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C. Mathematics

Summary

Definition

Checklist - needs anchor

Application/Examples - needs anchor

Note.  Printed with permission from National Board for Professional Teaching Standards, (Middle Childhood through Early Adolescence/Mathematics Standards, 1998; Adolescence and Young Adulthood/Mathematics Standards, 1996), www.nbpts.org. All rights reserved.

Checklist 

1. Middle childhood through early adolescence -needs anchor

Commitment to all students

___a. Commitment to equity and access 

Knowledge of students, mathematics, and teaching

___b. Knowledge of students

___c. Knowledge of mathematics

___d. Knowledge of teaching practice

The teaching of mathematics

___e. The art of teaching

___f. Learning environment

___g. Using mathematics

___h. Technology and instructional resources

___i. Assessment

Professional development and outreach

___j. Reflection and growth

___k. Families and communities

___l. Professional community

2. Adolescence and young adulthood - needs anchor

Commitment

___a. Commitment to students and their learning

Knowledge of students, mathematics, and teaching

___b. Knowledge of students

___c. Knowledge of mathematics

___d. Knowledge of teaching practice

The teaching of mathematics

___e. The art of teaching

___f. Learning environment

___g. Reasoning and thinking mathematically

___h. Assessment

Professional development and outreach

___i. Reflection and growth

___j. Families and communities

___k. Contributing to the professional community

Applications/Examples

1. Middle childhood through early adolescence

Commitment to all students

___a. Commitment to equity and access

  • Teachers value and acknowledge the individuality and worth of each student.

  • They believe that all students can learn and should have access to the full mathematics curriculum.

  • They demonstrate these beliefs in their practice by systematically providing all students with equitable and complete access to mathematics.

  • They hold high expectations for each student.

  • They view mathematics as a way to open doors for students, not as a series of barriers for students to overcome.

  • Teachers use a variety of approaches to teaching the subject and modify the mainstream curriculum and use adaptive strategies that enable each student to contribute.

  • Teachers are aware of the various cultural backgrounds of their students and of the ways in which this could influence the learning process.

  • They make mathematics relevant to students by making connections to daily-life applications.

Knowledge of students, mathematics, and teaching

___b. Knowledge of students

  • Teachers recognize that students are shaped by a variety of educational, social, and cultural backgrounds and experiences that influence learning.

  • They draw on their knowledge of how students learn and develop in order to understand their students and to guide curricular and instructional decisions.

  • Teachers are able to merge the goals of the classroom and curriculum with students' knowledge.

  • They integrate their developmental knowledge of students into their instructional planning, blending it with knowledge about how students see mathematics and develop new mathematical understanding.

  • They build on their students' strengths, interests, and experiences to challenge each student to achieve high levels of mathematical understanding.

  • Teachers insightfully observe and listen to their students, whether in a formal classroom setting, an individual conference, or an informal conversation.

  • Teachers work collaboratively with specialists when they are available and needed.

___c. Knowledge of mathematics

  • Teachers understand significant connections among mathematical ideas and the application of those ideas in mathematics, in other disciplines, and in the world outside of school.

  • They have a broad knowledge of mathematical concepts, principles, techniques, and reasoning that they use to set curricular goals and shape their teaching.

  • Teachers are aware of where their students are headed--individually or as a group.

  • Teachers monitor and adjust their teaching continuously, directing students toward important understandings that arise naturally from students' work by asking questions and guiding discourse toward these understandings.

  • They are knowledgeable about the shifts in importance of numerical skills and procedures that result from powerful and accessible hand-held technological tools.

  • Teachers see the power of patterns in mathematics, as they recognize and generate patterns to demonstrate a variety of relationships and can identify and justify patterns they observe in complex situations.

  • Teachers know that mathematics can be applied in a variety of settings, including other school subjects.

  • They enliven and enrich the mathematics they teach by drawing out the connections between mathematics and other aspects of students' lives and experiences.

  • Teachers help students discover concepts and principles underlying important mathematical topics, detect important relationships connecting content strands, and use mathematical ideas and methods in significant applications.

