NBPTS+Mathematics


 * National Board for Professional Teaching Standards: Mathematics **

Today’s students will live and work in tomorrow’s world. Teaching them to use mathematics in new and meaningful ways is an exciting and challenging task. Today’s teachers are moving away from tedious drills on narrow skills; away from the notion that mathematics comes packaged in neat, self-sustaining compartments; and away from the attitude that only the privileged can understand and do mathematics. Students across the educational spectrum today find their analytic ability to analyze and frame situations they encounter every day in a mathematical context to be at a premium. These students __need an understanding of fundamental mathematical concepts and techniques__ in order to achieve a full command of state-of-the-art technologies to help them solve problems that come up in daily life.

Against this backdrop, the teacher of mathematics at the adolescence and young adulthood level plays a key role—a role that is evolving as more is learned about students and mathematics. The mathematics teacher leads students in an exploration of the world of mathematics, provides learning opportunities for all students, and serves as an advocate for each student in the mathematics classroom. The mathematics teacher opens doors to a wide range of learning possibilities and future options.

In recent years, mathematics has undergone many changes both as a discipline and as a tool for educational, social, and economic opportunity. The development of modern technology has profoundly affected the ways organizations and individuals around the world achieve their objectives. Such technological advances have substantially broadened the applications of mathematics, not only in scholarship, science, industry, business, and government, but also more prominently than before in such areas as agriculture, health, politics, and the arts.

The changing nature of the workforce in the United States, coupled with stiff international competition, requires that all citizens—not just a select few—become mathematically literate. All young adult Americans must master mathematical ways of thinking if they are to succeed and if they are to make informed decisions about issues central to their lives and to society. The mathematical principles and concepts that today’s young adults need to know to participate fully in the modern world are both different from and more complex than those required of earlier generations.

These transforming events and new realities are highlighted in several national reports that provide recommendations for the reform of school mathematics and its teaching.1 Each report asserts that high priority must be given to the development of students who will mature into __adults with strong mathematical knowledge and with skills and confidence in their abilities to do significant mathematics.__

The recent and continuing changes in mathematics, and in the field’s current vision of mathematics instruction, demand the most from the nation’s teachers. This conception of mathematics teaching is predicated on the belief that all students—regardless of background, gender, race, ethnicity, special needs, or language—must be afforded the opportunity to achieve the level of mathematical competence required to succeed in life and in work. Giving students access to the full range of school mathematics, such as that outlined by the National Council of Teachers of Mathematics (NCTM) in //Principles and Standards for School Mathematics,//2 will require a great deal more, especially more knowledge, from tomorrow’s teachers than is typically required from teachers today.

It is important to emphasize that these standards recognize, reflect, and imply varied approaches to teaching. The National Board understands that such factors as the contexts in which teachers practice, the backgrounds and experiences they bring to their role, and the professional choices they make during their careers contribute to a diverse population of accomplished secondary mathematics teachers. For example, a teacher who has spent the past 10 years teaching algebra, and who has developed special expertise in alternative methods of assessment, offers a distinctly different profile of accomplished teaching from that of a teacher who has taught the full spectrum of secondary mathematics courses, including calculus, and who has a special interest in cooperative learning. Both, however, may well be teaching at the highest levels described in these standards.

At the same time, however, __certain guiding principles relevant to mathematics content and pedagogical knowledge are widely agreed on among mathematics educators__. These guiding principles represent the common ground that unites accomplished teachers and distinguishes their practice without regard to their current assignment, school context, or past experiences. Included in these principles are not only understandings about the essentials of first-rate practice, but also understandings about how to recognize and avoid damaging and ineffective practices.

At present, many accomplished teachers are meeting these challenges. Those entering the teaching profession continue to join them in working to achieve the vision set by the NCTM standards. The Carnegie Task Force on Teaching as a Profession3 had a similar vision when it advanced the idea of publicly certifying the work of exceptional teachers, teachers who are making a difference in the lives and abilities of the children they teach. The task force’s report stimulated the formation of the National Board for Professional Teaching Standards, which ultimately led to the development of the standards in this document—standards intended to recognize teachers of adolescents and young adults who ensure their students’ progress toward high achievement in mathematics.

