DOCUMENT RESUME

ED 467 603 SE 066 623

AUTHOR Gardner, Maureen B., Ed.

TITLE Journeys of Transformation: A Statewide Effort by Mathematicsand Science Professors To Improve Student Understanding. CaseReports from Participants in the Maryland Collaborative forTeacher Preparation.

SPONS AGENCY National Science Foundation, Arlington, VA. Div. ofUndergraduate Education.

PUB DATE 1998-01-00

NOTE 156p.; For Part II, see SE 066 624. Editorial assistant wasDonna L. Ayres.

CONTRACT DUE-9255745

AVAILABLE FROM For full text: http://www.towson.edu/csme/mctp/Journeys/Journeys_of_Transformation.pdf.

PUB TYPE Collected Works General (020) Reports Research (143)

EDRS PRICE EDRS Price MF01/PC07 Plus Postage.

DESCRIPTORS Active Learning; Constructivism (Learning); EducationalChange; Elementary Education; Higher Education;Interdisciplinary Approach; *Mathematics Instruction; Models;*Science Instruction; *Teacher Education; *TeacherImprovement; Teaching Methods

ABSTRACTThis document presents the Maryland Collaborative for Teacher

Preparation (MCTP) faculty's reviews on instructional issues of differentdisciplines. Contents include: (1) "The Maryland Collaborative for TeacherPreparation"; (2) "Guiding Principles: New Thinking in Mathematics andScience Teaching"; (3) "Introduction: Parallel Journeys of Risk andReward" (Maureen B. Gardner); (4) "Elementary Geometry" (Karen Benbury); (5)

"Interdisciplinary Science and Mathematics: Geometry and the RoundEarth" (Kenneth Berg and James Fey); (6) "A Constructivist Approach to PlateTectonics" (Rachel J. Burks); (7) "A Mathematical Modeling Course forElementary Education Majors" (Don Cathcart and Tom Horseman); (8) "Adaptationof a Traditional Study of Enzyme Structure and Properties for theConstructivist Classroom" (Katherine J. Denniston and Joseph J. Topping); (9)

"The Air We Breathe: Is Dilution the Solution to Pollution?" (Thomas C.O'Haver); (10) "Constructing the Stream Community: The Ecology of FlowingWaters" (Joanne Settel and Tom N. Hooe); (11) "The Challenge of TeachingBiology 100: Can I Really Promote Active Learning in a LargeLecture?" (Philip G. Sokolove); (12) "A Shift in Mathematics Teaching:Guiding Students to Become Independent Learners" (Richard C. Weimer, KarenParks, and John Jones); and (13) "A Tale of Two Professors: FacultyProfiles." Appendices include: "References for Reading," "Highlights ofNational Recommendations," and "MCTP Advisors and Participants." (Contains 49references.) (YDS)

Reproductions supplied by EDRS are the best that can be madefrom the original document.

JOURNEYS OFTRANSFORMAI1ON

A Statewide Effort byMathematics and Science Professors to

Improve Student Understanding

U.S. DEPARTMENT OF EDUCATIONOffice of Educational Research and Improvement

EDUCATIONAL RESOURCES INFORMATIONCENTER (ERIC)

41A-document has been reproduced asreceived from the person or organizationoriginating it.Minor changes have been made toimprove reproduction quality.

Points of view or opinions stated in thisdocument do not necessarily representofficial OERI position or policy.

Case Reports from Participants in theMaryland Collaborative for Teacher Preparation

Funded by the National Science Foundation Division of Undergraduate Education

rBEST COPT AMIABLE

JOURNEYS OFTRANSFORMATION

A Statewide Effort byMathematics and Science Professors to

Improve Student Understanding

Maureen B. GardnerEditor

Donna L. AyresEditorial Assistant

Case Reports from Participants in theMaryland Collaborative for Teacher Preparation

Funded by the National Science Foundation Division of Undergraduate Education

3

Published in January 1998 by theMaryland Collaborative for Teacher Preparation (MCTP)

NSF Cooperative Agreement DUE 9255745

For information about ordering more copies ofJourneys of Transformation,

visit the MCTP World Wide Web site athttp://www.wam.umd.edu/-toh/MCTP.html.

Student handouts included hereinmay be downloaded from the MCTP Web site andduplicated without permission from the authors.

"This protftt

National:Opinions expendsand not necessa

t'Ati.

uppg1/4co, in part,

eundationose,,Ett the authorsof the Foundation

4BEST COPY AVAILABLE

Forego 1 III

TABLE OF CONTENTS

3

Context for the Journeys

The Maryland Collaborative for Teacher Preparation 7

Guiding Principles: New Thinking in Mathematics and Science Teaching 11

e Journeys

Introduction: Parallel Journeys of Risk and Reward 23

Maureen B. Gardner, MCTP Faculty Research Assistant

Elementary Geometry 29

Karen Benbuty, Bowie State University

Interdisciplinary Science and Mathematics: Geometry and the Round Earth 37

Kenneth Berg and James Fey, University of Maryland, College Park

A Constructivist Approach to Plate Tectonics 57

Rachel J. Burks, Towson University

A Mathematical Modeling Course for Elementary Education Majors:Instructors' Thoughts 71

Don Cathcart and Tom Horseman, Salisbury State University

Adaptation of a Traditional Study of Enzyme Structure and Propertiesfor the Constructivist Classroom 87

Katherine J. Denniston and Joseph J. Topping, Towson University

The Air We Breathe: Is Dilution the Solution to Pollution? 101

Thomas C. O'Haver, University of Maryland, College Park

5

Constructing the Stream Community: The Ecology of Flowing Waters 113

Joanne Settel and Tom N. Hooe, Baltimore City Community College

The Challenge of Teaching Biology 100: Can I Really PromoteActive Learning in a Large Lecture? 121

Phillip G. Sokolove, University of Maryland, Baltimore County

A Shift in Mathematics Teaching: Guiding Students to BecomeIndependent Learners 129

Richard C. Weimer, Karen Parks, and John Jones, Frostburg State University

A Tale of Two Professors: Faculty Profiles 139

Chemistry Professor Transforms EnergyHis Own 141

Profile of Pallassana Krishnan, Coppin State College

Student Questions Hold the Answer 145

Profile of Rich Cerkovnik, Anne Arundel Community College

Appendices

I. References for Reading 149

Bibliography Compiled by Genevieve Knight, Coppin State College,and John Layman, University of Maryland, College Park

II. Highlights of National Recommendations 152

Professional Standards for Teaching Mathematics

National Science Education Standards

American Psychological Association Learner-Centered Psychological Principles

III. MCTP Advisors and Participants 156

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FOREWORD

"lofty years ago, on October 4, 1957, the--t

Soviet Union launched the first artificialearth satellite, Sputnik 1, and Americanscience, mathematics, engineering, andeducation were changed forever. Spurred bymilitary and political competition with theSoviets, we began rapid escalation of ourinvestment in scientific research anddevelopment and in the educationalinfrastructure needed to provide brain powerfor those efforts.

The investment in both science and educationpaid admirable dividends. We soon matchedand then outstripped Soviet accomplishmentsin space and became world leaders in all majoraspects of scientific research. We revitalizedschool and university science and mathematicsprograms and produced outstanding studentsheaded toward careers in science, mathematics,and engineering.

As Cold War national security threatsdiminished over the past 15 years, they werereplaced by new concerns about world-wideeconomic competition. The challenges of newhigh-tech industrial and business competitionincreased demands for scientific literacy andtechnical skills in broader segments of ourpopulation. Public schools and universitieshave again been asked to enhance scienceeducation, but success has not been as easy toachieve.

The Problem

Although schools have increased enrollmentsin academic science and mathematics coursesfor all students, reports of national andinternational assessments indicatedisappointing progress of American students.The number of students continuing past highschool into undergraduate study has increased

dramatically over the past several decades, butmany of those students face extensive remedialwork in mathematics before they can beginwork toward degrees. An emerging body ofresearch on undergraduate science andmathematics education shows that evenstudents who are apparently successful incollegiate science and mathematics courseshave disturbing gaps and misconceptions intheir understanding of those subjects. Therecent National Science Foundation review ofundergraduate education concluded that:

America has produced a significant share ofthe world's great scientists while most of itspopulation is virtually illiterate inscience . . . Undergraduate science,mathematics, engineering, and technologyeducation in America is typically still toomuch a filter that produces a few highly-qualified graduates while leaving most of itsstudents "homeless in the universe."

(Advisory Committee to the NationalScience Foundation, 1996, pp. 3-4)

The challenges and shortcomings ofcontemporary science and mathematicseducation have led to dozens of majorconferences and policy advisory reports thatoutline directions for reform at the school level(National Council of Teachers of Mathematics,1989, 1991, 1995; American Association forAdvancement of Science, 1995; NationalResearch Council, 1995) and at theundergraduate level as well (MathematicalSciences Education Board, 1991; AdvisoryCommittee to the National ScienceFoundation, 1996). As a result, significantexperiments with new curricula and teachingin K-16 science and mathematics are usingcalculators and computers to explore newapproaches in traditional science and

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mathematics courses and to create newinterdisciplinary configurations of knowledge.These experiments are drawing on insightsfrom research on human learning to designnew models for teaching, and are testing newways of accurately assessing student skills andunderstanding.

The Response

Every major proposal to improve mathematicsand science education describes teachers as thecritical agent in the change process. Thus,changing the content or pedagogy of courses inelementary, middle, secondary, orundergraduate mathematics and scienceeducation requires increasing the knowledgeand classroom skills of teachers. In recognitionof this fact, the NSF initiated a program in1992 to support Collaboratives for Excellencein Teacher Preparation (CETP) that wouldsignificantly change teacher preparationprograms within a state or region and serve asnational models of comprehensive change. Thebasic premise of the CETP program was that:

The content and presentation of science,mathematics, engineering, and technologythat prospective teachers learn as part oftheir undergraduate and pre-certificationexperience will determine the quality of theirown teaching efforts; their interest inintegrating science, mathematics andtechnology in their classroom activities; andtheir ability to adopt and adapt creativeeffective teaching methods validated byrecent research on teaching and learning.

(Straley, 1996, p. vii)

In response to the NSF request for proposals,the teaching and research institutions of theUniversity System of Maryland joined withBaltimore City Community College, MorganState University, and three major Marylandpublic school systems to form such a teacherpreparation collaborative. Our Collaborative'splan to design, develop, implement, andevaluate a new kind of teacher preparationprogram for teachers of mathematics andscience in the middle grades was funded in thefirst round of NSF-CETP proposals.

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Over a five-year period of funded work, theMaryland Collaborative for TeacherPreparation (MCTP) engaged more than 120school and university faculty in professionaldevelopment activities that led to the creationand implementation of 89 new or modifieduniversity science, mathematics, and teachingmethods courses. When we developed thosecourses, our first aim was to help prospectiveteachers develop a confident understanding ofthe fundamental concepts, principles, andreasoning processes at the heart of science andmathematics, especially those that underlie theschool curriculum. Thus, we engaged inextensive cross-disciplinary discussions toarticulate our views about the fundamentalideas of mathematics and science and theconnections across traditional disciplinary lines.

The dominant lecture mode of undergraduateinstruction conveys neither an accurate imageof the way science and mathematics arecreated nor of the ways that the subjects aremost effectively taught, and learned.Therefore, our second aim in developing andrevising science and mathematics courses wasto create learning environments in whichstudents could actively investigate problemsthat help them construct personalunderstanding of key ideas.

While there is remarkable commonality indiagnoses of the problems and prescriptions forimprovement in school and undergraduatemathematics and science education,implementation of the reform proposals is farfrom a routine task. It is very hard to identify aset of scientific and mathematical principlesthat provide a necessary and sufficientfoundation for teaching; it is even moredifficult to create engaging instructionalactivities, that bring about understanding ofthose principles. In the face of those challenges,participants in the MCTP experienced someearly failures and some subsequent remarkablesuccesses. This publication reports theseexperiences, which we hope will encourageand inform others who are interested inimproving undergraduate science andmathematics, especially for prospectiveteachers.

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Credits

The accomplishments of the MCTP projectresulted from contributions by science,mathematics, and education faculty and keyadministrators in schools and universities acrossMaryland (see Appendix III). Our NationalVisiting Committee (see Appendix III) hasprovided us with timely and insightful adviceon goals and specific activities at all stages ofthe project. Financial support from theNational Science Foundation and theUniversity System of Maryland has allowed usto explore a variety of intriguing new ideas inteacher preparation.

Since the beginning of our work in 1993, theMCTP project has been guided by theenergetic and insightful leadership of ourexecutive director, Susan Boyer. Sheunderstands science and mathematicseducation and teacher education, and isrespected by educators throughout the State ofMaryland. The successes reported in thisvolume are a credit to her ability to translatethe ideas of our initial proposal and subsequentoperational plans into first-class activities thatengaged and transformed everyone involved inthe Collaborative.

The papers in this volume were contributed byMCTP faculty from a variety of disciplines.Selection of papers was guided by an editorialcommittee including Rebecca Berg, JosephHoffman, Joan Langdon, and Gilbert Ogonji.Coordinating and editing these diversecontributions has been no small task. Asmanaging editor, Maureen Gardner has done aremarkable job not only in helping the authorstell their interesting stories in engaging ways,but also in overseeing all phases of publicationdevelopment. Donna Ayres' exceptional skillsin editorial assistance, copy editing, andproduction management were vital both to thequality of the document and to its completionon deadline. In addition, other MCTP staffmembersKaren Langford, Carolyn Parker,and Amy Roth-McDuffiedeserveacknowledgment for their technical andeditorial support of the production effort.

As principal investigators for the project, wehave been challenged by the task of creating a

productive collaborative from institutions,disciplines, and individuals with a history ofcompetition rather than cooperation.Nevertheless, the experience has beenconsistently stimulating and rewarding. In thispublication, you will find both the spirit andthe substance of the work of the MarylandCollaborative. We hope the stories within willgive both stimulus and direction to innovativeefforts of your own.

MCTP Principal Investigators

James Fey

Genevieve Knight

John W. Layman

d(WA1/41)11

Thomas O'Haver

4,AS-1f, Jack Taylor

References

Advisory Committee to the National ScienceFoundation. (1996). Shaping the Future: NewExpectations in Science, Mathematics,Engineering, and Technology. Washington, DC:National Science Foundation.

American Association for Advancement ofScience. (1995). Science for All Americans(Summary). Washington, DC: AAAS.

Mathematical Sciences Education Board(1991). Moving Beyond Myths: RevitalizingUndergraduate Mathematics. Washington, DC:National Academy Press.

5

National Council of Teachers of Mathematics.(1989). Curriculum and Evaluation Standardsfor School Mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics.(1991). Professional Standards for TeachingMathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics.(1995). Assessment Standards for SchoolMathematics. Reston, VA: NCTM.

6

National Research Council. (1995). NationalStandards for Science Education. Washington,DC: NRC.

Straley, T H. and Woodin, T. (1996). Forewordby workshop organizers. In Millar, S. B. andAlexander, B.B. (Eds.). Teacher Preparation inScience, Mathematics, Engineering, andTechnology: Review and Analysis of the NSFWorkshop, November 6-8, 1994. Madison, WI:National Institute for Science Education.

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THE MARYLAND COLLABORATIVE FORTEACHER PREPARATION

The heart of this publication is a set of casereports and profiles of Maryland college anduniversity professors who have madesubstantial changes in the way they teachmathematics and science. Their work tocreate new courses and revise existing ones,for the sake of improving studentunderstanding, is one of several facets of astatewide program called the MarylandCollaborative for Teacher Preparation(MCTP). This section summarizes theMCTP's goals, program components, andaccomplishments.

The MCTP is an innovative, interdisciplinaryundergraduate program to prepare futureteachers of grades 4-8 who are confident inteaching mathematics and science and canprovide exciting and challenging learningenvironments for all students. The program hassought to:

O introduce future teachers to standards-basedmodels of mathematics and scienceinstruction;

O provide courses and field experiences thatintegrate mathematics and science;

provide internships that involve genuineresearch activities;

O develop the participants' ability to usecomputers as standard tools for research andproblem solving, as well as for imaginativeclassroom instruction (through training onhow to incorporate calculators,microcomputer-and calculator-basedlaboratories, and the Internet into theirinstructional practices);

O prepare prospective teachers to dealeffectively with the diversity of students inpublic schools today; and

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0 provide graduates with placement assistanceand sustained support during the criticalfirst years of their teaching careers.

BuiDding a Co00aboratuve

In the first year of the initiative (1993-94),project staff recruited 85 mathematics andscience faculty members from 10 collaboratingcolleges and universities. The faculty becamefamiliar with the project's goals throughprofessional development activities; designednew mathematics and science courses thatincorporated hands-on, cooperative learningactivities; and developed teams within eachinstitution. They also participated in sessions toraise awareness about diversity and equityissues. The use of technology has beenemphasized throughout the program in coursedevelopment and, through the project's listseryand Web site, as a central mode ofcommunication.

During its second year, 20 new MCTP courseswere offered and the first group of pre-serviceteachers was recruited. The project continuesto recruit freshmen and sophomores atcollaborating institutions who are interested inbecoming elementary or middle schoolteachers. The project also enlisted a cadre ofcommunity college faculty to provide insightsinto how to recruit students who transfer intothe University System of Maryland. This isespecially important, as a high percentage ofMaryland's teachers begin their collegeeducation in community colleges.

By the start of the third year, the Collaborativehad reached a new, more multifaceted level ofoperation. A clarity of mission and a sense ofcamaraderie were evident among theparticipants. In addition to ongoing work on

recruiting faculty and students and ondeveloping courses, key program activities wereunder way, including student researchinternships, mentor teacher training workshops,and teacher education research.

During the fourth and fifth years, facultymembers formed working groups to tackleremaining needs such as the development ofculminating "capstone" courses, an inductionprogram for graduates, and faculty professionaldevelopment plans.

New Courses

The MCTP has grown to include more than120 faculty members (listed in Appendix III) in13 Maryland colleges and universities (listed atthe end of this section). These professors havemade fundamental changes in 89 mathematics,science, and methods courses, having shifted toa more hands-on, interactive, student-centeredapproach to teaching. In contrast to thetraditional lecture format, these facultymembers employ cooperative learningstrategies and create environments in whichstudents explore mathematics and sciencequestions and discover the answers themselves.Although MCTP's central goal has been toimprove the science and mathematicseducation of future teachers of grades 4-8, itsreach extends to all students who take coursestaught by these "reformed" instructorswhichamounts to approximately 4,000 students persemester.

Research Internships

In the summer of 1995, 11 MCTP studentstook part in a pilot program of full-timeresearch internships in the "real world" ofscience and mathematics. The internship sitesincluded local museums, informal sciencecenters, and zoos, as well as researchlaboratories and federal agencies. A quote fromone student, who was an MCTP intern atNASA, helps to show the kinds oftransformations these students undergo as aresult of the internship experience:

I learned that research is not cut and dried. . . there's a lot of trial and error involved.

I also learned a lot this summer about my

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own learning. It involves so much more thanreading what someone else has found to betrue. It involves making connections,experiencing, doing, trying, and sometimesmaking mistakes. It's very important whenteaching that I remember thisthat mystudents can be involved in what they'relearning and that will be meaningful forthem.

Based on the success of the pilot year, 24additional interns were supported in thesummer of 1996 and 18 more in 1997.

Mentors for New MCTP Teachers

Another important component of the MCTP isthe preparation of mentor teachers who willguide MCTP students during their fieldexperiences and student teaching, as well asduring their first year of teaching. During thepast three summers, some 64 upper-elementaryand middle school teachers have been preparedto be mentors. They participated in two-week,intensive workshops to enhance theirknowledge and skills in areas such as coachingand mentoring pre-service teachers, usingtechnology in the classroom, and integratingscience and mathematics in lessons.

More than 40 MCTP students are expected tograduate in the spring of 1998; many of theseare among the first to complete a full, 4-yearprogram of MCTP courses as well as summerresearch internships. (Already, 30 students withfewer than four years of MCTP experienceshave graduated, and many are alreadyemployed in school systems in the region.)

Evaluation

Evaluation of the program is performed by acombination of internal evaluators and theproject's own research group. The formativeevaluation of the MCTP activities is carried outby the staff of CoreTechs, a private consultingfirm run by the former director of the Centerfor Educational Research and Development atthe University of Maryland, Baltimore County.A recent CoreTechs survey of 33 facultymembers shows a marked shift in howfrequently they used certain teaching strategiesafter joining MCTP. For example, compared to

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4

the use of these strategies prior to MCTP, therewas a two-to ten-fold increase in the numberof faculty members who regularly usedcooperative learning groups, hands-on learningactivities, constructivist methods, alternativeassessments by students and peers, learner-centered approaches, project-oriented learning,technology in the classroom, and other"reform" strategies.

From the perspectives of faculty and students,the MCTP Research Group continuallydocuments the unique elements of theprogram, particularly the instruction methodsthat model active, interdisciplinary teaching.Data collection strategies include regularsurveys of students in MCTP classes; audio-taped and videotaped interviews of MCTPfaculty and students; observations of selectedMCTP classes; and collection of coursematerials. To follow are the main areas ofresearch and some findings to date:

Student Attitudes and Beliefs aboutMathematics and Science. Compared withother teacher candidates, MCTP studentshold more positive attitudes towardsmathematics and science as well as morepositive beliefs about the nature andteaching of mathematics and science.

O Mathematicians and Scientists' Views of EachOthers' Disciplines. This study, whichexplored how these views impact classroomefforts to connect disciplines, found thatMCTP mathematics content faculty tend torefer to science as a discipline similar tomathematics in that it requires a sustained,focused study, while MCTP mathematicsmethods faculty tend to view science as acontext for doing mathematics. Moreover,MCTP science content faculty tend to viewmathematics as a tool that enables them tosolve problems in science, while MCTPscience methods faculty tend to viewmathematics both as a tool used in doingscience as well as a separate disciplineindependent of science.

O Benefits of Student Discussions on Pedagogy in"Reformed" Content Classes. A study of howfaculty attempt to model exemplaryteaching practices, and how their students

perceive those efforts, found that MCTPteacher candidates benefit from regularopportunities to discuss reform-basedpedagogy while taking mathematics andscience content classes. Discussions canoccur in those classes or in concurrentseminars.

0 Student Content Knowledge and Process Skills.Preliminary findings from a current studyindicate that students who have takenMCTP science and mathematics classesperform better than non-MCTP teachercandidates in assessments of science andmathematics content knowledge, scienceprocess skills, and depth of approaches toteaching science and mathematics topics.

For More Onfonroation

The MCTP has an award-winning World WideWeb site created by co-Principal InvestigatorThomas O'Haver. The site provides a projectsummary, essays on constructivism andeducation, types of technologies and softwareused by the project, a list of project courses,and much more. Readers are encouraged tovisit the site for more information about theMCTP project: http: / /www.wam.umd.edu/ toh/MCTP. html.

Paralcipatirrog Onstirautions

Listed below are the institutions of highereducation that participate in the MarylandCollaborative for Teacher Preparation. Alsolisted are the public school systems andresearch institutions that have been involved inshaping the program.

Nigher Education institutions

The University of Maryland, College ParkThe University of Maryland, Baltimore CountyBowie State UniversityCoppin State CollegeThe University of Maryland, Eastern ShoreFrostburg State UniversityMorgan State UniversitySalisbury State UniversityTowson UniversityBaltimore City Community CollegeAnne Arundel Community College

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Catonsville Community CollegePrince George's Community College

Public School Systems

Baltimore City Public SchoolsBaltimore County Public SchoolsPrince George's County Public Schools

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Research Institutions

The University of Maryland BiotechnologyInstitute

The University of Maryland Center forEnvironmental Science

The University of Maryland School ofMedicine

GUIDING PRINCIPLES: NEW THINKING INMATHEMATICS AND SCIENCE TEACHING

Compiled by James Fey, MCTP Project Director

Based on a Collaborative-Wide Effort to Define a Framework of Guiding PrinciplesLed by Genevieve Knight and John Layman, MCTP Co-Principal Investigators

f young people whose adult lives will beJispent entirely in the 21st century are to beactive and influential participants in theemerging social, political, economic, andscientific world, their education must providebroader and deeper knowledge of scientificconcepts, principles, reasoning processes, andhabits of mind than students typically acquiretoday. Unfortunately, there is convincingevidence that large numbers of students leavesecondary and collegiate education withinadequate information and understandingabout science and mathematics and littleinclination to apply or even value the methodsof those disciplines in the solution of problemsor in reasoning about important personal andsocietal questions. A recent National ScienceFoundation review of undergraduate educationconcluded that:

Too many students leave science,mathematics, engineering, and technologycourses because they find them dull andunwelcoming. Too many teachers enterschool systems under-prepared, withoutreally understanding what science andmathematics are, and lacking the excitementof discovery and the confidence and abilityto help children engage science, mathematics,engineering, and technology. Too manygraduates go out into the workforce ill-prepared to solve real problems in acooperative way, lacking the skills andmotivation to continue learning

(National Science Foundation, 1996, p. 4)

Faculty who are drawn to Journeys ofTransformation are already sensitive to the

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problem of providing more sophisticatedscientific and mathematical education to abroad and underachieving population. Thecentral question is: "If I want my students todevelop a deeper understanding of science andmathematics, where do I turn for new ideas?"In grappling with the same question, leadersand participants in the Maryland Collaborativefor Teacher Preparation (MCTP) found anumber of credible sources of guidance.

