Teaching and Learning Mathematics

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  • Dr. Terry BergesonState Superintendent of

    Public Instruction

    Using Research to Shift From the Yesterday

    Mind to the Tomorrow Mind

    Teaching and



    March 2000

  • Teaching and Learning MathematicsUsing Research to Shift From the YesterdayMind to the Tomorrow Mind

    Dr. Terry BergesonState Superintendent of Public Instruction

    Rosemary FittonAssistant Superintendent

    Assessment, Research, and Curriculum

    Pete BylsmaDirector, Research and Evaluation

    Beverly Neitzel and Mary Ann StineMathematics Specialists

    The Office of Superintendent of Public Instruction complies with all federal andstate rules and regulations and does not discriminate on the basis of race, color,national origin, sex, disability, age, or marital status.

    March 2000

  • About the Author

    This document was written by Dr. Jerry Johnson, Professor of Mathematics atWestern Washington University in Bellingham. He has a B.A. from AugsburgCollege, a M.S. from California Institute of Technology, a M.A. from theUniversity of California, Los Angeles, and a Ph.D. from the University ofWashington. Dr. Johnson began teaching at WWU in 1984 and currently teachesclasses in both mathematics and mathematics education. He is also part of theWWU faculty team working toward the integration of science, mathematics, andtechnology curricula. He can be reached by e-mail at Johnsonj@cc.wwu.edu.

    About This Document

    This document can be found on our Web site (www.k12.wa.us). A free copy ofthis document can be obtained by placing an order on the website, by writing theResource Center, Office of Superintendent of Public Instruction, PO Box 47200,Olympia, WA 98504-7200, or by calling the Resource Center toll-free at (888) 595-3276. If requesting more than one copy, contact the Resource Center to determinethe printing and shipping charges.

    This document is available in alternative format upon request. Contact theResource Center at (888) 595-3276, TTY (360) 664-3631, or e-mailerickson@ospi.wednet.edu.

    The contents of this document can be reproduced without permission. Referenceto this document would be appreciated. Funding for this project was provided bythe Excellence in Mathematics Initiative, a state-funded program supportingmathematics education. For questions regarding the content of this document,call (360) 664-3155.


    Various staff at the Office of Superintendent of Public Instruction helped preparethis document for publication. Pete Bylsma, Beverly Neitzel, and Mary Ann Stinereviewed the draft document and provided other assistance. Lisa Ireland andTheresa Ellsworth provided editing assistance.


    Chapter 1 Introduction 1

    Chapter 2 Overview of the Research: A Washington State Perspective 5

    Number SenseNumber and Numeration 5Computation 9Estimation 13

    MeasurementAttributes and Dimensions 15Approximation and Precision 17Systems and Tools 18

    Geometry SenseShape and Dimension 18Relationships and Transformations 19

    Probability and StatisticsChance 21Data Analysis 22Prediction and Inference 23

    Algebraic SenseRelations and Representations 26Operations 28

    Problem Solving 30Communication 33Mathematical Reasoning 35Connections 37

    Chapter 3 Mathematics in the Classroom: What Research Tells Educators 39

    Constructivism and Its Use 39Role and Impact of Using Manipulatives 40How Students Solve Word Problems Involving Mathematics 43Mastery of Basic Facts and Algorithms 47Use and Impact of Computing Technologies 49Culture of the Mathematics Classroom 52Impact of Ability Grouping 55Individual Differences and Equity Issues 56Teacher Attitudes and Student Attitudes 57Using Performance-Based Assessment 61

    Chapter 4 Other Research and Issues 64

  • Professional Development Programs for Mathematics Teachers 64Changes in How Teachers Teach Math and How Students Learn Math 68Next Steps: Using Research as Educators 69

    Step 1: Mathematics Teachers Accepting Responsibility for Change 69Step 2: The Reeducation of Mathematics Teachers 70Step 3: Mathematics Teachers in Their New Roles as Researchers 70

    Appendix A Further Explorations 73

    Appendix B List of References 75

    Index 101


    EALRs Essential Academic Learning RequirementsWASL Washington Assessment of Student Learning

  • Teaching and Learning Mathematics 1

    Chapter 1


    Welcome! In as friendly and useful manner as possible, our goal is to provide aresearch-based overview of the potential and challenges of teaching qualitymathematics (K12). Though the primary contexts are the Washington State essentialacademic learning requirements (EALRs) in mathematics and the correlatedWashington Assessment of Student Learning (WASL), each reader must interpret andreflect on the content within his/her own district or classroom situation. Without thisimportant step toward interpretation and reflection by each reader, this publicationbecomes yet one more resource to be piled on a shelf for reading on that rainy day thatnever seems to come in Washington.