Accomplished teachers offer all students the opportunity to learn. __Only by having a deep and broad understanding of mathematics can teachers organize and deliver instruction that helps students build their own broad and deep understanding of mathematics__. Only by knowing their students well can teachers consistently make instructional decisions that will further students’ learning. Further, only by skillfully combining their knowledge of students and mathematics with their knowledge about how to teach mathematics can teachers enable students to learn mathematics successfully. The following three standards form the foundation for the decisions and actions taken by accomplished mathematics teachers. They are the basis for the seven remaining standards.
 * Knowledge of Mathematics, Students, and Teaching **

__ Accomplished mathematics teachers have a broad and deep knowledge of the concepts, principles, techniques, and reasoning methods of mathematics, and they use this knowledge to set curricular goals and shape their instruction and assessment. They understand significant connections among mathematical ideas and the applications of these ideas to problem solving in mathematics, in other disciplines, and in the world outside of school. __
 * // Standard III: //****// Knowledge of Mathematics //**

__Mathematics is a discipline that develops intellectual power to solve problems; make decisions; and describe significant visual, quantitative, and symbolic patterns__. It is a fundamental tool in the persistent human effort to make sense of the world—its order, chaos, stability, and change. It has applications, for example, in scientific, technical, economic, and political work; in art and music; and in our personal lives and social interactions. Although it is one of the oldest disciplines of human knowledge and thought, the field of mathematics continues to grow and evolve. New concepts, principles, and methods become a part of the discipline each year. For example, the emphasis on computational efficiency, the discovery of new algorithms, the advent of linear programming, and the articulation of the number theory underlying cryptography all have been developed during the lifetime of many of today’s teachers.

Mathematics is commonly discussed in terms of computational procedures and proofs in the school subjects of algebra, geometry, and calculus. However, mathematics is a far more complex organism. It embraces input from theoretical physics, computer science, economics, and other applied fields. It strives for generality and, even in applications, makes effective use of abstractions as a source of power. The emergence of sophisticated technology also makes possible the approximate solutions of formerly inaccessible problems by virtue of computer simulations. Technology also contributes to explorations that uncover new phenomena. A full understanding of these new advances then stimulates even more abstract theoretical work.

__To make classroom decisions that support student learning, teachers must understand both the mathematics and their students, and they must continue to grow in their understanding__. (See Standard I—//Commitment to Students and Their Learning// for a definition of mathematical understanding.) To help students acquire and then build on the ideas, methods, and skills that underlie important mathematics and see relationships among these elements and make significant applications of them, accomplished teachers must have a broad and well-integrated knowledge of both classical and contemporary ideas, as well as the methods and techniques of mathematics. They must appreciate the richly interconnected nature of the discipline.

__Accomplished teachers also view the discipline from several perspectives. They know the central concepts, principles, facts, and techniques of important mathematical domains— algebra, geometry, discrete mathematics, data analysis and statistics, and calculus. They know the fundamental processes of mathematical thinking, including representation, modeling, conjecture, inference, interpretation, and analysis. They understand the fundamental role of proof in establishing and explaining the truth of mathematical statements and in providing a standard of logical connection that sets mathematics apart from other disciplines.__

They know the productive connections between mathematics and other fields of human endeavor—connections that have given mathematics a remarkable history of intellectual service to problem solving and decision making across time and cultures. They also understand the ways mathematics is contributing to the technological changes in society and the ways technology is changing the face of mathematics.

