Table Of Content

NUMBER & MEASUREMENT•GEOMETRY • • Y F U T I Focus in N L C I B T A I High School O B N O S R • P Mathematics A & L G S E C B I R T Reasoning A S I I C T A S and Sense Making T Y S M B O L S Focus in High School Mathematics Focus in High School Mathematics Reasoning and Sense Making Copyright © 2009 by THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, INC. 1906 Association Drive, Reston, VA 20191-1502 (703) 620-9840; (800) 235-7566; www.nctm.org All rights reserved Library of Congress Cataloging-in-Publication Data Focus in high school mathematics : reasoning and sense making. p. cm. Includes bibliographical references. ISBN 978-0-87353-631-8 1. Mathematics—Study and teaching (Secondary)—United States. 2. Curriculum planning—United States. I. National Council of Teachers of Mathematics. QA13.F63 2009 510.71’273—dc22 2009032015 The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. Focus in High School Mathematics: Reasoning and Sense Making is an official position of the National Council of Teachers of Mathematics as approved by its Board of Directors, April 2009. For permission to photocopy or use material electronically from Focus in High School Mathematics: Reasoning and Sense Making, please access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750- 8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. Permission does not automatically extend to any items identified as reprinted by permission of other publishers and copyright holders. Such items must be excluded unless separate permissions are obtained. It will be the responsibility of the user to identify such materials and obtain the permissions. Printed in the United States of America Contents List of Examples ............................................................................................................ viii National Council of Teachers of Mathematics High School Curriculum Project ......................................................................................................... ix Preface .............................................................................................................................. xi Background .................................................................................................. xi Purpose ......................................................................................................... xi Organization of the Publication .................................................................. xii Acknowledgments ......................................................................................................... xiii President’s Letter .......................................................................................................... xiv Section 1: Defining Reasoning and Sense Making ......................................................... 1 1 Reasoning and Sense Making ........................................................................... 3 What Are Reasoning and Sense Making? ...................................................... 4 Why Reasoning and Sense Making? .............................................................. 5 How Do We Include Reasoning and Sense Making in the Classroom? ........ 5 Conclusion ..................................................................................................... 7 2 Reasoning Habits ............................................................................................... 9 Progression of Reasoning ............................................................................ 10 Developing Reasoning Habits in the Classroom ......................................... 11 Reasoning as the Foundation of Mathematical Competence ....................... 12 Statistical Reasoning .................................................................................... 12 Mathematical Modeling ............................................................................... 13 Technology to Support Reasoning and Sense Making ................................ 14 Conclusion ................................................................................................... 14 Section 2: Reasoning and Sense Making in the Curriculum....................................... 15 3 Reasoning and Sense Making across the Curriculum ..................................... 17 vi Contents 4 Reasoning with Number and Measurement .................................................... 21 Reasonableness of Answers and Measurements .......................................... 21 Approximations and Error ........................................................................... 25 Number Systems .......................................................................................... 27 Counting ....................................................................................................... 29 5 Reasoning with Algebraic Symbols ................................................................ 31 Meaningful Use of Symbols ........................................................................ 31 Mindful Manipulation .................................................................................. 33 Reasoned Solving ......................................................................................... 34 Connecting Algebra with Geometry ............................................................ 36 Linking Expressions and Functions ............................................................. 37 6 Reasoning with Functions ............................................................................... 41 Multiple Representations of Functions ........................................................ 41 Modeling by Using Families of Functions................................................... 45 Analyzing the Effects of Parameters ............................................................ 51 7 Reasoning with Geometry ............................................................................... 55 Conjecturing about Geometric Objects ........................................................ 56 Construction and Evaluation of Geometric Arguments ............................... 59 Multiple Geometric Approaches .................................................................. 62 Geometric Connections and Modeling ........................................................ 64 8 Reasoning with Statistics and Probability ....................................................... 73 Data Analysis ............................................................................................... 74 Modeling Variability .................................................................................... 80 Connecting Statistics and Probability .......................................................... 86 Interpreting Designed Statistical Studies ..................................................... 87 Contents vii Section 3: Reasoning and Sense Making in the High School Mathematics Program .................................................................................................... 