Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. This resource was developed by CSMC faculty and doctoral students with support from the National Science Foundation under Grant No. ESI-0333879. The opinions and information provided do not necessarily reflect the views of the National Science Foundation. 12-19-05 1 Committees and Reports that Have Influenced the Changing Mathematics Curriculum This set of PowerPoint slides is one of a series of resources produced by the Center for the Study of Mathematics Curriculum. These materials are provided to facilitate greater understanding of mathematics curriculum change and permission is granted for their educational use. Curriculum and Evaluation Standards for School Mathematics National Council of Teachers of Mathematics Commission on Standards for School Mathematics • 1989 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. http://www.mathcurriculumcenter.org 2 Curriculum and Evaluation Standards for School Mathematics National Council of Teachers of Mathematics Commission on Standards for School Mathematics 1989 3 Prominent Issues and Forces • Poor student performance on national and international assessments • Too many students, including a disproportionate number from minority groups, leaving school without the mathematical proficiency necessary for productive lives • Increasing use of quantitative methods in business, economics, linguistics, biology, medicine, and sociology • Advances in technology and broadening of areas in which mathematics is applied resulted in growth and changes in mathematics itself • Emerging research and changing perspectives on how students learn mathematics 4 NCTM Commission on Standards for School Mathematics, est. 1986 Thomas A. Romberg, Chairman Iris M. Carl Christian R. Hirsch F. Joe Crosswhite Glenda Lappan John A. Dossey Dale Seymour James D. Gates Lynn A. Steen Shirley M. Frye Paul R. Trafton Shirley A. Hill Norman Webb 5 Purpose 1. Create a coherent vision of what it means to be mathematically literate both in a world that relies on calculators and computers to carry out mathematical procedures and in a world where mathematics is rapidly growing and is extensively being applied in diverse fields. 2. Create a set of standards to guide the revision of the school mathematics curriculum and its associated evaluation toward this vision. (NCTM, p. 1) • To ensure quality • To indicate goals • To promote change 6 New Societal Goals for Education • Mathematically literate workers • Lifelong learning • Opportunity for all • Informed electorate 7 New Goals for Students • Learn to value mathematics • Become confident in one’s ability to do mathematics • Become a mathematical problem solver • Learn to communicate mathematically • Learn to reason mathematically 8 Grades K-12 Process Standards Standard 1: Standard 2: Standard 3: Standard 4: Mathematics as Problem Solving Mathematics as Communication Mathematics as Reasoning Mathematical Connections Across the grade bands K-4, 5-8, 9-12, each of these standards appear and increase in level of sophistication and expectation in developmentally-appropriate ways. Habits of mind are developed so that students become mathematically powerful and mathematically literate as they learn to value mathematics. 9 Members of the Grades K-4 Working Group Paul R. Trafton, Chair Hilde Howden Mary M. Lindquist Edward C. Rathmell Thomas E. Rowan Charles S. Thompson 10 Assumptions Underlying the Grades K-4 Standards The Grades K-4 Standards were premised on the assumptions that mathematics curriculum and instruction should: • be conceptually oriented emphasizing mathematical concepts and understanding; • actively involve students in doing mathematics by exploring and discussing mathematical ideas; • emphasize the development of students’ thinking and reasoning abilities; • emphasize the application of mathematics; • include a broad range of content beyond arithmetic: measurement, geometry, statistics, probability, and algebra; • make appropriate use of calculators and computers. 11 Grades K-4 Content Standards Standard 5: Standard 6: Standard 7: Standard 8: Estimation Number Sense and Numeration Concepts of Whole Number Operations Whole Number Computation Implementation of these standards should increase attention to place-value concepts, meaning of operations, mental computation and estimation, thinking strategies for basic facts, and use of technology; and decrease attention to complex and isolated treatment of penciland-paper computation, standard algorithm for division, and use of rounding to estimate. 