OVERVIEW

This section presents six standards for the professional development of teachers of mathematics:

1. Experiencing Good Mathematics Teaching
2. Knowing Mathematics and School Mathematics
3. Knowing Students as Learners of Mathematics
4. Knowing Mathematical Pedagogy
5. Developing as a Teacher of Mathematics
6. The Teacher's Role in Professional Development

INTRODUCTION

Teaching mathematics is a complex endeavor. It demands knowledge of mathematics, students, and teaching as well as opportunities to apply this knowledge in a variety of field-based settings. It requires an understanding of the impact that socioeconomic background, cultural heritage, attitudes and beliefs, and political climate have on the learning environment. Above all, it entails developing a personal knowledge of oneself combines sensitivity and responsiveness to learners with the knowledge, skills, understandings, and dispositions to teach mathematics.

The standards for the professional development of teachers of mathematics address the needs of preservice and in-service teachers of mathematics at the K-1 2 levels. These standards apply to introductory rams that prepare teachers of mathematics; programs that provide advanced study for teachers of mathematics; and various continuing education seminars, workshops, and other learning experiences in which hers of mathematics participate throughout their careers.

Those who teach mathematics bring with them experiences as learners in mathematics classes from elementary school through their college university careers. These experiences influence the ways they think the teaching process, their choice of teaching as a career, and subsequent ways in which they are involved with professional development programs (Ball 1988).

In the early stages of their careers, preservice teachers of mathematics are often involved in developing their knowledge, skills, understandings, dispositions to teach mathematics. This development includes knowledge of mathematics, students, and teaching and involves having opportunities to integrate and apply this learning as practitioners. During time, they are involved in a number of clinical experiences in which interact with supervising or experienced teachers who function as mentors. This period is one in which new ideas can be tried, analyzed, questioned with support and encouragement.

The first few years of teaching present a very different period in the professional development process. Initial teaching assignments and support structures play a significant role in shaping beginning teachers' views of the profession and their commitments to it. The focus is on being a teacher-planning for instruction, managing students' learning, responding to the changing needs of the learning environment-and involves a comprehensive application of what teachers have learned and experienced as part of their preservice programs. New issues are confronted, and knowledge and skills are built daily, more often within the context of the teaching environment than through formal continuing education.

As teachers of mathematics become more experienced, collegial interactions increase and teachers assume a different role. Experienced teachers of mathematics become more involved in decisions about curriculum and staff development programs. Indeed, the identification of staff development needs and the development of programs to meet these needs can become part of the professional responsibilities of an experienced teacher. In addition, experienced teachers may become mentors to beginning or developing teachers at this time in their careers. There is an ebb and flow to their need for more formal continuing education. Their ability to engage in ongoing analysis of their own teaching and learning is often central to their seeking experiences that address knowing mathematics, knowing students, and knowing teaching.

ASSUMPTIONS

Several basic assumptions provide the foundation for the standards detailed in this section:

1. The Curriculum and Evaluation Standards for School Mathematics provides the vision of mathematics education at the K- 12 level that is the basis for the professional development standards. The Curriculum and Evaluation Standards, designed to create a coherent vision of what it means to be mathematically powerful in today's world, provides an in depth focus on what is valued in the learning of mathematics K-12 and in the evaluation of that learning. Teachers must have a thorough understanding of the message in this document. Their education should include the development of the knowledge, skills, understandings, and dispositions needed to implement the recommended standards.

2. Teachers are influenced by the teaching they see and experience. Teachers' own experiences have a profound impact on their knowledge of, beliefs about, and attitudes toward mathematics, students, and teaching. Teachers' thirteen years as learners of K-12 mathematics provide them with images and models-conscious or unconscious-of what it means to teach and learn mathematics. They add to this many other learning experiences such as formal college preparation, clinical and field-based observation and practice teaching, the influence of school culture and colleagues within their immediate teaching environments, and in-service and advanced educational experiences. All these experiences convey messages about what constitutes appropriate teaching and learning. Such powerful influences need to be addressed when helping teachers learn to teach in new ways.

3. Learning to teach is a process of integration. Although the standards for the professional development of teachers of mathematics address various components of teacher knowledge and practice separately, the final success for any teacher is the integration of theory and practice. Ideally, teachers should engage simultaneously in studies of mathematics content and mathematics pedagogy. Teachers should be able to comment and reflect on their own learning environments at the same time they are involved in clinical and field-based teaching experiences. As different instructional strategies are implemented, teachers should be discussing the research that supports their choices of these strategies. Such integration is not easily achieved. However, it is a goal to strive for as reform of teacher education is pursued.

