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.
[edit] History
Illustration at the beginning of a 14th century translation of Euclid's
Elements.
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.
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 math 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.
In the
Renaissance, the academic status of mathematics declined, because it was strongly associated with trade and commerce. Although it continued to be taught in European universities, it was seen as subservient to the study of
Natural,
Metaphysical and
Moral Philosophy.
This trend was somewhat reversed in 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. However, it was uncommon for mathematics to be taught outside of the universities.
Isaac Newton, for example, received no formal mathematics teaching until he joined
Trinity College, Cambridge in 1661.
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 the 20th century, the cultural impact of the "
electric 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'."
[1]
[edit] 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:
Methods of teaching mathematics have varied in line with changing objectives.
[edit] Research
The following results are examples of some of the current findings in the field of mathematics education:
- Practice
Practice is an essential part of learning, and the
time on task is one of the best predictors of the level of competence. The importance of practice, embracing in-class activities and homework, has been slighted in recent writings on "constructivist" teaching, according to leading researchers in
cognitive psychology.
[2]
[The] criticism of practice (called 'drill and kill,' as if this phrase constituted empirical evaluation) is prominent in constructivist writings. Nothing flies more in the face of the last 20 years of research than the assertion that practice is bad. All evidence, from the laboratory and from extensive case studies of professionals, indicates that real competence only comes with extensive practice.... In denying the critical role of practice one is denying children the very thing they need to achieve real competence. The instructional task is not to 'kill' motivation by demanding drill, but to find tasks that provide practice while at the same time sustaining interest.
- Formative assessment[3]
- 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. Only short-term (within and between lessons) and medium-term (within and between units) assessment is effective. 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.
- Effective instruction[3]
- The two most important criteria for helping students gain conceptual understanding are making connections and intentionally struggling with important ideas. Research in the 70s and 80s concluded that skill efficiency is best attained by rapid pacing, direct instruction, and students practicing at least until mastery level. According to summaries of recent research, students who learn skills in conceptually-oriented instruction are better able to adapt their skills to new situations.
- Homework
- The NCTM has stated the following about homework. Homework which leads students to practice past lessons or prepare future lessons are more effective than those going over today's lesson.[3] Assignments should be a mix of easy and hard problems and ideally based on the student's learning style. Students benefit from feedback. Students with learning disabilities or low motivation may profit from rewards. Homework helps simple skills, but not broader measures of achievement.[3] The NCTM states that shorter homework is better than long homework,[3] and group homework is sometimes effective,[3] though these findings depend on grade level, it cautions.[3]
- Students with difficulties[3]
- 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.
- Bilingualism[3]
- Most bilingual adults switch languages when calculating. Such code-switching has no impact on math ability and should not be discouraged.
- Learning statistics[3]
- When studying statistics, children benefit from exploring, studying, and sharing reasoning about centers, shape, spread, and variability. The ability to calculate averages does not signify an understanding of population and sample mean. Students have difficulty remembering to use the sample median as a robust estimator of location, which is less sensitive than the sample mean or the sample mode when the data-set has outliers or when the population is skewed or heavy-tailed.
- Algebraic reasoning[3]
- 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...."
- What can we learn from research?[citation needed]
- Studies can examine the interaction of treatments with covariates and with blocking factors, instead of just estimating the main effects of treatments. Such examinations of interaction give insight into the contexts in which treatments work. Even well-done randomized experiments can yield different results, so it is important to read reviews of many well-done randomized experiments studies when deciding on practical implications.
[edit] Methodology controversy
In the scholarly literature in mathematics education, relatively few articles report random experiments in which teaching methods were randomly assigned to classes.
[4][5] Randomized experiments have been done to evaluate non-pedagogical programs in schools, such as programs to reduce teenage-pregnancy and to reduce drug use.
[6] Randomized experiments were commonly done in educational psychology in the 1880s at American universities.
[7][8] However, randomized experiments have been relatively rare in education in recent decades. In other disciplines concerned with human subjects, like biomedicine, psychology, and policy evaluation, controlled, randomized experiments remain the preferred method of evaluating treatments.
[9][10] Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods.
[5] On the other hand, many scholars in educational schools have argued against increasing the number of randomized experiments, often because of philosophical objections.
[4]
Both randomized experiments and observational studies are useful for suggesting new hypotheses, which can eventually be tested by new randomized experiments. The scientific community recognizes that observational studies remain valuable in education—just as observational studies remain valuable in
epidemiology, political science, economics,
sociology,
anthropology, etc.
[11]
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.
[12] 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.
[13]
[edit] 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. In the USA, the National Governors Association Center for Best Practices and the Council of Chief State School Officers have published the national mathematics
Common Core State Standards Initiative.
Ma (2000) summarised the research of others who found, based on nationwide data, that students with higher scores on standardised math tests had taken more mathematics courses in high school. This led some states to require three years of math instead of two. But because this requirement was often met by taking another lower level math course, the additional courses had a “diluted” effect in raising achievement levels.
[14]
In North America, the
National Council of Teachers of Mathematics (NCTM) has published the
Principles and Standards for School Mathematics. In 2006, they released the
Curriculum Focal Points, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards are not nationally enforced in US schools.
[edit] 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. In the
United States fractions are typically taught starting from 1st grade, whereas in other countries they are usually taught later, since the metric system does not require young children to be familiar with them. Most countries tend to cover fewer topics in greater depth than in the United States.
[15]
In most of the US, 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 US 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.
[edit] 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:
- 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.
- 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.
- 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.
- 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.
- 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.
- 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."
- Historical method - teaching the development of mathematics within an historical, social and cultural context. Provides more human interest than the conventional approach.[16], [17]
- 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 math and helps students understand why they need to know it as well as helping them to apply math to real world situations outside of the classroom.