Over the past decade, mathematics and scienceeducators have engaged in spirited criticalexamination of current practice and haveundertaken creative research and developmentactivities. While the greatest energy has beenfocused on elementary and secondaryeducation, a significant number of college anduniversity faculty are now engaged in basicresearch and applied curriculum developmentprojects aimed at improving undergraduatescience and mathematics education. Theircritical and creative work is producingremarkably consistent recommendations andintriguing resources.

For the Maryland Collaborative as a whole, thechallenge was sorting through therecommendations and deciding which to adoptas a package that could be called "the MCTPapproach." For the individual instructors, agreater challenge was deciding whichinnovations to try in their own classrooms. Thissection summarizes the underlying principlesand recommendations that have guided theirjourneys of transformation. It describes: (1)curriculum issues, including content coverage,connections among disciplines, and materials

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(i

emphasizing scientific and mathematicalinvestigation; (2) instructional issues, includingconstructivism, active learning, hands-onexperiences, collaborative learning, and writingto enhance learning; (3) new approaches toassessment; and (4) technology as an impetusfor changing course goals and classroomactivities and for enhancing communicationamong faculty and students. (See alsoAppendix II for highlights of nationalrecommendations that guided MCTPparticipants.)

(Unica=For most scientists and mathematiciansconcerned with improving education, the firsttarget of attention is usually the content andorganization of topics in school and collegiatecurricula. Quite naturally, they ask themselvesand others whether current instruction focuseson the most important facts, concepts,principles, and methods of the disciplines andwhether those topics are presented in the mosteffective sequences for students with differentinterests, aptitudes, and prior achievement (e.g.Leitzel, 1991).

Less us More. Given the explosive growth ofscientific knowledge and the application ofthat new knowledge to a wide range of humanactivities, it is natural to expect thatrecommendations for reform would focus onthe addition of topics to existing courses andan acceleration of the pace at which thosetopics are covered. In fact, there are strongarguments for quite different answers toquestions about appropriate curriculum. Forexample, a report from the AmericanAssociation for the Advancement of Science(AAAS) Project 2061 argues that:

Present curricula in science andmathematics are overstuffed andundernourished. . . . They emphasize thelearning of answers more than theexploration of questions, memory at theexpense of critical thought, bits and pieces ofinformation instead of understanding incontext, recitation over argument, reading inlieu of doing.

(AAAS, 1990, p. xvi)

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The sentiments in that influential AAASproposal are echoed in many other recentdiscussions of K-16 curriculum issues.Scientists and mathematicians have urgedcritical reexamination of their disciplines toidentify the truly fundamental ideas andreasoning processes that most students mustmaster and to cut away details that areimportant only for specialists. Lynn Steen(1988) has spoken eloquently about anevolving view of mathematics as the science ofpatterns, and he has proposed a framework ofstructures, actions, abstractions, attitudes,behaviors, and attributes that are useful indescribing and reasoning about patterns ofmany different kinds. The Project 2061recommendations include similar themessystems, models, constancy, patterns of change,evolution, and scalethat are useful inthinking about science, technology, andmathematics in the worlds that we experienceand seek to understand.

While students undoubtedly will need somespecific factual knowledge and skills as a basisfor learning the proposed broader concepts and"habits of mind" (Cuoco, et al., 1996), acommon thread in recent curriculum advicesuggests that students would be better off if wewould "organize curricula around profoundexploration of a few basic ideas, rather thanbasic exploration of many profound ideas"(Stanley, as quoted in Millar & Alexander,1996, p. 65). This point of view is oftenexpressed with the catchy phrase, less is more.

Connections. The growth of scientific andmathematical knowledge has also beenaccompanied by increasing specialization inresearch fields. Science and mathematicscurricula in secondary school andundergraduate education tend to be organizedin ways that honor those specializations. Onthe other hand, recent developments havedemonstrated that progress on major scientificproblems usually requires integration ofprinciples and methods from several traditionaldisciplines. For example, problems inenvironmental science often require conceptsand strategies from the biological and physicalsciences and mathematics as well as significantinsights from economics, public policy, and law.

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The case for better-connected science andmathematics curricula rests on pedagogicalgrounds as well. For example, recent critiquesof school and undergraduate mathematics havepointed out that typical syllabi and textbooksconsist of hundreds of exercises "detached fromthe life experiences of students and seen bymany students as irrelevant" (MathematicalSciences Education Board, 1991, p. 17). A solidbody of research in cognitive sciencedemonstrates that learning with understandingrequires learning how concepts, principles, andprocedures are connected (Hiebert &Carpenter, 1992; Redish, 1994). Theseconnections can relate ideas within a subjectarea, across other subject areas, and to real-lifesituations in which the scientific ormathematical principles are at work.

Students can develop such interdisciplinaryperspectives through theme-based curricularmaterials such as Teaching IntegratedMathematics and Science (TIMS) at theelementary level, Event-Based Science in themiddle grades, and even the ApplicationsReform in Secondary Education (ARISE) andSystemic Initiative for Montana Mathematicsand Science (SIMMS) projects at the highschool level. (See references to follow forpublisher information on curricular materials.)Not surprisingly, as one gets into advancedsecondary school and undergraduate curricula,specialization by traditional discipline becomesmore common. However, most of the recentdevelopments in calculus and linear algebracurriculum materials give a prominent role tomodeling of scientific and economic situations(Tucker & Leitzel, 1995; Carlson, et al., 1997).Moreover, at the college level, NSF hassponsored the development of curricularmaterials that honor the less-is-morephilosophy and provide modules thatexemplify current views of teaching andlearning. One such project is Powerful Ideas inPhysical Science, published by the AmericanAssociation of Physics Teachers (College Park,MD). This project had a major influence onhow MCTP structured its summer facultydevelopment programs. Another example oftheme-based materials at the college level isChemistry in Context (Schwartz, et al., 1994),

described by Dr. Thomas O'Haver in hiscontribution to this publication.

Scientific and ilgathereaticai Thinking. Whilecurriculum design has traditionally focusedsolely on the selection and sequencing of topicsto be 'covered' in a course, there is growingsupport for the principle that how we teach isas important as what we teach. The NationalCouncil of Teachers of Mathematics (NCTM)articulates this notion in its Curriculum andEvaluation Standards:

Students' ability to reason, solve problems,and use mathematics to communicate theirideas will develop only if they actively andfrequently engage in these processes.Whether students come to view mathematicsas an integrated whole instead of afragmented collection of arbitrary topics andwhether they ultimately come to valuemathematics will depend largely on how thesubject is taught.

(National Council of Teachers ofMathematics, 1989, p. 244)

That NCTM position has been echoed in manyrecent comments on science curriculum andteaching as well. For example, at a 1994conference on the preparation of science andmathematics teachers, Jaleh Daie suggested:

Science is best learned as a way of knowing,not a s a collection o f facts . . . (C)ertainscientific facts are . . . needed before largerconcepts can be understood and ideas canbe constructed. (But) learning the process ofscience (logic, methods, quantitative skills,and cause and effect relationships) is moreimportant than having a collection of facts.This approach emphasizes attainment ofintellectual skills (rather than routinememorization) to reinforce student interestand increase motivation.

(Daie, 1996, p. 70)

This point of view implies that curriculummaterials should emphasize investigation andproblem solving more than reading andimitation of examples, and that students shouldhave an opportunity to experience authenticscientific research environments.

13

Destruction

When content goals and curricular structureshave been agreed upon, it is natural to turn toconsiderations of learning and instruction. Howdo students with various interests andaptitudes acquire the understanding we aimfor? What kinds of instructional materials andactivities stimulate that learning? Consistentthemes have emerged from a substantial bodyof research on these questions, including muchwork directly related to undergraduate scienceand mathematics education.

Coestrectivison. The traditional pattern of K-16mathematics and science teaching includesteacher explanation and demonstration of newideas and skills followed by guided and thenindependent student work on routine exercises.Some students are able to learn from thispattern of interaction with their teachers andcurriculum materials, but many areunsuccessful. Teacher-focused class meetingsinduce student inattention and, even whenteacher explanations are clear and complete,students often report frustration when theyface homework tasks away from the teacher'simmediate guidance.

In response to the well-known difficulties thatstudents have in learning science andmathematics, extensive research has attemptedto clarify the mechanisms of learning instudents of various ages and aptitudes. Overthe years, this research, from a combination ofcontent discipline and psychologicalperspectives, has led to a number of generaltheories about learning and those theories havethen been used as bases for models ofinstruction. The theories that currently holdmost promise for explaining and facilitatinglearning with understanding in science andmathematics are described by the general labelof constructivism.

With primary roots in the research andthought of John Dewey, Jean Piaget, and LevVygotsky, modern constructivism has severalkey tenets:

® Knowledge is actively created, not passivelyreceived.

® Students construct new understanding by

14

reflecting on and modifying their priorknowledge structures.

6 Knowledge is personal, an amalgam ofindividual interpretations of experiences andobservations that are shaped by priorconceptions and social interactions.

G Learning is a social enterprise in whichscientific ideas are established by membersof a culture through shared observations andsocial discourse involving explanation,negotiation, and evaluation.

From a constructivist perspective, the goal oflearning is the formation of internalrepresentations of objects and relationships.Those representations can then be manipulatedmentally to explain and make predictionsabout the structure of modeled situations.While there is little direct evidence of howrepresentations are stored in the brain, there isgeneral consensus that students will developeffective representations of mathematical andscientific concepts and principles only if theyactively examine those ideas in varied forms,on a continuum from tactile to symbolic.

Active Learning. Constructivism is basically atheory of knowledge and how knowledge isacquired and used. However, it has beentranslated in various ways to give guidelines formathematics and science instruction as well.The fundamental premise of constructivistlearning theory is that knowledge is acquiredonly through direct involvement of the learner.Theories of teaching built on that principlehave emphasized some of the followingobjectives:

O Instruction should engage students inexperiences that challenge their priorconceptions and beliefs about Mathematicsand science.

® Instruction should encourage studentautonomy and initiative. The instructorshould be willing to let go of control overclassroom discourse.

O The instructor should encourage the spirit ofquestioning by posing thoughtful, open-ended questions and encouraging discussionamong students.

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O Instruction should not separate knowingfrom the process of finding out.

O The instructor should allow studentresponses to drive lessons and seekelaboration of students' initial responses.

O The instructor should allow students somethinking time after posing questions.

A comprehensive listing of "Learner-CenteredPsychological Principles" developed by theAmerican Psychological Association can befound in Appendix II.

Of course, it is one thing to set these admirablegoals and quite another to create the desiredclassroom learning community. It seems fair tosay that development has just begun oninstructional materials (problems, laboratoryinvestigations, etc.)and classroom strategiesthat will constitute a "wisdom of practice"comparable to the patterns of traditionalinstruction that have been passed fromgeneration to generation of teachers. Furtherdiscussion of constructivist teaching and itsfoundation in theory and research on learningappears in a number of contemporary researchand expository journals and books (forexample, see Brooks & Brooks, 1993; NCTM,1990; Simon, 1995; Driver, et al., 1994; Redish,1994).

Hands-On Laboratory Experience. Interpretationsof constructivist learning theory also generallyinfer that the most effective path to conceptualunderstanding is one that proceeds from activeengagement with concrete embodiments toprogressively more abstract and efficientrepresentations. The implication for educationis that science and mathematics instructionshould make extensive use of interactivephysical materials and "real-world" data fromprimary sources. While laboratory experienceshave been standard features in secondary andcollegiate science instruction for years, therehave always been challenges to the authenticityof those experiences. They tend to be exercisesin demonstrating or confirming principleslearned from classroom lectures, rather thanopportunities to discover answers to genuinescientific puzzles. Until very recently,mathematics classrooms have only rarely

19

engaged students in genuine investigation ofdata or modeling results from experiments.

Over the past decade, creative scienceeducators have demonstrated ways to makelaboratory investigations the centralcomponent of instruction in physics. Thepervasive influence of computers andcalculators on mathematics has transformedmany school and college classrooms intolaboratory environments as well. Furthermore,calculator-and computer-based laboratoryequipment has facilitated a blending of sciencelaboratory and mathematical modeling/analysisactivity. Many innovative instructionalmaterials and case reports of laboratory-styleclasses are now available, particularly forcourses in physics, statistics, and calculus. Muchof that work began at the Technical EducationResearch Center (TERC) under the leadershipof Robert Tinker, and descriptions of thedevelopment are in reports from physicseducators (see Thornton & Sokoloff, 1990;Laws, 1991; Krajcek & Layman, 1992).Moreover, evidence that these methods reallywork is building (see Redish, et al., 1997).

Collaborative Learning. The dominant publicimage of a scientist or mathematician is that ofa solitary figure in a private world of laboratoryexperimentation or hypothetical reasoning.Indeed many stunning examples of scientificand mathematical discoveries have developedfrom one individual's sustained deep thoughtabout complex problems. However, progress inscience and mathematics is also cumulativethe work of any individual builds on theobservations, conjectures, and reasoning ofmany others with similar interests.Furthermore, the power of teamwork inproblem solving has been demonstrated notonly in science and mathematics but also in ourincreasingly complex and technicalenvironments in business and industry.

In addition to the payoff of collaborative workin professional science, mathematics, andengineering, recent research on learning andteaching has demonstrated the effectiveness ofengaging students in structured cooperativelearning activities. Consistent results suggestthat student learning will be impressive if the

15

classroom is organized to put students in smallgroups working on challenging problems.Common guidelines for teaching throughcooperative problem solving include (fromJohnson, Johnson & Smith, 1991, andDavidson, 1990):

Simultaneous Interaction. Students shouldinteract in team structures that worksimultaneously to optimize theirengagement during classroom time.

Positive Interdependence. Activities should bestructured to require contributions from allteam members.

Individual Accountability. While learningtakes place in a team environment, studentsmust demonstrate individual achievement.

Social Skill Development. Since learning issocially negotiated from multipleperspectives, students must develop thesocial skills necessary for effective groupinteraction. Those skills must be taught andmonitored along with academic objectives.

Reflecting. At the conclusion of an activity,groups should be brought together to sharetheir findings and reflect on their meaning.

Studies show that when these guidelines areeffectively implemented, students have higherachievement, increased retention, increasedmotivation to learn, increased ability to considermultiple perspectives, more positiverelationships with others, more positive attitudestoward learning, better self-esteem, andimproved social skills.

If one is interested in acquiring skill in teachingthat employs cooperative small-group problemsolving in science and mathematics, advice isreadily available (see, for example, Artzt &Newman, 1990; Davidson, 1990). Furthermore,new curriculum materials generally include avariety of group problem-solving activities thatstimulate positive interdependence, allowmultiple solution approaches, and pose reflectionquestions that help groups to articulateimportant scientific and mathematical principlesrepresented in the problems.

Writing. When educators ask representatives ofbusiness and industry about skills they look for

16

in future employees, they often mention theability to work effectively as part of problem-solving teams. But representatives of technicalfields also frequently mention the importanceof the ability to communicate clearly inspeaking and writing.

One of the truisms of education asserts thatone never truly understands a subject untilrequired to teach it to others. Thus it is notsurprising that active participation incollaborative problem-solving activitiesfacilitates learning. It also appears to enhancethe development of student communicationskills. Furthermore, one of the standard aspectsof laboratory and small-group problem-solvinginstruction is the writing of reports thatsummarize and reflect on group work.Experimental K-16 science and mathematicscurricula are now including substantial writingtasks, and they seem to help students tosolidify and become more articulate abouttheir knowledge (see, for example,Countryman, 1992).

Many science and mathematics teachers havebegun using regular journal writing as a tool forcommunication with their students. On a dailyor weekly basis, students are asked to writeabout their understanding of ideas in science ormathematics courseswhat they are confidentabout and what they are still puzzled aboutand their reflections on class activities. Formany students this sort of opportunity to writeabout their learning is a powerful tool fordeveloping understanding as well as a way tomake personal contact with their teachers.

Assessment

Closely related to questions of learning andinstruction are questions about techniques forassessing students' skills and conceptualunderstanding. In K-16 science andmathematics classrooms, the most commonstrategy for assessing student learning isthrough competitive, timed, written quizzesand tests that require individual students toanswer a collection of specific short questionsor to perform routine calculations to solvewell-defined problems. Some students do verywell in this sort of testing, but for many others,

20

the conventional testing paradigms do not giveaccurate readings of their knowledge.Furthermore, even students who are"successful" on standard tests often haveembarrassing gaps in their understanding of keyscientific and mathematical ideas.

As curriculum developers have focused moreon developing student skills and conceptualunderstanding and less on memorization androutine procedural skills, they have beencompelled to devise assessment approachesthat reflect the same shift in goals. For bothmathematics and science, many prototypes fornew assessment strategies are available (seeHestenes, Wells & Swackhammer, 1992;Mathematical Sciences Education Board, 1993;Schoenfeld, 1997).

To make student assessment more authentic,the tasks used in testing are increasingly set inrealistic, open-ended contexts; may involvesmall groups of students; and allow multiplesolutions or multiple paths to solutions.Responses to these assessment tasks are beingscored using a range of new techniques,including looking at the work holistically,rather than by analyzing it as a collection ofmany discrete "right/wrong" responses.Assessment of portfolios of student work,which show the development ofunderstanding over time, can also play a role ina more complete evaluation of each student'sprogress.

Advocates of group and portfolio assessmentstrategies often emphasize their interest inlearning what students do know as much aswhat they do not know. They also stress theimportance of making assessment an integralpart of instruction, not simply a sorting tool toidentify winners and losers in the game ofscience and mathematics learning.

As with other new ideas in mathematics andscience teaching, the opportunities forinnovative assessment are described innumerous publications. The NCTM haspublished a Standards volume focusing onassessment in mathematics (NCTM, 1995),the National Science Education Standards(NRC, 1996) contain similar guidance to

21

options in science assessment, and there arepractical suggestions in dozens of books (e.g.,Stenmark, 1991), journal articles, and Websites.

l'echnogogy

Curriculum, learning, teaching, and assessmentare long-standing concerns in mathematics andscience education. But consideration of newapproaches to those problems is influencedtoday by fundamental changes in thetechnology available for doing and learningabout science and mathematics. Fromcalculators and computers to video disks andthe World Wide Web, science and mathematicseducators can employ many new tools in theirteaching practice. These tools influence thechoice of content goals in mathematics andscience courses, strategies for organizingclassroom and laboratory instruction, andoptions for communication with students andcolleagues about course materials andassignments.

Changing Course Goes. When calculators andcomputers are available as standard tools formathematical problem-solving, it makes senseto rethink the goals for student learning inmathematics. Curriculum developmentprojects at every level are designing newmathematics courses that assume access toarithmetic and graphic computing softwarereducing attention to training students inpaper and pencil execution of routineprocedures and increasing attention tostrategies for intelligent use of electronic tools.

The explosion of information resourcesavailable on CD-ROM and World Wide Webnetworks raises similar questions about thebalance between conceptual learning andacquisition of specific facts in science courses.When one can search the libraries of the worldfrom a home computer terminal, what sort ofbroad understanding is needed to guide theretrieval of information from those resources,and what knowledge must still be retained"locally"? It is still too early to tell how scienceand mathematics education will use theseinformation resources most effectively, but agreat deal of activity is under way.

17

Changing Classroom Activities. Hand-heldelectronic technologies of various kinds arereshaping the possibilities for classroominstruction in science and mathematics infundamental ways. The popular calculator- andcomputer-based laboratory instruments (CBL)are supporting exciting new kinds ofinvestigations in which data are collected inreal-time and then represented and modeled innumerical, graphic, and symbolic forms. Theseelectronic tools for data collection and analysisare helping to integrate science andmathematics more deeply than ever before.Mathematical concepts are developed throughmodeling of real-life activities and scientificexperiments are analyzed mathematically witha variety of convenient statistical and algebraictools.

For example, motion detectors can be used tomonitor the movement of people or objectsand transform and record that information,usually in real time, directly into computers orcalculators. The computers or calculators thendisplay the data in tabular or graphic formats.Students can then analyze the data to studyposition, velocity and acceleration for theseexperiments. In addition, devices are beingused with computers and calculators thatmeasure force, temperature, pressure, voltage,current, and kinetic energy for a wide varietyof "real-time" experiments. Laboratory materialis available for these devices (see Thornton &Sokoloff, 1990).

When mathematics students have access tohand-held calculators with powerful numeric,graphic, and symbolic capabilities, they caninvestigate algebraic expressions, functions, andgeometric shapes in search of interestingpatterns. The mathematics classroom canbecome an experimental laboratory in whichstudents construct their own understanding ofkey ideas and then share results with otherstudents and the teacher as co-investigator. Theuse of multiple representations formathematical ideas opens doors to the subjectfor students who are more adept at visual ornumeric reasoning than the conventionalsymbol manipulation. These attractive optionsare being built into a range of new K-16curricula.

18

Communication. Electronic communication viae-mail and the World Wide Web has made theworld seem physically smaller butintellectually larger. Recent projects have alsoexplored the power of electronic informationtechnologies for creating scientificcommunities that join individuals from manyschools and geographic areassharingexperimental data and mathematical problemsacross state and national borders. Althoughwork has only begun on projects to provideaccess to rich data-bases and on-demandassessments, the promise is great. Many schooland university faculty are already using e-mailto communicate with their students andcolleagues, as well as using the World WideWeb as an information resource and as amedium for publishing class materials andstudent projects.

The Challenge and the Opportunity

It is easy to identify the problems of teachereducation in mathematics and science. Butreaching agreement on ways to proceed towardsolutions for those problems is a moreimposing challenge. It took science,mathematics, and education facultyparticipating in the Maryland Collaborative forTeacher Preparation nearly two years to reach acomfortable consensus on the principles thatwould characterize MCTP courses. The keychanges from traditional practice that weagreed would be required to reach our goalsare summarized well by the following table,adapted from Johnson et al. (1991, p. 7) andWright & Perna (1992, p. 35).

Science, mathematics, and education facultyfrom across Maryland accepted the challengeof creating courses that reflect these desireddepartures from traditional practice. Workingtogether across disciplines and institutions,they have developed, tested, refined, andimplemented dozens of new courses. Their casereports describe both successes and failuresalong the way. They also illustrate ways thatsuch collaboration can facilitate facultyprofessional development by forming aninvisible college of interacting scientists,mathematicians, and educators who stimulateand support each other in change.

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Traditional Courses MCTP Courses

Science and Mathematics for Some Science and Mathematics for All

Science and Mathematics as Separate andDistinct Subjects

Science and Mathematics as Part of anInterdisciplinary World

Many Topics Covered with Little Depth Few Topics Covered in Greater Depth

Focus on Factual Knowledge Focus on Conceptual Knowledge

Behaviorist Learning Theory Constructivist Learning Theory

Teacher Imparts Knowledge and StudentsReceive and Store it

Teacher is a Facilitator of Learning and aLearner, Too

Passive Students Active Students

Text-Based Instruction Hands-on/Minds-on Investigation

Confirmatory Investigations Problem-Solving Investigations

Teacher Demonstration Laboratory and Field Experiences

One-Way Communication Networks of Communication in a LearningCommunity

Limited Use of Technology Full Integration of Appropriate Technology

Competitive Learning Cooperative Learning

Testing to Assign Grades Multi-dimensional Assessment Integratedwith Instruction

References

American Association for the Advancement ofScience (AAAS). (1990). Science For AllAmericans. Washington, DC: AAAS.

Artzt, A. F. & Newman, C. M. (1990). How ToUse Cooperative Learning in the MathematicsClassroom. Reston, VA: National Council ofTeachers of Mathematics.

Brooks, J. G. & Brooks, M. G. (1993). In Searchof Understanding: The Case for ConstructivistClassrooms. Alexandria, VA: Association forSupervision and Curriculum Development.

