Perspectives on School Mathematics

Measuring Up

Prototypes for Mathematics Assessment

Mathematical Sciences Education Board

National Research Council

NATIONAL ACADEMY PRESS
Washington, DC
1993



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Measuring Up: Prototypes for Mathematics Assessment Perspectives on School Mathematics Measuring Up Prototypes for Mathematics Assessment Mathematical Sciences Education Board National Research Council NATIONAL ACADEMY PRESS Washington, DC 1993

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Measuring Up: Prototypes for Mathematics Assessment NATIONAL ACADEMY PRESS 2101 Constitution Avenue, NW Washington, DC 20418 NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance. This report has been reviewed by a group other than the authors according to procedures approved by a Report Review Committee consisting of members of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy's purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of Medicine. Dr. Frank Press and Dr. Robert M. White are chairman and vice chairman, respectively, of the National Research Council. The Mathematical Sciences Education Board was established in 1985 to provide a continuing national overview and assessment capability for mathematics education and is concerned with excellence in mathematical sciences education for all students at all levels. The Board reports directly to the Governing Board of the National Research Council. Library of Congress Catalog Card Number 92-62904 International Standard Book Number 0-309-04845-1 Copyright 1993 by the National Academy of Sciences. All rights reserved. Permission for limited reproduction of portions of this book for educational purposes but not for sale may be granted upon receipt of a written request to the National Academy Press, 2101 Constitution Avenue, NW, Washington, DC 20418. Copies of this report may be purchased from the National Academy Press, 2101 Constitution Avenue, NW, Washington, DC 20418. Printed in the United States of America B081

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Measuring Up: Prototypes for Mathematics Assessment Preface As the United States moves resolutely towards standards-based education, we must learn anew how to measure quality — of students' learning, of teachers' performance, of school and district accomplishment. Assessment demonstrates the real meaning of "standards." We can see in the tasks children are expected to perform just what they must learn to meet our national goals. It is all too clear that current tests used for assessment of educational performance fail to measure adequately progress toward national standards. This is especially true in mathematics, where curriculum and teaching standards recommended by the National Council of Teachers of Mathematics have earned nationwide consensus. Yet commonly used tests continue to stress routine, repetitive, rote tasks instead of offering children opportunities to demonstrate the full range of their mathematical power, including such important facets as communication, problem solving, inventiveness, persistence, and curiosity. In Measuring Up, the National Research Council seeks to demonstrate in one specific instance — fourth grade mathematics — one approach to standards-based assessment. The prototypes in this volume are intended to bring the ethos of the standards to life in concrete assessment tasks. A superficial glance at the tasks will convince any reader that the new mathematics standards call for a significantly different type of education. But these prototypes do more than illustrate standards. They contribute to a public debate on significant redirection of U.S. traditions concerning the role of assessment in education, including these key issues:

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Measuring Up: Prototypes for Mathematics Assessment Should assessments set goals for learning, or merely sample the present curriculum? Typically, nations that routinely outperform the United States in international comparisons link assessment to goals for learning, and make both goals and assessments known in advance to parents, teachers, students, and the public. In the United States, in contrast, our standardized tests are largely hidden from public view and measure only a very limited part of school curricula. Should assessments tell us what students cannot do or what each student can do? The new standards stress broad and flexible approaches to problem solving, making use of many different approaches and mathematical tools. Yet most tests continue to emphasize set questions with a limited number of options for student responses and rarely provide sufficient flexibility for students to display the full range of their knowledge or abilities. Should students be judged only on their individual work, or also on their ability to work together for the benefit of a larger group? Both in business and in society, real problem solving requires individuals to work together. The new standards for mathematics education stress communication, cooperation, and group performance as an important component of mathematical power. Yet traditional testing in the United States measures only the individual, elevating competition over cooperation as if education were a zero-sum game. How can assessment encourage and recognize inventive, imaginative responses that, although unexpected, are constructive and appropriate? In traditional testing, the examiner determines what topic an item is "intended" to measure. But in assessment attuned to the holistic nature of the new standards, as are the prototypes in this volume, the measure of students' responses is not so much how well they anticipate what the examiner had in mind, but how well they apply their own minds to the task at hand.

