Beliefs about Teaching and Learning Mathematics: Unproductive Beliefs Productive Beliefs Current Beliefs Action to Take Mathematics learning should focus on practicing procedures and memorizing basic number combinations. Mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning, and discourse. Students need only to learn and use the same standard computational algorithms and the same prescribed methods to solve algebraic problems. All students need to have a range of strategies and approaches from which to choose in solving problems, including, but not limited to, general methods, standards algorithms, and procedures. Students can to apply mathematics only after they have mastered the basic skills. Students can learn mathematics through exploring and solving contextual and mathematical problems. The role of the teacher is to tell students exactly what definitions, formulas, and rules they should know and demonstrate how to use this information to solve mathematics problems. The role of the teacher is to engage students in tasks that promote reasoning and problem solving and facilitate discourse that moves students toward shared understanding of mathematics. The role of the student is to memorize information that is presented and then use it to solve routine problems on homework, quizzes, and tests. The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to
Mathematics is a static, unchanging field. Mathematics is a dynamic field that is ever changing. Emphases in the curriculum are evolving, and it is important to embrace and adapt to appropriate changes. The availability of mathematics curricula means that every teacher should design his or her own curriculum and textbook.. Open-source curricula are resources to be examined collaboratively and used to support the established learning progressions of a coherent and effective mathematics program
Beliefs about tools and technology in learning mathematics: Unproductive Beliefs Productive Beliefs Current Beliefs Action to Take Calculators and other tools are at best a frill or distraction and at worst a crutch that keeps students from learning mathematics. Students should use these tools only after they have learned how to do procedures with paper and pencil. Technology is an inescapable fact of life in the world in which we live and should be embraced as a powerful tool for doing mathematics. Using technology can assist students in visualizing and understanding important mathematical concepts and support students’ mathematical reasoning and problem solving. School mathematics is static. What students need to know about mathematics is unchanged (or maybe even threatened) by the presence of technology. Technology and other tools not only change how to teach but also affect what can be taught. They can assist students in investigating mathematical ideas and problems that might otherwise be too difficult or to explore. Physical and virtual manipulatives should be used only with very young children who need visuals and opportunities to explore by moving objects. Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas. Technology should be used primarily as a quick way to get correct answers to computations. Finding answers to a mathematical computation is not sufficient. Students need to understand whether an answer is reasonable and how the results apply to a given context. They also need to be able to consider
Beliefs about mathematics assessment: Unproductive Beliefs Productive Beliefs Current Beliefs Action to Take The primary purpose of assessment is accountability for students through report card marks or grades. The primary purpose of assessment is to inform and improve the teaching and learning of mathematics. Assessment in the classroom is an interruption of the instructional process. Assessment is an ongoing process that is embedded in instruction to support student learning and make adjustments to instruction. Only multiple-choice and other “objective” paper-and-pencil tests can measure mathematical knowledge reliably and accurately. Mathematical understanding and processes can be measured through the use of a variety of assessment strategies and A single assessment can be used to make important decisions about students and teachers. Multiple data sources are needed to provide an accurate picture of teacher and student performance. Assessment is something that is done to students. Assessment is a process that should help students become better judges of their own work, assist them in recognizing high-quality work when they produce it, and support them in using evidence to advance their own learning. Stopping teaching to review and take practice tests improves students’ performance on high-stakes tests. Ongoing review and distributed practice within effective instruction are productive test preparation strategies.
Beliefs professionalism in mathematics education: Unproductive Beliefs Productive Beliefs Current Beliefs Action to Take Teachers arrive from teacher preparation programs prepared to be effective teachers. Developing expertise as a mathematics teacher is a career-long process. The knowledge base of effective mathematics teaching and learning is continually expanding. A deep understanding of mathematics content is sufficient for effective teaching. Teachers of mathematics continue to learn throughout their careers in the areas of mathematical knowledge for teaching, mathematical pedagogical knowledge, and knowledge of students as learners of mathematics. Effective teachers can work autonomously and in isolation. As long as the students in one’s own classroom are successful, all is well. Teachers who collaborate with colleagues inside and outside their school are more effective. All mathematics teachers are collectively responsible for student learning, the improvement of the professional knowledge base, and everyone’s effectiveness. Instructional coaching is unnecessary and a luxury in a school’s budget. However, novice teachers might benefit from some general coaching support. All professionals, even experienced teachers, can benefit from content-focused instructional coaching. Teachers should be in direct contact with students for all or A priority for schools and districts is to establish regular content-focused
almost all of each school day. collaborative planning time for teachers at the same grade level or teachers of the same course and to schedule time periodically for vertical articulation. Highly effective teachers have an innate and natural ability to provide innovative instruction that results in high levels of student achievement. Highly effective teachers become master teachers over time by continually improving mathematical knowledge for teaching, mathematical pedagogical skills, and knowledge of students as learners of mathematics. The textbook and digital resources provide all the necessary lesson plans and activities, so teachers have no need to engage in detailed unit and lesson planning. Effective mathematics teaching results from purposeful planning. Effective mathematics teaching results from purposeful planning. Highly effective teachers collaborate to design detailed mathematics lessons and then reflect on the effectiveness of those plans for student learning, in a cycle of continuous improvement.