Increasingly, in today’s classrooms, students engage in mathematical investigations— from simple explorations to open-ended problems aimed at forming conjectures to intensive or long-term researchlike activities. In this environment, teachers should anticipate or respond to the questions and ideas students may develop and to the disruptions or unexpected connections they might encounter. They should acknowledge how long a worthwhile journey might take. Good classroom judgment—knowing when or whether to intervene, deciding which of many possible avenues will be most fruitful to follow, and knowing when to bring closure to an activity and move on—depends not only on teachers’ sensitivity to students’ learning, but also on profound mathematical knowledge that allows them to recognize and pursue mathematically significant ideas.

An increased emphasis on applications also puts new demands on teachers’ mathematical knowledge. Sometimes responding to these demands means entering unfamiliar mathematical territory. But even when the mathematics itself is not new to the teacher, a shift of curricular organization from mathematical topics to contexts or applications increases the need for teachers to distinguish between those mathematical ideas that are new and important for students and those that are relatively routine.

One of the strongest forces in the contemporary growth and evolution of mathematics— and of mathematics teaching—is the power of modern computational technology. Entirely new mathematical fields are emerging. For example, understanding recursive structures and being able to analyze algorithms become more important for students to know, placing new demands on what teachers must know. Also, some problems and topics are becoming more accessible to students, along with new ways to represent and manipulate mathematical information. This gives teachers new choices about both content and pedagogy. To make wise choices about how to use calculators, computers, and appropriate software in the service of students’ mathematical learning, teachers need not only the mathematical knowledge that guides their pursuit of mathematical goals, but also knowledge of the potential of the technology—including the emerging software relevant to their discipline—and fluent expertise with its use.

__ Accomplished teachers understand the major ideas in the core domains of mathematics. Although their expertise may vary in degree for particular domains, they have a fundamental knowledge base from which to build student mathematical understanding. This knowledge includes algebra, geometry, discrete mathematics, data analysis and statistics, and calculus. __
 * Core Mathematical Knowledge **

Accomplished teachers recognize algebra as a language for describing abstract mathematical structures and for generalizing and extending ideas from arithmetic. They know how to apply these aspects of algebraic content and thinking to classroom teaching in appropriate ways throughout the mathematics curriculum. Substantive mathematical reasoning applies to such algebraic ideas as identities, inverses, closure, linearity, and the distributive law. These ideas and the systems that use them (e.g., groups, rings, and fields) are foundations for understanding the mathematics of everyday classrooms.
 * // Algebra //**

Accomplished teachers also recognize algebra’s role as a language for modeling problem situations and for reasoning and drawing inferences about functions and relations. They understand the interplay among numerical, symbolic, verbal, and graphical representations of quantitative relationships and the role and means of transforming and simplifying these representations. They are proficient in using concepts and symbolic expressions for working with families of functions, such as polynomial, exponential, rational, logarithmic, trigonometric, and those that depend on parameters or are recursively defined. They connect these functions to important applications through their special properties. Many ideas that are central to secondary mathematics have their roots in a more abstract algebra that is not always evident in the standard curriculum. To maintain a coherent view of the standard material, teachers need to know its underpinnings. For example, the fact that the decimal expansions of rational numbers either terminate or repeat can be seen from an examination of the division algorithm, and the never-divide-by-zero rule derives from the algebraic equivalence of two statements, //a/b// = //c// and //a// = //bc//.

Accomplished teachers understand that geometry provides a repertoire of techniques for studying spatial objects and patterns as well as a setting for the representation of seemingly purely analytic or algebraic ideas. Just as accomplished teachers use algebraic methods to support reasoning about geometric situations, they use visual models and methods (concrete and virtual, in two and three dimensions) or ideas from non- Euclidean geometry to provide insight into patterns expressed in numbers or other symbols. Using their knowledge of the properties of Euclidean spaces, teachers solve problems in a variety of fields, such as art, architecture, and engineering.
 * // Geometry //**

They use approaches that involve measurement (of distance, area, volume, and angle measure, for example ideas of similarity (proportion and trigonometric ratios, for example), vectors, and coordinates. They also use drawing techniques, including paper/ pencil and software, to investigate and analyze properties of geometric objects and to study which properties change and which remain invariant as the objects are transformed in various ways. Accomplished teachers should also be familiar with other areas of mathematics opened up by technology, such as self-similarity in fractal geometry.