  • Teachers have extensive knowledge and connected understandings of the major ideas in the core domains of mathematics: number and operation sense, algebra and functions, geometry and measurement, statistics and data analysis, concepts of discrete mathematics, and concepts of calculus.

___d. Knowledge of teaching practice

  • Teachers rely on their extensive pedagogical knowledge to make curricular decisions, select appropriate instructional strategies, develop instructional plans, and formulate assessment plans.

  • Teachers are knowledgeable about techniques for working with students in groups and about selecting tasks appropriate for group work.

  • They acknowledge that individual work is important while recognizing the importance of learning how to work together on mathematical problems.

  • They take into account individual needs and developmental levels of students when designing instruction and know how to pull students' ideas together and build on them.

  • In selecting tasks and planning instruction, teachers are adept at connecting mathematics to science, social studies, arts, physical education, and other fields.

  • They maintain a delicate balance between skills practice and deeper understanding, frequently embedding such skills practice into rich problem-solving opportunities that in turn reflect the world of their own students.

  • Teachers focus on motivating students on the basis of individual needs, interests, and intrinsic motivation.

The teaching of mathematics 

___e. The art of teaching

  • Teachers create elegant and powerful approaches to instructional challenges.

  • Their practice reflects a highly developed personal synthesis of their caring for students, passion for teaching and mathematics, understanding of mathematical content, ability to apply mathematics, and rich knowledge of established and innovative educational practices.

  • They design intellectually challenging lessons that enable students to learn important mathematics and they involve students in the learning process.

  • They carefully guide, direct, and encourage decision making, mathematical reasoning, and insight through questioning, discussion, and written communication.

  • They listen to students and are ready to adapt instruction to accommodate unexpected tangents.

  • Teachers encourage students to formulate and test mathematical conjectures.

  • They capitalize on opportunities generated by unexpected but relevant student questions or discussions to pursue important, challenging mathematical ideas.

  • They observe and listen to students in order to know when a student misunderstands, adapting their instructional plan to clarify misconceptions and facilitate mathematical learning.

  • They modify and adjust the pacing and mode of instruction to accommodate their students and promote their learning.

  • Teachers appreciate the diversity of their students by building on students' individual strengths, background, world view, and intuitive knowledge to guide them to higher and more sophisticated levels of mathematical thinking and understanding.

  • They ask interesting and thought-provoking questions that stimulate students' interest in mathematics, and they provide students with engaging tasks that are well matched to the students' level of mathematical literacy.

  • Tasks focus on meaningful, challenging mathematics, sometimes centered on applications and sometimes designed to build an understanding of important mathematical ideas and relationships.

  • Teachers may assume a variety of roles--facilitator of student inquiry, information provider, or collaborator with students.

  • They give students frequent opportunities for multiple forms of communication and to write and speak mathematically in order to explain their thinking, make generalizations about their explorations, justify their conclusions, and describe their feelings about mathematics.

___f. Learning environment 

  • Teachers create stimulating, caring, and inclusive environments.

  • They develop communities of involved learners in which students accept responsibility for learning, take intellectual risks, develop confidence and self-esteem, work independently and collaboratively, and value mathematics.

  • Teachers create classrooms where students work on engaging mathematical tasks in small groups, in large groups, and individually, with a focus on the importance of learning to work with others on mathematical problems.

  • Teachers provide students with direct opportunities to construct their own mathematical meanings and even their own procedures for mathematical operations.

  • They maintain high expectations for student behavior, demanding that it support the learning of mathematics; posters, student work, mathematical models, and manipulative materials that are likely to pique students' interest and encourage their involvement in mathematics are evident in the classroom.

  • The physical arrangement of space and furniture, along with the teachers' use of space is purposeful and designed to foster mathematical discourse and support both collaborative and independent student work.

___g. Using mathematics 

  • Teachers help students develop a positive disposition toward mathematics and foster the development of all students' ability to use mathematics as a way to understand the world around them.

