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Mathematics education

 
Education Encyclopedia: Mathematics Education
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Mathematics teachers are educated in diverse ways, depending to a great extent on the context in which the education occurs. Typically, pre-service teacher preparation occurs at the baccalaureate level, while in-service education occurs at the graduate level or is conducted by the local school systems in which the teacher is employed. There are, however, some pre-service programs in which participants acquire a masters' degree prior to beginning their teaching career.

In preparation for teaching at the elementary level, most undergraduate students take two mathematics courses that are either part of the institution's core liberal arts program or are designed specifically for elementary teaching majors. Additionally, it is likely that prospective elementary teachers will have one or two courses that deal specifically with the teaching of elementary school mathematics. Concerns that teachers at the elementary level need more background in mathematics have resulted in recent trends toward upgrading the mathematical education of prospective elementary teachers. Secondary teachers typically have a major in mathematics, or a closely related field, with an additional course (or courses) in mathematics education. Smaller programs are more likely to offer only a single course in mathematics education. The education of prospective middle school teachers is very dependent on the type of institution. In some schools, middle school teaching majors follow a program similar to the elementary majors, but with extra courses in mathematics; in others, they take a program for secondary school pre-service teachers specializing in mathematics, with one or more additional courses in middle school education. A few larger universities have programs designed specifically for the prospective middle school mathematics teacher. The following sections focus on the intent and foci of the different programs.

The Evolution of Mathematics Teacher Education

Before 1960 most teacher education programs for secondary school mathematics teachers consisted of training in mathematics, a methods course of some kind, and student teaching. Smaller programs at colleges or universities tended to have generic methods courses that addressed the needs of secondary teachers of all subjects. One can glean an understanding of the content-specific methods courses by considering the methods texts of that time. For example, the popular 1960 methods text by Charles Butler and Frank Wren (first published in 1941) consisted of two sections. The first section dealt with general issues such as planning for instruction. The second section was decidedly mathematical, with specific suggestions for teaching topics such as arithmetic, algebra, geometry, and trigonometry. There was a clear distinction between these two sections. Donovan Johnson and Gerald Rising's innovative 1967 text was based on what mathematics teachers do in the classroom. As such, it addressed issues specific to the teaching and learning of mathematics. A 1975 text by Thomas Cooney, Edward Davis, and Kenneth Henderson for secondary mathematics teachers also had a very distinct pedagogical orientation based on research on how teachers teach mathematics. Whereas the Johnson and Rising text was based primarily on teachers' daily responsibilities, the Cooney, Davis, and Henderson text was based on a theoretical analysis of teachers' verbal actions, called moves, and the way those moves were used to teach mathematical concepts, generalizations, and skills.

During the 1960s and 1970s educators began to see the value in studying the teaching and learning of mathematics more specifically. Out of this new focus on research grew an interest in developing a psychological basis for understanding why some students learned but others did not, and what kind of teaching methods and curricula could affect student learning. This growing knowledge base contributed to mathematics teacher education as well.

The Evolution of Mathematics Education As a Field of Inquiry

Prior to 1960 there was little research on how children learn mathematics and how teachers teach mathematics. The teacher's job was seen primarily as a matter of telling students the mathematics they were expected to learn. But as research in mathematics education matured, questions arose about how students understand mathematics. Consider, for example, the variation in understanding of mathematics conveyed in the responses of two students to the following questions:

Are there any numbers between 440 and 450 that are divisible by 7? Why or why not?

Response of Student 1: There must be a number because 7 is less than 10. So in every 10 numbers there has to be at least one that is divisible by 7. (Student elaborates for entire page.)

Response of Student 2: There is no number because 440 and 450 is not divisible by 7 - 44 is not, 45 is not, and 0 is not.

The response of student 1 reveals a deep understanding of how numbers work, while the response of student 2 demonstrates some understanding of divisibility, since 44 and 45 are not divisible by 7, but fails to capture the mathematical essence of the question. If the interest of teacher educators in evaluating these two responses goes beyond one student having gotten it right and the other student not, then they can begin to ask how a teacher could enable the second student to better understand divisibility. Indeed, teacher education today focuses, in part, on enabling teachers to create and use such questions so that they can better analyze their students' understanding of mathematics. Simply put, the education of mathematics teachers entails a certain kind of knowledge that involves mathematics, psychology, and ways of teaching mathematics that are more effective than simply telling students what mathematics is and what the answers to various problems are. This knowledge base has grown substantially over the past decades because of the extensive research in mathematics education.