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Measuring Up: Prototypes for Mathematics Assessment As our nation needs standards in curriculum and teaching, so too we need standards for assessment. Without such standards, we will continue, unwisely, to measure what is convenient rather than what is most important. In 1991 the Mathematical Sciences Education Board called attention to the need for new thinking about assessment through a National Summit on Assessment. The National Council of Teachers of Mathematics is now working, in cooperation with the MSEB, to develop needed standards for assessment. Our nation's drive towards standards-based education will not be complete until joined with appropriate assessment. Measuring Up is an important step in the process of encouraging widespread debate about the long-term direction of assessment in U.S. education. Frank Press, Chairman National Research Council

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Measuring Up: Prototypes for Mathematics Assessment Foreword The mathematics assessment prototypes in this document were developed by a writing group consisting of the following individuals: Lowell Carmony (Lake Forest College); Robert Davis (Rutgers University); Susan Jo Russell (TERC); Joan Rutherford (Haynesbridge, Georgia Public Schools); Lourdes Santiago (Boston Public Schools); and Paul Shoecraft (University of Houston at Victoria). The group began its work in the summer of 1991 and was involved in pilot testing of tasks throughout the project. On behalf of the MSEB, I want to thank each member of the writing group for his or her thoughtful contributions to this challenging task. The tasks themselves, without the surrounding explanatory materials, were read by a number of people with long-standing interest in mathematics assessment: Philip Daro (California Mathematics Project); Bonnie Hole (Princeton Institute for Research); Roberta Flexer (University of Colorado at Boulder); Jo Ann Mosier (Kentucky State Department of Education); Thomas Romberg (University of Wisconsin at Madison); Joel Schneider (Children's Television Workshop); and Jean Kerr Stenmark (Lawrence Hall of Science). We are grateful for their many thoughtful comments and insight. Nonetheless, the tasks that appear in this manuscript are the responsibility of the MSEB and do not necessarily reflect the position or views of these readers. We thank the many people who were involved in the pilot testing of the prototypes at various stages of the project: Alice Alston, Amy Boyd, Pam Brown, Vanessa Burns, Deanna Callahan, Lowell Carmony, Robert Davis, Doris Ann Edwards, Lisa Johnson, Carolyn Maher, Suzanne Rogers, Mary Russ, and

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Measuring Up: Prototypes for Mathematics Assessment Paul Shoecraft. We are also grateful to the principals, mathematics supervisors, and teachers associated with the trial classrooms, for allowing the pilot testing to interrupt their normal activities. Financial support for the project was provided by the National Research Council and the National Science Foundation. Additionally, the New Standards Project assisted with the first meeting of the Writing Group, which was held concurrently with a New Standards Project meeting in Snowmass, Colorado. We gratefully acknowledge the support of these organizations. We owe a special thanks to Edward Esty, who managed endless details of this complex project with patience and wisdom, and to Linda P. Rosen, who coordinated the overall effort as staff officer for the project. Finally, we thank the students who worked on these tasks. The written work of some of them appears as examples in this volume, and all of them provided invaluable feedback. Lynn Arthur Steen, Executive Director Mathematical Sciences Education Board

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Measuring Up: Prototypes for Mathematics Assessment Contents     INTRODUCTION         The Challenge   4     The Criteria   5     The Caveats   7     The Audience   8     The Prototypes   9     The Tryouts   12     The Format   13     The Protorubrics   14     The Standards   16     The Future   19     THE PROTOTYPES         Mystery Graphs   23     The Checkers Tournament   31     Bridges   43     Hexarights   53     Bowl-A-Fact   65     Point of View   75     The Quilt Designer   85     How Many Buttons?   95     The Taxman   101     Lightning Strikes Again!   115     Comparing Grizzly Bears and Black Bears   125     The Towers Problem   133     The Hog Game   141

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