In March 1989, the National Council of Teachers of Mathematics (NCTM) released its Curriculum and Evaluation Standards for School Mathematics. These standards were the result of three years of planning, writing, and consensus-building among the membership of NCTM and the broader mathematics, science, engineering, and education communities, the business community, parents, and school administrators. The document describes what a high-quality mathematics education for North American students, K­12, should comprise. Central to the Curriculum and Evaluation Standards is the development of mathematical power for all students. Mathematical power includes the ability to explore, conjecture, and reason logically; to solve nonroutine problems; to communicate about and through mathematics; and to connect ideas within mathematics and between mathematics and other intellectual activity. Mathematical power also involves the development of personal self-confidence and a disposition to seek, evaluate, and use quantitative and spatial information in solving problems and in making decisions. Students' flexibility, perseverance, interest, curiosity, and inventiveness also affect the realization of mathematical power.

To reach the goal of developing mathematical power for all students requires the creation of a curriculum and an environment, in which teaching and learning are to occur, that are very different from much of current practice. The image of mathematics teaching needed includes elementary and secondary teachers who are more proficient in

• selecting mathematical tasks to engage students' interests and intellect;
• providing opportunities to deepen their understanding of the mathematics being studied and its applications;
• orchestrating classroom discourse in ways that promote the investigation and growth of mathematical ideas;
• using, and helping students use, technology and other tools to pursue mathematical investigations;
• seeking, and helping students seek, connections to previous and developing knowledge;
• guiding individual, small-group, and whole-class work.

This is a considerable change from the descriptions of mathematics classes drawn from the NSF case studies (Welch 1978, p. 6):

In all math classes that I visited, the sequence of activities was the same. First, answers were given for the previous day's assignment. The more difficult problems were worked on by the teacher or the students at the chalkboard. A brief explanation, sometimes none at all, was given of the new material, and the problems assigned for the next day. The remainder of the class was devoted to working on homework while the teacher moved around the room answering questions. The most noticeable thing about math classes was the repetition of this routine.

Even though these observations were made over ten years ago, there is little indication that the situation is different today. The routine described continues (NCTM,1989; National Research Council 1989; Weiss 1989).

There are many persistent obstacles to making significant changes in mathematics teaching and learning in schools. Among these are the beliefs and dispositions that both students and teachers bring to the mathematics classroom, as well as the assumptions held by school administrators, parents, and society in general about mathematics curriculum and instruction. In order to change our perspective about mathematics teaching and learning, we need direction on how mathematics can be taught and learned to enhance the development of mathematical power.

In early 1989, NCTM established a commission to produce a set of Professional Standards for Teaching Mathematics as a companion to the Curriculum and Evaluation Standards. The goal of this second set of standards was to provide guidance to those involved in changing mathematics teaching. Together, these two sets of standards are a part of NCTM's long-term commitment to provide direction for the reform of school mathematics. As teachers and administrators, school districts, states, provinces, certification boards, university faculty, and other groups propose solutions to curricular, teaching, and evaluation issues in mathematics education, these two sets of standards can be used as criteria against which their ideas can be compared and judged.

The Professional Standards for Teaching Mathematics rests on the following two assumptions:

• Teachers are key figures in changing the ways in which mathematics is taught and learned in schools.
• Such changes require that teachers have long-term support and adequate resources.

Educational research findings from cognitive psychology and mathematics education indicate that learning occurs as students actively assimilate new information and experiences and construct their own meanings (Case and Bereiter 1984; Cobb and Steffe 1983; Davis 1984; Hiebert 1986; Lampert 1986; Lesh and Landau 1983; Schoenfeld 1987). This is a major shift from learning mathematics as accumulating facts and procedures to learning mathematics as an integrated set of intellectual tools for making sense of mathematical situations (Resnick 1987). This view of learning is summarized in Everybody Counts (National Research Council 1989, pp. 58­59):

Effective teachers are those who can stimulate students to learn mathematics. Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding. To understand what they learn, they must enact for themselves verbs that permeate the mathematics curriculum: "examine," "represent," "transform," "solve," "apply," "prove," "communicate." This happens most readily when students work in groups, engage in discussion, make presentations, and in other ways take charge of their own learning.

All students engage in a great deal of invention as they learn mathematics; they impose their own interpretation on what is presented to create a theory that makes sense to them. Students do not learn simply a subset of what they have been shown. Instead, they use new information to modify their prior beliefs. As a consequence, each student's knowledge of mathematics is uniquely personal.