  • Instruction focuses on developing students' mathematical understanding by providing opportunities for students to investigate, explore, and discover structures and relationships, demonstrate flexibility and perseverance in solving problems, create and use mathematical models, formulate problems of their own, and justify and communicate their conclusions.

  • Teachers interweave mathematics concepts throughout the curriculum, using a rich and varied set of tasks, often beginning with manipulative activities and connecting to symbolic ones.

  • Teachers skillfully use visual and kinesthetic tools (interlocking cubes, pattern blocks, etc.) to help students model real-world problems and make mathematical ideas.

  • Teachers notice when a middle childhood or early adolescent student is having difficulty at an abstract level, and he/she might therefore choose to use a concrete or representational approach.

  • Teachers use a variety of tools to help students develop reasoning skills, including the ability to make and justify simple conjectures, explain the thinking behind solutions to problems, and formulate convincing arguments for generalizations and conclusions.

  • Teachers regularly have students convince one another of the validity of particular representations, solutions, conjectures, and answers through both writing and discussion.

  • Teachers ensure that students use both written and oral language to describe and discuss their mathematical thinking and understanding, providing them frequent opportunities to listen to, respond to, and question the teacher and one another in the process of discussing mathematical ideas, developing mathematical understanding, and solving mathematical problems.

  • Teachers provide students with specific guidance on how to read their mathematics books and on how to access mathematical reference materials.

  • They provide students with opportunities to talk with each other and work together in solving problems.

  • Problem solving relates to students' everyday lives, as they use mathematics to make sense of their world.

  • Teachers encourage investigation, cooperation, and communication in order to promote problem solving and problem posing.

___h. Technology and instructional resources 

  • Teachers are knowledgeable about and, whenever possible, use current technologies and other resources to promote student learning in mathematics.

  • They select, adapt, and create engaging instructional materials and draw on human resources from the school and community to enhance and extend students' understanding and use of mathematics.

___i. Assessment 

  • Teachers integrate assessment into their instruction to promote the learning of all students.

  • They design, select, and employ a range of formal and informal assessment tools to match their educational purposes.

  • They help students develop self-assessment skills, encouraging them to reflect on their performance.

  • Teachers skillfully incorporate opportunities for assessing students' progress into daily instruction, observing students as they produce a project or complete a task, interviewing a group of students as a final step in an assignment, or watching students present their work to a panel of citizens or parents.

  • Teachers are knowledgeable about a variety of approaches to assessment, including testing, performance assessment, observation and analysis of student behavior, interview techniques, etc.

  • Assessment is appropriate to the students' needs, the mathematics being studied, and the instructional approach used.

  • Assessment could focus on groups or individuals.

  • Teachers provide students with regular opportunities to reflect on what they have learned and are learning (entries in a journal, discussions with other students, or notes to the teacher).

Professional development and outreach 

___j. Reflection and growth

  • Teachers regularly reflect on teaching and learning.

  • They keep abreast of changes in mathematics and mathematical pedagogy, continually increasing their knowledge and improving their practice.

  • Teachers devise and employ a variety of strategies to regularly gather information about their teaching (videotapes of lessons, observations of students' responses to particular topics and teaching methods, conversations with students, test scores, etc.).

  • Teachers actively work with colleagues to expand their knowledge and understanding of mathematics and incorporate this knowledge and understanding into the design of learning activities.

___k. Families and communities

  • Teachers work to involve families in their children's education, help the community understand the role of mathematics and mathematics instruction in today's world, and, to the extent possible, involve the community in support of instruction.

___l. Professional community 

  • Teachers collaborate with peers and other education professionals to strengthen the school's program, promote program quality and continuity across grade levels, advance knowledge in the field of mathematics education, and improve practice within the field.

2. Adolescence and young adulthood Commitment to all students
___a. Commitment to students and their learning

  • Teachers value and acknowledge the individuality and worth of each student, believe that all students can learn and use significant mathematics, and demonstrate these beliefs in their practice.