Carlson, D., Johnson, C. R., Lay, D. C., Porter,A. D., Watkins, A., & Watkins, W. (Eds.) (1997).Resources for Teaching Linear Algebra.Washington, DC: The MathematicalAssociation of America.

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Countryman, J. (1992). Writing to LearnMathematics. Portsmouth, NH: Heinemann.

Cuoco, A., Goldenberg, E. P., & Mark, J.(1996). Habits of mind: An organizingprinciple for mathematics curricula. TheJournal of Mathematical Behavior, 15, (4), 375-402.

Daie, J. (1996). Response to Elizabeth Stage. InMillar, S. B. & B. B. Alexander (Eds.). Teacherpreparation in science, mathematics, engineering,and technology: Review and analysis of the NSFworkshop, November 6-8, 1994. Madison, WI:National Institute for Science Education, 69-71.

Davidson, N. A. (1990). Cooperative Learning inMathematics: A Handbook for Teachers.Reading, MA: Addison-Wesley.

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Driver, R., Asoko, H., Leach, J., Mortimer, E., &Scott, P. (1994). Constructing scientificknowledge in the classroom. EducationalResearcher; 23, 5-12.

Hestenes, D., Wells, M., & Swackhammer, G.(1992). Force concept inventory. PhysicsTeacher, 30,141-158.

Hiebert, J. & Carpenter, T. P. (1992). Learningand teaching with understanding. In D. A.Grouws (Ed.). Handbook of Research onMathematics Teaching and Learning. New York:Macmillan.

Johnson, D.W., Johnson, R.T. & Smith, K.A.(1991). Active Learning: Cooperation in theCollege Classroom. Edina, MN: InteractionBook Company.

Krajcek, J.S. & Layman, J.W. (1992).Microcomputer-based laboratories in thescience classroom. In Research matters to thescience teacher. NARST Monograph, 5,101-108.

Laws, P. (1991). Calculus-based physicswithout lectures. Physics Today, 44 (12), 24-33.

Leitzel, J. R. C. (Ed.). (1991). A Call forChange: Recommendations for the MathematicalPreparation of Teachers of Mathematics.Washington, DC: The MathematicalAssociation of America.

Mathematical Sciences Education Board.(1991). Moving Beyond Myths: RevitalizingUndergraduate Mathematics. Washington, DC:National Academy Press.

Mathematical Sciences Education Board.(1993). Measuring Up: Prototypes forMathematics Assessment. Washington, DC:National Academy Press.

Millar, S. B. & Alexander, B. B. (Eds.). (1996).Teacher Preparation in Science, Mathematics,Engineering, and Technology: Review andAnalysis of the NSF Workshop, November 6-8,1994. Madison, WI: National Institute forScience Education.

National Council of Teachers of Mathematics.(1989). Curriculum and Evaluation Standardsfor School Mathematics. Reston, VA: NCTM.

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National Council of Teachers of Mathematics.(1990). Constructivist views of the teachingand learning of mathematics. Journal forResearch in Mathematics Education, MonographNo. 4.

National Council of Teachers of Mathematics.(1995). Assessment Standards for SchoolMathematics. Reston, VA: NCTM.

National Research Council. (1996). NationalScience Education Standards. Washington, DC:National Academy Press.

National Science Foundation. (1996). Shapingthe future: New expectations for undergraduateeducation in science, mathematics, engineering,and technology (executive summary).Washington, DC: NSF.

Redish, E. F. (1994). Implications of cognitivestudies for teaching physics. American Journalof Physics, 62, 796-803.

Redish, E. F, Saul, J. M., & Steinberg, R. N.(1997). On the effectiveness of active-engagement microcomputer-based laboratories.American Journal of Physics, 65, 45-54.

Schoenfeld, A. (Ed.). (1997). StudentAssessment in Calculus. Washington, DC:Mathematical Association of America.

Schwartz, A.T., Bunce, D.M., Silberman, R.G.,Stanitski, C.L., Stratton, W.J., & Zipp, A.P.(1994). Chemistry in Context: ApplyingChemistry to Society. Dubuque, IA: Wm. C.Brown.

Simon, M. A. (1995). Reconstructingmathematics pedagogy from a constructivistperspective. Journal for Research in MathematicsEducation, 20, 114-145.

Steen, L. A. (1988). The science of patterns.Science, 240, 611-616.

Stenmark, J. (1991). Mathematics Assessment:Myths, Models, Good Questions, and PracticalSuggestions. Reston, VA: National Council ofTeachers of Mathematics.

Thornton, R. K. & Sokoloff, D. R. (1990).Learning motion concepts using real-timemicrocomputer-based laboratory tools.American Journal of Physics, 58, 858-867.

if

Tucker, A. & Leitzel, J. R. C. (1995). AssessingCalculus Reform Efforts. Washington, DC:Mathematical Association of America.

Wright, E.L. & Perna, J.A. (1992). Reaching forexcellence: A template for biology instruction.Science and Students, 30 (2), 35.

References (Curricadar lEateninOs)

Applications Reform in Secondary Education(ARISE). Cincinnati, OH: SouthwesternPublishing.

Event-Based Science. Menlo Park, CA: Addison-Wesley.

Powerful Ideas in Physical Science. College Park,MD: American Association of Physics Teachers.

Systemic Initiative for Montana Mathematicsand Science (SIMMS). Needham, MA: Simon-Shuster Custom Publishing.

Teaching Integrated Mathematics and Science(TIMS). Institute for Mathematics and ScienceEducation. Chicago: University of Illinois atChicago.

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ct

4;

INTRODUCTION:PARALLEL JOURNEYS OF RISK AND REWARD

Maureen B. GardnerMCTP Faculty Research Assistant

University of Maryland, College Park

e teach the way we were taught," theadage goes. This is no longer true for

dozens of mathematics and science professorsin Maryland who have taken part in theMaryland Collaborative for TeacherPreparation (MCTP), a statewide experimentto improve student understanding. Thesefaculty members had the courage to take risks,to transform their teaching practices, to trynew and vastly different methods ofinstruction.

Breaking long-held traditions has beendemanding for the faculty and their studentsalike. But it has also been rewardingmanystudents are grasping mathematics and scienceconcepts in deeper, more meaningful ways thanthey ever have before. For the professors, whocare profoundly about their areas of expertise,this payoff has made the extra effortor whatone even called "the agony"of changing theirteaching approaches worthwhile.

In their own schooling days, these facultymembers learned through lectures, drills, andstep-by-step laboratory exercises. Figuring thatthese methods "worked for them," they taughtthat way themselves, some for 20 to 30 years.But what "worked for them" wasn't workingfor many of their students. The professors weredisturbed by their students' lack of interest,meager participation in class, and inability toconnect concepts across disciplines. Too manystudents just weren't "getting it" and didn'tseem to care. As one mathematics professorput it, "the overall atmosphere of my classroombordered on 'siesta time.'"

Now these professors have shelved theirlecture notes, preferring to have studentsdevelop concepts through hands-on, problemsolving activities done in small groups. Theymight cover less material, but do so in moredepth. Generally they refrain from answeringstudent questions directly; rather, they guidestudents to finding their own answers throughdiscussions and investigations. No longer "siestatime," classrooms buzz with activity anddiscovery. Instead of simply testing students'abilities to recall memorized material, theprofessors assess students' progress by havingthem apply knowledge in new situations.Recognizing that learning takes time andreflection, they have students write in journalsto express their understandings,misunderstandings, and thoughts on theirlearning.

The net result has been a quiet revolution ofinstructional change in Maryland, involvingmore than 120 science and mathematicsfaculty members in 13 institutions who havemodified 89 courses in the past three years.Their individual journeys of transformationhave taken different twists and turns, but thereare many parallels in the risks they took andthe rewards they reaped.

Journeys of Transformation captures the storiesof 17 of these professors. This introductorysection previews some highlights of thesejourneys, pulling together the commondifficulties and the payoffs. It also describes thescope of this movement in Maryland and itsnational roots, why many of the professorsjoined this innovative program, how they were

26

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introduced to the new concepts, and why theystuck with it when times got tough.

For any mathematics or science professor whois concerned about students' understanding,and who has the courage to depart fromtradition, Journeys of Transformation willprovide practical advice as well as inspiration.

C00000

Until the MCTP program came along, manymathematics and science faculty in Marylandprobably did not know where to turn foradvice in improving their students' depth ofunderstanding. Fortunately, the groundwork forreform in Maryland was laid in the precedingyears by movements at the national level thatseeded this program and provided guidelinesfor its cultivation.

Action at the National Level

At the national level, the problem of "studentsnot getting it" fueled a growing concern aboutundergraduate education in mathematics andthe sciences. In particular, poor US rankings ininternational mathematics and science tests ofgrade school students called into question thecollege preparation of K-12 teachers in thesesubjects. Too few teachers, especially at theelementary and middle school levels, had morethan a cursory exposure to mathematics andscience as part of their undergraduate work.Too few teachers had college courses thatconnected the science disciplines andmathematics, or engaged students in theexcitement of the science process, orincorporated technology such as computersand graphing calculators into lessons. And toofew teachers were prepared to help anincreasingly diverse population of students tobecome critical, independent thinkers. TheNSF responded by initiating large-scaleprograms such as MCTP to reform the ways inwhich college mathematics and science aretaught to teachers-to-be.

Also at the national level, mathematics andscience education communities mobilized inrecent years to develop standards for teachingthese subjects at the K-12 levels. Keyelements of the new standards call for

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0

teachers to actively engage students in theirlearning, to connect mathematics and sciencein "real world" problem solving, and tointegrate technology into instruction. Thesestandards and other recommendations fromthe mathematics and science educationcommunities became guidelines for theMCTP faculty as they revised their coursesand teaching strategies. (For more details seethe section "Guiding Principles" andAppendix II.)

Maryland Breaks the Cycle

Like most school systems nationally,Maryland's educational leadership wants itsK-12 teachers to be guided by the newmathematics and science education standards.So a central goal for MCTP was to encouragecollege and university faculty members toadopt these "standards-based" teachingmethods as well, particularly when they taughtfuture teachers. This way, the old cycle of"teaching the way we were taught" would bebroken, and replaced with a new one thatrevolved around active learning.

To help faculty members make this changeadrastic one for manythe project heldsummer workshops for three years, starting in1993. Faculty members took part in manyhands-on sessions on topics such as integratingmathematics, science and technology inlessons; cooperative learning strategies;alternative assessment techniques; and other"reform-style" teaching methods. Throughreadings and discussions, the participants wereintroduced to the philosophy of"constructivism" that underscores the newmethods. The philosophy recognizes thatstudents do not learn passively but insteadactively construct knowledge, filtering newinformation through earlier learningexperiences, values and attitudes.

Not all faculty members bought the "wholepackage" of reform-style teaching, at least notat first. Each needed to sort through the newinformation and decide which elements toincorporate into his or her instructionalstrategy. Over time, though, many of themmade significant shifts in their teaching

methods. Here are some examples fromcontributors to Journeys of Transformation:

At Frostburg State University in themountains of western Maryland, cooperativelearning has replaced lecturing as thestandard mode of operation in themathematics classes of professor DickWeimer and his colleagues. No longer justnote-takers, students actively exploremathematical concepts by working in smallgroups on hands-on activities.

At Coppin State College in inner cityBaltimore, chemistry professor PallassanaKrishnan resists the urge to answer studentquestions. Instead, he opens the questions toclassroom discussions, from which theanswers "evolve." "It takes a little time," hesays, "but then this learning takes place,which is exciting."

At suburban Towson University, geosciencesprofessor Rachel Burks reverses thetraditional method of introducing the topicof plate tectonics. Instead of telling studentsthe major land mass movements and givingsupporting examples, she has createdactivities in which students discover andconstruct key concepts themselves. Forassessment, Dr. Burks has the students applytheir new knowledge to explain relatedphenomena that they've never discussed inclass.

At the University of Maryland, BaltimoreCounty, biology professor Phillip Sokoloveincorporates small group activities into hislecture classes of 240 students, having themdevelop responses to question prompts. Nolonger nameless and voiceless in a lecturehall, his students wear name tags so that allcan be identified, and use wirelessmicrophones to contribute to large groupdiscussions.

At Bowie State University, students in KarenBenbury's elementary geometry coursecommunicate mathematical ideas not onlywith numerals and other symbols, but also bywriting in journals. Her students often have"aha" experiences upon writing theirreflections, while Dr. Benbury herself gains

2

unprecedented insights from journal entriesin which students convey both theirfrustrations and their breakthroughs incomprehension.

0 On the eastern shore of the Chesapeake Bay,professors Don Cathcart and Tom Horsemanresist the natural tendency to correctstudents instantly when they make faultygeneralizations in a mathematical modelingcourse. Instead, these Salisbury StateUniversity professors take the time to devisenew activities that allow students torecognize their misconceptions themselves.

Having made these and other remarkablechanges, the professors tend to stick with them.A 1997 survey of 33 MCTP faculty membersshows that they have not only markedly shiftedtheir instructional strategies, but they have alsoretained the changes over the three years or sosince revising their courses. Many even employsome of the new methods in their upper levelcourses for studentsmajoring in scienceand mathematics. Theprofessors have "madea commitment to thenew way of providinginstruction," accordingto project evaluatorGilbert Austin, whoconducted the survey.He adds, "It's a veryimpressive finding."

In the Classroom:Adjustment Pains

When the first MCTPcourses were offeredin the fall of 1994, thenew courses posed many challenges forprofessors and students alike. For theprofessors, it generally took more time to planactivities that actively engaged students, and tofigure out ways for students to discover thingson their own instead of just "telling them."Responding to student journal entries anddevising new methods of assessment tookadditional time and thought as well. Acommon source of tension was the urge to

A 1997 survey of33 MCTP faculty

members shows thatthey have not only

markedly shifted theirinstructional strategies,

but they have alsoretained the changes

over the three years orso since revising

their courses.

25

cover the traditional amount of content despitethe greater time needed for activities thatpermit students to develop deeperunderstandings. As stated by Tom Hooe andJoanne Settel, biology professors at BaltimoreCity Community College:

We realized early on that we were not goingto be able to cover all the content we hadoriginally planned. . . . We came to therealization that simply "covering" materialdoes not improve student learning. Instead,we adopted the 'less is more' concept, whichis hard for many professors to accept. Butwe believe that when assessments show thatstudents aren't learning effectively when youjust 'cover' the information, then you mustsearch for new ways to facilitate studentunderstanding.

Students, too, must make their own journeys oftransformation in these new courses. At first,most students expected professors to respondimmediately to questions; few were used totaking the initiative to find answers themselves.Some found it very difficult to adjust whenprofessors basically handed the questions back tothem. It was not uncommon to hear commentssuch as, "I would just kill for a formula." A fewstudents even responded with hostility, assertingthat they were not getting what they (or, moreoften, their parents) paid for.

As semesters progressed, however, the studentsbegan to accept responsibility for theirlearning. Having shifted their expectations,they typically became active learners andwilling participants. In later MCTP courses,many became role models for reluctantstudents with no prior experience in activelearning classrooms.

The Payoff: Discovery and Deep Understanding

As MCTP professors gained proficiency in newmethods, and as the students graduallyaccepted responsibility for their learning,wonderful things began to happen. "Aha"experiences occurred during hands-onactivities and discussions that could not havehappened when lecture format was used.Students gained understanding of science andmathematics concepts, often building the

26

29

confidence to overcome deep-seated fearsabout the subjects. Through reflective writingin journals, students gleaned insights into howthey learn, an invaluable experience for futureteachers. Some quotes from student journals:

I am beginning to lose some of the anxiety Ifirst had about venturing into the unknown.The labor involved in our pursuit is wherethe real learning occurs and this seems farmore valuable to me as a future educatorthan merely obtaining the actual "answers"to this assignment.Frostburg StateUniversity mathematics student

Before taking this course, the only way Icould identify a rock or mineral wasbecause I had seen it before andremembered what it looked like. Now I canlook at a rock I've never seen before andfind identifying characteristics in it. To me,it's so neat to be able to do that. And if I'mexcited about it, and find it fun, when I getinto the classroom I can pass thatexcitement on to my students.TowsonUniversity physical sciences student

After completing the paper onconstructivism, I feel that it is the bestpossible way to teach math wheneverpossible. Sometimes there is more than oneway to reach a solution, and by letting thestudent try by trial and error, it builds theirconfidence and critical thinking skills. Ithink that is a part of how this course wastaught, and it worked! It has helped me overa large part of my 'math phobia'!"BowieState University geometry student

Student responses such as these, many ofwhich are included in Journeys ofTransformation, serve as strong reinforcementto professors who have taken the risk to trysomething new. Support has also come fromstudies conducted by the MCTP's researchgroup, which is co-directed by scienceeducation professor J. Randy McGinnis at theUniversity of Maryland, College Park andmathematics education professor TadWatanabe at Towson University. The group hasfound that, compared to other teachercandidates, those who have taken MCTPcourses hold more positive attitudes towards

mathematics and science as well as morepositive beliefs about the nature and teachingof mathematics and science. Preliminary resultsfrom new studies indicate that MCTP scienceand mathematics courses do, as expected,enable students to develop deeper contentunderstanding as well. (Visit the MCTP Website for updates on MCTP research: http://www.inform.umd.edu/UMS+State/UMD-Projects/MCTP/WWW/MCTPresearch. html)

Re-Writing the Textbook

Thanks to their professors' student-centeredmethods, some MCTP students evendeveloped enough confidence in their contentunderstanding to make improvements in acollege laboratory text. At Towson Universityin 1995, in an introductory course thatintegrated biology and chemistry, studentsstruggled with some enzyme experiments thatdid not deliver clear-cut results. Instead ofstepping in with a standard protocol to savethe day, professors Katherine Denniston andJoe Topping, who team-taught the course, hadthe students discuss the problems in depth.The students were able to recognize flaws intheir abilities to observe, describe results, andkeep records, which led them to devise a muchmore reliable and efficient method.

Even more remarkable, the students were ableto draw upon their own newly developed, butsolid, understandings of chemical structures tosuggest that a different chemical be used toreplace one that they suspected had cloudedthe initial experiments. The students'modification to the experiment was sosuccessful that Dr. Denniston discussed it withthe editor of their text, and it is nowincorporated into the accompanying labmanual. Says Dr. Denniston, "This was veryimportant reinforcement of their sense thatthey have the ability to contribute significantlyto the sciences."

Courage and Community Suppora

Although joining the MCTP program offeredprofessors hope for improved studentunderstanding, it also guaranteed transitiondifficulties. So why did so many professors join

30

MCTP, and why did they persevere? Asmentioned previously, many were dissatisfiedwith the status quo, with their students not"getting it." Others had resigned themselves tothe lack of student participation, but joinedMCTP after being coerced by colleagues orperhaps lured by the prospect of a summerstipend. Some professors signed up a few yearslater, after noticing excited, engaged students ina neighboring classroom taught by an MCTPparticipant.

Whatever the motivation to join, sticking withthe program and making changes in long-heldteaching practices was risky. It took courage forthese professors to try something new anddrastically different, and to continue in the faceof uncertainty and initial student resistance.The urge to give up and go back to the "oldways" was often strong and compelling.Fortunately, the MCTP program offered solidsupport from a community of kindred spirits.

Chemistry professor Pallassana Krishnan, forexample, said it took him one entire, difficultsemester to adjust to a student-centeredapproach after more than 20 years of lecturingat Coppin State College. "The first time Itaught an MCTP course it was a disaster," saidDr. Krishnan. "If the professor is confused, thestudents sense that so easily, and that causes alot of disturbance in the students." When askedwhy he didn't just "bag it" when times gottough, he replied, "'Bagging it' flew through mymind several times, but bag it and do what? Ifyou go back to the old way, that isn't anybetter; it's equally bad."

Dr. Krishnan persevered, and the followingsemesters got better and better. Now he says,"Even if someone were to question it, I stillwould not go back to the lecture. The studentsare learning and excited; that's all I need." Hesaid he survived his tumultuous start because"MCTP was the backing; I know at least thatsomebody was behind me to help me out." Hisinstitution's formal support of this programand this new style of teaching was essential.Beyond that, the "somebody to help out"included five other science and mathematicsfaculty members at his own institution whoparticipated in MCTP, as well as dozens of

27

other participants in his field and otherdisciplines from across the state.

In the early project years, as all the facultymembers delved into the job of re-structuringtheir courses, they came to each others' aid inmyriad ways, offering both practical assistanceand moral support. Although professors fromdifferent disciplines and institutions initiallyheld each other suspect, this distrust eventuallygave way to mutually beneficial relationshipsthat have endured since. Many conversed onthe project's listserv, helping each other designinterdisciplinary courses, locate resources, andsort out reactions to class activities. Somedecided to form partnerships for team-teaching. When statewide "course de-briefings"

were held at the endof each semester, theyshared inspiring storiesof their challenges andtriumphs.

It took time, but acommunity withunprecedented cross-disciplinary and cross-institutional tiesformed. A sense ofsolidarity took hold."In the long run, oneof the most markedmeasures of thesuccess of this projectis this collaboration,"according to Dr.Austin, the projectevaluator. "It really haspanned outpeoplein differentinstitutions now knoweach other as

colleagues, and not just in a passing way but ina working way. I think that will remain andcontinue to be fostered."

"In the long run,one of the most

marked measures ofthe success of this

project is thiscollaboration. It really

has panned outpeople in different

institutions now knoweach other as

colleagues, and not justin a passing way butin a working way."

Dr. Gilbert Austin

Better Undergraduate Science and MathematicsEducationFor All

The MCTP program came into existence toimprove teacher education in mathematics andscience, but its effects will extend far beyond

28

its initial intent. In the coming years, theprogram will meet its goal of producinghundreds of new teachers for elementary andmiddle schools who will know first-hand thethrill of discovery, the connections betweenmathematics and science, and ways to enhancelearning with technology. Their confidence andexpertise will benefit countless school children,enabling them to understand their world byseeking their own answers and, as they grow, toparticipate more fully in a society increasinglyinfluenced by science and technology.

Beyond this vital goal, the program will benefitthousands of undergraduate students who donot intend to teach, but who still need betterinstruction in science and mathematics. Arecent NSF report, Shaping the Future, calls forcolleges and universities throughout the nationto improve the way mathematics and scienceare taught to all students. Noting the pressingneed for a more scientifically literate public,the report urges science and mathematicsprofessors to "model good practices thatincrease learning" (Advisory Committee to theNational Science Foundation, 1996, p. iv). Thereport's recommendations align directly withthe teaching strategies now used by MCTPprofessors not only in their classes for futureteachers, but also, in many cases, in their classesfor other undergraduates.

Thus, the MCTP program has given Marylandan enormous head start in meeting the nationalgoal of improving undergraduate education inmathematics and science for all students. In theyears to come, more and more students willbenefit from deeper understandings, as MCTPprofessors today are spreading the word totheir colleagues about the risks, the rewards,and the wisdom of no longer "teaching the waythey were taught."

Reference

Advisory Committee to the National ScienceFoundation. (1996). Shaping the Future: NewExpectations in Science, Mathematics, Engineeringand Technology. Washington, D.C.: NationalScience Foundation.

31

ELEMENTARY GEOMETRY

Karen BenburyDepartment of Natural Sciences, Mathematics and Computer Science

Bowie State UniversityBowie, MD 20715

When Bowie State University decided in1994 to require a geometry course of

all elementary and early childhood educationmajors, I created "Elementary Geometry"under the auspices of the MarylandCollaborative for Teacher Preparation(MCTP). The course was first offered in thefall of 1994; as of this writing I have taught itthree times. This paper describes the coursecontent, the difficulties and rewards I haveencountered in changing my teachingapproach, and student reflections on theirexperiences.

Course Content and Evaluation

In developing the course content, I consultedthe NCTM Standards (NCTM, 1989) as wellas my colleagues in the MCTP program, whohad many helpful suggestions. The outline ofthe new course included critical thinking andlogical reasoning, basic ideas of geometry(points, lines, planes, triangles, polygons, etc.),some theorems from Euclidean geometry,symmetry, tessellations, geometry in 3D-space,measurement, motions, similarity, topology, andsome non-Euclidean geometries. In addition,since one MCTP goal is to incorporatetechnology wherever possible, I planned toinclude a little LOGO and Geometer'sSketchpad. (As described later, not all topicswere covered, and difficulties with technologyat our institution at the time allowed us to doonly a little with it.)