Accomplished teachers are proficient in constructing mathematical proofs and in explaining proofs to students. Teachers are able to use the axiomatic structure of geometry to construct proofs in varied forms (e.g., paragraph, indirect, two-colum),

Accomplished teachers’ understanding extends beyond the mathematics of continuous phenomena. In the context of this document, discrete mathematics includes such topics as algorithms; sequences, formal series, finite differences, binomial theorem; recursion and iteration; proof by induction; graphs and networks; and counting techniques (e.g., combinatorics, arrangements, partitions). It also includes elements of number theory (e.g., modular systems, divisibility and counting of divisors, integer solutions to problems) and fundamental notions in set theory and logic. Accomplished teachers also understand the foundations and concepts of finite probability (e.g., mutually exclusive, independent, and compound events; conditional probability; expected value).
 * // Discrete Mathematics //**

Accomplished teachers also can use representations appropriate to these domains, including matrices, Venn diagrams, and tree diagrams. They are able to use the ideas and methods of discrete mathematics in such contexts as social choice (e.g., voting, apportionment), finance, population dynamics, optimization (e.g., choosing best paths), and computer science as well as in the analysis of problems within mathematics itself.

Accomplished teachers use both quantitative and qualitative approaches when answering questions involving data. To do so, they collect, organize, represent, and reason about data, using a variety of numerical, graphical, and algebraic concepts and procedures, and they look for ways to describe and model patterns in data. They know how to interpret and draw inferences from data to make decisions in a wide range of applied problem settings as well as how to use simulations to investigate situations.
 * // Data Analysis and Statistics //**

Teachers understand the connection between simulations and experimental probability. They also see how probability contributes to an understanding of sample spaces, distributions, and the foundations of inferential statistics.

Accomplished teachers understand the mathematical underpinnings of such basic inferential techniques as confidence intervals and hypothesis testing. In addition, they are aware of the advantages, limitations, and appropriateness of each technique. They understand that statistical inference goes beyond describing data and involves using formal methods to support or refute generalizations about populations based on samples, using the methods and language of probability, and using statistical reasoning to make or modify decisions on the basis of data. Teachers understand that a goal of data interpretation is to help students become more informed consumers of information.

Accomplished teachers appreciate the historical development and significance of calculus and know that it provides methods for modeling dynamic change in such areas as the physical, biological, and social sciences, as well as in business applications. They are knowledgeable about the theoretical foundations of calculus, including the rigorous development of calculus concepts such as limits, continuity, differentiation, and integration.
 * // Calculus //**

They understand and can explain how to use limits, derivatives, integrals, and infinite series as tools to measure and analyze rates of change, optimization, and accumulation of continuously varying quantities. They apply the ideas and techniques of calculus, using numerical methods to analyze data-based problems. They understand how to use technology appropriately to assist in visualizing and solving problems.

Mathematics is often described by naming important concepts, facts, and operations in its major topic strands. However, characteristic mathematical thinking processes are used to solve problems in all the topical strands. Accomplished teachers of mathematics understand and are able to demonstrate such mathematical processes as these:
 * Mathematical Thinking Processes **

• Discovering, describing, and reasoning about patterns represented in visual, numerical, and symbolic form— including such processes as classification, representation, and deductive and inductive reasoning and such concepts as symmetry, similarity, stability, recursion, and continuity • Using methods for formal verification of mathematical conjectures—including rules of logical inference and proof strategies • Modeling mathematical relations in problem situations by using symbolic expressions—representing important relationships, operating on symbolic expressions to gain understanding of the situation or to draw inferences about it, and applying mathematical analysis to solve problems and make decisions • Using heuristics to solve mathematical problems—such as testing extreme cases, conducting an organized search of specific examples, and using visual problem representations • Using technology to search for patterns and formulate generalizations • Applying strategies for communicating mathematical information in verbal, numerical, graphic, and symbolic forms and through physical models of mathematical principles