In-Service and Staff Development Programs

An appreciation of the complexity of teaching has led teacher educators to move toward programs in which teachers are provided with extensive training and support to implement new practices - such as problem-solving techniques or infusing technology into their teaching. There is mounting evidence that teachers need support and time if they are to reform their practice. For example, the successful professional development program by Raffaella Borasi, Judith Fonzi, Constance Smith, and Barbara Rose not only emphasizes having teachers interact with materials designed to foster student inquiry but also provides teachers with support as they use the materials in their classroom. Some in-service programs engage teachers in deep experiences with the mathematics they are teaching, thereby giving them new insights into their students' understanding of that mathematics. Programs that encourage teachers to reflect on the types of experiences they have and are providing to their students are becoming increasingly popular.

Trends, Issues, and Controversies

Perhaps the single most significant force affecting mathematics teacher education today has been the development of standards for school mathematics by the National Council of Teachers of Mathematics (NCTM). Through these standards, the NCTM has taken the view that mathematics is a subject suitable for inquiry and not just memorization, a subject that can be learned by all students and should be taught with an emphasis on processes such as problem solving, reasoning, communicating mathematically, and connecting mathematics to the real world. One way or another, most teacher education programs today embody the NCTM standards. Controversies about this approach stem from several questions, including: What constitutes mathematics? and, Should mathematics teacher education programs be about reform or about maintaining the status quo?

The Nature of Mathematics

Different segments of society possess different views about what constitutes mathematics. Some think of mathematics as a collection of rules and procedures to be learned and applied for basic living. From this perspective, the teaching of mathematics relies on those methods best suited to promote the acquisition of skills. Others see mathematics as a basis for developing critical thinking and problem-solving skills. From this second perspective, which is closely aligned to the NCTM Standards, teacher education encourages reflection and promotes attention to problem solving and critical thinking. How a community defines mathematics affects what, and how, mathematics gets taught in the local schools. It can also have an impact on how teachers are trained to teach in those schools.

The Intent of Teacher Education Programs

There is always a certain tension between the intellectual preparation of teachers and the practice of teaching as manifested in student teaching. Those from outside the field of mathematics education often take the position that teacher education should be modeled after an apprenticeship program. That is, one learns mathematics and then works in the schools to acquire the necessary pedagogical skills to be a successful teacher. This type of program tends to promote the status quo, as young teachers model those methods of teaching that they experienced as students. Teacher educators, however, usually take the position that a greater part of the program should be devoted to transforming the teaching of mathematics from a "teaching is telling" approach to an inquiry-based teaching style that is student centered. The notion of constructivism is often used to describe this latter kind of teaching; that is, children construct their own mathematical ideas, and teachers need to be aware of these constructions in order to effectively teach the children.

The preparation and education of mathematics teachers, like any educational endeavor, exists in a sociopolitical environment that ultimately shapes the enterprise. Conditions of the workplace also shape what transpires in classrooms. These circumstances affect mathematics teacher education programs as well. Schools today are run much as they were in yesteryear, thus perpetuating a certain conservatism with respect to reform. This approach strengthens the position of those who advocate an apprenticeship form of teacher education. Evidence suggests that the United States is experiencing, and will continue to experience, serious teacher shortages, particularly in mathematics. Such shortages usually preclude more extensive training in favor of short, intense programs that are less demanding on the schools' staffing resources.

On the other hand, reform-based teacher education programs enjoy the support of such national organizations as the NCTM and are rooted in the thinking of scholars such as John Dewey. Dewey's notion of reflective thinking, albeit adapted and modified, is part and parcel of most current teacher education programs. Indeed, if the position is taken that education is about educating young people to become thinking citizens in a democratic society, then the education of teachers to infuse problem solving, reasoning, and critical thinking into their teaching should be of paramount importance. In some sense, the notion of what constitutes a good teacher education program is dependent on what one values regarding society's education of its young people.

Bibliography

Borasi, Raffaella; Fonzi, Judith; Smith, Constance F.; and Rose, B. J. 1999. "Beginning the Process of Rethinking Mathematics Instruction: A Professional Development Program." Journal of Mathematics Teacher Education 2:49 - 78.

Butler, Charles H., and Wren, Frank L. 1960. The Teaching of Secondary School Mathematics. New York: McGraw-Hill.

Cooney, Thomas J. 1994. "Research and Teacher Education: In Search of Common Ground." Journal for Research in Mathematics Education 25:608 - 636.

Cooney, Thomas J.; Davis, Edward J.; and Henderson, Kenneth B. 1975. Dynamics of Teaching Secondary School Mathematics. Boston: Houghton Mifflin.

Davis, Philip, and Hersh, Reuben. 1981. The Mathematical Experience. Boston: Birkhauser.

Dewey, John. 1933. How We Think: A Restatement of the Relation of Reflective Thinking to the Educative Process. Boston: Heath.

Donovan, Brian F. 1990. "Cultural Power and the Defining of School Mathematics: A Case Study." In Teaching and Learning Mathematics in the 1990s, ed. Thomas J. Cooney and Christian R. Hirsch. Reston, VA: National Council of Teachers of Mathematics.