For evaluation, I decided to base half of eachstudent's grade on assignments done in and outof class, individually and in groups.Assignments included logic puzzles, a worksheet entitled "What is a Math Fact?" (in which

I tried to communicate the idea thatmathematics is almost alone in requiringformal proof to establish a statement'svalidity), geometric constructions usingcompass and straight edge, a symmetry unit,and a unit on Eratosthenes' measurement ofthe Earth's circumference. Students also wrotea short paper on constructivism, and appliedwhat they learned to produce a lesson plan fora mathematics topic for a grade level of theirchoice. Since most students had not yet takentheir mathematics methods courses, the pointof the lesson plan was entirely to demonstratethe salient features of a constructivistapproach. In addition, the students keptjournals, which I read a few times during thesemester. I purposely required a lot of writing,since many prospective teachers seem to feelcomfortable with that mode of expression, andalso because I felt they needed experienceexpressing mathematical ideas and relations.

The other half of the grade was based on thestudent's performance on tests. Two in-class,one-hour tests covered material from the textand handouts. I tried a new assessment strategyin which 25% of each exam was done as agroup. The groups were comprised of studentswho finished the individual part of each examat roughly the same time. Although I hadconcerns that this arrangement might result inthe brighter students automatically workingtogether, in practice this did not happen. Thefinal was a take-home, cumulative exam, whichincluded some essay questions.

Approach to Teaching

Like many college professors, my mainexposure to teaching has been in the form of

29

lectures. Some of the best lecturers I haveheard use a question-and-answer format inwhich students are naturally led to asking theappropriate questions or noting similarities toprevious ideas. I have always tried newapproaches with this type of presentation inmind.

Before I joined the MCTP project, I had a littletraining and experience in group learning andsome types of active learning. I had been toworkshops in which group work and learningstyles were explored; these were often led byexperienced people who made it look easy. Theworkshops gave the participantsfacultymembers like myselfthe experience ofworking in a group that really does cooperate.They did not, however, provide the experienceof facilitating such work among students, whoare not necessarily cooperative.

When I had tried these methods in theclassroom, I found itconsiderably harder toinduce students towork together on asustained project. Thesame factors thatcaused failure in other

situationsmissing classes, not performingassigned tasks, and lack of respect for theopinions of their peersled to problems withgroup performance. Also, to many of thestudents, working in groups felt partly likecheating, and partly unfair. (One commoncomplaint went along the lines of: "If thegenius were in my group, I would get a bettergrade also.") At first, my own inexperience infacilitating group learning was a drawback, too.With practice, however, and with contact withother MCTP faculty who had been verysuccessful with cooperative learning, I startedto get better at it. As described later, dependingon the mix of students in my classes, Ieventually achieved a good deal of success withgroup work.

Through MCTP I also gained exposure toteaching in the constructivist vein. What Ifinally attempted, however, was only partiallyconstructivist, at least as I understand the term.I decided to start new material either with a

A decision to trya new approach is not

an easy one.

30

class discussion to elicit what students alreadyknew or with a work sheet to stimulatethinking in the new direction. This wasfollowed by reading, possibly some explanation(explanation was better if given by thestudent), and more work in class to consolidateand test understanding. Finally, out-of-classwork was given, with the following rules: (1)collaboration on figuring things out wasencouraged, but (2) each individual was towrite his or her own answers, and (3)explanations and solutions were expected, evenwhen the question had a "yes or no" answer.

In practice, the first class was quite small,which meant that class discussions tended toinclude everyone. Although I tried to avoidlecturing to this small group, I did lecture onoccasion, partially in response to the students'feelings about what I "should" be doing.Another traditional classroom activity in whichI indulged was having them answer questionsfrom the book or from the activities.

Much of what ordinarily would have beenlecture, however, was given over to my askingleading questions to guide student discussions(and sometimes their thinking), and havingthem work on projects in groups or as a class.They liked the group work, especially on theexams; it reduced their anxiety. A later classwas appalled when I told them that somestudents do not like group work.

For me, the consequences of abandoninglecture presentation are many. Lecturing allowsa planned amount of material to be presentedover a planned duration of time. Mostmathematics courses do not allow much classtime for student exploration if the syllabus is tobe completed. (This is a recurring problem; itis my belief that not everything needs to betaught, but many professors and studentsdisagree.) Another problem is that most facultyteach a fairly heavy load, with little timeavailable for developing new presentations ofideas. When we lack time, we revert to our oldways. Finally, the students themselves areaccustomed to lectures, and resent thefrustration inherent in figuring things outinstead of being told "the answer." A decision totry a new approach is not an easy one. .

3

Student kge and Math Anxiety

During my first semester with the course, mystudents were older than the traditional collegestudent. This had some distinct advantages, inthat they were serious about learning to teachand wanted to be prepared to do a good job.They also accepted the idea that they neededto understand the mathematics.

A disadvantage with this group, however, wasthat math anxiety was prevalent. At least at thebeginning, some students "froze" as soon asthey were presented with anything challengingor out of the ordinary. Over the years, I havenoticed that the math anxious are oftenperfectionists. A person without math anxietymight be happy with an 85 on a test, but manymath anxious people are not happy withanything less than 100. For example, oneassignment for this class involved graphingsome measurements. In plotting the graph,which turned out to be linear, one studentfound that one of her points was less than aneighth of an inch off. She was so upset aboutthis that she redid the exercise.

Math anxious students also tend to be unhappywith questions having multiple, approximate,or incomplete answers. I tried to bolster theconfidence of these students by giving them alot more time on assignments than I wouldnormally give, then waiting for them to achievean understanding. Some points had to becovered several times, and many had to beexplained in detail more than once. Iencouraged the students to work together onassignments outside class, as long as eachperson wrote his or her own responses. My goalwas to model what their task as teachers willbe; I certainly am not used to waiting sopatiently for results.

Course Chronology

Almost everything took more time than I hadallotted for it, so we did not complete thesyllabus. Since this course had never beenoffered before, this may also have been amiscalculation on my part. In contrast to myoriginal plans, non-Euclidean geometries weretouched on only briefly, rather than in anydetail. The topics of motions in geometry and

34

similarity were not covered at all, nor wastopology. Because technology was a problem atthe time in my institution, we were not able todo as much as I would have liked with thecomputer.

On the plus side, however, we did cover unitson the nature of geometry; critical thinking andlogic; some theorems and proof concerninglines, triangles and other polygons; straight-edge and compass constructions (from theappendix of O'Daffer and Clemen's [1992]text); symmetry; tessellations; measurement ofthe circumference of the Earth; measurements(the metric system, areas, volumes); Platonicsolids; and an introduction to non-Euclideangeometry. To follow are summaries of the keyunits interspersed with excerpts from studentjournals.

Crnitical Thinking. In the "critical thinking" (logic)part of the course, the students were assigned awork sheet to do out of class. Collaborationwas allowed. They were to decide whethercertain conclusions were justified given thehypotheses, and write both their answer andtheir reasoning. These were everyday situations,some of which were taken from the text.

We then discussed the propositional calculus inclass, together with truth tables. We did some"knights and knaves" problems from Smullyan'sWhat Is the Name of This Book? (1986). (Theseproblems are based on the interaction betweenthe truth or falsity of statements and the typeof speaker making each statement; knightsalways tell the truth, knaves always lie, andnormals could do either. For example, only anormal can say "I am a knave." Both knightsand knaves must say "I am a knight.")

The students were also given work sheets ofsimilar problems; they had some trouble withthis, but we went over some of the problems inclass, and other problems were collected andgraded. Finally, the last assignment was to goback over the first work sheet, and change anyanswers they now thought were wrong. Theywere asked to hand in their final answer,together with a short statement of what hadchanged in their reasoning as a result of ourdiscussions in class. We also discussedquantifiers, with special attention to the

31

negations of universally and existentiallyquantified statements. In the following journalexcerpts, students express a range of reactionsto this unit:

I'm having a hard time grasping what this[critical thinking] has to do with math. Myimpression of math has always been of adiscipline where one answer is correct, andthere are rigid procedures to follow step-by-step to achieve the result.

000000

Still on critical thinkingI'm getting a graspon it thanks to the "knights and knaves"work sheet. . . . [Dr. Benbuty] clarifies andsimplifies the ideas and makes it fun (formath, anyway!)

000000

The knights and knaves activities wereenjoyable. I have always had difficulty withthese types of puzzles. It was thrilling tofinally understand how to do them.

Role of Logic in Mathematics. After the criticalthinking section, we spent some timediscussing the role of logic in mathematics.Since they had little prior knowledge of this, Itried to find out what they knew by havingthem consider some questions about facts.In groups, they were asked to list three factsfrom different fields of knowledge, such ashistory, geography, biology, or whatever theycame up with. Then they were asked howthese facts had been validated (how do we"know" they are facts?). They were thenasked to do the same for math (they all wrote,for example, 1+1=2). We were then able todiscuss definitions, postulates, and theorems.

Basic Geometry. Next we studied the basicideas in the geometry chapter in the text.Students enjoyed doing the straight-edge andcompass constructions, probably in partbecause they are tactile (although somestudents had never used a compass andencountered difficulties at first). Someremarked that going over these helped themwith their understanding of geometricobjects. The following journal excerptsdemonstrate how this hands-on approach

32

35:

helped to build the students' confidence inbasic geometry:

Making a figure really gives you a feel for it.

000000

I felt very clumsy with the compass at first,but once I got the hang of it, I reallyenjoyed doing them. Therefore, I wassurprised to have trouble with some ofthem.

000000

I did not know how to use a compassbefore coming to class. . . . It took me threetimes . . . to bisect a line segment usingthe compass and straight edge. However, Idid it! It was a great feeling to be able todo something with the compass

When students tackled some of the laterconstructions that lacked step-by-stepinstructions, they had difficulty. It seemedbeneficial to have students do some of theseharder problems in groups so that thestudents could brainstorm together to arriveat a solution.

Symmetry. We next studied symmetry, usingthe text and a draft module created byanother MCTP faculty member for reference.The text covers the different types ofsymmetry for plane and solid figures. Wediscussed figures that exhibit certain types ofsymmetry and how to complete a figure sothat it would have certain symmetries. TheMCTP module had students compute thesymmetry groups for a rectangle and anequilateral triangle. This led into symmetrygroups in a natural way, so we continued witha brief discussion of groups and gave somearithmetic examples of groups as well (forexample, the integers modulo 3 and thegroups related to "clock arithmetic"). One ofthe points I tried to have them discover wasthat geometry and algebra are related in someways that they had not previously known.

During this unit, one student wrote thefollowing journal entry, which encapsulates aproblem that persisted for many students inother units as well:

I1

Each of the questions on the rectangle asksyou to justify your answer. I have discoveredthat with some of these activities I can dothem and know I'm right, but don't have aclue as to why I'm right.

I find that students often have troubleexpressing mathematical reasoning; part of theproblem is lack of familiarity with expressingmathematical concepts. When I talk withstudents, their response is "Oh, I didn't know Icould write that as a reason." Often, theproblem is more subtle in that they are notpaying attention to their own thinkingprocesses. They arrive at an answer, but are notsure how. Sometimes I can straighten thingsout by talking about it, and sometimes I do notseem to be able to help.

Tessellations. The material on tessellations camefrom the text, and also some related topicsusing excerpts from Marvin Gardner's TheRiddle of the Sphinx (1987) and Ah Ha Gotcha!(1982).

The value of this unit to a future teacher isshown in the following journal excerpt by astudent who had been doing observations in anelementary classroom:

I watched during the school year as studentsdid beautiful tessellations and I was cluelessas to how they did it. After this chapter I feelthat I can have a conversation with thesestudents and I have developed anunderstanding.

Globe Module. The best material in the course wasthe "Where Are We?" module, developed by KenBerg and Jim Fey for MCTP (included in thisvolume). It is carefully developed to lead thestudent through Eratosthenes' estimate of theEarth's circumference. In so doing, students learnthe significance of longitude and latitude, andgain a clearer picture of how the Earth rotatesabout its axis and revolves around the Sun. Mostof these students did not previously understandthese things. Some of them made remarks like:"We covered this in geography, but I had no ideawhat the professor was talking about."

The first part of the exercise involves findinglongitudes and latitudes for various cities. One ofthe globes we used had a built-in device that

gave the longitude and latitude of an illuminatedspot on the globe. The students who used thisparticular globe discovered that the device wasnot working, as it gave Tokyo's coordinates indegrees West as opposed to East. Although theyhad to redo some of their work, the fact thatthey used their own powers of reasoning todetermine that the globe was brokenunderscores the value of this kind of hands-on,investigative activity.

Then the module asks students to supplyevidence for the ideas that the Earth is spherical,that it rotates, and that it revolves about theSun. They had sometrouble distinguishingbetween evidence forrotation (like sunriseand sunset) andevidence forrevolution about theSun on a tilted axis

e the seasons).They also haddifficulty identifyingthe geometrytheorems used toderive thecircumference of theEarth, even thoughthey are provided in the module. The strugglewas worthwhile, however, as indicated in thestudents' journal excerpts below:

I understand the reasons for the seasonsbetter now. I never really thought how usefulgeometry was for geography and astronomy. Ialso liked the way the unit gave backgroundinformation on the people who made thesemathematic/geometric discoveries.

. . . the factthat they used

their own powers ofreasoning . . .

underscores thevalue of this kind

of hands-on,investigative

activity.

36

000000

We worked together on the 'Where Are We?'handout. I really enjoy working with others. Itook Geography over the summer . . . I feelthat working on the handout has helped meunderstand about the globe better

000000

I wasn't prepared for a geography-type lesson.Math is everywhere!

000000

33

The globe module is an interestingapplication of geometry and angle measures.I never really knew how latitude/longitudewere used or how they were reallycalculated.

In my opinion, this module (or something likeit) should be a part of every elementaryeducation major's course work. The surprisethe students almost uniformly showed whenexposed to these concepts leads me to believethat their understanding of many phenomenais weak at best. Moreover, this kind of unithelps them to recognize that even though wecannot observe directly the workings of someportions of our world, we can still model andpredict them, and in this sense, understand.

Measurements. The last part of the coursefocused on measurements. We covered someof the standard area and volume measuresand, briefly, the metric system. The studentsfound conversion of measurements verychallenging. They were also asked to explorehow doubling or tripling the side of a square(or the radius of a circle) affected the area andhow doubling or tripling the side of a cube (orradius of a sphere) affected its volume.Students were often uncomfortable whenasked to state a general principle that theyhad discovered.

Platonic Solids and Geometry of a Sphere. Duringthe first semester, we only had time to touchon Platonic solids and the geometry of asphere. During the summer of 1996, however,we had time for the students to write briefexplorations on "great circles"the curves onthe surface of a sphere that are the "straightlines." This caused considerable consternation,as many of the students had never heard ofsuch a concept before. But some good papersresulted, which reinforced my view that futureteachers in particular benefit from expressingmathematical ideas. The following journalquote captures one student's frustration as wellas recognition of the value of doing the paper:

This was a nightmare for me. . . . Finally, Irealized that I understood what was beingsaid but [writing the paper] forced me to beable to articulate it. . . .

34

Paper on Constructivism. At the end of thecourse, each student submitted a paper onconstructivism as well as a lesson plan. Tocomplete the project, the students needed todiscover the educational resources for teachingelementary mathematics available at thecampus library. (There is a considerablecollection, but many of the students wereunaware of this at the outset of the course.)While some students at first hunted until theyfound "the book" they thought I wanted themto use, most were comfortable with multiplesources once they explored the availablematerial. Each presented his or her lesson planto the class as a whole. A surprising andwonderful outcome of the paper and lessonplan assignment was that most of the studentsfelt that they could now learn and figure outhow to present at least some mathematics ontheir own. The following journal excerpt showshow one student gained confidence inconstructivism as a teaching philosophy, andmore importantly, in herself as a learner:

After completing the paper onconstructivism, I feel that it is the bestpossible way to teach math wheneverpossible. Sometimes there is more than oneway to reach a solution, and by letting thestudent try by trial and error, it builds theirconfidence and critical thinking skills. Ithink that is a part of how this course wastaught, and it worked! It has helped me overa large part of my 'math phobia'!!

Reflection

Probably my biggest surprise was the extent towhich the students' confidence increased as aresult of doing the lesson plan. They found thatthe educational resources in mathematicsavailable in the library were not overwhelming,but instead comprehensible. This may havebeen their first experience reading andunderstanding mathematics (no matter whatthe level) on their own.

My next biggest surprise was the extent towhich the elementary education majors arewilling to work. (I have since found that someof them are not so energetic; my first class wasnot completely typical.) I was quite dismayed

3'7

ii

at the lack of knowledge in some of thestudents, particularly when we studied themeasurement of the Earth's circumference, butwas pleased at the time and energy they putinto answering the questions and, in the main,the understanding they were finally able toexhibit.

Although the students often struggled withmaterial, through their journal entries Ilearned that they appreciated my efforts tomake mathematics relevant to them. Forexample, journal reactions to the first material(what is geometry, where do geometric objectsappear, etc.) indicated surprise that geometryhad any relevance to anything. So far I havetaught the class three times, and each time thestudents have had this same reaction. (Thosewho took a standard high school geometrycourse seem most prone to this response.) Inaddition, one student appreciated that the texttook "into account that many cultures had animpact on what we know as geometry today."For some types of learners, relevant non-mathematical content must be included in thecourse.

The first class was so good about keeping theirjournals that I did not realize the extent towhich students need to be guided. They wroteabout their frustrations and theirbreakthroughs of comprehension; they oftenpasted in relevant news articles andcommented about them. They commentedfrequently about rote learning versus thinking,and related this to their own experiences inthe course. Unfortunately, I caused some briefinterruptions in the recording of theseexperiences because I had to collect theirjournals to read and comment upon them. Thesecond time, we used e-mail, so that feedbackwas more immediate, but they were lessreflective, and I was not sure what to do aboutit. I think that this was mainly the problem ofthe younger students, and I did not recognizefor some time that I had to give them moreleading and focused questions. The third time,we were back to notebooks again becausee-mail was once more not available; once againthe class was a bit more mature, so thejournals offered more insight.

Much of the group work in the first elementarygeometry class seemed to go more smoothlythan with my other students (both generaleducation and math majors). I believe that thestudents in this class were older and moreresponsible. My second class seemed lessresponsible than average, and the groups wereless successful. The third class said that theyhad always heard that group work was better,but that this was the first time that working ina group seemed natural to them. It seems thata certain type of student really likescollaborative learning (I've found that mathmajors, in the main, do not); I think many ofthe more mature elementary education majorsfit this profile.

The grades in this class were better than usualin a lower level mathematics class, probablybecause I tried to give the students more timewhen they appeared to need it, and explainedsome concepts many (three or four or more)times in class. I madea conscious choice todo whatever possibleto decrease theiranxiety, since withoutconfidence andknowledge, they willnot be able to copewith mathematics inthe classroom. Severalof my students havestarted the course in astate of tenor, andhave finished it with afair degree ofconfidence. One ofthem, for example, went on to apply andparticipate successfully in an MCTP summerresearch internship. Considering that sheliterally shook with fear at the beginning of thecourse, this is something I doubt she wouldhave considered without having gainedconfidence in herself through the course.

For future semesters, I plan to use portfolios aspart of the assessment, and manipulatives forthe section on Platonic and semi-semi-regularsolids. There are ways that folding papers(including origami) can illustrate geometricconcepts that I would like to explore. I also will

Although thestudents often struggledwith material, through

their journal entriesI learned that they

appreciated my effortsto make mathematics

relevant to them.

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be incorporating technology in a moreconsistent way.

In closing, I'd like to note that one of the often-overlooked aspects of learning is the timebetween classes. For a student who is willing towork consistently on material, this time isperhaps the most critical, for it is in theseperiods that the concepts come together. I havefound that, even though I have more of atendency to "tell them" than a trueconstructivist should, I still had to wait for theknowledge to be assimilated. Students'understanding too soon slips away, particularlyif we go on to different material; the first "ahha" is not enough.

References

Gardner, M. (1987). The Riddle of the Sphinx.New Mathematical Library, Volume 32.Washington, DC: Mathematical Association ofAmerica.

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Gardner, M. (1982). Ah Ha, Gotcha! NewYork: W. H. Freeman.

Henderson, D. (1996). Experiencing Geometryon Plane and Sphere. Upper Saddle River, NJ:Prentice Hall.

National Council of Teachers of Mathematics.(1989). Curriculum and Evaluation Standardsfor School Mathematics. Reston, VA: NCTM.

O'Daffer, P. and Clemens, S. (1992). Geometry,an Investigative Approach (2nd ed.). Reading,MA: Addison-Wesley.

Smullyan, R. (1986). What is the Name of thisBook? New York: Simon & Schuster.

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INTER it ISCIPLINARY SCIENCE ANDMATHEMATICS: GEOMETRY AND

THE ROUND EARTH

Kenneth Berg and James FeyDepartment of Mathematics

University of Maryland, College ParkCollege Park, MD 20742

Tt's hard to imagine doing important science!without using some mathematics, andmathematics would certainly be a much lessimportant subject without its applications inthe sciences. But, in traditional science andmathematics courses, the connectionsbetween these subjects are poorlyrepresented. Mathematical ideas andtechniques are generally taught with littlereference to their uses in scientific problemsolving. A typical mathematics course mightoften use scientific formulas as contexts forpractice of mathematical calculation, butwould seldom show how importantmathematical principles are suggested byabstraction of patterns observed in scientificphenomena. Conversely, science educatorstypically think of mathematics only as atoolbox to which they turn for techniques ofdata analysis and problem solvingoncescientific principles have led to well-definedplans for calculation.

Over the past several decades, the increasingmathematization of all sciences has beenaccompanied by the emergence of a new pointof view about the relation betweenmathematics and science. Instead of thinkingabout mathematics as only a collection ofconvenient computational techniques, it isincreasingly seen as a source of models fordescribing and reasoning about all sorts ofstructural patterns that occur in the naturalworld. Mathematical modeling helps scientiststo find patterns in data and explanations forthose patterns, as well as predictions of the

ways that changes in conditions of anexperiment will lead to changes in results. Thismodeling point of view about the relationbetween mathematics and science has alsostimulated intense interest in using scientificexperiments and data as starting points for thestudy of important mathematical structures.

With the objective of conveying to liberal artsstudents and prospective elementary schoolteachers some breadth of understanding aboutthe relation between mathematics and thephysical, life, and management sciences, theUniversity of Maryland, College Park requires afundamental studies course titled ElementaryMathematical Models. Even with the moderntitle alluding to mathematical modeling, thestandard offering of MATH 110 is commonly apretty traditional finite mathematics coursethat trains students in calculations required byfamiliar applications in linear programming,probability, game theory, and mathematics offinance. To give students a more authenticexperience with the processes of mathematicalmodeling, we used support from the MCTPproject to develop new materials andinstructional approaches in that course.

In designing the new version of the modelingcourse, we wanted students to analyzeproblems by:

e beginning with data from experiments orrealistic situations;

® producing numeric, graphic, and symbolicrepresentations of significant data patterns;

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o using mathematical models to makepredictions;

® interpreting those predictions in the originalproblem contexts;

® examining the models and their implicationsfor sensitivity to changes in modelingassumptions; and

o looking for cause-and-effect explanations ofmodel results.

We wanted students to experience use of thetechnological modeling tools like the dataplotting, curve-fitting, and random numbergenerating routines of graphing calculators.Furthermore, we wanted students to gaininsights into mathematical modeling throughstudent-centered classroom activities, notlectures.

We chose to include material that would leadto function models (primarily linear andexponential) in both iterative (yn+1 = yn + kand yn+i = kyn) and closed forms (y = kx + band y = akx) and probability models (usingsimulation and formal analysis). But we also

chose to extend thestandard list of finitemathematics topicsby including asubstantial unit thatdemonstrated theusefulness ofgeometric models.

The report thatfollows describes theunit on geometric

modeling that we developed and have used innearly a dozen sections of MATH 110 over anumber of semesters. We've included the fullstudent text, which is essentially a narrativeand a series of problems to be solved bystudents in collaborative groups, and somesample assessment probes. We've alsointerspersed some commentary on ourobjectives and experiences in using the unit.

. . . we wanted studentsto gain insights into

mathematical modelingthrough student-

centered classroomactivities, not lectures.

Geometry and the Round Earth

One of the most striking examples ofinterdisciplinary science and mathematics is

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the long history of interplay between physicsand mathematics for work on astronomicalquestions. Unable to stand outside the solarsystem or galaxies to see how the observablepieces fit together into a system, scientists havebeen forced to make extensive records ofvisible patterns and then to use analogy,metaphor, and scientific imagination to buildphysical and mathematical models of theobjects and their interaction.