12 Grades K-4 Content Standards Standard 9: Geometry and Spatial Sense Standard 10: Measurement Standard 11: Statistics and Probability Standard 12: Fractions and Decimals Standard 13: Patterns and Relationships Implementation of these standards should increase attention to geometric properties and relationships, spatial sense, measuring and the concepts of measurement, collecting and organizing data, recognizing and describing patterns, using variables to express relationships, engaging in real-world problems and developing problem solving strategies; and decrease attention to naming geometric figures, memorizing unit of measurement equivalencies, and using clue words to determine operations to use in problem solving. 13 Members of the Grades 5-8 Working Group Glenda Lappan, Chair Daniel T. Dolan Joan F. Hall Thomas E. Kieren Judith E. Mumme James E. Schultz 14 Assumptions Underlying the Grades 5-8 Standards The Grades 5-8 Standards were premised on the assumptions that mathematics curriculum and instruction should: • focus on basic topics in algebra, geometry, probability and statistics, rather than computational skills. • be available to everyone, not simply those who have demonstrated proficiency with calculation and pencil-and-paper computation. • provide students with new problem solving opportunities that renew motivation for learning and provide context for the mathematical skills they are learning. • include allowing students to wrestle with problems that are not welldefined. • include hands-on activities in tactile, auditory, and visual instruction modes. • incorporate the cultural background and unique characteristics of each student into the learning environment. 15 Grades 5-8 Content Standards Standard 5: Standard 6: Standard 7: Standard 8: Number and Number Relationships Number Systems and Number Theory Computation and Estimation Patterns and Functions Implementation of these standards should increase attention to exploring whole numbers, integers, and rational numbers, developing number sense, identifying and using functional relationships, and creating and using tables, graphs and rules; and decrease attention to memorizing procedures, rules and algorithms, finding exact forms of answers, and tedious paperand-pencil computations. 16 Grades 5-8 Content Standards Standard 9: Algebra Standard 10: Statistics Standard 11: Probability Standard 12: Geometry Standard 13: Measurement Implementation of these standards should increase attention to developing an understanding of variables, expressions and equations, using statistical methods to analyze and make decisions, creating experimental models, using geometry in solving problems and estimating and using measurement; and decrease attention to manipulating symbols, memorizing procedures, formulas, facts and relationships and converting within and between measurement systems. 17 Members of the Grades 9-12 Working Group Christian R. Hirsch, Chair Sue Ann McGraw Gerald R. Rising Harold L. Schoen Cathy L. Seeley Bert K. Waits 18 Assumptions Underlying the Grades 9-12 Standards • Students entering 9th grade will bring experience with the broad integrated curriculum described in the grades K-8 standards. • Students will not be denied access to the broad, rich curriculum proposed for high school because of lack of paper-and-pencil computational facility. • Graphing calculators will be available for students at all times and at least one computer will be available in every classroom. • A three-year core curriculum will be studied by all students with differentiation in terms of depth and breadth of topics. • College-intending students will be expected to study mathematics each year of high school with calculus no longer viewed as the capstone of high school mathematics. • All students will study appropriate mathematics during their senior year. 19 Grades 9-12 Content Standards Standard 5: Standard 6: Standard 7: Standard 8: Standard 9: Algebra Functions Geometry from a Synthetic Perspective Geometry from an Algebraic Perspective Trigonometry Implementation of these standards should involve increased attention to technology, communication, real-world applications, mathematical modeling, multiple representations, and connections between strands; and decreased attention to symbolic manipulation, by-hand graphing of functions, development of synthetic geometry as a complete axiomatic system, and two-column proofs. 20 Grades 9-12 Content Standards Standard 10: Standard 11: Standard 12: Standard 13: Standard 14: Statistics Probability Discrete Mathematics Conceptual Underpinning of Calculus Mathematical Structure Implementation of these standards should involve increased opportunities to study contemporary mathematics largely influenced by the explosion of technology and to develop a deeper understanding of change and the broad underlying themes and logical consistency of mathematics; and decreased attention to topics and skills only needed as preparation of some for calculus. 