OVERVIEW

This section presents six standards for the teaching of mathematics organized under four categories.

Tasks

1. Worthwhile Mathematical Tasks

Discourse

2. Teacher's Role in Discourse
3. Students' Role in Discourse
4. Tools for Enhancing Discourse

Environment

5. Learning Environment

Analysis

6. Analysis of Teaching and Learning

INTRODUCTION

The Curriculum and Evaluation Standards for School Mathematics represents NCTM's vision of that students should learn in mathematics classrooms. Congruent with the aims and rhetoric of the current reform movement in mathematics education (e.g., National Research Council 1989, 1990), the Standards is threaded with a commitment to developing the mathematical literacy and power of all students. Being mathematically literate includes having an appreciation of the value and beauty of mathematics as well as being able and inclined to appraise and use quantitative information. Mathematical power encompasses the ability to "explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods effectively to solve nonroutine problems" and the self-confidence and disposition to do so (National Council of Teachers of Mathematics 1989, p. 5). It also includes being able to formulate and solve problems, to judge the role of mathematical reasoning in a real-life situation, and to communicate mathematically.

The vision of the Curriculum and Evaluation Standards is that mathematical reasoning, problem solving, communication, and connections must be central. Computational algorithms, the manipulation of expressions, and paper-and-pencil drill must no longer dominate school mathematics. Beyond the standard fare of number concepts and operations, the school curriculum must include serious exploration of geometry, measurement, statistics, probability, algebra, and functions. Whether working individually or in small or large groups, students should encounter, develop, and use mathematical ideas and skills in the context of genuine problems and situations. In so doing, they should develop the ability to use a variety of resources and tools, such as calculators and computers and concrete, pictorial, and metaphorical models. They must know be able to choose appropriate methods of computation, including estimation, mental calculation, and the use of technology. As they explore and solve problems, they must engage in conjecture and argument.

In setting these goals for school mathematics, the Curriculum and Evaluation Standards implies a significant departure from the traditional practices of mathematics teaching. It suggests changes in not only what is taught but also how it is taught. Teachers and students have different roles in such classrooms and different notions about what it means to know and to do mathematics. The purpose of this section is to make explicit and expand the images of teaching and learning implicit in the Curriculum and Evaluation Standards for School Mathematics, to elaborate a vision of instruction that can light the path toward such change.

Six standards encompass the vision's core dimensions. These standards are grouped under four headings: tasks, discourse, environment, and analysis-major arenas of teachers' work that are logically central shaping what goes on in mathematics classes.

• Tasks are the projects, questions, problems, constructions, applications, and exercises in which students engage. They provide the intellectual contexts for students' mathematical development.

• Discourse refers to the ways of representing, thinking, talking, and agreeing and disagreeing that teachers and students use to engage in those tasks. The discourse embeds fundamental values about knowledge and authority. Its nature is reflected in what makes an answer right and what counts as legitimate mathematical activity, argue and thinking. Teachers, through the ways in which they orchestrate discourse, convey messages about whose knowledge and ways of thinking and knowing are valued, who is considered able to contribute and who has status in the group.

• Environment represents the setting for learning. It is the unique interplay of intellectual, social, and physical characteristics that shapes the ways of knowing and working that are encouraged and expected the classroom. It is the context in which the tasks and discourse are embedded; it also refers to the use of materials and space.

• Analysis is the systematic reflection in which teachers engage. It entails the ongoing monitoring of classroom life-how well the tasks, discourse, and environment foster the development of every studentâs mathematical literacy and power. Through this process, teachers examine relationships between what they and their students are doing and what students are learning.

In deciding how to present and elaborate the ideas underlying each of the six standards, we confronted two basic dilemmas. First, teaching is an integrated activity. Although we can analyze the practice of teaching into these four arenas of teachers' work- tasks, discourse, environment, and analysis- they are in fact interwoven and interdependent. The quality of the classroom environment, for example, is both a function of and an influence on the classroom discourse. Alternatively, tasks are shaped by the discourse that surrounds them and the environment in which that work takes place. Our second dilemma was that professional standards for mathematics teaching should represent values about what contributes to good practice without prescribing it. Such standards should offer a vision, not a recipe.