Astronomy today is a deep and complexsubject that employs sophisticatedmathematics, physics, and chemistry inattempts to explain the origin and evolutionaryprocesses of the entire universe. However,many significant questions about our Earth andsolar system can be studied with elementarymathematics and science. In the process,prospective teachers and other students cangain valuable knowledge and insight intoscientific method.

Contrary to popular belief, the spherical shapeand approximate size of the Earth weredetermined more than two thousand years ago,though direct physical verification of thesefacts was not achieved until many centurieslater. Unraveling the story of these earlyastronomical inferences is a fascinatingopportunity to review fundamental concepts ofgeometry (proportionality, similar triangles, andcentral angles in circles), geography (latitude,longitude, equator, Tropics of Cancer andCapricorn, Arctic/Antarctic Circles, and changeof seasons), history (Mediterranean region inthe time of Euclid and later when Columbuswas seeking support for his voyages ofdiscovery to the Americas), and scientificmethod (the interplay of empirical andtheoretical work and the profound challengesposed by astronomical questions whenobservational tools and vantage points werelimited).

To help-our students gain some of the scientificand mathematical insights that underlie thoseprofound questions and their importanteveryday consequences, we developed plansand instructional materials for aninterdisciplinary unit of instruction which wetitled "Where Are We?" Our goal was to guide

students to construct or refine their personalunderstanding of several fundamentalquestions:

O How are latitude and longitude determinedon our Earth and why are several latitudelines designated as especially significant?

o How does the Earth's relation to the Sundetermine seasons and the length of daysand nights?

O How could scientists over 2000 years agohave known that our Earth is a sphere andestimated the circumference of that sphereto a high degree of accuracy?

The answers to these questions are widelyknown today. In any event, it would clearly beeasy to "cover" the facts in a few short lectures.But there is also a growing body of evidencethat many educated adults have very shaky andsuperficial understanding of the "facts" and thatlistening to a lecture or two (no matter howlucid and well illustrated) will not begin toproduce the deep understanding that onewould hope a science or mathematics teacherought to have.

We planned the unit so that it would beginwith an assessment of students' priorconceptions on these issues, engage students inactive investigations that involved hands-onmodeling of important scientific phenomena,encourage students to communicate theirquestions and findings clearly to each otherand the instructor, demonstrate the power ofmathematical modeling as a tool in scientificreasoning, and cause students to reflect on thecomplex paths of reasoning that lead toscientific discoveries.

In a typical classroom launch of the unit, webegin with a short quiz and discussion to assessstudent prior knowledge and conceptions onthe issues raised here. For many students thissimple assessment of prior conceptions revealshow little they understand about lines oflatitude and longitude. In a subsequent journalentry, one student wrote:

I learned this week that I knew virtuallynothing about the lines of latitude andlongitude, nor the reasons for theirplacement on the globe/map. I was surprisedat how little I knew. So, I guess this was abit of an eye-opening exercise.

Geometric Modeling Unit

Student Text

WHERE ARE WE?

In the search for scientific theories that explain our observations and experiences of the worldaround us, few things have sparked as much interest as the simple question, "Where are we?"Throughout recorded history (and probably long before), people in all places and cultures havepuzzled over the size and shape of our planet and the location of the Earth among the planets andstars of the heavens. What do you know or believe about:

O the size and shape of planet Earth?

o systems for locating points and travel routes on the Earth?

O the size, location, and motion of the Moon, the Sun, and the other planets of our solar system?

As fascinating and important as are many facts of life on Earth and in the solar system, it is equallyintriguing to ask how and when we came to current understandings of our place in the universe.What do you know or believe about the timing and methods of thought used in the discoveriesthat:

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Our planet is round like a ball, not flat like a table top?

The circumference of the Earth is about 25,000 miles, and the distance from the Earth to the Sun isabout 93,000,000 miles?

The Earth rotates around a tilted axis and revolves around the Sun to cause days and nights andseasons of the year?

The answers to these fundamental scientific questions are generally considered part of the disciplinecalled astronomy. However, the methods used by astronomers rely heavily on numerical andgeometric ideas from mathematics. This unit explores scientific and historical connections betweenmathematics and astronomy in four investigations titled:

Latitude and Longitude

Days and Nights, Years and Seasons

Shape and Size of the Earth

Where in the World Is Christopher Columbus?

The goal is to explore the basic ideas and processes that have been involved in the long and fruitfulsearch for mathematical models of our Earth and the solar system.

Latitude and Longitude

This first investigation reviews and extendsstudent understanding of the basic globalpositioning system based on latitude andlongitude. In our experience, most studentscertainly remember something about these twoideas. However, they've often forgotten whichis which, they don't know how the locations ofthe lines are actually determined, they don'tknow the connection between latitude anddistance of points directly north/south of eachother, and they don't know the fundamentaland fascinating connection between thelatitudes of the Tropics and the Arctic/Antarctic Circles. In the hands-on activitymapped out to help students reconstruct theirknowledge of these things, many students aregenuinely challenged by the problem ofdetermining spherical radius or diameter fromcircumference. The group project workinvolved in constructing a planar section of aglobe works well to get students cooperating,because there are significant roles for severalpeople at each stage. Also, while an instructorcould lecture (even with demonstration for theclass) through this investigation, we find thatwatching students work on the problem

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themselves gives us insight into theconceptions and puzzles of individual studentsthat we would never get in a lecture setting.

In their journals, many students commented onboth the mathematical substance and theclassroom investigative style of the activities.One student's remarks are particularly vivid:

Math and its exactness have always beenhard to get used to. It carries a connotativemeaning of 'One false move and you'redead' concept. It was a horrible way to cometo class. So my idea of learning was that Ihad to just be right, no matter what I did.Scan,. In our class, and with ourexperiments, I make mistakes all over theplace, end up with the right answer afterbouncing it off of my group members, and Iam not downgraded for that. Ourexperiment on latitude and longitude gaveme the opportunity to say, and even think,that latitude was north to south andlongitude was east and west distance, andhave that idea changed and changed againbefore it was correctly stated, and I wasn'tgoing to be condemned to death as a result.

Student Text

LATITUDE AND LONGITUDE

If you had to tell someone how to locate College Park, Maryland, it's unlikely that you would say,"It's at 77° west longitude and 39° north latitude." But as an international scheme for describingpositions on Earth, the (longitude, latitude) coordinate system has proven itself as an invaluable toolfor navigation on land, sea, or air. It's a system that we take for granted, but:

What do you know about the rules for assigning longitude and latitude coordinates or the specificcoordinates locating important places on the Earth?

® Do you know when the current system was developed and what sorts of prerequisite knowledgewere necessary for building the system?

It's hard to recreate the human experience of exploration and discovery that led to our currentglobal positioning system. But by analyzing the way that lines of longitude and latitude are drawnon a model globe, we can make some guesses at the origins of this important mapping system. Toanswer the following questions, and those of subsequent investigations, you'll need a globe map ofthe Earth, several feet of string, and rulers for length and protractors for angles.

1. To get started, use the globe to find approximate map coordinates for the following major citiesaround the Earth:

a) Chicago, Illinois b) Honolulu, Hawaii c) Tokyo, Japan

d) Sydney, Australia e) Kinshasa, Zaire f) Sao Paulo, Brazil

g) London, England h) Cairo, Egypt i) Moscow, Russia

2. One natural question about map coordinates is whether you can tell the distance between twopoints from the difference of their coordinates. Collect some data from your globe to explorethis question:

O Locate the Equator and several lines of latitude in the northern hemisphere.

O Use a tape measure to find the distance from the Equator to each of the lines of latitude onyour globe. Don't worry right now about the actual distance on the Earth; use inches orcentimeters to measure the distance on your globe model of the Earth.

o Make a table and a graph plot of the (latitude, distance) data.

a) Describe the pattern of data relating latitude and distance from the Equator. How, if at all,does it allow you to predict north-south distance between two points from the latitudecoordinates?

b) Locate the two special lines of latitude in the northern hemispherethe Tropic of Cancer andthe Arctic Circle.

(1) Measure the distance on your globe from the Equator to each of those two latitude lines.

(2) Use the pattern of (latitude, distance) data from your measurements in (a) to estimate thelatitude of the two special lines.

(3) If you see any interesting connection between the Tropic of Cancer and Arctic Circlelatitude numbers, see if you can come up with an explanation of why that relationship

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occurs. If you don't see anything noteworthy right now, keep on the lookout in theinvestigations ahead.

3. You know, or have noticed from the globe, that latitude is reported as a measurement in degrees.That suggests that angles are involved in establishing the lines of latitude. What angles do youbelieve are being measured?

O Figure out the diameter of your globe (Do you remember the relation betweencircumference and diameter?).

In a piece of poster board paper, cut a circular hole with the same diameter as your globeand slip the resulting circular collar over the globe from pole to pole. It represents a planepassing through the sphere.

O Around the circular collar, mark the points where lines of latitude touch it. Then take the"planar section" off the globe and lay it flat for analysis.

a) Think again about the possible angles being measured when lines of latitude are given theirdegree measures. Record your ideas.

b) Any more thoughts about the Tropic of Cancer and the Arctic Circle latitudes?

4. The circles on the globe that pass through both North and South Poles are called lines oflongitude. Those lines are also measured in degrees.

a) How do you think degree measures are assigned to lines of longitude?

b) Do some experimentation to see if you can tell the east-west distance between two points onthe globe from their longitude coordinates.

5. Having studied the relation between measures of latitude and longitude and distance on yourglobe, scale up your findings to facts or principles about distance on the whole Earth.

a) If one point is directly north or south of another, how can you predict the distance from oneto the other, based on knowledge of the latitude of each?

b) If one point is directly west or east of another, how can you predict the distance from one tothe other, based on knowledge of the longitude of each?

6. A family in the northern hemisphere sets out from their home and walks 3 miles due south, then2 miles due west, and then returns home by walking 3 miles due north. Along the way they see abear.

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a) What color was the bear?

b) Could a family in the southern hemisphere make a similar trip?

Conclusions and ConnectionsNow that you've reviewed the structure of our familiar latitude andlongitude coordinate system for locating positions on the Earth, think about the ideas that had to bedeveloped before the current system could be organized and what problems remained fornavigation even with the notion of latitude and longitude developed.

1. What big ideas about the size and shape of our Earth seem prerequisites for defining longitudeand latitude, and in what order do you think those ideas were discovered?

2. The system of latitude and longitude coordinates is only one of many locator systems used in awhole variety of tasks. Think about and describe several other coordinate locator systems thatyou are familiar with. Then compare them with the basic features of Earth longitude andlatitude coordinates.

3. What is it about 90°, 180°, and 360° that makes them common in angle measurement?

4. The Equator is the line (circle) of latitude with measure 0; the prime meridian is the line (circle)of longitude with measure 0. Which of these two great circles pretty much has to have acoordinate of 0° and which has that coordinate pretty much by historical accident?

5. How do you think satellite dishes for television are "tuned in"?

6. The U. S. Defense Department developed a global positioning system that uses orbiting satellitesand a portable piece of apparatus on the ground that can be carried by an individual. It canpinpoint the carrier's location on the Earth to within 1 meter! How do you suppose it does that?

Days and Flights Years and Seasons

As the title suggests, this investigation moveson to the connection between latitude, the tiltof the Earth's axis, the orbit of the Eartharound the Sun, and other related topics. Inour experience, these questions are much moreforeign ground than the simple latitude/longitude questions. Many students really don'thave much sense of what it means for theEarth to orbit the Sun in such a way that itsaxis of rotation at any time is always parallel toits position at the beginning of a year.Explaining how the Earth's axis tilt and orbitalmotion around the Sun cause seasonal changesin daylight and typical temperatures at variousplaces on the Earth is not a trivial problem. Forexample, one common misconception is thatsummer in either hemisphere is warmerprimarily because that part of the Earth istilted "closer" to the Sun. That misconception isvery resistant to modification.

The interplay between the perceived size ofthe Sun and Moon and their distance from the

Earth is a challenging question of scaling,similar triangles, and other geometry. Thequestion of howpeople livingthousands of yearsago could figure outastronomical sizesand distances is awonderful puzzle toget students thinkingabout.

In this unit weencourage students toget their hands onmodels of the Earthand the Sun byproviding them withglobes, flashlights, andmeasuring tools. Atsome points weactually hang a lightbulb Sun from the classroom ceiling and havestudents "walk" the Earth around that Sun.Keeping the Earth axis in the requisite parallel

For many mathematicsinstructors, such hands-on activities are oftenseen to be somethingappropriate only for

elementary or middleschool students.

However, we've foundthat appropriate hands-on work is appreciated

by undergraduatesas well.

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position has been a surprisingly tough task foreven our stronger students, highlighting theimpressive insights of those who conceived andconvinced others of the proper heliocentricmodels of planetary motion, all from an Earth-bound vantage point.

For many mathematics instructors, such hands-on activities are often seen to be somethingappropriate only for elementary or middleschool students. However, we've found thatappropriate hands-on work is appreciated byundergraduates as well. In the process werealize our objective of teaching in ways thatour prospective candidates might themselvesteach in future K-12 classrooms. The effect ofhands-on work with challenging problems isendorsed in these journal reflections by manystudents. One student wrote:

The proportional model of the sun and theearth that we put together helped the peoplein my group understand how far away thesun really is. It is sometimes hard tocomprehend the distances that we use whentalking about planets or stars. Using aproportional model helps put things on asmaller scale so that it is easier to grasp theconcept.

Another student wrote:

In this class activity, I learned more aboutthe specific motions of our solar system. Theuse of the flashlight and globe in thisexperiment was a great way to visualize themathematical aspect of this project. Thesemodels helped me understand the conceptmuch more. It was interesting to study howthe sunlight hours are different at eachparticular season.

Student Text

DAYS AND NIGHTS . . . YEARS AND SEASONS

In the search for mathematical models of our solar system, one of the toughest problems wasfiguring out how (if at all) the Earth, the Moon, the Sun, the other planets, and the stars are moving.

© What do you know or believe about the motion of planets and the Sun in our solar system?

© Try to imagine yourself thinking about this question 1000 or even 2000 years ago. What evidencewould you have that something was moving, and what clues would you have that would lead tothe present understanding of solar system motion?

What practical consequences could there be to understanding motion of the heavenly bodies?

These are very hard questions, but scientists (and lay people also) conjured many theories and hadheated debates about them.

You can get an understanding of the present-day theories by simulating the Earth and the Sun witha globe and a flashlight. Of course, those crude simulation tools only suggest what is going on. Youhave to imagine the experiments scaled up many times to get a picture of what is really happening.

1. To get a rough idea of the kind of scale model that would be needed to do accurate Earth-Sunsimulations, consider these data:

o The diameter of the Earth is about 8000 miles; the diameter of our Sun is about 865,000miles;

G The Sun is typically about 93,000,000 miles from the Earth;

o Light from the Sun travels 186,000 miles per second.

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a) If your globe is 1 foot in diameter; what diameter would you need for a model Sun?

b) How far from your Earth model should your Sun model be placed to have the model beproportional to the real thing?

2. Your answers to question (1) might convince you that a globe and a flashlight will tell nothingabout the real Earth-Sun relations. But with some care and thought, you'll see illustrations ofmany important facts of the solar system. Hold your flashlight several feet from the globe andexperiment with movement of those Earth-Sun models to simulate various times of day andnight and various seasons of the year.

It was not until the late 1950's that we were able to observe the solar system from orbits farabove our Earth's surface'. Think about how an earlier Earth-bound observer might reasonablyget the following ideas about solar system phenomena:

a) That the Sun is revolving around the Earth

b) That the Earth itself is rotating on an axis

c) That the Earth is revolving around the Sun

d) That the Earth moves around the Sun with its center in a fixed plane

e) That the Earth's axis is tilted at an angle to the orbital plane

3. Although it took thousands of years for scientists to realize (believe?) that the Earth revolvesaround the Sun, not the other way around, let's assume some modern knowledge of Earth-Sunrelations to figure out why days and nights and years and seasons behave the way they do. TheEarth moves in an elliptic orbit and the plane of that orbit cuts through the center of the Sun. Atany point in the orbit (any time of year), the Earth's axis of rotation is parallel to its position atany other point in the orbit and tilted to the plane of the orbit.

Use this model of Earth-Sun relations and your flashlight-globe simulation to study the followingquestions about familiar seasonal phenomena. Move the globe around to different positions inrelation to the flashlight "Sun" and observe differences in the way the Sun illuminates the Earthas it rotates on its own axis. Record your ideas with sketches that help to describe events andtheir explanations.

a) At noon on March 21 and September 21 (the spring and fall equinoxes), the Sun is directlyoverhead at points on the Equator. Where is the Earth in its orbit around the Sun on thosedates, and why do all points on the Earth experience equal amounts of daylight and darknesson those days?

b) At noon on June 21 (our summer solstice in the northern hemisphere), the Sun is directly

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overhead at points on the Tropic of Cancer. Where is the Earth in its orbit around the Sun onthat date and why do some points on the Earth experience 24 hours of sunlight and others 24hours of darkness?

c) At noon on December 21 (our winter solstice in the northern hemisphere), the Sun is directlyoverhead at points on the Tropic of Capricorn. Where is the Earth in its orbit around the Sunon that date and why do some points on the Earth experience 24 hours of sunlight and others24 hours of darkness?

d) Why is it that in the northern hemisphere, June, July, and August are generally the warmestmonths, while December, January, and February are the coldest?

e) How are the latitude measurements of the Tropic of Cancer and the Tropic of Capricornrelated to each other and to the tilt of the Earth's axis, and what connection does thisrelationship have to seasonal variations in daylight and darkness on the Earth? What is thecomparable answer for the Tropic of Capricorn and the Antarctic Circle?

Conclusions and ConnectionsThe flashlight and globe experiments use a physical model to study thesolar system. It gives a general qualitative idea of how the system works. To describe things moreprecisely and to make predictions, it helps to have a model of the Sun and planets that can bedescribed with geometric shapes and equations.

1. What geometric shapes and measurements play key roles in a solar system model and inexplaining days and nights and years and seasons?

2. Because the Sun is so much larger than the Earth and so far away from the Earth, it is hard tomake true scale model drawings of the relationship between them. However, the followingdrawing (inaccurate as it is in some respects) can be used to describe some important aspects ofthe relationship. Imagine that you are looking at a side view of the Sun and Earth with your eyedirected along the orbital plane.

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Sun

a) On sketches like this, draw in the Earth's axis and Equator as you would see them from such aside view at each of the four key points in the year: the summer solstice, the fall equinox, thewinter solstice, and the spring equinox.

b) Explain how the sketches demonstrate the different sunlight hours at different seasons. Inparticular, how do parts of the northern hemisphere have 24 hours of sunlight at the summersolstice and 24 hours of darkness at the winter solstice.

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c) Explain how the sketches show the special properties of the Tropics of Cancer and Capricornand their relation to the Equator and the Arctic and Antarctic Circles.

Shape and Size of the Earth

This investigation gets to the two big questionsthat drive the entire unit: How and when didscientists and explorers know that the Earth isa ball? How and when were they first able tomake good estimates of the size of the Earth?

Consistent with our focus on studentconceptions, we usually begin the investigationwith a discussion of student prior notionsabout these questions. When time is allowedfor students to work in groups to formulatepreliminary conjectures about the issues, we'vefound that many groups engage in very livelydiscourse of the sort that must certainly havetypified scientific debates over long periods oftime. We believe that this experience, when itis called to their attention, gives students avaluable insight into the nature of scientificwork. Instead of simply listening to (andcopying) well-polished theories explained byan expert, the students grapple with thequestions themselves.

The student text material is written as a seriesof questions, offering progressively greaterguidance or scaffolding to help studentsformulate ideas that can be used to reconstructthe amazing work attributed to Eratosthenes.In our judgment, the reasoning involved in hisestimation of the Earth's circumference is oneof the truly impressive examples ofmathematical modeling at work to makescientific predictions that could not beconfirmed by physical measurement untilmuch later. While few student groups are ableto integrate the various pieces of knowledgeneeded to replicate Erathosthenes' methodwithout several steps through the hints, it isalways exciting to hear the, "Oh, I get itit'salternate interior angles," from students when

they have progressed far enough to recognize afamiliar relationship that is the key.

Again, we are sure that a clear lecturepresentation of the reasoning could beprovided in much less than one class period.But we are convinced that until students havestruggled with the problem themselves, eventhe best exposition will not connect with theirprior conceptions in a way that producespermanent conceptual change. By teachingthrough investigations, we see clear instances ofthe students thinking deeply about theproblems, and raising questions as to what theydo and do not understand,rather than simply acceptinginformation as the truth. Forexample, in the followingjournal entry one studentshared how he was havingdifficulty understanding aconcept:

So why is it that the sun'srays can be considered to beparallel lines striking theearth? I have trouble withthe concept. I understandthat it works, but it alwaysseems to me that light travelsin all directions out from the surface of thesun . . . so why doesn't some light which, forexample, let's say it leaves the sun at aminutely small anglewhy doesn't this lighthit the earth's surface, also at an angle,throwing off the "parallel rays" theory? Isthere no angle small enough to accomplishthis (meaning that, by the time it had comethe distance of the earth, it would haveangled too far left/right/up/down)? I justdon't know.

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Instead ofsimply listening to(and copying) well-polished theories

explained by an expert,the students grapplewith the questions

themselves.

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Student Text

SHAPE AND SIZE OF THE EARTH

The Earth is round. Are you sure? Why do you think so? Why would a resident of Nebraska (one ofour flatter states) think so? When and how do you think astronomers, geographers, and explorerscame to believe that the Earth is round?

When Columbus proposed his westward trip from Spain to China, via the Canary Islands off thecoast of North Africa, his estimate of the distance was not very accurate and there was a large landmass blocking his planned route. However, he was right that the Earth was round. Isabella andFerdinand of Spain financed the voyage, over the objections of their royal advisors who believed theEarth was round but thought Columbus was mistaken about the distance . . . which he was

As early as 240 B. C. it was commonly believed that the Earth was round and various people hadgiven estimates, or perhaps their guesses, as to its size. In fact, in 240 B. C., Eratosthenes, thelibrarian of Alexandria, produced a well-reasoned estimate placing the Earth's circumference atabout 25,000 miles (using more modern units).

Eratosthenes' Bright Idea

The city of Alexandria in Egypt was founded in 332 B. C. by Alexander the Great. After Alexander'sdeath nine years later, Ptolemy I emerged from the conflict among potential successors to establishcontrol in Egypt with a government based in Alexandria. A succession of Ptolemys built Alexandriainto an enormous cultural center. A center of learning was established where, for example, Euclidtaught mathematics. Eratosthenes was the librarian responsible for building a collection of the bestthought from around the world. Among the information he collected were some facts that led to hisimpressive deduction about the size of the Earth.

The questions that follow give a series of challenges and hints that allow you to trace Eratosthenes'reasoning about the size of the Earth and then to move farther back in history to see how one mightfirst reckon the Earth to be round.

1. Imagine that the year is 240 B. C. and you live in Alexandria. You accept the general idea thatthe Earth is round, so you can imagine it marked up with lines of latitude and longitude, but youdon't know its size. First bright idea: If you knew the connection between change in latitude anddistance on the earth's surface, you could estimate the circumference of the Earth.

If a one-degree change in latitude corresponds to a known distance d, howwould you calculate the circumference of the Earth?

2. Imagine next that someone phones you from a place due south of your home and says, "Whew, isit hot? I just went outside and the Sun is directly overhead." You hang up and go outside to findthat the Sun is not directly overhead where you live.

a) How would your friend know that the Sun was directly overhead?

b) How would you know that the Sun was not directly overhead where you live?

c) How could you use the observations by you and your friend to estimate the change in latitudebetween your homes?

This is a hard problem . . . it's the key to Eratosthenes' discovery. However, if you think about yourflashlight Sun shining on a globe and make some sketches of the situation, you might be able to

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guess (if not prove) the answer. A good guess about the truth is usually the essential first step infinding a logical argument.

3. One observation that might help your reasoning is the fact that, because the Sun is so large andso far from Earth, its light comes in a beam that can be thought of as parallel light rays.

Sun

How, if at all, does this assumption change your thinking about using Sun rays to measure change inlatitude between two spots on the Earth?

4. The following sketch shows a side view of the Earth with Sun rays falling at two places, onedirectly north of the other. The Sun is directly overhead at point A, but not directly overhead atpoint B. Suppose that it is you standing outside in the Sun at point B and your friend at point A.

a) Do you see anything in the sketch that you could measure to find the change in latitude frompoint A to point B?

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b) The next sketch shows two points, neither of which lies directly under the Sun's rays. Is it stillpossible to find the change in latitude from point A to point B by suitable clevermeasurement in this case?