Certain fundamental thinking processes and mathematical structures apply across all topic strands, giving coherence to the subject and powerful support for teaching, learning, and applied problem solving. For example, matrices are invaluable tools for recording and reasoning about complex data sets in algebra (systems of equations), geometry (transformations), graph theory (edge connections), and probability (Markov chains). The concept of limit—central to and made explicit in calculus—appears in the curriculum prior to calculus in such diverse areas as geometry and functions. The linear algebra concepts of vector space and linear transformations provide representations of concepts, problems, and techniques in both algebra and geometry. The algebraic structures of groups, rings, and fields illuminate the common and fundamental properties of operations in number systems, families of functions, and geometric patterns like symmetry. Teachers must be familiar with such core patterns in the discipline if they are to make the power of these structural connections available to students.

In one sense, mathematics is among the most abstract of disciplines. For centuries, it has been cultivated for its intrinsic beauty and merit as an intellectual discipline. But its most interesting abstract concepts, structures, and operations have arisen from or found embodiment in patterns of objects and actions in scientific, technical, economic, or other practical situations. Accomplished teachers understand the roots of abstract concepts and techniques in concrete cases, and __they use this understanding to make wise curricular and instructional decisions and to help students make connections across disciplines__.
 * Contexts for Mathematics **

They appreciate the historical course through which mathematical ideas have developed and the ways different cultures have influenced and contributed to that development.

__An accomplished teacher’s knowledge of the context within which mathematics has evolved__ and is useful includes the following: • Knowledge of the major threads in the historical development of key mathematical ideas—the conceptual stumbling blocks and insights that provided important breakthroughs—and the contributions of various individuals and cultures to those developments. • Knowledge of the ways mathematical ideas have been and are today fundamental to practical and scientific progress in fields related to mathematics. This includes applications for the major concepts, principles, and techniques of core content topics in the school curriculum as well as the modeling processes that are fundamental to effective applications of mathematics. Such applications provide a basis for thinking about and using mathematics. Effective use of technology is an essential part of this modeling and application process. • Knowledge of a set of analytical and representational techniques, the ability to recognize when the techniques are appropriate, and the ability to apply them in real situations.

Knowledge of mathematics by itself does not guarantee that an individual will become an accomplished teacher of mathematics. However, it __provides an essential foundation that supports all other teaching standards and meets the content-based challenges that will occur in classrooms. Accomplished mathematics teachers’ knowledge of mathematics is constantly growing in a way that encourages the integration of new facts, concepts, procedures, technologies, and applications into their teaching repertoire__. They employ this solid knowledge base to design instruction that reflects the diverse historical and cultural roots of mathematics. They convey to their students the spirit of mathematics as a human endeavor that has evolved from the contributions, values, and social perspectives of a wide variety of people across thousands of years.
 * Applying Mathematical Knowledge in Teaching **

Their love and enthusiasm for mathematics permeates the classroom environment. They convey the power and fascination of the discipline to their students as they engage them in exploring and discovering the intriguing patterns and processes of mathematics and in applying mathematical ideas in realistic settings. They place a high value on doing mathematics, take joy in it, and communicate that joy and excitement to their students.

Accomplished mathematics teachers design their lessons with important mathematical goals in mind. They articulate these goals clearly, and they select instructional techniques and activities that enable students to meet them. They know their students, and they know the ideas and procedures of mathematics. They also know that mathematical thinking evolves in the minds of individual students based on their experiences.
 * // Standard V: Knowledge of nTeaching Practice //**

They know the problems and difficulties that students commonly encounter when studying various mathematical topics. Accomplished teachers apply this knowledge to make judgments about content choice, sequence, emphasis, and instruction. Their repertoire of teaching strategies— including inquiry, cooperative learning, discovery, directed instruction, individualized instruction, and group instruction— engages students in exploring, discovering, and using mathematical ideas. The tasks and activities they select are structured deliberately to facilitate student understanding, communication, and reasoning.

