# Mathematics Education, Teacher Preparation

# MATHEMATICS EDUCATION, TEACHER PREPARATION

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.

** See also: ** Mathematics Learning; National Council of Teachers of Mathematics.

## 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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