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Guidelines for Assessment & Instruction in Mathematical GUIDELINES FOR ASSESSMENT & INSTRUCTION IN MATHEMATICAL MODELING EDUCATION CONSORTIUM FOR MATHEMATICS AND ITS APPLICATIONS (COMAP) SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS (SIAM) GAIMME IS FUNDED AND PUBLISHED JOINTLY BY: Consortium for Mathematics and Its Applications Society for Industrial and Applied Mathematics COMAP, Inc. SIAM 175 Middlesex Turnpike, Suite 3B 3600 Market Street, 6th Floor Bedford, MA 01730 USA Philadelphia, PA 19104 USA [email protected] [email protected] ADDITIONAL SUPPORT AND COOPERATION PROVIDED BY: National Council of Teachers of Mathematics Moody’s Mega Math (M3) Challenge 1906 Association Drive c/o SIAM Reston, VA 20191 Philadelphia, PA 19104 [email protected] [email protected] WRITING TEAM: What is Mathematical Modeling?: Karen Bliss*, Jessica Libertini Early grades: Rachel Levy*, Rose Mary Zbiek, Ben Galluzzo, Mike Long High school: Dan Teague*, Landy Godbold, Joe Malkevitch, Henk van der Kooij Undergraduate: Frank Giordano*, Karen Bliss, Katie Fowler, Henry Pollak, Jessica Libertini Resources: Heather Gould Editors: Sol Garfunkel*, Michelle Montgomery * Lead writers PRODUCTION: First Edition: April 2016 Printed and bound in the United States of America No part of this report may be reproduced or stored in an online retrieval system or transmitted in any form or by any means without the prior written permission of the publishers. All rights reserved. ISBN: 978-1-611974-43-0 eISBN: 978-1-611974-44-7 2 CONTENTS PREFACE V CHAPTER 1 WHAT IS MATHEMATICAL MODELING? 9 CHAPTER 2 MATHEMATICAL MODELING IN THE EARLY GRADES: PREKINDERGARTEN THROUGH GRADE 8 25 CHAPTER 3 MATHEMATICAL MODELING IN HIGH SCHOOL: GRADES 9 THROUGH 12 47 CHAPTER 4 MATHEMATICAL MODELING AT THE UNDERGRADUATE LEVEL 73 CHAPTER 5 WHAT IS MATHEMATICAL MODELING, THE ART AND FLAVOR 97 APPENDIX A MATHEMATICAL MODELING RESOURCES 105 APPENDIX B MODELING EXAMPLES, ELEMENTARY AND MIDDLE GRADES 121 APPENDIX C EXTENDED EXAMPLES 149 APPENDIX D ASSESSMENT TOOLS 195 REFERENCES 209 3 4 PREFACE INTRODUCTION In 2015, leaders from SIAM and COMAP came together to produce a report we have named GAIMME — Guidelines for Assessment and Instruction in Mathematical Modeling Education. The name gives homage to the fine work of the American Statistical Association’s impressive GAISE report.1 Like that document our primary audience is you the teacher. While we hope that test and policy makers will read this document and use it in their decision-making, it has been written for the front-line teacher. We hope and intend that these guidelines will be of help as you incorporate the practice of mathematical modeling into your classrooms. A major reason for the creation of GAIMME was the fact that, despite the usefulness and value in demonstrating how mathematics can help analyze and guide decision making for real world messy problems, many people have limited experience with math modeling. We wanted to paint a clearer picture of mathematical modeling (what it is and what it isn’t) as a process and how the teaching of that process can mature as students move through the grade bands, independent of the mathematical knowledge they may bring to bear. HOW TO USE THIS While we very much hope that you have the time and interest to read this document DOCUMENT cover to cover, it has been written to permit a more focused reading as well. The opening section, “What is Mathematical Modeling?” is intended for every reader and contains the main arguments for the inclusion of mathematical modeling across all grade levels. It is a necessary introduction to each of the succeeding sections. This chapter is followed by three grade-level chapters: early grades, high school, and undergraduate. As a teacher you may wish, naturally, to proceed from “What is Mathematical Modeling?” to the relevant grade level section. However, we encourage you to look at preceding and/or subsequent sections to get a clearer idea of how the modeling process can progress from grade level to grade level. v While these grade level sections have naturally a somewhat different voice, they have important common elements. Each begins by making the case for teaching mathematical modeling at that level. Where relevant, we discuss how teaching the mathematical modeling process at a particular level differs from how that process was treated at the prior level. This is independent of the mathematical topics brought to bear in creating the models. Each section contains a collection of exemplary tasks. These tasks have been chosen for one or more of the following reasons: – They are interesting and/or important for students to experience. – They exemplify specific components of the modeling cycle. – They are doable by real students in real