21 Members of the Evaluation Working Group Norman Webb, Chair Elizabeth Badger Diane J.Briars Thomas J. Cooney Tej N. Pandey Alba G. Thompson 22 Assumptions Underlying the Evaluation Standards • Student assessment should be integral to instruction. • Multiple means of assessment methods should be used. • All aspects of mathematical knowledge and its connections should be assessed. • Instruction and curriculum should be considered equally in judging the quality of a program. 23 Evaluation Standards Organized in three sections: General Assessment Standards: Recommends principles relevant to any form of assessment and program evaluation. Standard 1: Alignment Standard 2: Multiple Sources of Information Standard 3: Appropriate Assessment Methods and Uses Student Assessment Standards: Identifies aspects of mathematical knowledge that should be assessed, as derived from the Curriculum Standards. Standard 4: Mathematical Power Standard 5: Problem Solving Standard 6: Communication Standard 7: Reasoning Standard 8: Mathematical Concepts Standard 9: Mathematical Procedures Standard 10: Mathematical Disposition 24 Evaluation Standards Program Evaluation Standards: Examine the assessment of the extent to which a mathematics program is consistent with the Standards. Standard 11: Indicators for Program Evaluation Standard 12: Curriculum and Instructional Resources Standard 13: Instruction Standard 14: Evaluation Team 25 Significance: Curriculum and Instruction • Stimulated major school mathematics curriculum development projects funded by NSF Three elementary school curriculum projects: – Everyday Mathematics (University of Chicago School Mathematics Project) – Investigations in Number, Data, and Space (TERC) – Math Trailblazers (Teaching Integrated Mathematics and Science Project) Five middle school curriculum projects: – Connected Mathematics (Connected Mathematics Project) – Mathematics in Context (Wisconsin Center for Education Research) – MathScape: Seeing and Thinking Mathematically (Education Development Center) – MATHThematics (STEM) (University of Montana) – Pathways to Algebra and Geometry (MMAP) Five high school curriculum projects: – Contemporary Mathematics in Context (Core-Plus Mathematics Project) – Interactive Mathematics Program (IMP) – MATH Connections: A Secondary Mathematics Core Curriculum (CBIA) – Mathematics: Modeling Our World (COMAP) – SIMMS Integrated Mathematics (SIMMS) 26 Significance: Curriculum and Instruction • Stimulated development of standards for other school subjects. • NSF funded numerous professional development projects to support the necessary work with teachers, districts, and communities to successfully implement these Standards and the new Standards-based instructional materials. • State-level curriculum frameworks and assessments were developed that reflected the content of the Standards. • General Mathematics, Consumer Mathematics, and other remedial high school courses were replaced by Standards-based courses. • Changes were made in instructional practices toward more studentcentered approaches. • Use of technology, particularly hand-held calculators, in teaching increased dramatically. • “Math wars” emerged as implementation of these standards were construed in ways unintended by the Standard’s authors. 27 Significance: Evaluation • Alternatives to pencil-and-paper tests, such as writing assignments, projects, portfolios, and classroom dialogue began to be used by teachers to assess student understanding. • Local and statewide testing moved away from strictly short answer and multiple-choice questions to include constructed response questions to assess student understanding. • Standardized tests such as the ACT and SAT and AP exams began to allow calculator use, to reflect the trend in high school classrooms. • Curriculum evaluation began to be viewed as a K-12 initiative, rather than a grade-level or building level issue. • Many schools and individual teachers evaluated the alignment of their curriculum and instructional methods with the NCTM Standards and later with state frameworks. 28 References National Council of Teachers of Mathematics Commission on Standards for School Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: The Council. http://www.standards.nctm.org/index.htm McLeod, D. B., Stake, R. E., Schappelle, B. P., Mellissinos, M., & Gierl, M. J. (1996). Setting the Standards: NCTM’s role in the reform of mathematics education. In S. A. Raizen & E. D. Britton (Eds.), Bold ventures: Case studies of U.S. innovations in mathematics education. (Vol. 3, pp. 13-132). Dordrecht, The Netherlands: Kluwer Academic. 29