  • Teachers care about students as individuals and demonstrate this concern through their words and actions.

  • They are alert and sensitive to the variability that exists in students' prior learning experiences, individual learning approaches, family and cultural backgrounds, interests, and special needs.

  • They recognize and work to overcome barriers that can prevent women, minorities, or any students, including those with disabilities, from achieving success in mathematics.

Knowledge of students, mathematics, and teaching

 ___b. Knowledge of students

  • Teachers use their knowledge of adolescents and of adolescent development, as well as their knowledge about the effects of this development, on the learning of mathematics to guide curricular and instructional decisions.

  • They understand the impact of home life, cultural background, individual learning differences, student attitudes and aspirations, and community expectations and values on the learning of their students.

  • Teachers create situations that encourage students to explore and build upon previous knowledge and understanding.

  • They structure lessons--often using manipulatives, technology, and activities--that enable students to recognize connections among concrete, symbolic, and graphical representations.

  • Teachers insightfully observe and listen to their students in whatever setting students use to express themselves (classroom setting, individual conference, or informal conversation).

___c. Knowledge of mathematics 

  • Teachers understand significant connections among mathematical ideas and the application of those ideas in mathematics, in other disciplines, and in the world outside of school.

  • They have a broad knowledge of mathematical concepts, principles, techniques, and reasoning that they use to set curricular goals and shape their teaching.

  • Teachers are aware of where their students are headed--individually or as a group.

  • Teachers monitor and adjust their teaching continuously, directing students toward important understandings that arise naturally from students' work by asking questions and guiding discourse toward these understandings.

  • They are knowledgeable about the shifts in importance of numerical skills and procedures that result from powerful and accessible hand-held technological tools.

  • Teachers see the power of patterns in mathematics, as they recognize and generate patterns to demonstrate a variety of relationships and can identify and justify patterns they observe in complex situations.

  • Teachers know that mathematics can be applied in a variety of settings, including other school subjects.

  • They enliven and enrich the mathematics they teach by drawing out the connections between mathematics and other aspects of students' lives and experiences.

  • Teachers help students discover concepts and principles underlying important mathematical topics, detect important relationships connecting content strands, and use mathematical ideas and methods in significant applications.

  • Teachers have extensive knowledge and connected understandings of the major ideas in the core domains of mathematics: number and operation sense, algebra and functions, geometry and measurement, statistics and data analysis, concepts of discrete mathematics, and concepts of calculus.

___d. Knowledge of teaching practice 

  • Teachers have an extensive base of pedagogical knowledge and use it to make curriculum decisions, design instructional strategies and assessment plans, and choose materials and resources for mathematics instruction.

  • Goals are clearly articulated and instructional techniques and activities are selected to allow students to meet the goals.

  • Teachers' repertoire of teaching strategies (including cooperative learning, discovery, individualized instruction, and large-group instruction) allows students to explore, discover, and use mathematical ideas.

  • Teachers have collected or identified a variety of activities and materials (manipulative tools, printed materials, human resources, historical materials, and library and media resources) that are particularly valuable to help them reach their mathematical goals and that can be drawn on to meet the students' needs.

  • Teachers strive to provide students with opportunities to use educational technology (videos, computers, calculators, or CD-ROMs).

The teaching of mathematics 

___e. The art of teaching

  • Teachers stimulate and facilitate student learning by using a wide range of formats and procedures, and assuming a variety of roles to guide students' learning of mathematics.

  • Teachers modify classroom plans and activities in response to student needs and unexpected opportunities for learning.

  • They recognize the mathematical potential of student comments, and they pursue ideas of interest that emerge in classroom discussion.

  • They vary standard practice, use conventional methods in unexpected ways, and use unconventional methods and activities to further mathematical understanding.

  • Teachers adjust the pace of the class by moderating it to give students time to internalize concepts and build perspective.

  • They employ teaching strategies such as whole-class discussion, small-group work, individual study, and one-on-one sessions that allow students to explore, discover, and use mathematical ideas.