Accomplished teachers understand that not all instructional formats and materials are appropriate for—nor will they appeal to—all students, and they structure their teaching in a variety of ways to address this diversity. Because they understand different types of representational models and the strengths and weaknesses of each, they can select those best suited for different students and for different teaching situations. Thus, they provide students with a variety of tools to solve problems, and they know how to use those tools appropriately. They help students learn about learning mathematics.

Accomplished teachers have developed a framework of mathematical goals and instructional strategies that allows them to identify, assess, adapt, and create instructional resources to support and enhance student learning. They use a variety of activities and materials to reach their mathematical goals, and they draw on these resources to meet the mathematical needs of their students. These resources include manipulative tools, printed materials, human resources, historical material, appropriate technology, and library and media resources.

The visual, computational, and interactive power of modern technology can be used to influence both what is taught and how students learn mathematics. Accomplished teachers recognize the opportunities afforded by the new tools—access to new ideas and new ways of representing and manipulating them—and effectively use the tools to deepen and enrich students’ mathematical learning. They continually improve their own skills and fluency with these tools and reexamine their teaching practice in light of what the tools make possible.

Accomplished teachers attend to the primary goal—helping students develop sound mathematical knowledge, understanding, and ways of thinking. These teachers make necessary changes in accord with their best judgment about students’ needs. They give students opportunities to develop expertise with the technology that will best help them learn good mathematics and solve problems mathematically. Where access to such technology is limited or nonexistent, accomplished teachers seek ways to acquire the appropriate instructional and computational tools. (See Standard XI—//Families and Communities//.) They may work actively within the school community to advance knowledge about the learning opportunities afforded by such technology.

The ways a teacher makes decisions and implements plans in the classroom provide the most visible and—arguably—the most important demonstrations of accomplished practice. The next four standards describe the kinds of tasks teachers construct and select, the ways teachers facilitate classroom discourse, and the practices teachers use to assess and monitor learning. Accomplished mathematics teachers successfully perform these functions through the roles they assume, the organizational schemes they use, the decisions they make, and the ways they adjust their plans from moment to moment.

Accomplished mathematics teachers value mathematics highly. They take joy in it. They are excited by the ideas they explore with students. And they communicate that joy to their students. For example, they share with students the remarkable fact that the ratio of the circumference to the diameter of any circle is always the same number and that the number shows up in probability, statistics, number theory, and many other seemingly unrelated contexts.
 * // Standard VI: The Art of Teaching //**

Mathematics has the power to fascinate students—for example, the concept of infinity, the patterns found in Pascal’s triangle, and the connection between Fibonacci numbers and the Golden Spiral. Accomplished teachers provide their students with opportunities to discover mathematical delights and to experience the intellectual satisfaction that comes from finding a solution to a problem or justifying a conjecture with a well-considered argument. They notice the light in the eyes of the students who are “turned on” to mathematics, and they seek ways to elicit the same excitement in those who have not yet been captivated.

Teachers modify classroom plans and activities in response to student needs, interests, and unexpected opportunities for learning. They demonstrate flexibility, insight, and responsiveness in dealing with the flow of the classroom. They recognize and respond to the mathematical potential of student questions and comments, and they pursue ideas of interest that emerge in classroom discussion. They also help students reflect on and extend their learning, and they expect them to take responsibility for their learning. There is no recipe for what teachers do at any given moment. Their choices are governed by their immediate and long-term goals, the progress and interests of their students, the instructional opportunities that present themselves, and the particular dynamics and tone of the day. However, there are important dimensions to accomplished teaching.

Teachers know that classroom interactions can develop a life of their own, that no plan should be followed simply for its own sake, and that they must adapt their plans where appropriate. They are prepared to adjust instruction—either because unforeseen diffi- culties suggest that a path they had planned to take will not succeed, or because a classroom discussion points to a beneficial alternative. They are able to anticipate misunderstandings and provide instruction that will help as ideas unfold. Furthermore, they choose topics for discussion wisely, relying on their understanding of what is appropriate and important.