The kind of teaching envisioned in these standards is significantly different from what many teachers themselves have experienced as students in mathematics classes. Because teachers need time to learn and develop this kind of teaching practice, appropriate and ongoing professional development is crucial. Good instructional and assessment materials and the latitude to use them flexibly are also keys to the process of change.

For teachers to be able to change their role and the nature of their classroom environment, administrators, supervisors, and parents must expect, encourage, support, and reward the kind of teaching described in this set of standards. We cannot expect teachers to respond simultaneously to several different calls for change or other new demands. Change is difficult and will take time and reliable, systematic support.

Woven into the fabric of the Professional Standards for Teaching Mathematics are five major shifts in the environment of mathematics classrooms that are needed to move from current practice to mathematics teaching for the empowerment of students. We need to shift

• toward classrooms as mathematical communities - away from classrooms as simply a collection of individuals;
• toward logic and mathematical evidence as verification - away from the teacher as the sole authority for right answers;
• toward mathematical reasoning - away from merely memorizing procedures;
• toward conjecturing, inventing, and problem solving - away from an emphasis on mechanistic answer-finding;
• toward connecting mathematics, its ideas, and its applications - away from treating mathematics as a body of isolated concepts and procedures.

As teachers shift toward the vision of teaching presented by these standards, one would expect to see teachers asking, and stimulating students to ask, questions like the following:

* Helping students work together to make sense of mathematics

"What do others think about what Janine said?"
"Do you agree? Disagree?"
"Does anyone have the same answer but a different way to explain it?"
"Would you ask the rest of the class that question?"
"Do you understand what they are saying?"
"Can you convince the rest of us that that makes sense?"

"Why do you think that?"
"Why is that true?"
"How did you reach that conclusion?"
"Does that make sense?"
"Can you make a model to show that?"

"Does that always work?"
"Is that true for all cases?"
"Can you think of a counterexample?"
"How could you prove that?"
"What assumptions are you making?"

"What would happen if . . .? What if not?"
"Do you see a pattern?"
"What are some possibilities here?"
"Can you predict the next one? What about the last one?"
"How did you think about the problem?"
"What decision do you think he should make?"
"What is alike and what is different about your method of solution and hers?"

"How does this relate to ?"
"What ideas that we have learned before were useful in solving this problem?"
"Have we ever solved a problem like this one before?"
"What uses of mathematics did you find in the newspaper last night?"
"Can you give me an example of?"

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, K­12. 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 can do and that we will have to strive to create learning environments in which raised expectations for children can be met.

The kind of instruction needed to implement the NCTM Standards requires a high degree of individual responsibility, authority, and autonomy - in short, professionalism on the part of each teacher. To give guidance to the development of such professionalism in mathematics teaching, the Professional Standards for Teaching Mathematics consists of five components:

1. Standards for teaching mathematics
2. Standards for the evaluation of the teaching of mathematics
3. Standards for the professional development of teachers of mathematics
4. Standards for the support and development of mathematics teachers and teaching
5. Next steps

Standards for Teaching Mathematics. This section develops a vision of what a teacher at any level of schooling must know and be able to do to teach mathematics as envisioned by the NCTM Curriculum and Evaluation Standards for School Mathematics and the Professional Standards for Teaching Mathematics. The standards in this section are organized around a framework emphasizing the important decisions that a teacher makes in teaching

• Setting goals and selecting or creating mathematical tasks to help students achieve these goals;
• Stimulating and managing classroom discourse so that both the students and the teacher are clearer about what is being learned;
• Creating a classroom environment to support teaching and learning mathematics;
• Analyzing student learning, the mathematical tasks, and the environment in order to make ongoing instructional decisions.

The statements of standards focus on the major aspects of good mathematics teaching across all grade levels. Guides for teaching at different levels are suggested in the elaborations of each standard and in the annotated vignettes that are used to provide examples. These vignettes show a range of situations in which good mathematics teaching and learning can take place. A high-quality mathematics experience is not determined simply by the presence of computers or calculators or the use of small groups, manipulatives, or student discussions. The nature of the mathematical task posed and what is expected of students are critical aspects against which to judge the effectiveness of the lesson. Although each vignette could illustrate many of the standards, the guiding comments focus on the particular standard being elaborated.

