health of the undergraduate SME&T enterprise. There will not be enough students who choose to, and are adequately prepared to, participate in SME&T, without an effective K-12 teaching force.

This report is about the preparation of teachers of K-12 mathematics. It raises questions about directions for future work in critical areas of mathematics education. It does not address more general topics in teacher preparation and development, such as school reform, the reform of teacher preparation, teacher learning and program philosophy, and the professional education of prospective teachers. In some cases the assumptions and issues raised are equally relevant to the preparation of teachers of science, and to the preparation of the professoriate more generally.

CONSIDERATIONS

Below we explicitly highlight some considerations to be kept in mind while reading this report.

Teacher preparation—the formal undergraduate and graduate experience of prospective teachers —is only one part of a continuum of experiences which contribute to the process of learning to teach. Prospective teachers are deeply influenced by their own background as K-12 students (Lortie, 1975). Their professional development continues through the field experience, the induction years into teaching, and in ongoing formal and informal staff development throughout their careers (Loucks-Horsley et al., 1987). The undergraduate preparation of a teacher provides a bridge from the prospective teacher's precollege experiences to beginning teaching, and a foundation for subsequent professional development. Preservice experiences at the undergraduate level need to be coordinated, in both a practical and theoretical sense, with teachers' initiation into the profession in schools, as well as with their continuing education.

Educational research, about K-12 mathematics teaching and learning, and about the preparation and continuing education of teachers, is a critical component in the improvement of mathematics teacher preparation. There is an increasingly coherent body of research about mathematics teaching and learning (Grouws, 1992), and about the preparation and development of teachers of mathematics (Brown & Borko, 1992; Grouws & Schultz, 1996). Reform of K-12 mathematics teacher preparation programs can be based on such investigations. Dewey (1910/91) commented on the limitations of unanalyzed empirical impression, and we are positioned in teacher preparation to move beyond this (Lampert, 1988). Scholarly inquiry in a field not only challenges assumptions and beliefs, it is essential to systematic change.

Many of the mathematical preparation issues for prospective elementary school teachers, middle school teachers, and secondary school specialists differ considerably. Preparation programs differ substantially depending on whether they are for generalist teachers in elementary and middle schools, or for specialists in middle and secondary schools (National Center for Education Statistics [NCES], 1993, 1995; Nelson, Weiss & Conaway, 1992). This is partly because of the organization of schools, of universities, and of state credentialling and certification programs. Moreover, K-8 teachers face a need to integrate mathematics with other content areas that their secondary school colleagues do not face to the same degree. There is some evidence that people



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THE PREPARATION OF TEACHERS OF MATHEMATICS: CONSIDERATIONS AND CHALLENGES: A LETTER REPORT health of the undergraduate SME&T enterprise. There will not be enough students who choose to, and are adequately prepared to, participate in SME&T, without an effective K-12 teaching force. This report is about the preparation of teachers of K-12 mathematics. It raises questions about directions for future work in critical areas of mathematics education. It does not address more general topics in teacher preparation and development, such as school reform, the reform of teacher preparation, teacher learning and program philosophy, and the professional education of prospective teachers. In some cases the assumptions and issues raised are equally relevant to the preparation of teachers of science, and to the preparation of the professoriate more generally. CONSIDERATIONS Below we explicitly highlight some considerations to be kept in mind while reading this report. Teacher preparation—the formal undergraduate and graduate experience of prospective teachers —is only one part of a continuum of experiences which contribute to the process of learning to teach. Prospective teachers are deeply influenced by their own background as K-12 students (Lortie, 1975). Their professional development continues through the field experience, the induction years into teaching, and in ongoing formal and informal staff development throughout their careers (Loucks-Horsley et al., 1987). The undergraduate preparation of a teacher provides a bridge from the prospective teacher's precollege experiences to beginning teaching, and a foundation for subsequent professional development. Preservice experiences at the undergraduate level need to be coordinated, in both a practical and theoretical sense, with teachers' initiation into the profession in schools, as well as with their continuing education. Educational research, about K-12 mathematics teaching and learning, and about the preparation and continuing education of teachers, is a critical component in the improvement of mathematics teacher preparation. There is an increasingly coherent body of research about mathematics teaching and learning (Grouws, 1992), and about the preparation and development of teachers of mathematics (Brown & Borko, 1992; Grouws & Schultz, 1996). Reform of K-12 mathematics teacher preparation programs can be based on such investigations. Dewey (1910/91) commented on the limitations of unanalyzed empirical impression, and we are positioned in teacher preparation to move beyond this (Lampert, 1988). Scholarly inquiry in a field not only challenges assumptions and beliefs, it is essential to systematic change. Many of the mathematical preparation issues for prospective elementary school teachers, middle school teachers, and secondary school specialists differ considerably. Preparation programs differ substantially depending on whether they are for generalist teachers in elementary and middle schools, or for specialists in middle and secondary schools (National Center for Education Statistics [NCES], 1993, 1995; Nelson, Weiss & Conaway, 1992). This is partly because of the organization of schools, of universities, and of state credentialling and certification programs. Moreover, K-8 teachers face a need to integrate mathematics with other content areas that their secondary school colleagues do not face to the same degree. There is some evidence that people