    We are fully aware of the ominous nature of the word research and its associatedbaggage. The mere inclusion of the word in the title of articles or workshop offeringsoften causes teachers and administrators to look for an escape route, whether it isphysical or mental. Yet, our intent is to counter this attitude by constructing aresearch-based perspective that helps both teachers and administrators further themathematics education reform efforts in Washington at all grade levels. As CharlesKettering, an American engineer and inventor (18761958), once said:

    Research is a high-hat word that scares a lot of people. It neednt. It is rather simple.Essentially, research is nothing but a state of minda friendly, welcoming attitudetoward change going out to look for change instead of waiting for it to come. Research is an effort to do things better and not to be caught asleep at the switch. It is theproblem solving mind as contrasted with the let-well-enough-alone mind. It is thetomorrow mind instead of the yesterday mind.

    From Ketterings words, we pull the guiding theme for this book: to use research-basedinformation to support the necessary shift from a yesterday mind to a tomorrowmind in the making of the many decisions as to how mathematics is taught or learnedin Washington.

    An introductory road map through this text can be useful, especially for thosereadersreluctant to make the trip.

    First, we discuss what research in mathematics education can and cannot do.This section is important because it helps orient the tomorrow mind in apositive direction while also ensuring that teachers and administrators areaware of potential misuses of research.

  • Chapter 1 Introduction

    Teaching and Learning Mathematics 2

    Second, we overview some of the research results related to each of the essentiallearning academic requirements in mathematics. The key word here is some,as the volume of research available in mathematics education is quite large andvaried (in both quality and applicability). Though an attempt was made to sortthrough and select research results in a fair manner, the goal of supporting thetomorrow mind was always in full view. If we omitted mention of researchresults that you have found useful, we apologize for their omission and suggestthat you share them with your colleagues. Also, some of the research resultsmentioned may seem dated but was included because it contributes in somefashion to our current situation and concerns.

    Third, we address specific questions in mathematics education as raised byteachers, administrators, or parents. These questions range from the classroomuse of calculators or manipulatives, to the role of drill and algorithmic practice,to the best models for the professional development of teachers. In mostinstances, the research evidence is not sufficient to answer the questions raisedin a definitive manner. We suggest that even small insights or understandingsare better than teaching in the dark.

    And fourth, we outline a plan that a teacher, district, or state can follow tomaintain relevancy relative to this document and the issues it addresses. Thatis, the text should be viewed as another small step forward for WashingtonState teachers and administrators. Combined with the other efforts of the Officeof Superintendent of Public Instruction (OSPI), districts, administrators,teachers, and professional groups, these steps forward help us both gain andmaintain momentum in adopting the tomorrow mind in mathematicseducation.

    Given that road map, we ask you to now join us on this trip through the field ofmathematics education research and hope that you find the journey useful. As ourultimate goal is to support teachers and administrators in their efforts to improvestudent learning in mathematics, we know that an increased awareness of researchresults is an important form of support. Our apologies are offered if we have eithermisrepresented or misinterpreted the research results as reported. Also, we apologizein advance for any misleading interpretations or summaries of the research conclusionsof others; these lapses were not intentional.


    Think of the many things that can be investigated in mathematics education; it is easyto be overwhelmed. Four key ingredients can be identified:

    The students trying to learn mathematicstheir maturity, their intellectualability, their past experiences and performances in mathematics, their preferredlearning styles, their attitude toward mathematics, and their social adjustment.

    The teachers trying to teach mathematicstheir own