5. Eratosthenes did not receive a phone call! However, he knew some facts that allowed him toreason with much the same information that your phone call from a friend could provide.

a) Locate the cities of Alexandria and Aswan in Egypt on a map or globe. In 240 BC. the city ofSyene was located at the site of present-day Aswan. Residents of Syene noticed that on June21 of every year at about mid-day the Sun's rays were reflected from the bottom of a deepwell. 'What does that tell you about the latitude of Syene?

b) On June 21 at mid-day in Alexandria, Eratosthenes measured the angle between the Sun'srays and "straight up" by looking at a triangle formed by the Sun's shadow on a tall building.

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OM'A'INN Main'

IIMI OMOM MOW

MIN ENI MN WM MIM:

MEE MIIMIMI= elUM10

IMMEIM I=NM OM

IM MEM'all NM In=IMM MU==11MIMMUNE,=I= InI NM OM I=MI MN=UM NM%MINIM

He found the angle to be about 71/4°. What does that tell you about the latitude of Alexandria, inrelation to Syene?

c) Eratosthenes then learned that the distance on land from Alexandria to Syene was a 50-daycamel trip and that a camel could cover about 10 miles (in modern units) per day. What doesthis information imply about the distance from Alexandria to Syene and the distance aroundthe Earth?

6. Now turn to the even older question of determining the shape of the Earth. How could peoplehave become convinced that it is a ball and not a flat disk?

a) What happens in a lunar or solar eclipse and how would the images seen in such an eventsuggest that the Earth, Moon, and Sun are round?

b) Perhaps you are not a stargazer . . . or maybe you are. People spent more time outdoors twothousand years ago. They were fascinated by star patterns. Using your twentieth centuryknowledge, what might observant stargazers have noticed as they traveled? How would thoseobservations suggest something about the shape of the Earth?

c) Imagine a ship with a tall mast sailing directly outward from land over a relatively calm sea. Ifyou watch for a long time from sea level, what would you notice? How would thisobservation suggest a round Earth to people living thousands of years ago?

Conclusions and ConnectionsThe questions of this investigation have been designed to help youfollow the course of discovery that led to good estimates of the shape and size of our Earth. Testyour grasp of the main ideas by answering these questions:

1. The latitude and longitude of Washington, DC are about 39°N and 77°W; the location of Nassauin the Bahamas is about 25°N and 77°W. How far apart (in miles) are the two cities?

2. Chicago, Illinois is located at about 42°N and 87°W while Managua, Nicaragua is at about 12°Nand 87°W. How far apart (in miles) are those two cities?

3. The city of Madison, Wisconsin is located at about 89° west longitude and New Orleans,Louisiana at about 90° west longitude. How could you get friends in those two cities tocollaborate with you to make the measurements needed in Eratosthenes' method of estimatingthe circumference of the Earth, using the fact that it is about 1000 miles on land from Madisonto New Orleans?

4. Without access to a modern clock or telephones to communicate, how could observers at Syeneand Alexandria know that their Sun shadow observations were occurring at the same time onthe same day?

5. Without knowledge of the magnetic compass, how could people have known about directions ofNorth, South, East, and West?

6. On the following sketch of parallel lines cut by a transversal, what pairs of angles are congruentand how do those facts play a critical role in Eratosthenes' method for estimating thecircumference of the Earth?

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Where in the World is Christopher Columbus(Going)?

In some sense this investigation is the frostingon the cake. It asks students to put togethereverything that they've learnedlatitude/longitude, Sun shadows, estimation of Earthcircumferenceto reconsider one of the oldestand most common historical/geographic/scientific misconceptions. We have most oftenused this investigation (at least part 1) as aperformance assessment project that studentsdo outside of class. While we don't discourage

students from consulting each other, wegenerally ask for individual written reports.Even when there has been collaboration on theproblem, the written reports reveal a lot aboutpoints in the unit that remain unclear forindividual students. At the same time, thosereports offer an opportunity for individual stylein expressing ideas. Many students really getinto the spirit of writing a report to Ferdinandand Isabella and embellish their scientific ideaswith stylistic creativity that demonstrates realengagement with the problem.

Student Text

WHERE IN THE WORLD ISCHRISTOPHER COLUMBUS (GOING)?

A lot happened between 240 B. C. and 1492 A. D. During the Renaissance, the various nations ofwestern Europe, particularly Portugal, Spain, France, and (the city states of) Italy, found trading withChina and India desirable. However, between those European nations and their prospective Asiantrading partners lay several other countries inclined to block the trade. So, since the Earth is round,why not go west instead of east to reach the same spot?

Of course, there were other possibilities. Working out a friendly and mutually beneficial relationshipwith the other countries did not seem to be considered. Perhaps it would be possible to sail aroundthe southern tip of Africa and then up through the Indian Ocean. At the beginning of the fifteenthcentury Europeans were not sure that there was a southern tip of Africa. (If this sounds silly,remember that explorers were later to spend much effort trying to find a Northern Passage over thetop of North America. It was perfectly reasonable to think that Africa might extend into thesouthern polar regions.) But the tip was reached in 1488, and then Vasco de Gama successfully,reached India by that route in 1498. Thus it was de Gama who succeeded in doing what wasactually intended. Nonetheless, Columbus, in error about the size of the Earth and believing that hewas off the coast of Asia when he was really in the Caribbean, changed history. Such is often theway of major discovery and change.

Accounts vary as to exactly how Eratosthenes' excellent estimate of the size of the Earth gotreduced to 18,000 miles by Columbus. Columbus also planned to start in the Canary Islands to saildue west at the latitude of about 30°making the projected trip considerably shorter than it wouldhave been at the Equator. Further, he thought that the eastward distance to the coast of China waslonger than it actually was, and this led him to a corresponding reduction in the westward distanceestimate. At any rate, Columbus had it figured that Japan was about 2700 miles west of the CanaryIslands. Even if, as many Europeans believed, Japan was a fictional place made up by Marco Polo, itwouldn't be much farther to China.

Columbus' proposed trip was controversial. He had been turned down by Portugal, France, andEngland when he turned to Ferdinand and Isabella in Spain. The royal advisors in Spain were

52

opposed to the trip. Remember, the issue was whether this was a practical way to get to China andIndia, not whether it was a way to explore a whole new continent.

Rdvise the King and Queen

Imagine that you are a Royal Advisor to Their Majesties Queen Isabella and King Ferdinand ofSpain. The Queen calls you in one day and requests that you design an experiment to discover thetrue size of the Earth.

1. How could you reproduce the experiment of Eratosthenes without leaving Spain? Use whatevercurrent globe or atlas information you need and write up a careful explanation of your methodand conclusions.

2. Use your result from (1) and further reasoning to figure the distance around the Earth at alatitude of 30°. The following sketch might be a helpful guide in calculating the circumferenceof that circle of latitude.

Again, write up a careful explanation of your method and results.

What Do You Know?

As mentioned above, the previous investigationis often used as a performance assessment for

the unit. But we have also included questionsabout the unit on more conventional quizzesand examinations. A few typical questionsfollow.

Student Text

WHAT DO YOU KNOW?

I Latitude, Longitude, and Distance on the Earth

The cities of New Orleans, Memphis, and St. Louis are all located very close to the 90° west lineof longitude.

a) New Orleans is at 30° north latitude and Memphis is at 35° north latitude. What does thatimply about the distance in miles from New Orleans to Memphis?

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b) St. Louis is 250 miles north of Memphis. What does that imply about the latitude of St.Louis?

c) At noon on June 21 two friends observed their shadows cast by the Sun, in Memphis andNew Orleans respectively. If the friends were the same height, what difference would benoted in their shadows?

2. Angles, Circles, Sunlight, and Seasons

Imagine that you are looking at a side view of the Sun and Earth with your eye directed alongthe orbital plane.

a) Make two sketches like this. On one draw in the Earth's axis as you would see it from such a sideview on June 21. On the other draw in the Earth's axis as you would see it on December 21.

Sun

Earth

b) Explain how the sketches show that parts of the northern hemisphere have 24 hours ofsunlight at the summer solstice and 24 hours of darkness at the winter solstice.

c) What are the latitudes of the Equator, the Tropic of Cancer, the Arctic Circle, and the NorthPole?

d) How are those special latitudes related to each other and to the inclination of the Earth's axisof rotation?

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At the center of our campus is a veryspecial kind of sundial that shows, inaddition to time of day, the shadow lengthexpected in each month of the solar year.We have used this extended sundial as thebasis of another major performanceassessment project in the course final exam.

We usually initiate the project with a classvisit to the sundial for orientation to the

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problem and some initial measurements ofshadow lengths and angles. Despite the factthat the problem posed is really astraightforward application of the methodof Eratosthenes, we've found thatrecognition of this fact and the ability tovisualize the mathematical modelembodied in the sundial is a genuinechallenge to the students.

Student Text (continued)

3. As you know, lines, angles, circles, and spheres can be used to construct geometric models of thesolar system, including detailed positioning systems on our own planet Earth. Using thesemodels, you can develop scientific explanations of fundamental phenomena like days and nightsor seasons and years, and to estimate distances on and around our Earth.

At the center of our campus mall is a large sundial with an arm extending northward on whichmarkings are labeled with dates in each month of the year. Apply what you know about Sunshadows and latitude and distance to develop answers to the following questions:

a) How can you use the sundial arm and the date marks for the summer or winter solstice toestimate the latitude of the campus, given these facts: Tropic of Cancer is 23.5° north latitude,Equator is 0° latitude, Tropic of Capricorn is 23.5° south latitude.

b) Suppose that another sundial was located exactly 700 miles south of our campus with similarappropriate markings for shadow lengths at various times of the year. How could you usemeasurements from the two sundials to estimate the circumference of the Earth?

Write up your conclusions in explanations that could be understood by someone with knowledge ofhigh school geometry. Use sketches as needed to help clarify the ideas.

Summary and Conduding Comments

The "Where Are We" unit has been wellreceived by varied groups of studentswithsome local modifications by individualinstructors. Students are genuinely interestedto understand some facts of life on Earth thatthey had only vaguely comprehended before.They do seem to gain insight into both thehistory of science and scientific methods aswell as the interplay of science andmathematical modeling.

When we have the patience to let studentsreally grapple with open questions in the unit(instead of succumbing to the urge to "cover"more material) we find spirited discussionsbreak out frequently. Students who've neverthought of themselves as mathematicallycapable are excited to figure things out andreally understand some big ideas. One studentexpressed the feelings shared by many studentswhen she summarized her learning experiencein this unit:

I was just thinking today about the way I'vebeen learning in class. I think that it isworking well the way we need to figurethings out by thinking about them. It is alsocool the way the questions in a section lead

5S

up to more difficultthings as they goalong. I might notknow anythingabout the subject weare talking about,but with thinkingabout the questionsand bouncing ideasoff other people inthe group, I learn. Ilike this way oflearning by figuringout things for myselfrather than justbeing told theanswer. It does getfrustrating at timeswhen the answerdoes not comequickly, but I don'tthink that I willforget it as easily.I've been learning alot about models andhow to use them asa tool. As with manypeople in the class, I did not see how math

When we have thepatience to let students

really grapple withopen questions in the

unit (instead ofsuccumbing to the urge

to "cover" morematerial) we find

spirited discussionsbreak out frequently.

Students who've neverthought of themselvesas mathematically

capable are excited tofigure things out and

really understand somebig ideas.

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really entered into anything when we firststarted the class, but now I do.

Of course, not all students retain (or even gain)complete and confident grasp of themathematical and scientific ideas involved inthe problems of the unit. In fact, most facultyteaching the unit find themselves questioningprior inadequate or incomplete understandings

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themselves? This can be a disquietingexperience for faculty accustomed to masteryof the material they teach, but if one becomescomfortable with a genuine response of "That'san intriguing question; I'll have to think aboutit myself," there are many days on which deepstudent questions add an exhilaratingspontaneity to class sessions.

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A CONSTRUCTIVIST APPROACHTO PLATE TECTONICS

Rachel J. BurksGeosciences Program, Department of Physics

Towson UniversityTowson, MD 21252

have twice taught the course "Physical_Science II" for the MCTP program at TowsonUniversity. The first year I team-taught withJoe Topping, a chemist, and the syllabusreflected a blend of geological and chemicaltopics. The second year I taught it solo andrevised the curriculum to include about two-thirds earth science and one-third topics fromphysical optics (my most recent syllabus isattached as Appendix A). My choice ofsubjects within each field was guided by boththe AAAS Benchmarks (AAAS, 1993) and theNational Science Education Standards (NRC,1996), as well as those topics which lendthemselves particularly well to a"constructivist" pedagogy. In truth this selectionrepresents my favorite topics in the courses inphysical geology and the physics of light andcolor that I have taught in an increasinglyconstructivist manner at TU for several years.

Most of the activities I used in Physical ScienceII have been somewhat modified from what Ido in my other courses, where the earthscience content is more in-depth. In thefollowing text I present a sample of one suchactivity. After presenting this "module," I closethis chapter with some personal observationsof my MCTP experience and some excerptsfrom the students' final journal entries, inwhich they reflect on their learningexperiences throughout the course.

Primary Goal: Student Discovery

This unit focuses on the essential principles ofplate tectonics, the most significant unifyingtheory of modern geology. The primary goal of

the unit is to provide students with anopportunity to discover and construct forthemselves the rudimentary elements of platetectonics movements that have created theearth's present-day surface. This approachessentially reverses the traditional method ofpresenting this material, in which students aretold what major movements have occurredover the last 200 million years since the break-up of the great landmass of Pangaea, and arethen shown specific examples that illustratethe relative motions.

Given the opportunity to observe resourcessuch as detailed maps (especially the Heezenand Tharp ocean floor map) and globes (suchas the National Geographic Physical Globe),my students have been able to formulate someof the fundamental relationships betweengeologic features, such as mid-ocean ridges andcoastal mountain belts, and the processes thatform them. They have also been able tocalculate general movement rates and to gainan appreciation for the enormity of timerequired for such large-scale movements.

Creating opportunities for discovery for mystudents required that I choose from among anumber of physiographic areas that showevidence of plate movements. For example, therelatively straightforward illustration I chosefor a central activity in this unit is thesymmetric seafloor spreading of the mid-Atlantic ridge, which has been occurring sincethe opening of the north Atlantic Oceanapproximately 100 million years ago. Datedrock core samples from specific locations in theAtlantic seafloor enable students to measure

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spatial distance of the cores from the ridgeaxis, where basaltic rock material is presentlyforming. Dividing this distance by the timeelapsed since the rocks' formation (i.e., theage) generates the rate of movement of thatsegment of oceanic plate from the ridge. Sincethe other side of the plate is also moving awayfrom the ridge, the total movement rate istwice this value. Such data are readily availablefrom numerous locations across ridges on theseafloor (Brice, Levin, & Smith, p. 52-53).

Although my students often struggled duringthese activities, their journal entries have

shown recognitionthat, as one studentwrote, " . . . thebeauty is that whenyou figure it out withsome effort, youunderstand it better."Moreover, studentswho plan to becometeachers have drawnconclusions such as,"When I see my kids

frustrated, I will guide them to discovering theanswer because it is more rewarding to solve aproblem than to have the material dictated."

Overall, the plate tectonics unit was designedto engage students in the following:

® observing the distribution and geometricrelationships of major physiographic featuresof the earth's surface;

applying previous knowledge of basicgeography to the development of broaderthemes;

manipulating scale, involving measurementof both enormous spatial dimensions as wellas inexorably slow rates of motion; activitiesinclude:

constructing a scaled representation ofthe 4.6 billion years of geologic time(plotting specific events on this scalerequires real comprehension rather thanmemorization of the use of exponents);

determining movement rates andconversion of these quotients to

. . . as one studentwrote, " . . . the beautyis that when you figureit out with some effort,

you understandit better."

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meaningful expressions, generally on theorder of centimeters per year;

- utilizing measuring skills, as studentsexpress the distances between locationswhere rock samples have been dated; and

performing authentic assessments ofknowledge through meaningful analysis ofnew data sets and comparison with theirprevious results.

In designing the unit with these goals, itbecame essential for me to integrate instructionin mathematics. The students' facility withexponents was addressed in several guises:labeling of a time scale with billion- andmillion-year divisions, converting rates forkilometers per million years to centimeters peryear, and construction of the Richter scale.

Context ler the Mate 'Tectonics Unit

This unit is designed to follow several student-centered investigations wherein they discoverthe essential characteristics of minerals androcks from their own observations, andascertain that the earth's surface is divided intocontinents and ocean basins. One of thesepreliminary activities involves thedetermination of densities of materials fromthe continental crust, oceanic crust, andmantle. A hypothesis students may develop toexplain the density relationships between thesematerials is that the earth may be layered, andthe lighter layers "float" on the heavier layers.Class discussion of this may lead students tospeculate about the earth's core, which wouldbe comprised of the heaviest material. Since ofcourse we do not have samples of this, a smallpolished piece of a iron meteorite, presumablyfrom the core of an ancient planetary body,usually amazes students at this point.

Students then develop hypotheses for heattransfer from the molten outer core outwardthrough the mantle, the process that driveslarge-scale plate tectonic movements of earth'splates. From this point, using the unit describedin this paper, students investigate fundamentalprocesses and plate movement directions andrates, eventually verifying their hypotheses anduncovering for themselves evidence for the

enormity of geologic time. Students then beginto explore aspects of seismic waves as atransition to the physical optics section of thecourse.

Pedagogical Considerations

In developing and teaching this unit with aconstructivist approach, I took the followingkey factors into consideration:

Engagement. To spark students' interest andstart them thinking about physical changes ona large spatial scale, I presented them withcutouts of some of the major continents,especially Africa and South America, andallowed them time to explore. (Photographicslides of satellite views of parts of the earth canalso illustrate this.) After some minutes ofmanipulation, the students "discovered" thatsome of the coastlines fit together.

Prior Knowledge. On the first day of this unit, Ihad the students fill out an anonymousquestionnaire where they can record theirpreconceptions about the earth's surface andinterior (see Appendix B). I have found thatmost students have sufficient geographicknowledge to identify the landmasses, and haveheard of "continental drift," which is theantiquated forerunner of modern plate tectonictheory. Our class discussions elicitedconnections between tectonic movement andgeologic processes like earthquakes andvolcanoes. A likely area of frustration involvesstudents' manipulation of exponents, requiredboth in the constructions of the geologic timescale and in the expression of tectonicmovement rates.

Wionitoring Beliefs. By encouraging continualclass discussion and questioning, I have foundthat students have ample opportunity toexpress skepticism and sometimes awe at themagnitude of the physical changes recorded onthe earth's surface. I have also found thatletting students "discover" plate tectonicsgradually in this manner, rather than firstlecturing on it as a "fact," allows those studentswith more fundamentalist religiousbackgrounds to appreciate and understand thisgeological theory with more open minds.

Concept Development. I encourage students todevelop multiple working hypotheses as theywonder about the origin and nature of thevarious features they encounter in examining adetailed physiographic map of earth's surface,both continental and ocean floor (Dolgoff,1996, p. 256-257). I created a "travelogue"(Appendix C) to guide their observations asthey cruised along mid-ocean ridges fromIceland all the way around to the San Andreasfault in California. Photographic illustrations offeatures such as volcanic island arcs cansupplement student investigations of theselandforms.

Mahan Connections. This unit gives students theopportunity to recognize that a geographicpattern of "disasters" can be seen as theyconnect localities they have heard of in thenews with their geologic positions relative toactive plate boundaries. To facilitate theirmaking these connections, we explore severalInternet sites, such as the U.S. GeologicSurvey's Earthquake Information page, to letthe students "wander and wonder" through theWeb's geological resources.

Group Dynamics. Collaborative learning wasemployed throughout the course, from grouplaboratory activities to group work on parts ofthe performance assessments in the courseexaminations. Groups were rearrangedthroughout the semester to give these futureteachers exposure to a variety of learning stylesand changing group dynamics.

Student Reactions. Expressions of bewilderment,frustration and confusion are fairly common inthe beginning of this unit, and students areencouraged to share and to listen to these, asthey only validate the enormity of the scale ofthe processes that the students are in theprocess of discovering.

Materials

The various maps used in this unit may befound in earth science supply catalogs (likeNatural Science, Ward's annual catalog) andlaboratory manuals in either physical orhistorical geology. Materials for the earthquakeseismogram activity can also be from publishedcase studies. The geologic dates of major events

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for the time scale construction can also befound in lab manuals. Other basic materialsinclude rulers and calculators.

The most significant change I made when Itaught this course for the second time, otherthan topic choices, was to incorporate a fewCD-ROMs (such as The Theory of PlateTectonics [Tarbuck, 1994]) for geophysicaldatabases and interactive plate tectonicreconstructions. Several Internet sources werealso used, such as sites for seafloor rock agesand current earthquake and volcano data.

Unit Chronology and Reflection

The first semester this course was taught, theunit on plate tectonic movements andreconstructions lasted for three class days (eachclass meeting time was 2'4 hours). My team-instructor and I realized then that a fourth daywould have provided more opportunity forstudents to practice their calculation skills andextend their comprehension. During thesecond semester, I added two days to the unit,and introduced earthquake data, determinationof epicenter locations, and seismic wavebehavior, which made the transition to the nextunit on light waves and physical optics.

Day 1

As an introduction to the concept of platetectonics, students examined continentalshapes and their positions on a largephysiographic map of earth's surface. Using thetravelogue mentioned earlier, they took a"guided tour" with question prompts thatfollowed mid-ocean ridges from Iceland aroundto Central America and up into the SanAndreas fault of California. Observations ofphysiographic features such as volcanic islandswere correlated with geologic activities thatthey knew about (earthquakes, volcanoes).During a class discussion, we connected apreviously assigned reading (from Trefil &Hazen, 1995) with these discoveries, althoughprior readings are not necessary nor probablyeven desirable. A follow-up "homework"activity was to reassemble Pangaea asaccurately as possible using cutout continents;problems encountered in the reconstructionswere to be discussed in the next class.

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Day 2

The class began with the film Powers of Ten(Demetrios, 1989). Students then constructed ageologic time scale using 15-meter lengths ofregister tape and a meterstick, with 1 meterbeing equal to 1 billion years. Students were tolabel the eras (Precambrian, etc.) and"discover" a conversion that would enablethem to plot important dates onto the tapesuch as the extinction of dinosaurs 67 millionyears ago (lab manuals containing thisinformation include Fletcher and Wiswall,1990,p. 91-92). Students struggled a bit as theyall discovered that they did not understandexponents as well as they thought they did.(This became a recurring mathematical themein the course.) Student discussion and journalentries indicated that this was an eye-openingand thought-provoking experience.

Day 3

Students continued discussion of platemovements by manipulating a three-dimensional globe and examining "flip books"(Scotese, 1977) showing the plate positions forthe last 700 million years. This led to the ideathat plate movements can be quantified fromage determinations in seafloor core samples asa function of distance from a mid-ocean ridge.Students' activities involved carefullymeasuring map distances of dated cores andcalculating the spreading rate, being eventuallyguided into doubling the values because of thesymmetry of the ridge. Expressing these ratesin centimeters per year proved challenging, andwe spent time manipulating exponents onceagain. Students were astonished to discoverfrom their own calculations that the Atlantic isopening at the rate of several centimeters peryear. They were then asked to convert this tosomething that they could visualize, such asrate of fingernail growth. This was undoubtedlymuch more meaningful than if they were toldthe plate movement rates as a given fact.

Day 4

Since the most dramatic evidence of platemovements is earthquakes, we discussed thebasics of earthquakes, which diverted into anillustrated mini-lecture format to explain basic

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seismology terminologyepicenter, focus,magnitude, P-and S-waves. Longitudinal andtransverse wave motions were illustrated withsprings. Frequency, velocity, amplitude andenergy were also explored and discussed.

Day 5

An earthquake epicenter location exerciseinvolved measuring P-and S-waves arrival timesfrom a seismogram, converting to distance, andplotting intersecting arcs on a map to pinpointlocation. Various Web sites can be visited toexamine some aspects of the occurrence of andparameters associated with earthquakes.Earthquakes provide a transition into the laterstudy of waves.