Accomplished teachers are willing to take risks in their classrooms. They vary standard practice, using conventional and unconventional methods to further their students’ mathematical understandings. For example, they might help their students lead some aspect of instruction, or they might design a technology activity without first having all the answers themselves. They are not afraid to take calculated risks if they see better instructional opportunities in a new course of action. Their knowledge of mathematics and their understanding of different experiences help them move students toward important mathematical goals.

Accomplished teachers adjust the pace of the class. They moderate it to give students sufficient time to internalize concepts and build perspectives, to deepen and extend students’ mathematical understanding, or to approach a new topic. They employ such teaching strategies as whole-class discussion, small-group work, individual study, and oneon- one sessions that allow students to explore, discover, and use mathematical ideas. They engage students in myriad activities, such as experiments, demonstrations, projects, games, puzzles and contests, writing, presentations, discussion, and debates. They make sound judgments about the use of time and pacing, and they know when to adjust a classroom format to optimize learning. Accomplished teachers assume different roles to accomplish their complex tasks, such as acting as a facilitator of student inquiry, an information provider, and a collaborator with students in solving problems. Accomplished teachers foster learning by choosing imaginative examples, problems, and situations designed to interest and motivate students, illuminate important ideas, or reveal the growth of student understanding. They work with small groups of students, asking clarifying or leading questions when necessary. They involve students in decisions about mathematical topics or ways to study those topics. They provide students with opportunities to reflect on their learning. And, they serve as a catalyst in launching student investigations.

Accomplished teachers promote meaningful discourse through the well-conceived questions they pose and through the rich tasks they provide. They demonstrate their use of appropriate questioning strategies by knowing how, when, and why to question students about their understanding of mathematics, and they provide a safe arena in which students can counter the arguments of others. They encourage students to pursue learning on their own.

Accomplished mathematics teachers use their knowledge of how students learn to create a stimulating environment in which students are empowered to do mathematics and to foster a respectful, engaging, and cooperative atmosphere for learning. They encourage students to develop a good work ethic and to assume ownership and responsibility for the learning process.
 * // Standard VII: Learning Environment //**

Accomplished teachers create a positive learning environment. From the beginning of the school year, they engage their students in creating a community of learners in which students are willing to take intellectual risks. Such an environment is evident when teachers and students share ideas in a positive and productive manner; when students question mathematical ideas and concepts; when students realize that struggling and making errors are part of the process of learning; and when teachers invite students’ discussion of mathematical activities. In such an environment of trust, students feel safe to communicate different points of view, to conduct open-ended explorations, to make mistakes and learn from them, and to admit confusion or uncertainty. Creating and maintaining such a learning environment require skill and planning, a variety of instructional methods, flexibility, good judgment, and discretion.

Accomplished teachers consider the mathematical understanding needs, interests, and working styles of their students and the mathematics they are studying. They are sensitive to the needs of students with exceptionalities. They create a climate in which each student learns to value mathematics and experience success in doing significant mathematics. They lead by example, and they convey to students the delight that comes with the command of a mathematical tool or principle. They help students develop the ability to work both independently and collaboratively on mathematics.

__Mathematics is a discipline of concepts, principles, procedures, and reasoning processes.__ Its tools include representation, modeling, proof, experimentation, questioning, classification, visualization, and computation. Its practice has been profoundly affected and extended by technology—especially calculators and computers. In the classrooms of accomplished teachers, students are engaged in identifying patterns, solving problems, reasoning, forming and testing conjectures, and communicating results. They search for connections and solve problems while reflecting on both the mathematics and their own thought processes.
 * // Standard VIII: Ways of Thinking Mathematically //**

Accomplished teachers of mathematics __recognize that important general concepts and reasoning methods undergird the development of mathematical power__. They model mathematical reasoning as they work with students, and they encourage their students to question processes and challenge the validity of particular approaches. Their students make conjectures and justify or refute them, formulate convincing arguments, and draw logical conclusions. Sound reasoning—not an edict from the teacher—is the arbiter of mathematical correctness. In short, students become mathematically empowered as they learn to think, reason, and communicate mathematically.