classrooms in real time. Because modeling problems can be extremely rich and most often require reexamining our assumptions, they can require time (and pages) to explore. For this reason we have included several of the longer tasks and explanations in Appendices B and C. In each grade level section we spend time discussing both the teaching of mathematics through the modeling process and the teaching of the modeling process itself. In addition we discuss the characteristics of appropriate assessments. Because of this structure you may find some duplication of ideas. This is intentional. While there are certain differences in how one teaches mathematical modeling from grade to grade, there are a number of important features they share in common and that deserve emphasis. Following the educational level sections we revisit the question of “What is Mathematical Modeling” through a series of quotes and comments from the author team. This section, intended for all readers, is included to emphasize the art and flavor of modeling from the perspective of practitioners and mathematics educators. The report includes a Resources Appendix A, designed to sample some of the important modeling assessments and curricula materials in wide usage. A discussion of what makes a good modeling assessment is also included in the Resources section, along with a discussion of how to adapt assessments to meet different modeling goals. We conclude with an Appendix D, which contains helpful examples of rubrics and assessment tools which may be used when teaching mathematical modeling or teaching mathematics through modeling. A few things that the GAIMME report is not: It is not intended as a curriculum. It does not contain a full set of modeling problems. It is not designed as a collection of lesson plans for immediate classroom use, nor a complete set of classroom assessments. Rather, the purpose of these guidelines is to give a sense what mathematical modeling is and isn’t and practical advice on how to teach modeling through the grade levels. We have tried to demonstrate the importance of mathematical modeling and how and why it should be an essential part of every student’s mathematics experience throughout their education. vi vii 1. viii WHAT IS MATHEMATICAL MODELING? INTRODUCTION In the course of a student’s mathematics education, the word “model” is used in many ways. Several of these, such as manipulatives, demonstration, role modeling, and conceptual models of mathematics, are valuable tools for teaching and learning. However, they are different from the practice of mathematical modeling. Mathematical modeling, both in the workplace and in school, uses mathematics to answer big, messy, reality-based questions. Speaking in this report to our colleagues as teachers, supervisors, and teacher educators, we share our experience in the classroom so that, with appropriate facilitation from teachers, students can engage in genuine modeling activities and use mathematics to answer questions that they find meaningful and will enhance their futures. The authors of this report firmly believe that mathematical modeling should be taught at every stage of a student’s mathematical education. After all, why does society give us so much time to teach mathematics? In part, it is because mathematics is important for its own sake, but mostly because mathematics is important in dealing with the rest of the world. Certainly mathematics will help students as they move on through school and into the world of work. But it can and should help them in their daily lives and as informed citizens. It is crucial that students’ experiences with mathematical modeling, as they progress through the grades, give them exposure to a wide variety of problems — how do we determine the average rainfall in a state? Where’s the best place to locate a fire station? What is a fair voting system? How can I hang pictures along a staircase so they look straight? As we demonstrate in subsequent sections of this report, students can learn and appreciate the importance of 1. modeling in their lives at all educational levels. 9 BUT WHAT IS In the sections below we will provide several examples of what modeling is and isn’t. We MATHEMATICAL will give a more detailed working definition of mathematical modeling. But to describe it in a MODELING? nutshell: Mathematical modeling is a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena. Most short definitions we find emphasize this most important aspect, namely the relation between modeling and the world around us. – Using the language of mathematics to quantify real-world phenomena and analyze behaviors. – Using math to explore and develop our understanding of real world problems. – An iterative problem solving process in which mathematics
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