Assessment

Several methods for assessment of studentcomprehension were utilized, including theweekly journals. The midterm exam containedtwo authentic assessment activities of studentunderstanding of the features andmeasurement of plate tectonic movement. Thefirst involved a physiographic map of theSouth Sandwich Islands; students interpretedthis area's plate tectonic setting from thetrench-island arc features, and then predictedwhat type of geological materials would beencountered there (the exam must followigneous rock section). The second assessmentproblem included several maps of theHawaiian Islands taken from a variety ofsources. Each map displayed radiometric agesof basaltic rock from several islands, whichhave been moving over a stationary "hot spot"in the mantle for the last 100 million years.From the pattern of rock ages, which getprogressively older to the northwest along thechain, students accurately concluded that platemovement over the hot spot produced thisstring of islands and submerged seamountsacross the Pacific floor toward the Aleutiantrench. They then measured the distancesbetween rock localities and calculated themovement rate; expressing this in centimetersper year always proved to be a challenging test

of their facility with exponents. Students thencompared these with rates determined for theAtlantic, discussed the variations, andexplained their significance (i.e., the Pacificrates are much faster). This process alsorequired them to evaluate which of thepublished maps were most useful for thisexercise, and to explain why.

Instructor's Reflections on e N1 Experience

During the first semester instructors as well asstudents kept journalsof their thoughts andobservationsthroughout the courseand many of thesewere included in theprevious description ofthe unit activities. Ihave since this timealso followed a moreconstructivist approachin my "regular"physical geologycourse, and have notedsimilar student surpriseand satisfaction whenthey "discover" forthemselves evidencefor plate movementand those processesthat create mountains.However, I find it isgenerally difficult formany introductory students to realistically graspand internalize either complex sequences ofgeologic events or time periods of more than amillion years.

I have gradually become less devoted to"covering" a certain amount of content in myintroductory geology courses, but rather to usinga selection of thorough, student-centeredinvestigations of key concepts and processes. It isstill difficult for me, however, to whittle downthe "core" of the physical geology course anyfurther to include more constructivist units.

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I have sincethis time also followeda more constructivist

approach in my"regular" physical

geology course, andhave noted similar

student surprise andsatisfaction when they

"discover" forthemselves evidence forplate movement andthose processes thatcreate mountains.

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Students' Reflections

After my sense of initial frustration, I beganto desire to understand. . . .

The students' experiences in this course weregenerally good, although frustration levels andgrade anxiety were higher than usualthroughout the course. Weekly journal entrieswere one way I tried to assuage their fears andprovide them with encouragement throughresponses to their journal questions andsuggestions for further explorations. I foundthat some students remained shy about askingquestions in class, although their intellectual

involvement in thecourse material andlearning process wasevident in theirjournal entries. I alsocame to preferwritten journals toelectronic onesbecause I felt I couldgauge students'anxiety levels fromtheir writing; theycould also makesketches of geologicfeatures they'd seenand wanted to askabout. I havecontinued these"learning journals" inall of myintroductory coursesbecause, despite the

time involved, I get invaluable feedback andcan keep some students focused on improvingtheir performance rather than losing them infrustration during the semester.

For this semester described in this report, thefinal journal entry was a reflection on theentire course and an assessment of their ownefforts and participation. Students were to readthrough their entire journals prior to thisexercise. I have excerpted below some of theirmost revealing comments, both positive andnegative, and I think they tell us more aboutthe success of the MCTP program thananything I could write.

I have continuedthese "learning

journals" in all ofmy introductory courses

because, despite thetime involved, I getinvaluable feedbackand can keep somestudents focused on

improving theirperformance ratherthan losing them infrustration during

the semester.

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Reflections on ways of learning:

I just finished reading all my journal entriesfrom the past classes and I noticed a verystrong trend: at the beginning of units Isounded baffled and unsure of myself andthe material, by the mid-to-end of units itsounded like I was actually getting the hangof what we were doing during the class. . . .

000000

When it comes to testing, if you havelearned the material rather than justmemorized it, you can usually apply it toany situation.

000000

In the beginning, with the classification ofminerals, I wanted to start the lab bymeasuring and calculating rather than justlooking at what was there. But, by the end, Iwas able to just look at something and thinkabout what it meant.

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After the first exam, I began to focus onunderstanding concepts instead ofmemorizing.

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These journals have been very helpful to meas well. I was able to write down questions,concerns, and new learning. After havingread through my journal again, it is evidentthat I got better at figuring out the toughconcepts within my group and on my own. Ialso got better at asking questions. Beforethis class I was embarrassed to askquestions. This has been tough yetrewarding.

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I learned much more from demonstrationsthan anything else. The hands-on approachmakes me remember more for longer.

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It was a good idea having us figure outabout plate movement instead of telling us.It really made me think about it and putdifferent ideas together

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On group work:

Group work was very helpful for me. I havealways learned best through discussion withmy peers. However, I feel that we should havedeserved more credit for our groupassignments.

000000

I feel that group work really enhances one'ssense of knowing Working with peers allowsyou to actually discuss the concepts out loudand also allows you to help each other whenone member of the group stumbles upon aproblem they cannot solve by themselves.

On how tough the course was for somestudents:

[This course] was a challenge and frustratingand at times downright confusing. . . . I havelearned quite a bit about physical science(although a different pedagogy would havemade a less stressful learning experience).It's like the course was harder than it neededto be.

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I found myself taking great interest in thematerial for a couple of reasons: 1) the actualtopics blew my mind, i.e., plate tectonics, andI wanted to know more about it for personalreasons; and 2) this was really the first time(in a science course) I had trouble grasping aconcept upon initial introduction.

On the course's value for future teachers:

There are a lot of things that I have learnedfrom this course that I can use in my ownclassroom. I learned that science can be fun!This is essential for children to realize, itsparks their interest. We did a lot of neatthings with rocks, prisms, and rainbows, all ofwhich kids in general like to learn about.

000000

This course not only taught me aboutphysical science, but it taught me how Ilearn. . . . By understanding the process oflearning, I feel I can be a much more effectiveteacher.

C00000

I think the method of doing an experiment,without a lot of instruction, seeing whatchildren gain from the experiment bydiscussing it, and then building and refiningtheir experiment by discussing it, is awonderful way to teach.

000000

This course successfully taught me physicalscience, but it also has impacted the way Ilearn and the style of teaching that I will usein my classroom.

000000

Another outstanding attribute I am takingfrom this course comes directly from yourteaching: I will not give the students theanswers. I will stick beside them and aidthem in coming upon the answer forthemselves.

000000

Before taking this course, the only way Icould identify a rock or mineral wasbecause I had seen it before andremembered what it looked like. Now I canlook at a rock I've never seen before andfind identifying characteristics in it. To me,it's so neat to be able to do that. And if I'mexcited about it, and find it fun, when I getinto a classroom, I can pass that excitementonto my students.

And my personal favorite:

It's like the old saying: You can catchsomeone a fish and they can eat for a day.But if you teach them how to fish you canfeed them for a lifetime. Throughout thiscourse I have learned how to fish.

Motes and References

The laboratory manuals listed below are widelyused in introductory college geology courses;other texts would be appropriate as well. One ofthe best maps is the classic Heezen and Tharp1977 Ocean Floor Map; large laminated versionsof this map are available from Ward's, or smallerversions may be found in lab manuals likeFletcher and Wiswall (p. 256-257). RecentlyWeb sites have become a good source of plate

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tectonics information and other earth sciencesubjects; addresses may be found in the newestgeology texts like Press and Siever.

American Association for the Advancement ofScience. (1993). Project 2061, Benchmarks forScience Literacy. New York: Oxford UniversityPress.

Brice, J.C., Levin, H.L., & Smith, M.S. (1993).Laboratory Studies in Earth History (5th ed.).Dubuque, IA: Wm. C. Brown.

Demetrios, L.E. (Exec. Producer). (1989). TheFilms of Charles & Ray EamesVolume 1Powers of Ten [Film]. Santa Monica, CA:Pyramid Film & Video.

Dolgoff, A. (1996). Physical Geology.Washington, DC: D.C. Heath and Co.

Fletcher, C.H., & Wiswall, C.G. (1990)Investigating the Earth: A Geology LaboratoryText. Dubuque, IA: Wm. C. Brown.

Heezen, B.C., & Tharp, M. (1997) World OceanFloor [Scale 1:23,230,000]. Palisades, NY:Lamont Doherty Geologic Observatory.

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National Research Council. (1996). NationalScience Education Standards. Washington, DC:National Academy Press.

Natural Science (Annual Catalog). (Availablefrom Ward's Natural Science Est., Inc., 5100 W.Henrietta Rd., P.O. Box 92912, Rochester, NY14692).

Press, F., & Siever, R. (1997). UnderstandingEarth (2nd ed.). New York: W.H. Freeman &Co.

Scotese, C.R (1977). Phanerozoic Plate TectonicReconstructions: Paleoceanographic Mappingproject (Tech. Rep. No. 90). Austin: Universityof Texas, Institute for Physics. [Currentavailability unknown.]

Tarbuck, E.J. (1994). The Theory of PlateTectonics [CD-ROM]. Albuquerque, NM: TasaGraphic Arts, Inc.

Trefil, J. & Hazen, R.M. (1995). The Sciences:An Integrated Approach. New York: Wiley andSons.

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4

Appendix A

COURSE SYLLABUSPHYSICAL SCIENCE IIPHSC103

Coarse Schedulle

Date Topic/Activity Text

1/30 Introduction; Questionnaires; Observation of physical objects2/4 Description, measurement and classification of physical objects 2

2/6 Introduction to floating versus sinking2/11 Determining density: assessment activity 4, 52/13 Continued buoyancy, specific gravity measurements2/17 Physical properties of minerals: hardness, cleavage, etc.; 28

Solid geometry of crystals, investigation of symmetry elements2/19 Solids: dissolution; crystallization from solution (cont'd at home)225 Crystallization from a melt: inference of cooling rate from grain size_2/27 Exam 13/4 Igneous rocks: interpretation of textures, classification3/6 The Rock Cycle: weathering, transportation, deposition, lithification, etc.3/11 Soil: physical and chemical characteristics3/13 Earth's differentiated structure; Introduction to Plate Tectonics 273/18 Films: "Powers of Ten"; Geologic time scale; Tectonic plate movements 31

3/20 Measurement of plate movements4/1 Earthquake waves: epicenter location4/3 Waves and wave behavior; Interference4/8 The electromagnetic spectrum 16

4/10 Exam 24/15 Color mixing: additive and subtractive4/17 Theatrical lighting: trip to Fine Arts422 Young's double slit demonstration: diffraction of blue versus red light 15

4/24 Atoms: emission spectra, identification of gases, solar spectroscopy429 Refractive index of solids and liquids5/1 Dispersion and the white light spectrum5/6 Rainbows, halos, and coronas5/8 Scattering; Why the sky is blue5/13 Mirages; Course evaluations

BEST COPY AVAILABLIF,

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Genera(' Course Onformagon

Meeting limes acrod Teaching Methodology

Class meets in Smith Hall 473 on Tuesdays and Thursdays from 12:30 to 2:45 p.m. The sessions willgenerally be laboratory-type activities with little formal "lecture." The pedagogical approach in thiscourse employs a series of student-centered collaborative exercises that hopefully lead to students'discovery of and construction of fundamental concepts. We will be integrating fundamental topicsfrom the fields of earth science and physics, along with some chemistry and mathematics.

This course has been developed as part of a state-wide program called the Maryland Collaborativefor Teacher Preparation (MCTP) that is funded by the National Science Foundation. Although theultimate goal of the MCTP program is to prepare pre-service teaching specialists in science andmath for middle grades, we strongly believe that other students will benefit from such a course thatstrives to integrate science topics in an innovative way. The essential' prerequisite is Physical ScienceI, along with enough mathematics to handle the calculations and data manipulations we will bedoing. This is a "content" course, rather than a "methods" course; the pedagogy will employ aspectsof "constructivism" similar to what you might be using as in-service teachers. The content isappropriate to university students, although most of the topics are also designated for middle gradesin both AAAS Benchmarks (AAAS, 1993) and the National Science Education Standards (NRC,1996).

Text

The Sciences: An Integrated Approach: Laboratory Manualby Robert M. Hazen & James Trefil (1995), Wiley and Sons.

This text will serve as a supplement to the laboratory handouts, and may prove a useful resource inyour future profession. Additional readings will be assigned throughout the semester.

Grading

Exercises approximately 250 points

Journal 100

Exams 300

Portfolio 50

Participation 50

750 points

Exercises

The in-class activities and at-home assignments (homework, writing, Internet exercise, etc.)comprise the majority of your learning experiences in this course, and it is vital that you undertakethese in a serious manner. Be prepared to turn in each day's exercise at the end of that class period.In-class writing will be done periodically as well. Keep all your course materials together in aportfolio that you will turn back in at the end of the semester, along with additional self-evaluations.

Journal

The journal is intended as a means for you to reflect on your learning experience: what yourpreconceptions were about a topic, a very brief synopsis of what you did in class, what worked for

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you in that activity, what didn't, what you discovered, what surprised you, what you still areconfused or frustrated about. You should make at least one full-page entry for each class meeting.Unless otherwise instructed, the journals will be turned in each Thursday at the end of class. I preferyour entries be made on loose-leaf paper in a manila folder (so you can make an entry even when Ihave your journal). Please turn in your entire journal each time so that I may refer back to monitoryour progress (this is one reason why I prefer written journals to electronic ones).

Examinations

The exams in this course will reflect its goals and methodology, requiring you to develop anddisplay the higher-order thinking skills of application, analysis, synthesis, and evaluation, rather thanjust recall and comprehension. The exams consist of authentic and/or performance assessments andproblem-solving exercises that you will work on alone and/or in a group.

Attendance and Participation

Because of the "hands on" nature of the activities, you (and your hands) must be here each classperiod. The instructor's schedule this semester rules out your making up a 2+ hour class without alegitimate, documented excuse involving some type of personal emergency. Your physical andmental participation are absolutely essential for this kind of pedagogy to work, as you and yourcolleagues learn from each other. Good participation includes asking and answering questions inclass (reflecting your preparation), undertaking and completing the assigned activities in an engagedand thoughtful manner, cleaning up materials as instructed, and being in general a good classroomcitizen.

Portfolio

At the end of the semester, you will assemble all of your course work (including your journals)neatly into a portfolio, which will include some new metacognitive reflections and self-evaluations.These portfolios may be displayed to other faculty participants in the MCTP program, and journalentries may be photocopied (all without your names). They will be returned to you after thesemester ends.

Academic integrity

Group work is of course collaborative by nature, but should not involve copying. You are expectedto contribute your own ideas, and will be required to perform individually as well as a teammember of your group. Groups will be reassigned periodically throughout the semester. Cheating onclassroom activities will result in zero credit to all involved parties. Cheating on an exam will resultin failure of the course and will be reported to the Physics Chair and the College Dean.

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Appendix B

PHYSICAL GEOLOGYPRELIMINARY QUESTIONNAIRE

What are the basic types of rocks and how do you tell them apart?

What exactly causes earthquakes? Why do they occur where they do?

How old is the earth and how do we know this?

What is the largest mountain chain on earth? Why are there mountains?

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No name please

Appendix C

TRAVEL GUIDE TO THE TECTONIC FEATURESOF THE EARTH'S OCEAN FLOOR

Start at Iceland. What do you hear about geologic hazards in Iceland?

Draw a general sketch of the prominent feature you see running south from Iceland on the floor ofthe Atlantic Ocean:

Follow this mid-ocean ridge south. Compare the shapes of the coastlines of South America and_Africa._How does the ridge relate to these coastlines? What process do you think could create thismid-ocean ridge? Now add arrows to your sketch.-

Follow the mid-ocean ridge around the southern tip of Africa and into the Indian Ocean, notingthat it branches. Explore the branch that goes under the seas that bound the Arabian Peninsula andnortheastern Africa. From what you observe on the seafloors, how do you think these seasoriginated?

Now go back and follow the mid-ocean ridge across the Indian Ocean and all the way across thePacific. Trace its end northward, where it becomes the San Andreas Fault. What are some of thegeological events you hear about that occur here?

You have just traced an important plate tectonic boundary. Not all volcanoes happen along plateboundaries. Locate the Hawaiian Islands in the Pacific. Note that as you go northwest, the "islands"become submerged. Why do you think they do that?

The only active volcanism within this chain is at the southeastern edge of the "big island" of Hawaii,although all of these islands and seamounts are made of the volcanic rock basalt. Whatinterpretations can you make about the origin of this chain?

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This entire continuous chain is called the Hawaiian-Emperor seamount chain. What happens to theorientation of the seamount chain as you go northwest? What might cause this?

Draw the general map pattern of the chain, including arrows to illustrate your interpretations:

Now examine the islands of Japan. What geological activities do you hear about happening inJapan?

Mt. Fiji is a famous volcano in Japan that you've probably seen pictures of. How does its shapediffer from the broad shield volcanoes of Hawaii? What does its steeper shape signify about itseruption style?

Note that the Japanese islands form an arcuate shape in contrast to the linear trend of the Hawaiianislands. What deep undersea feature lies adjacent to the islands?

How is it positioned relative to the archipelago of islands, that is, is the trench on the concave orconvex side of the island arc?

Draw the general map pattern of Japan and the adjacent undersea trench:

Where else in the western Pacific do you see a similar pattern?

What can you predict about the geological hazards of these areas?

Now compare the eastern and western coasts of North America, particularly offshore. How are theydifferent, and what might be some geological reasons for this?

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4

A MATHEMATICAL MODELING COURSEFOR ELEMENTARY EDUCATION MAJORS:

INSTRUCTORS' THOUGHTS

Don Cathcart and Tom HorsemanMathematics and Computer Science Department

Salisbury State Universitynol Camden AvenueSalisbury, MD 21801

ntroduction to Mathematical Modeling is thefirst mathematics course in the sequence of

MCTP courses taken at Salisbury StateUniversity by prospective elementary andmiddle school teachers of mathematics andscience. It is taken concurrently with an-interdisciplinary science course. The -followingcase report is based on our experiences indesigning and teaching the course and onmaterial in our Instructor's Guide (Cathcartand Horseman, 1995), which is used tointroduce other instructors to the MCTPphilosophy and to provide a framework for thecourse. We first taught two sections of thiscourse during the spring semester of 1996.

We designed the course to focus on problem-solving processes used in mathematics andscience. It uses perspectives, knowledge, data-gathering skills, and technological tools relevantto those disciplines. Our underlying principlewas to integrate mathematics and science usinga constructivist, activity-based approach.Specific assignments are described herein, andone is covered in detail.

Students are required to use calculators,spreadsheets, and microcomputer-basedlaboratories (MBLs) for collecting their owndata, generating graphs, and analyzing theirresults. Working together, students areencouraged to construct physical conceptsfrom their observations and make connectionsbetween science and mathematics. At no timeare the students permitted to use thetechnology for a "black box" solution. In other

words, they are not permitted to use the curve-fitting capability of graphing calculators orspreadsheets.

Activities in the course focus on helpingstudents to: (a) see connections betweenmathematics and other disciplines; (b)represent and-analyze-real-world phenomenausing a variety of mathematicalrepresentations; (c) develop strategies andtechniques for applying mathematics to solvereal-world problems; and (d) explain andjustify their reasoning, using appropriatemathematical and scientific terminology, inboth oral and written expression.

The need to improve the mathematicalpreparation of pre-service elementary schoolteachers is well documented in recent reportssuch as On the Mathematical Preparation ofElementary School Teachers (Cipra and Flanders,1992), and Curriculum and Evaluation Standardsfor School Mathematics (NCTM, 1989) andProfessional Standards for Teaching Mathematics(NCTM, 1991) by the National Council ofTeachers of Mathematics. The NCTM Standardscall for increased emphasis on thinking,reasoning, and problem solving. The NCTMBoard of Directors, in a 1995 statement oninterdisciplinary learning, stated, " . . . thecurriculum must encourage the use of theperspectives, knowledge, and data-gatheringskills of all disciplines. .. . " (NCTM, 1995, p. 1).

Currently the mathematics requirement forelementary education majors varies widely

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from college to college (Cipra and Flanders,1992). In many cases, prospective teachers take asingle, one-year course covering a variety oftopics such as number systems, a little numbertheory, a look at probability and descriptivestatistics, and informal geometry. While somecolleges require more than a single, two-coursesequence, few prospective teachers are placed insituations where they are expected to discoverand express relationships found in the worldaround them. Moreover, few are currentlyexposed to course content that integratesmathematics and science while placing specialemphasis on the concept of a "function as amodel" and on "model building" as a process.

In our experience teaching upper-levelmathematical modeling courses for mathematics

majors, we facedtension between ourwish to cover specificmathematical contentand our desire todevelop the students'ability to solve realproblems usingmathematics. Ourtypical approach wasto state a problem insome quasi-realisticcontext, introduce atechnique oralgorithm for solvingthe problem, and thenrequire students to

employ the technique or algorithm inisomorphic contexts.

That is not the approach employed in thiscourse. We decided to focus on processes used inapplying mathematics to address questions inother disciplines. In our experience, we havefound that students have difficulty focusing onproblem solving as a process whensimultaneously trying to understandmathematical techniques and concepts. So, thecourse described here was not intended to serveas a vehicle for teaching specific mathematics orscientific content. We assumed that pre-or co-requisite mathematical or scientific content hadbeen, or would be, addressed in other courses.

. . . as students wereinvolved with activities

in this course, theyactively engaged inconfronting their

misconceptions andcreating their own

knowledge. So, our rolewas that of facilitator

or guide.

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Poimany Expected Outcomes

In completing the exercises and activities weexpected that students would:

O see and value connections betweenmathematics and other disciplines;

O represent and analyze real-worldphenomena using a variety ofrepresentations;

O use their own observations,experimentation, computers or calculators,and references to find answers to questions;

O develop strategies and techniques forapplying mathematics to solve real-worldproblems;

work with others while solving problemsand learning;

O become skillful in explaining and justifyingtheir reasoning, using appropriatemathematical and scientific terminology, inboth oral and written expression; and

O critically evaluate their own work and thework of others.

Pedagogica0 Approach and Course Content

We are convinced of the validity of aconstructivist approach for teaching integratedmathematics and science. Therefore, asstudents were involved with activities in thiscourse, they actively engaged in confrontingtheir misconceptions and creating their ownknowledge. So, our role was that of facilitatoror guide. We determined the degree to whichstudents met the expected outcomes andaltered their preconceptions by examiningtheir written work, listening to their oralexplanations, and observing them duringworking sessions.

Our intent was to integrate mathematics andthe sciences. We selected applications fromboth physical and life sciences. A fewapplications from sports and managementscience were included. Some microcomputer-based laboratory (MBL) activities werecoordinated with a companion integratedscience course taken concurrently with thiscourse.

A sample course syllabus (Appendix A) andtwo sample exercises are provided (one withinthe body of this paper and another inAppendix B). The syllabus form is based onthat of O'Haver (1994). The guidelines forexercises are adapted from those of Shannonand Curtin (1992), and the portfolio guidelinesfrom Abruscato (1993).

Part I of this course focused on functions asmodels for phenomena and on thedevelopment of a repertoire of techniques tobe used in modeling. Functions are used torepresent relationships of some particularinterest: A change in the value of one variablecauses a change in the value of anothervariable. Students investigated the manner ofthat change in both qualitative andquantitative ways. Change and rate of changewere illustrated and given meaning. Additionalmathematical topics addressed in exercisesincluded the following: difference equations,difference tables, difference quotients, recursivefunctions; secants, least-distances_criterion,_andleast-squares criterion.

The focus of Part II was on the concept of amathematical model and on mathematicalmodeling as the essential element in applying

mathematics to real problems (e.g., studies ofmotion, heat loss, and light intensity). Thephrase "applications of mathematics" usuallymeans useful connections betweenmathematics and other fields. Although theapplication of mathematics in different fieldsrequires a variety of different mathematicaltechniques, there is a common unifyingelement in applying mathematics to real-worldproblems. That unifying element is the conceptof a mathematical model. The activity ofconstructing, analyzing, interpreting, andevaluating mathematical models is the centralprocess in applying mathematics to realproblems. Our view of the modeling process isbased largely on that of Maki and Thompson(1973), Roberts (1976), and the Curriculumand Evaluation Standards for SchoolMathematics (NCTM, 1989). This modelingprocess is illustrated in idealized form below(see Figure 1).

Step 1: Simplificationfidealization. The modelingprocess usually begins when one is faced with asituation and is concerned about some aspectof that situation. Often the situation is very"fuzzy" and the concern is vague. So, the initialstep in modeling is to define the situation or

SimplificationIdealization

Interpretation

MathematicalTheories andTechniques

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MathematicalTranslation

MathematicalModel

Figure 1. The Modelling Process

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problem as carefully as possible. Theformulation of a specific problem is important,but often difficult, and sometimes requirescreative effort. Attempts at precisionfrequently involve making idealizations,simplifications, and approximations becausemost real problems are too complex ornebulous to be treated mathematically in a waythat includes all real-world aspects of theproblem (Maki, 1975). In this problemformulation step essential variables are isolatedand the critical questions identified.