The format of this section grew out of consideration of these issues. Because teaching is an integrated activity and because we wanted to provide concrete images of a vision, we have chosen to use illustrative annotated vignettes of classroom teaching and learning. The statement of each of the six standards is first elaborated with an explanation of its main ideas. Each explanation is then followed with illuminating cases that show these ideas embedded in actual teaching contexts. Drawn from transcripts, observations, and experiences in a wide variety of real classrooms, the vignettes were selected to illustrate a range of teaching styles, classroom contexts, mathematical topics, and grade levels. These vignettes were gathered from actual classrooms in a wide variety of settings, with students of diverse cultural, linguistic, and socioeconomic backgrounds. We included examples of teachers facing problems as well as cases of accomplished practice. The italicized commentaries focus on issues pertinent to that standard only, although many other features of the vision of teaching are apparent in the descriptions. For example, the vignettes in the Environment section are annotated from the perspective of the learning environment only.

ASSUMPTIONS

The standards for teaching are based on four assumptions about the practice of mathematics teaching:

1. The goal of teaching mathematics is to help all students develop mathematical power. The Curriculum and Evaluation Standards for School Mathematics furnishes the basis for a curriculum in which mathematical reasoning, communication, problem solving, and connections are central. Teachers must help every student develop conceptual and procedural understandings of number, operations, geometry, measurement, statistics, probability, functions, and algebra and the connections among ideas. They must engage all students in formulating and solving a wide variety of problems, making conjectures and constructing arguments, validating solutions, and evaluating the reasonableness of mathematical claims. Along with all this, teachers must foster in students the disposition to use and engage in mathematics, an appreciation of its beauty and utility, and a tolerance for getting stuck or sidetracked. Teachers must help students realize that mathematical thinking involves dead ends and detours and encourage them to persevere when confronted with a puzzling problem and to develop the self-confidence and interest to do so.

2. WHAT students learn is fundamentally connected with How they learn it. Students' opportunities to learn mathematics are a function of the setting and the kinds of tasks and discourse in which they participate. What students learn-about particular concepts and procedures as well as about thinking mathematically-depends on the ways in which they engage in mathematical activity in their classrooms. Their dispositions toward mathematics are also shaped by such experiences. Consequently, the goal of developing students' mathematical power requires careful attention to pedagogy as well as to curriculum.

3. All students can learn to think mathematically. The goals described in the Curriculum and Evaluation Standards for School Mathematics are goals for all students. Goals such as learning to make conjectures, to argue about mathematics using mathematical evidence, to formulate and solve problems-even perplexing ones-and to make sense of mathematical ideas are not just for some group thought to be "bright" or "mathematically able." Every student can-and should-learn to reason and solve problems, to make connections across a rich web of topics and experiences, and to communicate mathematical ideas. By "every student" 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 part the majority;

• students who are female as well as those who are male;

• students who have not been successful as well as those who have been successful in school and in mathematics.

This assumption is supported by the vignettes, which were drawn from classrooms with students of diverse cultural, linguistic, and socioeconomic backgrounds.

4. Teaching is a complex practice and hence not reducible to recipes or prescriptions. First of all, teaching mathematics draws on knowledge from several domains: knowledge of mathematics, of diverse learners, of how students learn mathematics, of the contexts of classroom, school and society. Such knowledge is general. However, teachers must also consider the particular, for teaching is context-specific. Theoretical knowledge about adolescent development, for instance, can only partly inform a decision about particular students learning a particular mathematical concept in a given context. Second, as teachers weave together knowledge from these different domains to decide how to respond to a student's question, how to represent a particular mathematical idea, how long to pursue the discussion of a problem, or what task to use to engage students in a new topic, they often find themselves having to balance multiple goals and considerations. Making such decisions depends on a variety of factors that cannot be determined in the abstract or governed by rules of thumb.

Because teaching mathematics well is a complex endeavor, it cannot be reduced to a recipe for helping students learn. Instead, good teaching depends on a host of considerations and understandings. Good teaching demands that teachers reason about pedagogy in professionally defensible ways within the particular contexts of their own work. The standards for teaching mathematics are designed to help guide the processes of such reasoning, highlighting issues that are crucial in creating the kind of teaching practice that supports the learning goals of the Curriculum and Evaluation Standards for- School Mathematics. This section circumscribes themes and values but does not-indeed, it could not-prescribe "right" practice.