.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 which leads students to practice
past lessons or prepare future lessons are more effective than those going over today’s
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 It is important for elementary school children
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 in order 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” 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. 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 randomized trials provide clear, objective
evidence on “what works”, policy makers often take only those studies into consideration. Some scholars have pushed for more random
experiments in which teaching methods are randomly assigned to classes. In other disciplines concerned with human
subjects, like biomedicine, 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, the National Mathematics Advisory Panel (NMAP)
published a report in 2008 based on studies, some of which used randomized assignment of
treatments to experimental units, such as classrooms or students. The NMAP report’s preference for randomized
experiments received criticism from some scholars. In 2010, the What Works Clearinghouse (essentially
the research arm for the Department of Education) responded to ongoing controversy by extending
its research base to include non-experimental studies, including regression discontinuity
designs and single-case studies.==Mathematics educators==
The following are some of the people who have had a significant influence on the teaching
of mathematics at various periods in history: Euclid (fl. 300 BC), Ancient Greek, author
of The Elements Felix Klein (1849 – 1925), German mathematician
who had substantial influence on math education in the early 20th Century, Inaugural president
of the International Commission on Mathematical Instruction
Andrei Petrovich Kiselyov (1852 – 1940) Russian and Soviet mathematician. His textbooks on basic arithmetics, algebra
and geometry were the standard for Russian classrooms since 1892 well into the 1960s,
when Russian mathematics education got embroiled in the New Math reforms. In 2006 these textbooks were re-printed and
became popular again. David Eugene Smith (1860 – 1944) American
mathematician, educator, and editor, considered one of the founders of the field of mathematics
education Tatyana Alexeyevna Afanasyeva (1876–1964),
Dutch/Russian mathematician who advocated the use of visual aids and examples for introductory
courses in geometry for high school students Robert Lee Moore (1882–1974), American mathematician,
originator of the Moore method George Pólya (1887–1985), Hungarian mathematician,
author of How to Solve It Georges Cuisenaire (1891–1976), Belgian
primary school teacher who invented Cuisenaire rods
William Arthur Brownell (1895–1977), American educator who led the movement to make mathematics
meaningful to children, often considered the beginning of modern mathematics education
Hans Freudenthal (1905–1990), Dutch mathematician who had a profound impact on Dutch education
and founded the Freudenthal Institute for Science and Mathematics Education in 1971
Caleb Gattegno (1911-1988), Egyptian, Founder of the Association for Teaching Aids in Mathematics
in Britain (1952) and founder of the journal Mathematics Teaching. Toru Kumon (1914–1995), Japanese, originator
of the Kumon method, based on mastery through exercise
Pierre van Hiele and Dina van Hiele-Geldof, Dutch educators (1930s–1950s) who proposed
a theory of how children learn geometry (1957), which eventually became very influential worldwide
Bob Moses (1935–), founder of the nationwide US Algebra project
Robert M. Gagné (1958–1980s), pioneer in mathematics education research. David Tall (1941 – ), most cited mathematics
education researcher in modern times . Established the “Advanced Mathematic Thinking” working
group. Has contributed to the education and learning
process of advanced mathematics.==Mathematics teachers==
The following people all taught mathematics at some stage in their lives, although they
are better known for other things: Lewis Carroll, pen name of British author
Charles Dodgson, lectured in mathematics at Christ Church, Oxford. As a mathematics educator, Dodgson defended
the use of Euclid’s Elements as a geometry textbook; Euclid and his Modern Rivals is
a criticism of a reform movement in geometry education led by the Association for the Improvement
of Geometrical Teaching. John Dalton, British chemist and physicist,
taught mathematics at schools and colleges in Manchester, Oxford and York
Tom Lehrer, American songwriter and satirist, taught mathematics at Harvard, MIT and currently
at University of California, Santa Cruz Brian May, rock guitarist and composer, worked
briefly as a mathematics teacher before joining Queen
Georg Joachim Rheticus, Austrian cartographer and disciple of Copernicus, taught mathematics
at the University of Wittenberg Edmund Rich, Archbishop of Canterbury in the
13th century, lectured on mathematics at the universities of Oxford and Paris
Éamon de Valera, a leader of Ireland’s struggle for independence in the early 20th century
and founder of the Fianna Fáil party, taught mathematics at schools and colleges in Dublin
Archie Williams, American athlete and Olympic gold medalist, taught mathematics at high
schools in California.==Organizations==
Advisory Committee on Mathematics Education American Mathematical Association of Two-Year
Colleges Association of Teachers of Mathematics
Mathematical Association National Council of Teachers of Mathematics==Related==
“. . . our students of mathematics would profit much more from a study of Euler’s Introductio
in Analysin Infinitorum, rather than of the available modern textbooks.” (Andr´e Weil 1979, quoted by J.D.Blanton
1988, p. xii) “..maxim of Niels Henrik Abel, “I learned
from the masters and not from the pupils” .. ”
(Extracted from Alexander Ostermann • Gerhard Wanner, Geometry by Its History )==See also

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