Standards for the Evaluation of Teaching of Mathematics. This section presents NCTM's vision for the evaluation of mathematics teaching. The assumption is that the major purpose for such evaluation is the improvement of teaching. The standards include the following aspects of the evaluation of teaching:

• The process of evaluation
• The foci of evaluation

In March 1987, the NCTM Board of Directors approved a position statement on the Evaluation of Teacher Performance, which includes the following:

Evaluation includes the identification of goals by the teacher and the evaluators, the collection of information, and a collaborative dialogue between the teacher and evaluators to reformulate, redirect, and refine goals for the future. Goals for personal and professional growth may include some that are mandated by the state or province, district, teacher education institution, or individual school, but the teacher must be an active participant in identifying goals of a more specific nature.

The use that is made of the information gained through the evaluation process is as important as the act of evaluation itself. The appropriate outcome of this ongoing process is a collaborative dialogue between the teacher and others involved in the process, resulting in a mutually agreed-on plan for professional growth.

The standards in this section are consistent with this position statement on evaluation and support the assumption that the evaluation of teaching should result in the professional growth of teachers. These standards give guidance to teachers seeking self-improvement, to colleagues mentoring others, and to supervisors and others who are involved in the evaluation of teaching.

The vignettes in this section show a range of assessment activities and personnel involved in evaluation. They illustrate the substance and process of assessing teaching and of the results of evaluation and are not meant to be exhaustive of the possibilities.

Standards for the Professional Development of Teachers of Mathematics. This section expresses NCTM's vision for well-prepared teachers of mathematics from the time prospective teachers of mathematics take their first courses in collegiate mathematics throughout their career-long development. These standards focus on what a teacher needs to know about mathematics, mathematics education, and pedagogy to be able to carry out the vision of teaching discussed in the first component of this document. The following aspects of both the preservice and in-service phases of the professional development of teachers are addressed:

• Modeling good mathematics teaching
• Knowing mathematics and school mathematics
• Knowing students as learners of mathematics
• Knowing mathematical pedagogy
• Developing as a teacher of mathematics
• Teachers' roles in professional development

These teaching standards provide essential guidance to colleges, universities, and schools; state departments and provincial ministries of education; public and private schools; and all who are a part of the preparation and professional development of teachers. These standards focus attention on the roles of faculty in college and university departments of education and mathematics and school officials responsible for professional development. They also emphasize the need for dialogue among these partners in nurturing excellence in mathematics teaching.

The current reform movement in mathematics education, and in education in general, has as a strong underlying theme the professionalism of teaching. This view recognizes the teacher as a part of a learning community that continually fosters growth in knowledge, stature, and responsibility. The standards in this section provide a guide to the preparation, support, and career development of teachers. The standards themselves are meant to be general principles that can be used to improve the quality of teacher preparation programs as well as school and university professional-development activities. Applications of these general principles to levels of preparation and phases of career development are illustrated in the elaborations and the vignettes.

Standards for the Support and Development of Teachers and Teaching. The standards in this section spell out the responsibilities of those who make decisions that affect teaching mathematics. The responsibilities of the following groups are addressed:

• Policy makers in government, business, and industry
• Schools and school systems
• Colleges and universities
• Professional organizations

Decisions made by others can enable teachers to move toward the vision of teaching described in these standards or can constrain the mathematics program in ways that cripple efforts to improve teaching. The environment in which teachers teach is as important to their success as the environment in which students learn is to theirs. These standards highlight responsibilities and ways in which others can support teachers in shifting toward the vision of teaching needed to support the implementation of the Curriculum and Evaluation Standards for School Mathematics.

Next Steps. The final section of this document discusses some of the issues and next steps that we can take to move toward our goal of mathematical power for all students.

These teaching standards are not intended to be an exhaustive checklist of specific concepts, skills, and behaviors that teachers must have. Instead, these standards are a set of principles accompanied by illustrations or indicators that can be used to judge what is valuable and appropriate. They give direction for moving toward excellence in teaching mathematics. They furnish guidance to all who are interested in improving teaching, including teachers, universities, state departments of education and provincial ministries of education, local school districts, private schools, teacher organizations, the National Council for Accreditation of Teacher Education, the National Board on Professional Teaching Standards, and others who license or certify teachers or who evaluate teaching or teacher education programs.