  • They engage students in experiments, demonstrations, projects, games, puzzles and contests, writing activities, presentations, discussions, or debates.

  • They assume different roles in their relationships with the students: facilitator of student inquiry, information provider, or a collaborator with students in solving problem.

  • They foster leaning by choosing imaginative examples, problems and situations designed to interest and motivate students, working with small groups of students (asking clarifying or leading questions when necessary), using questions and probes skillfully to assist student learning, involving students in decisions about mathematical topics or ways to study these topics.

___f. Learning environment 

  • Teachers create stimulating, caring, and inclusive environments.

  • They develop communities of involved learners in which students accept responsibility for learning, take intellectual risks, develop confidence and self-esteem, work independently and collaboratively, and value mathematics.

  • Teachers create classrooms where students work on engaging in mathematical tasks in small groups, in large groups, and individually, with a focus on the importance of learning to work with others on mathematical problems.

  • Teachers provide students with direct opportunities to construct their own mathematical meanings and even their own procedures for mathematical operations.

  • They maintain high expectations for student behavior, demanding that it support the learning of mathematics.

  • Posters, student work, mathematical models, and manipulative materials that are likely to pique students' interest and encourage their involvement in mathematics are evident in the classroom.

  • The physical arrangement of space and furniture, along with the teachers' use of space, is purposeful and designed to foster mathematical discourse and support both collaborative and independent student work.

___g. Reasoning and thinking mathematically 

  • Teachers develop students' abilities to reason and think mathematically--to investigate and explore patterns, discover structures and relationships, formulate and solve problems, justify and communicate conclusions.

  • Students are engaged in problem solving, mathematical communication, reasoning and searching for connections, and reflecting both on mathematical ideas and their own thought processes.

  • Students make and justify conjectures, formulate convincing arguments, and draw logical conclusions.

  • Teachers provide settings that allow students to test mathematical ideas, discover principles, and apply their growing knowledge to both pure mathematical and everyday problems.

  • Teachers provide opportunities for students to recognize and formulate their own problems stemming from their own personal interests or experiences.

  • Teachers provide opportunities for students to talk with each other and work together in solving problems, and they also have students use both writing and oral discourse to describe and discuss their mathematical thinking and understanding.

___h. Assessment 

  • Teachers employ a range of formal and informal assessment methods to evaluate student learning in light of well-defined goals.

  • They use the results to inform the teaching process and provide opportunities for students to reflect on the strengths and weaknesses of their individual performance.

  • They check for student understanding prior to a lesson and at the close of a lesson.

  • Teachers engage students in discussions and presentations, and direct them in projects.

  • The assessment techniques used are varied and they include open-ended problems, group investigations, projects, and portfolios.

  • Teachers modify the lessons, opportunities, and activities they offer students based on assessment results, thus resorting to backtracking or designing strategies for peer tutoring.

  • Assessment is fair and unbiased with regard to gender, ethnic, and cultural differences of students.

  • Teachers help students use various performance measures as tools for self-reflection, and they offer students opportunities for self-reflection.

  • Students are also encouraged to sensitively assess their classmates' work.

Professional development and outreach 

___i. Reflection and growth

  • Teachers regularly reflect on teaching and learning.

  • They keep abreast of changes in mathematics and mathematical pedagogy, continually increasing their knowledge and improving their practice.

  • Teachers devise and employ a variety of strategies to regularly gather information about their teaching (videotapes of lessons, observations of students' responses to particular topics and teaching methods, conversations with students, test scores, etc.).

  • Teachers actively work with colleagues to expand their knowledge and understanding of mathematics and incorporate this knowledge and understanding into the design of learning activities.

___j. Families and communities 

  • Teachers work to involve families in their children's education, help the community understand the role of mathematics and mathematics instruction in today's world, and, to the extent possible, involve the community in support of instruction.

___k. Contributing to the professional community 

  • Teachers collaborate with peers and other education professionals to strengthen the school's program, promote program quality and continuity across grade levels, advance knowledge in the field of mathematics education, and improve practice within the field.