Teachers recognize that mastering mathematical facts and procedures is only a part of what it means to learn mathematics. They also know the importance of developing students’ understanding of and disposition to do mathematics. They realize that teaching students to “think mathematically” means helping them develop a mathematical point of view; recognize situations in which mathematical reasoning might be useful; and have the ability, skill, and confidence to think through a situation. Consequently, they provide settings that allow students to test mathematical ideas, patterns, and conjectures; discover principles; synthesize evidence; and apply their growing knowledge to a variety of problems. Teachers know and use the overarching themes of mathematics that help students understand and appreciate the powerful relationships between mathematical ideas and problems, as in making students aware of the relationship between diverse fields, such as algebra and geometry or geometry and probability.

Teachers know multiple ways to represent mathematical ideas, and they organize tasks so that students will learn that a single problem may have many representations. Accomplished teachers encourage students to distinguish between these representations and to select a compelling and efficient representation for a given problem or situation. They teach students to recognize and seek alternative ways to solve mathematical problems. They provide opportunities for students to understand that multiple solutions can be informative, useful, and interesting. Mathematical reasoning permeates the tasks, actions, and discourse of accomplished teachers’ mathematics classes.

Accomplished teachers provide students with problems and applications that will allow them to explore new mathematical content, to reflect on the problem-solving process, to extend and refine their thinking, and to make generalizations about the procedures they have used and link those generalizations with what they have learned previously.

Accomplished teachers provide many rich opportunities for students to apply mathematics to interesting problems. In doing so, they point out the interrelated domains of mathematics. They not only choose tasks related to everyday life—to the sciences, to economics, to politics, or to business—but they also choose tasks that will extend understanding within mathematics. Their choice of problem contexts reflects the breadth of mathematics and its applications.

Teachers also provide opportunities for students to recognize and formulate their own problems—problems that stem from their personal interests or experiences or that build on other work they are doing in mathematics. Accomplished teachers deliberately structure opportunities for students to use and develop appropriate mathematical discourse as they reason and solve problems. These teachers give students opportunities to talk with one another and to work together in solving problems, and they have students use both written and oral discourse to describe and discuss their mathematical thinking and understanding. As students talk and write about mathematics—as they explain their thinking—they deepen their mathematical understanding in powerful ways that can enhance their ability to use the strategies and thought processes gained through the study of mathematics to deal with life issues.

Accomplished teachers are successful in using technology effectively to develop students’ reasoning, mathematical thinking, and discourse. They are able to use applications such as graphing technology, interactive geometry software, and computer algebra systems not as “black boxes” that produce answers with little insight, but rather as tools for supporting student inquiry, conjecture, and proof.

Accomplished mathematics teachers encourage students to confront and challenge ideas and to question peers as they discuss mathematical ideas, develop mathematical understanding, and solve mathematical problems. They monitor what students do, using mathematical communication regularly to help students build understanding. They use probing and supportive questions to advance students’ thinking about the use of available resources, the methodological choices they make to address problems, and the approaches they might pursue.

Accomplished teachers also tackle curriculum issues related to reasoning and mathematical thinking in their departments, schools, and districts. When appropriate, they make an effort to influence the way mathematics is taught at the lower grades. They inform themselves about how students develop reasoning ability. They consider carefully how different curriculum materials or issues might affect mathematical reasoning and thinking. For instance, they consider how an integrated mathematics curriculum might require special attention to the development of reasoning, or how scheduling strategies, such as block scheduling, might influence the emphasis on thinking or reasoning in different courses. Accomplished teachers are aware of the many influences on the process of learning and the ways curriculum decisions might have an intended or unintended impact on the instructional goals.

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