4. The education of teachers of mathematics is an ongoing process. Teachers are in a constant state of "becoming." Being a teacher implies a dynamic and continuous process of growth that spans a career. Teachers' growth requires commitment to professional development aimed at improving their teaching on the basis of increased experience, new knowledge, and awareness of educational reforms. Their growth is deeply embedded in their philosophies of learning, their attitudes and beliefs about learners and mathematics, and their willingness to make changes in how and what they teach. Their growth is also affected by numerous external agents including school administrators, educational policymakers, college and university faculty, parents, and the students themselves.

5. There are level-specific needs for- the education of teachers of mathematics. Although certain knowledge, skills, and abilities are common to the preservice and continuing education of all teachers of mathematics, level-specific needs-such as the learning needs of elementary, middle, and secondary school students-must be addressed. For example, in considering the development of teachers' knowledge of mathematics, different expectations about the breadth, depth, and scope of the content knowledge are needed for teaching at each grade level. These different needs must be addressed when considering teachers' knowledge of students and teaching.

ALL STUDENTS

Throughout these standards the phrase all students is used often. By this phrase we mean to set the mathematical education of every child as the goal for mathematics teaching at all levels, K-12. In April 1990, the NCTM Board of Directors endorsed the following statement:

As a professional organization and as individuals within that organization, the Board of Directors sees the comprehensive mathematics education of every child as its most compelling goal.

By "every child" we mean specifically-

• students who have been denied access in any way to educational opportunities as well as those who have not;
• students who are African American, Hispanic, American Indian, and other minorities as well as those who are considered to be a part of the majority;
• students who are female as well as those who are male; and
• students who have not been successful in school and in mathematics as well as those who have been successful.

It is essential that schools and communities accept the goal of mathematical education for every child. However, this does not mean that every child will have the same interests or capabilities in mathematics. It does mean that we will have to examine our fundamental expectations about what children can learn and do and that we will have to strive to create learning environments in which raised expectations for children can be met.

ORGANIZATION

In deciding how to present and elaborate the ideas underlying each of the six standards in this section, we faced two dilemmas. First, these standards seek to answer the question: What is the nature of the professional development experiences and opportunities needed in order for teachers to teach mathematics as described in the first section, "Standards for Teaching Mathematics"? Depending on one's view, this is a deceptively simple or an obviously complex question. In our efforts to be succinct yet comprehensive, we struggled with both the breadth and depth needed to provide a map for those interested in possible paths to answer this question. There are a variety of perspectives that must be considered in identifying what constitutes standards for the professional development of teachers of mathematics. We believe these standards help the professional community address many of these perspectives.

Our second dilemma evolved as we struggled to balance our message with that of the first section. The distinctions between the two sections are not always clear; they are intertwined in mutually supportive ways. Consequently, we chose not to address directly what is meant by tasks, discourse, environment, and analysis, although we make frequent reference to these ideas and integrate them throughout these standards. We urge readers of this section to carefully review the first section as part of their efforts to consider what constitutes appropriate opportunities for professional development.

The statement of each of the six standards is first elaborated with an explanation of its main ideas and occasionally highlighted by quotations from mathematicians, mathematics educators, teachers, and students. Then, for each standard, we follow with annotated vignettes that show and extend these ideas through a variety of contexts related to the preservice and continuing education of teachers of mathematics. Drawn from transcripts, observations, and experiences, the vignettes are selected to illustrate a range of professional development opportunities and issues. The commentaries in the outside column focus on issues pertinent to that standard and, in some instances, include additional detail that elaborates as well as annotates.

SUMMARY

The standards in this section focus on what we believe are essential components for the professional development of teachers of mathematics. They comprise the threads that are woven as the fabric of successful mathematics teaching: personal experiences in contexts that model and value good mathematics teaching; ongoing development of knowledge about mathematics, students, and teaching; numerous and diverse opportunities to apply knowledge and experience through practice; and the gradual assumption of responsibilities for professional growth and change. Ideally, the weave of the fabric will evolve and change, reflecting the numerous stages to be explored in the career-long development of mathematics teachers.