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THE PREPARATION OF TEACHERS OF MATHEMATICS: CONSIDERATIONS AND CHALLENGES: A LETTER REPORT entering teaching at these different levels bring different expectations, experiences, and professional goals (Ball, 1988). The trend is that elementary teachers tend to enter the profession more deeply committed to children, and less committed to particular content areas. CHALLENGES IN THE PREPARATION OF K-12 TEACHERS OF MATHEMATICS We identify five challenges to improving the preparation of teachers of K-12 mathematics. Both through applied work in mathematics teacher preparation, and through research, there have been efforts to better understand each. Many practical problems and theoretical dilemmas remain, however. We discuss “where we are now” -- highlighting what is happening in practice, what is known from research, and where there is consensus. We then address “what is needed” by pointing out where consensus does not exist, where movement toward consensus might be possible, and where informed debate among researchers, practitioners and policymakers might be especially productive. Issue I: What Mathematics Should Teachers Know? The mathematics community has a long history of supporting strong mathematics content preparation for prospective teachers. Current publications of the professional societies continue to make this case, emphasizing that the new K-12 reforms require teachers to have increased mathematical breadth. A Call for Change (Leitzel, 1991, preface) notes: “The content of collegiate level courses must reflect the changes in emphases and content of the emerging school curriculum and the rapidly broadening scope of mathematics itself. In general, current requirements for certification of teachers of school mathematics, particularly at the elementary and middle school levels, and the learning experiences of prospective teachers within college mathematics classes fall far short of these goals.” With the publication of the NCTM Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) and the ensuing development of curriculum materials reflecting the mathematical emphases of the Standards, teachers face subjects, such as data analysis and discrete mathematics, not traditionally included in preservice preparation programs. Sources which describe contemporary mathematics (Peterson, 1988; Steen, 1990) should be considered in re-thinking content issues. New curriculum materials and standards also raise issues about the depth of mathematical understanding needed by teachers. The NCTM Professional Standards, for example, suggest that teachers “orchestrate discourse by deciding what to pursue in depth from among the ideas that students bring up in a discussion” (NCTM, 1991, p. 35). Teachers may also need deeper mathematical understanding in order to promote mathematical sense-making, problem solving, reasoning, and justification. Ball observes “elementary teachers, most of whom experienced school knowledge as given—and who acquired facts and memorized rules—must invent a teaching that engages students in complex reasoning in authentic contexts” (Ball, in press, p. 14). Lampert foreshadowed the current content need in her choice of the single indicator of an ideal mathematics teacher: “whether that teacher could give students at the grade level he or she is teaching a mathematically legitimate and comprehensible explanation for why the procedures students are using are appropriate or not, or why the answers they are giving are correct or not”