Dossey, John A. 1992. "The Nature of Mathematics: Its Role and Its Influence." In Handbook of Research on Mathematics Teaching and Learning, ed. Douglas A. Grouws. New York: Macmillan.

Johnson, Donovan A., and Rising, Gerald R. 1967. Guidelines for Teaching Mathematics. Belmont, CA: Wadsworth.

National Council of Teachers of Mathematics. 1989. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. 1991. Professional Standards for Teaching Mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. 1995. Assessment Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

Schifter, Deborah. 1998. "Learning Mathematics for Teaching: From a Teacher's Seminar to the Classroom." Journal of Mathematics Teacher Education 1:55 - 87.

Simon, Martin A. 1997. "Developing New Models of Mathematics Teaching: An Imperative for Research on Mathematics Teacher Development. In Mathematics Teachers in Transition, ed. Elizabeth Fennema and Barbara Scott Nelson. Mahwah, NJ: Erlbaum.

— THOMAS J. COONEY

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Wikipedia: Mathematics education
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Mathematics education is the practice of teaching and learning mathematics, as well as the field of scholarly research on this practice. Researchers in mathematics education are in the 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 of mathematics, has developed into a fully fledged field of study, with its own characteristic concepts, theories, methods, national and international organisations, conferences and literature. This article describes some of the history, influences and recent controversies concerning mathematics education as a practice.

A mathematics lecture at Helsinki University of Technology.

Contents

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.

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 eighteenth and nineteenth 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
  • A new interest in mathematics education emerged in the 1960s, and the 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 years

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]

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.

Research

An increasing amount of research has been done in the area of mathematics education in the last few decades. The National Council of Teachers of Mathematics has summarized the state of current research in mathematics education in nine areas of current interest, as follows.[2] (Though the NCTM has special interest in American education, the research summarized is international in scope.)

What can we learn from research?
Instead of just looking at whether a particular program works, we must also look at why and under what conditions it works. Teachers can adapt tasks used in studies for their own classrooms. Individual studies are often inconclusive, so it is important to look at a consensus of many studies to draw conclusions. Theory can put practice in a new perspective. For example, research shows that when students invent their own algorithms first, and then learn the standard algorithm, they understand better and make fewer errors. Such findings can have an impact on classroom practice.
Homework
Homework which leads students to practice past lessons or prepare future lessons are more effective than those going over today's lesson. Assignments should be a mix of easy and hard problems and ideally based on the student's learning style. Students must receive feedback. Students with learning disabilities or low motivation may profit from rewards. Shorter homework is better than long homework and group homework is sometimes effective, though these findings depend on grade level. Homework helps simple skills, but not broader measures of achievement.
Student learning
Most bilingual adults switch languages when calculating. Such code-switching has no impact on math ability and should not be discouraged.
When studying statistics, children need time to explore, study and share reasoning about centers, shape, spread and variability. The ability to calculate averages does not mean students understand the concept of averages, which students conceptualize in a variety of ways—from a simplistic "typical value" to a deeper idea of "representative value." Learning when to use mean, median and mode is difficult.
Algebra
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...."
[American] Curriculum
The US National Research Council has found it difficult to evaluate any given program, but two general patterns have become clear from large-scale studies: (1) Students achieve greater conceptual understanding from standards-based curricula compared to traditional curricula. (2) Students achieve the same procedural skill level in both types of curricula as measured by traditional standardized tests.
Effective instruction
The two most important criteria for helping students gain conceptual understanding are making connections and intentionally struggling with important ideas. Skill efficiency is best attained by rapid pacing, direct traditional teaching and a smooth transition from teacher modeling to error-free practice. Students who learn skills in conceptually-oriented instruction are better able to adapt their skills to new situations.
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.
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. 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.
Mathematics specialists and coaches
Little research has been done so far on mathematics coaches and the studies that have been done are hard to evaluate because coaching is usually part of larger programs. What research has been done seems to show that coaches can improve teaching, but the coaching program must be well designed.

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.

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. [3]

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.

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.[4]

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 their final year of secondary school.

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 classical education syllabus 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. Parrot Maths was the title of a paper critical of rote learning. Within the conventional approach, rote learning is used to teach multiplication tables.
  • 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.
  • 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.

Mathematics teachers

The following people all taught mathematics at some stage in their lives, although they are better known for other things:

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:

See also

References

  1. ^ Marshall McLuhan (1964) Understanding Media, p.13 [1]
  2. ^ Research clips and briefs
  3. ^ Ma, X. (2000). A longitudinal assessment of antecedent course work in mathematics and subsequent mathematical attainment. Journal of Educational Research, 94, 16-29.
  4. ^ http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf (E.g., p. 20)
  5. ^ Freddie Mercury Interview, Melody Maker, May 2, 1981

Further reading

External links

Teacher organisations and others associated with mathematics education

Scholarly journals: print

Scholarly journals: on-line

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