Teaching and Learning Mathematics ?· Teaching and Learning Mathematics Using Research to Shift From…

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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 understanding ofmathematics, their beliefs relative to both mathematics itself and how it is

  • Chapter 1 Introduction

    Teaching and Learning Mathematics 3

    learned, their preferred styles of instruction and interaction with students, theirviews on the role of assessment, their professionalism, and their effectiveness asa teacher of mathematics

    The content of mathematics and its organization into a curriculumitsdifficulty level, its scope and position in possible sequences, its requiredprerequisite knowledge, and its separation into skills, concepts, and contextualapplications.

    The pedagogical models for presenting and experiencing this mathematicalcontentthe use of optimal instructional techniques, the design of instructionalmaterials, the use of multimedia and computing technologies, the use ofmanipulatives, the use of classroom grouping schemes, the influences oflearning psychology, teacher requirements, the role of parents and significantothers, and the integration of alternative assessment techniques.

    All of these ingredients, and their interactions, need to be investigated by carefulresearch. Again, it is easy to be overwhelmed (Begle and Gibb, 1980).

    Our position is that educational research cannot take into account all of thesevariables. The result we must live with is acceptance that educational research cannotanswer definitively all of the questions we might ask about mathematics education. Atbest, we can expect research in mathematics education to be helpful in these ways:

    It can inform us (e.g., about new pedagogical or assessment techniques). It can educate us (e.g., about the pros/cons of using different grouping models). It can answer questions (e.g., about the potential impact of professional

    development models for teachers). It can prompt new questions (e.g., about the impact of using the Internet to

    make real-world connections). It can create reflection and discussion (e.g., about the beliefs that students and

    teachers hold toward mathematics). It can challenge what we currently do as educators (e.g., about our programs for

    accommodating students with differing ability levels or learning styles). It can clarify educational situations (e.g., about how assessment can inform

    instruction). It can help make educational decisions and educational policy (e.g., about

    student access to calculators or performance benchmarks).

    Yet, research in mathematics education can also be counterproductive or fall short ofwhat we would expect in these ways:

    It can confuse situations (e.g., about which math curriculum is the best). It can focus on everything but your situation (e.g., about your classroom, your

    specific students, and their learning of mathematics). It can be hidden by its own publication style (e.g., its scholarly vocabulary and

    overwhelming statistics). It can be flawed (e.g., about the interpretation of the research data). It can be boring and obtuse (e.g., its technical jargon, its overuse of statistics

    and graphs, and its pompous style).

  • Chapter 1 Introduction

    Teaching and Learning Mathematics 4

    Above all, despite the wishes of many teachers and administrators, educationalresearch cannot PROVE anything! At best, educational research provides informationthat the community of educators can use, misuse, or refuse.

    It is a well-established notion that research results tend not to be used by educatorsand at times are purposely ignored. For example, Reys and Yeager (1974) determinedthat while 97.5 percent of the elementary teachers frequently read general educationjournals, 87.5 percent of these same teachers seldom or never read the research-flavored articles. When asked why, 80 percent of the teachers replied with lack oftime or lack of direct classroom implications. In contrast and on a more positive side,Short and Szabo (1974) found that mathematics teachers at the secondary level weremuch more knowledgeable about and favorable toward educational research than theircolleagues in English and social science.

    The situation needs to change, as research results must be reflected on and integratedas an important part of the mathematics education plan and process in Washington.The entire education communitymathematics teachers, administrators, legislators,parents, and college mathematics educatorsmust take and share in the responsibilityfor this reflection process and integration of research results, whether it occurs at theindividual learner level, the classroom level, the district level, the university level, orthe state level. This resource text is designed to serve as a catalyst for promotingreflection, discussion, and problem solving within this education community, helpingthis same community continue to shift from the yesterday mind to the tomorrowmind in its approach to mathematics education.

  • Teaching and Learning Mathematics 5

    Chapter 2


    Each of the EALRs will be considered within the context of some known researchresults. The search for relevant research results was broad but not exhaustive. Themajority of the research results have been omitted (fortunately or unfortunately for thereader), with the few results selected being those that can whet your appetite andillustrate best how research can inform and educate the education community. When itwas appropriate, research results that are conflicting or complementary have beenjuxtaposed to prompt further reflecti...


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