91 9 Equity ............................................................................................................. 93 Courses ......................................................................................................... 93 Student Demographics and Opportunities for Learning .............................. 96 High Expectations ........................................................................................ 98 Conclusion ................................................................................................... 98 10 Coherence ..................................................................................................... 101 Curriculum and Instruction ........................................................................ 101 Vertical Alignment of Curriculum ............................................................. 102 Assessment ................................................................................................. 104 Conclusion ................................................................................................. 105 11 Stakeholder Involvement .............................................................................. 107 Students ...................................................................................................... 107 Families ...................................................................................................... 108 Teachers ..................................................................................................... 109 Administrators ............................................................................................ 110 Policymakers .............................................................................................. 110 Higher Education ....................................................................................... 111 Curriculum Designers ................................................................................ 112 Collaboration among Stakeholders ............................................................ 112 Conclusion ................................................................................................. 112 Annotated Bibliography ............................................................................................... 115 References ...................................................................................................................... 124 viii Contents List of Examples Example 1: Around the World ........................................................................................ 22 Example 2: Fuel for Thought .......................................................................................... 23 Example 3: Around Pi .................................................................................................... 25 Example 4: A Model Idea ............................................................................................... 27 Example 5: Horseshoes in Flight .................................................................................... 32 Example 6: Distribute Thoroughly ................................................................................. 33 Example 7: Finding Balance ........................................................................................... 35 Example 8: Squaring It Away ......................................................................................... 36 Example 9: More Than Meets the Eye ........................................................................... 38 Example 10: Patterns, Plane and Symbol ......................................................................... 42 Example 11: Take As Directed ......................................................................................... 45 Example 12: Money Matters ............................................................................................ 49 Example 13: Tidal Waves ................................................................................................. 52 Example 14: Picture This ................................................................................................. 57 Example 15: Circling the Points ....................................................................................... 59 Example 16: Taking a Spin ............................................................................................... 62 Example 17: Clearing the Bridge ..................................................................................... 64 Example 18: Assigning Frequencies ................................................................................ 70 Example 19: Meaningful Words, Part A ........................................................................... 74 Example 20: What Are the Chances? Part A .................................................................... 81 Example 21: What Are the Chances? Part B .................................................................... 86 Example 22: Meaningful Words, Part B .......................................................................... 87 National Council of Teachers of Mathematics High School Curriculum Project Writing Group W. Gary Martin, Chair Judith Reed Quander, Staff Liaison Professor of Mathematics Education Director of Research Auburn University National Council of Teachers of Mathematics John A. Carter, NCTM Board Liaison William McCallum Assistant Principal for Teaching and Learning Professor of Mathematics Adlai E. Stevenson (Illinois) High School University of Arizona Susan Forster Eric Robinson Mathematics Department Chair and Teacher Professor of Mathematics Bismarck (North Dakota) High School Ithaca College Roger Howe Vincent Snipes Professor of Mathematics Professor of Mathematics Education Yale University Director, Center for Mathematics, Science, and Technology Education Gary D. Kader Winston-Salem State University Professor of Mathematical Sciences Appalachian State University Patricia Valdez Mathematics Department Chair and Teacher Henry (Hank) Kepner Pájaro Valley (California) High School President, National Council of Teachers of Mathematics Professor, Mathematics Education University of Wisconsin—Milwaukee Planning Group W. Gary Martin, Chair Ken Krehbiel, Staff Liaison Judith Reed Quander, Staff Liaison Professor of Mathematics Education Associate Executive Director for Director of Research Auburn University Communications NCTM NCTM Henry Kepner Ruth Casey, NCTM Board Liaison President, National Council of Teachers (2007−2008) of Mathematics PIMSER Professor, Mathematics Education University of Kentucky University of Wisconsin—Milwaukee Fred L. Dillon, NCTM Board Liaison Jennifer J. Salls, NCTM Board Liaison (2008−2009) (2008–2009) Mathematics Teacher Mathematics Teacher Strongsville (Ohio) High School Sparks (Nevada) High School Kaye Forgione Richard Schaar Senior Associate, Mathematics Texas Instruments Achieve, Inc.