Step 2: Mathematical Translation. The realproblem is translated into the language ofmathematics. Symbols and mathematicaloperations are used to represent real quantitiesand processes. This process of mathematizationproduces a mathematical realization of the realproblem called a mathematical model.

Step 3: Application of Mathematics. Once theproblem has been expressed in mathematicalform, we study the resulting mathematicalsystem using concepts and techniques ofmathematics. If we are skillful, persistent, orfortunate, we will produce mathematicalconclusions or predictions.

Step 4: interpretation. Our mathematicalconclusions or predictions must next be"translated back from the language of ourmodel to the language of the real world andinterpreted as real-world conclusions orpredictions." (Roberts, 1976, p. 8)

Step 5: Validation. The conclusions andpredictions are compared with the real-worldphenomenon being considered. Only rarelywill our results from the mathematical theoryagree completely with real-world outcomes.Usually a mathematical model will not reflectsome important aspects of the situation beingstudied. Only when the results of ourmathematical study compare favorably withreal-world data will we have some confidencein our model. Otherwise, we must either makedo with a model we suspect is inadequate, orwe must refine the existing model by retracingthe modeling process until an acceptablemodel is found. (Maki, 1975).

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PolevequisOte Knowledge

The emphasis in the exercises and activitieswas for the students to use mathematics andscience they already knew in approaching atask. We would expect that students with twoyears of algebra, one year of life science, andone year of physical science would be able toperform and understand these activities andexercises. Of course, some review and study offorgotten material was at times required. Themathematical content required knowledge ofthe following terms and concepts: function;table of values for a function; graph of afunction; inverse, linear, power, polynomial,rational, exponential, and logarithmicfunctions; ratio; proportionality relations;variation; slope; and average (mean).

Students' Pveconcentions

Students enter mathematics courses with manypreconceptions concerning mathematics, itsusefulness, and their own abilities. Somestudent preconceptions that we needed toattend to are the following:

O Mathematics, beyond arithmetic, is of littlevalue to me.

O There is only one correct answer to a mathproblem, and there is only one correctmethod for finding that answer (theteacher's).

O I can only solve easy math problems.

O The only important thing in solving a mathproblem is the answer.

O In functions, x and y are just numbers; notanything real.

O I cannot tell much about the graph of afunction simply by looking at its rule.

O I cannot tell much about a function's rulesimply by looking at its graph or table ofvalues.

O We can easily tell who has the best answerto a math problem.

In drawing a graph, I must use the samescale on each axis.

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O Science is important, but lots of school mathis not of much use in science.

O The math that is useful in science is beyondmy capacity to learn.

O I don't know enough math to be able tosolve real-world problems using math.

Activities and Equipment

Many of the course's activities requiredstudents to gather or develop their own data,display that data in tables and graphs, and thentry to find a functional model fitting that data.Some of the activities required specializedequipment. In Part I for example, Cuisenairerods were used for one activity, the Tower ofHanoi puzzle in another, and MBL orcalculator-based laboratory (CBL) equipment(motion detector, photogate, and microphone)were used in several activities. In Part II also,some activities required equipment usuallyavailable in a chemistry or physics lab andMBL or CBL equipment (light sensor, _

temperature probe). We made graphingcalculators and computers with spreadsheetsoftware available to the students. Also,students were encouraged to use otherresources that can be found in a library or onthe Internet.

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Case Rel. at Summary off Some Student Work

Midway through the semester, we presented apopulation prediction problem (see StudentHandouts to follow). In one week, working ingroups of two or three, the students completedthe assignment. By this time in the semester,the students were expected to solve theproblem without the assistance of theinstructor. Since the students had progressed asanticipated, the following questions did notsurface:

O What do we do?

o Where do we get the data we need?

O How much data do we need?

O Are we using the right procedures?

o How do we know when we have a rightanswer?

Students gathered and plotted their own data.We expected that linear, polynomial, andexponential functions would be suggested asmodels for the data. Consistent with anemphasis on intuitive understanding ratherthan just "plugging numbers into formulas," wedecided not to cover techniques of curve fittingsuch as least-squares in this course. However,we expected that students had alreadydeveloped heuristics and "best fit" criteria oftheir own in critiquing solutions discussedearlier in the course.

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Student Handout

POPULATION PREDICTION (SESSION 1)

[Introduction

We are interested in predicting what the US population will be by the years 2000, 2010, and 2020.

Personal Popaalation Prediction

Take about one minute to record your best prediction for what the population of the US will be bythe year 2000.

!individual Plan

Take about five minutes to record how you will attempt to solve this problem.

Group Prediction

Take about five minutes to record the group's best prediction. Compare the group prediction toyour individual prediction.

Group Plan

Take about fifteen minutes to decide how the group will attempt to solve this problem. Comparethe group plan to your individual plan.

Ondividuall Assignment

Collect any data that will help to solve this problem and bring to the next class. Explain why theanswer to this problem could be important. Send your explanation to your instructor by e-mail.

Results of Session 1

To follow are some student responses andinstructors' comments on the first session ofthis activity:

Personal Population Prediction. This questionwas asked to encourage the students to drawon their preconceptions. Sample studentresponses:

I don't know what the population of the USis today but I would guess it to be 250million-350 million. I would guess that thepopulation will at least increase by 25% somy approximate guess for the population inthe year 2000 would be 380 million.

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The number of people living in most UScities is at least 2,000,000. I believe that thetotal US population would be in the billionsand I'm going to guess 200 billion for theapproximate number for the year 2000.

000000

By the year 2000 the population of theUnited States will be around 290 millionpeople. I am not sure what the population ofthe United States is now, but I think that itis around 250 million. I think that thepopulation will increase, but not as rapidlyas it has in the past because people arehaving fewer children than in the past.

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Comments on responses:

All of the students were willing to make anestimate even if they had no clearpreconception concerning the currentpopulation. Misconceptions at the beginning ofan activity are accepted without judgment sothat students will take chances and not beembarrassed by their lack of knowledge.

How will you attempt to solve this problem? Eachstudent was expected to formulate a plan forsolving this problem. Sample studentresponses:

To solve this problem I will graph datapoints from about the year 1600, usingincrements of either 25, 50, or 75 yearsdepending on how I space the graph. I willnext extrapolate that data to get the pointsfor the years 2000, 2010, and 2020.

000000

Go to the library and look at an almanacand find increases over last Several years.Look up articles in database and see if thereare any predictions.

000000

I plan on further investigating this problemin the library. By looking in almanacs andother reference materials, I can gather somestatistics from about the turn of the century.I would gather a point for about every 10years. These points would be graphed and Iwould try and come up with a formula.After I find a formula, I can find the valuesfor the years in question.

Comments on responses:

The students have become independentenough not to be paralyzed by this problem.They expect to find the data they need in thelibrary and they have a plan for analyzing thedata.

How will your group attempt to solve the problem?This was an opportunity for the students toresolve their differences and arrive at aconsensus. Sample student responses:

We agree that the population in the year2000 will be around 300-400 million. We

also agree that the best way to make ourprediction more accurate is to look atprevious year's population and find the rateof increase between each year. There are alsomaterials which list predictions of thepopulationwe think these would behelpful also. After we have collected ourdata, we can graph our data and attempt toconstruct a formula.

000000

Population for the year 2000 is predicted tobe 260-300 million (range from groupmembers).

1. We plan to go to the library and getcensus data (as far back as accurate,1900 ? .').

2. Graph the datawith years after1900 on the x-axis andpopulation on they-axis.

3. Find a model(function) for thegraph usingstandardfunctions as aguide.

4. Put in 100 for xand find y. This is the populationprediction for the year 2000. (110 for xfor 2010, etc.).

Comments on responses:

The group estimates narrowed the range ofpredictions. All three groups had definite plansfor collecting data. The particular years thatthey plan to use will probably need to bechanged but that should not be a problembecause flexibility in planning is stressedthroughout the course.

Explain why the answer to this problem could beimportant. This task required the students toreflect on the relevance of the problem.Sample student response:

The answer to the problem posed could beimportant for various reasons. First, it may

Misconceptionsat the beginning of anactivity are acceptedwithout judgment so

that students will takechances and not be

embarrassed by theirlack of knowledge.

77

be important to have an estimate of thepopulation of the United States for futureyears to plan for an increased use of energy.It is important to know how many peoplethere will be so that we can estimate theamount of energy that will be needed in thefuture. One question that depends upon thepopulation is 'When will our supply of fossilfuels run out?' Another estimate for thefuture that depends on population is livingspace. The more people, the more space isneeded for living area and waste. Ourcountry needs this information for planningpurposes. In addition, the country needs toestimate future population for such issues as

food supply, jobs, and government. Almostevery issue in our world is dependent uponthe future population, and if we were notable to estimate this number we may not beable to provide for the needs of everyone inour country. For these and many otherreasons, predicting the future population ofour country is extremely important!

Comment on response:

This student recognized the relevance of theproblem. The student's response shows aconcern for preparedness in many aspects ofsociety.

Student Handout

POPULATION PREDICTION (SESSION 2)

Group Revised Population Prediction

Based on your data, revise your estimate of the what the US population will be by the year 2000.

Group Plan for Knalyzing the Data

Develop a plan for selecting the data you will use and how you will use this data to predict thepopulation for the years 2000, 2010, and 2020.

Group Sununaoy and Discussion of Accuracy

Explain the "best" way to solve this problem, and indicate your "best" estimates. Discuss yourfeelings about the accuracy of your estimates. Send your response to your instructor by e-mail.

Results of Session 2

Students collected data individually and usedthis session to compare data and form a planfor analyzing the data and for producing agroup summary of their results.

Revise your previous population prediction. Thisactivity required students to reflect on thepredictions they had made before they hadcollected data, and gave them an opportunityto address their prior knowledge andmisconceptions. Sample responses:

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Our initial prediction for the year 2000 was300 million to 400 million. We based thison the fact that there were 250 millionpeople in the US in 1990. We felt that thepopulation had been increasing at fasterand faster rates. Our data does not supportthe idea of an increasing rate. We adjustedour prediction for the year 2000 to bebetween 270 million and 280 million.

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000000

After seeing the data that we have collected,we have changed our prediction quite a bit.We do not think that the population increasesby 5 to 10 million every year. Lately, thepopulation has been changing byapproximately 2.3 million every year. Ournew prediction for the year 2000 is 270million.

000000

After briefly examining the data that wascollected, our predictions were revised. Wehad found census information from 1900 to1990 and graphed it. The graphed datareveals the data to be almost a straight line.From this graph we realized that the USpopulation increases by approximately 20million people every ten years. We alsolearned that there are about 2 million morebirths than deaths in the US each year,making an increase in population over a tenyear period about 20 million people. Ournew prediction, therefore, became 270 millionfor the year 2000.

Comments on responses:

New predictions were based on new knowledgeof the situation. Students analyzed the dataintuitively. Either linear or exponential modelsmight have emerged following quantitativeanalysis.

What data will your group use, and how will you use it?Since, at this point, data collected had beenviewed only by individuals, it was important forthe group members to agree on a strategy foranalyzing the data. Sample student responses:

We noticed that the population data for thelast thirty years seemed to be a straight line.We calculated the change in population foreach of the ten year periods from 1960 to1990. During this time the population hasbeen increasing at approximately 2.3 millionevery year. A closer look at the data suggestsa slightly decreasing rate for each decade. Thetrend seems to be that the change inpopulation decreases by approximately onemillion each decade. We will use thisapproach to make our final predictions.

000000

With the gathered data, we decided to onlyuse the known population for the years 1960to 1990 because this information, whengraphed, is almost a perfect straight line.From this data we will determine anequation that fits the points for 1960 and1990. We can then use this line to predictthe population for the years 2000, 2010,and 2020.

C00000

We will calculate the average percent growthper decade, and then use that value todevelop a formula to predict the futurepopulation.

Comments on responses:

Each group developed a plan. Linear modelswere likely to evolve from two of the plans. Anexponential model was likely to evolve fromthe other.

Summarize your conclusions. Sample studentresponses: _ _

Since the data from 1960 to 1990 suggesteda straight line, we determined the equationof the straight line using the data for 1960and 1990. The equation is y = 2,300,000x+ 41,000,000 where x is the number ofyears from 1900 and y is the population forthat year Using this equation we calculatedthe population for the years 2000, 2010,and 2020 to be respectively 271,000,000,294,000,000, and 317,000,000. Since ourequation does not fit the data perfectly, ouranswers are only approximate. For aproblem such as this one, the 'best' solutionis to find the model that best fits the knowndata points and use that model to predictthe unknown data points. We feel verystrongly that our model will do this.

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000000

We developed two modelsone linear andone exponential. We found that the averagepopulation increase per decade from 1900 to1990 was about 19.2 million, and theaverage percent growth per decade from1900 to 1990 was about 14.1%. The 1900population was approximately 76 million.We decided to let the population in millions

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t decades after 1900 be represented by P(t).The equation for our first, linear, model isP(t) = 76 + 19.2t. The equation for oursecond, exponential, model is P(t) =76(1.141)t. We compared the graphs of bothequations to the actual data, and decidedwe liked the second (exponential) modelbetter. Substituting 10, 11, and 12successively for t in that equation wedetermined that the population of the US in2000, 2010, and 2020 will be about 284million, 324 million, and 370 millionrespectively. Since our equation's graphseems to follow the actual growth graph for1900 to 1990, we think our predictions arefairly good. (The graph these studentsproduced using a spreadsheet is shownbelow in Figure 2.)

Comments on conclusions:

Although students were able to develop a pairof models, linear and exponential, littleconsideration was given to the superiority ofone model over another. If the instructor hademphasized the importance of establishingboth qualitative and quantitative criteria forgoodness of fit before assigning this problem,students might have thought of a "compoundinterest" interpretation for the situation, andthey might also have compared deviations ofthe competing models' estimates from theactual data. (For these students, apparentgeometric closeness in the graphs was sufficientevidence to accept a model.) In the future, thisproblem will be assigned after the concept ofbest fit has been suggested and some criteriafor determining best fit have been developedby the students.

400

1.7.1 350

g 300250

0 200

150

100

Populle[ion -rends

°- 50

0

1900

< -CT

Actual

1920 1940 1960Year

-1=1" Linear model

1980 2000 2020

Exponential Model

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Figure 2. Student Graph of Popuiation Prediction

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ConciaBsions and Reflections

We attempted not only to address theexpected outcomes discussed earlier but alsoto follow the guiding principles stated byBrooks and Brooks (1993) by (a) posingproblems in such a way that kindle thestudents' interests and therefore are maderelevant; (b) presenting broad primaryconcepts that the students must "break intoparts that they can see and understand;" (c)seeking, addressing, and valuing students'points of view; and (d) adapting the learningenvironment to address the students'preconceptions. Our ultimate goal was toprovide prospective teachers with anunderstanding of the problem-solvingprocesses used in mathematics and scienceand a workable teaching/learning model fortheir own subsequent teaching careers. By de-emphasizing the teaching of specific algebraicand statistical content, special emphasis wasplaced on allowing students to constructpersonal knowledge and understanding of theconcept of a "function as a model" and on"model building."

We found it very difficult at the beginning ofthe course to stay away from the lecture modeand to not answer questions that would havebeen answered in a traditional course. Thestudents were also unsure of themselves at thebeginning. Those who had previouslycompleted other MCTP courses very quicklyrealized that they would have to accept theresponsibility for their own learning. Once theother students also accepted thatresponsibility, the atmosphere of the coursechanged drastically. We were able to avoidlecturing and instead became facilitators whilethe students took a larger part in directing theclass activities.

One consequence of this new atmosphere wasthat unanticipated topics had to be addressedand additional assignments prepared. Thestudents were often too quick to generalizeand consequently many of the classes endedwith them possessing misconceptions. Whenthis happened we needed to scramble beforethe next class to come up with anotherassignment that contradicted their erroneous

generalizations. This approach worked; theyquickly realized their misconceptions andmodified their concepts. This process ofmaking mistakes, realizing that something iswrong, and then adjusting thinking may be amajor strength of this new approach tolearning. It is very important to not penalizethe students for their misconceptions.

It became apparent very early in the coursethat most of the students liked to work ingroups. Even though the group compositionchanged for each activity, the group dynamicbecame a compelling force. The groupactivities allowed the students time to shareideas, preconceptions, and even theirapprehensions abouta particularassignment. Oncethey realized thatmost of theassignments would bediscussed in groups,the studentsdemonstrated verypositive attitudestoward theassignments. The onlydetrimental aspect ofthe group work wasthat it became verydifficult to get someof the students to do their individual activitiesbefore they started their group activities. Itwas necessary at times to collect individualefforts before initiating group activity.

We are still struggling with the assessmentaspect of the course. Students need to be givena variety of activities to assess their progress inproblem solving. Utilization of an e-mailjournal was an effective way to get immediatefeedback from the students. Misconceptionsand even attitude problems were quicklyrevealed in journal entries. The students wereboth serious and open with their journalentries. In grading group reports, it was difficultto determine the amount and quality ofindividual contributions. We still believe that aportion of the assessment should rely ontraditional in-class examinations.

This process ofmaking mistakes,

realizing thatsomething is wrong,and -then adjusting _ _

thinking may be amajor strength of this

new approachto learning.

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The students in this course recognized how thecourse differed from traditional mathematicscourses. Class discussions and courseevaluations suggest that they hope toimplement similar instructional practices intheir own classrooms. The intensity of theirfeelings and their confidence in their ability toteach in a different way was the mostsurprising aspect of this course. Theseprospective teachers have the potential to beinstrumental in transforming the waymathematics and science are taught inelementary and middle school.

References

Abruscato, J. (1993). Early Results andTentative Implications from the VermontPortfolio Project. Phi Delta Kappan, 74 ( 6),474-477.

Brooks, J. and Brooks, M. (1993). In Search ofUnderstanding: the Case for ConstructivistClassrooms. Alexandria, VA: Association forSupervision and Curriculum Development.

Cathcart, D. and Horseman, T. (1995).Mathematical Models and Modeling for MiddleSchool Teachers: Instructor's Guide. Availablefrom Authors.

Cipra, B. and Flanders, J. (1992). On theMathematical Preparation of ElementarySchool Teachers. Report of a Conference Held atthe University of Chicago. Chicago: TheUniversity of Chicago.

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Maid, D. and Thompson, M. (1973).Mathematical Models and Applications.Englewood Cliffs, NJ: Prentice-Hall.

Meir, S. (1992). Evaluating Problem-SolvingProcesses. The Mathematics Teacher, 85 (8),664-666.

National Council of Teachers of Mathematics.(1989). Curriculum and Evaluation Standardsfor School Mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics.(1991). Professional Standards for TeachingMathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics.(1995, January). NCTM News Bulletin. Reston,VA: NCTM.

O'Haver, T. (1994). CHEM 121/122Chemistry in the Modern World (MCTPPhysical Science), Course Syllabus. [On-line].Available: http://www.wam.umd.edu/-toh/MCTP.html [1997, Dec 18].

Roberts, F. (1976). Discrete MathematicalModels. Englewood Cliffs, NJ: Prentice-Hall.

Schaufele, C. and Zumoff, N. (1993). EarthAlgebra: College Algebra with Applications toEnvironmental Issues (Preliminary Version).New York: Harper Collins.

Shannon, K. and Curtin, E. (1992). SpecialProblems: An Alternative to Student Journalsin Mathematics Courses. PRIMUS, II (3), 247-256.

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Appendix A

COURSE SYLLABUS:INTRODUCTION TO MATHEMATICAL MODELING

Objectives

To help prospective middle school teachers of mathematics or science (a) see connections betweenmathematics and other disciplines; (b) present and analyze real-world phenomena using a variety ofmathematical representations; (c) develop strategies and techniques for applying mathematics tosolve problems; and (d) explain and justify their reasoning, using appropriate terminology, in bothoral and written expression.

Prerequisites

Three or four years of college preparatory mathematics (including at least two years of algebra) andtwo or three years of college preparatory science (including a life science and a physical science).

COIESe Outiine

Part I. Functions as ModelsCurve Fitting

Development of a Repertoire of Techniques to Be Used in Modeling:

o Recognizing Some Standard Functions

o Qualitative Aspects of Graphs

O Change, Rate of Change

O Scales Used in Graphs

O Finding Perfect Fits

O Looking for Good Fits

Part II. Mathematical Modeling

Focus on Modeling as the Process of Applying Mathematics to Solve Real Problems:

O Modeling Examples

O Modeling Activities

Activities

The course is built on a variety of activities, including classroom activities, laboratory experiments,group discussions and reports, and modeling projects. Usually students will be working in groups.

Notes

Students are expected to take and organize class notes. Since there is no conventional textbook forthis course, students will need to take good notes when terms, techniques, and concepts areintroduced during class discussions.

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lf

Journal

Each student will write a weekly e-mail message to the instructor based on the student's reflectionson the week's activities in the course. The instructor will reply to each student's message by e-mail.All messages and replies will be kept in an electronic journal.

Porto lio

Each student will develop a modeling portfolio providing (a) samples of the student's best work inproblem solving and mathematical communication, and (b) an indication of the range and quality ofmathematical and scientific skills and concepts acquired.

Assessment

Exercises, Group Activities, Reports 40%

Electronic Journal Entries 10%

Portfolio 10%

Examinations (2 @ 20% each) 40%

Calculators

Students are expected to bring calculators to class, the laboratory, and exams.

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Appendix B

SAMPLE MODELING PROBLEM:DESCRIBING THE MOTION OF A BOUNCING BALL

Develop a mathematical model to describe the motion of a bouncing ball. Create a modelappropriate for considering questions such as the following:

O How high will the ball bounce on the nth bounce?

O What will be the ball's velocity at time t seconds?

O When will the ball stop bouncing?

O Do basketballs and volleyballs exhibit the same behavior?

O Of what significance is the type of surface upon which the ball is bouncing?

Before starting on this task, take a moment right now to sketch a position-versus-time graphpredicting the ball's distance above the floor from the moment it is dropped until four or fiveseconds have elapsed. Also, sketch a velocity-versus-time graph predicting the ball's velocity at anytime from the moment it is dropped until four or five seconds have elapsed.

In writing up your solution, be sure your paper conforms to our guidelines by communicating yourefforts in:

O Problem Formulation

Mathematization of the Problem

O Solving Within the Model

O Interpreting Your Solution

O Validation of Your Model

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ADAPTATION OF A TRADITIONAL STUDY OFENZYME STRUCTURE AND PROPERTIES FOR THE

CONSTRUCTIVIST CLASSROOM

Katherine J. Denniston, Department of Biological SciencesJoseph J. Topping, Department of Chemistry

Towson UniversityTowson, MD 21204

At Towson University the MCTP Program in..Elementary Education includes a two-

sem ester sequence of introductory biologyclasses. The unit described in this paper wasused in the second semester course we taughttogether in the spring of 1995. We chose toteach as a team and to integrate concepts fromour two disciplinesbiology and chemistryas we wanted our students to appreciate theinterdisciplinary nature of science. This was anatural strategy for us since we had previouslycollaborated in writing a series of chemistrytexts that integrate inorganic, organic, andbiological chemistry, as well as biologicalapplications based on fundamental chemicalprinciples. The course first focused on theconcepts of chemical bonding and the linksbetween a molecule's structure and itschemical and physical properties. Theseconcepts were then applied to the study ofenzymes, photosynthesis, and aerobicrespiration.

Context and GoaOs

The topics addressed in this course are aselection of those one would find inintroductory biology and chemistry courses. Webegan with the structure of the atom, helpingstudents to construct the periodic table. Thisled naturally to an investigation of theprinciples of chemical bonding and how theyapply to biologically important molecules.Students designed a series of experiments toinvestigate the relationship between bonding,structure, and properties. These relationships

were then applied to the study of enzymes andbiochemical pathways.

In the past, we both had used a conventionallecture style to convey basic information.Laboratories were largely of the "cookbook"variety. In these laboratory exercises (noticethat we do not call them experiments)students served primarily in the role oftechnician. They benefited mainly by learningtechnique and manipulation. They were notchallenged to make predictions, designexperiments, or evaluate the experimentaldesign or conclusions.

There are several reasons why we chose adifferent path. We were dissatisfied with theconventional teaching methods just described,as well as with the lack of student involvementin that type of classroom. We were alsodisturbed by our students' inability toappreciate the interrelationships of theconcepts taught in various courses in theirundergraduate curriculum, including writing,mathematics, physics, chemistry, and biology.

All of the activities designed for this coursewere planned with a central goal: to give thestudents primary responsibility for their ownlearning and to allow them every opportunityto explore new relationships throughobservations and experimentation.

Enzyme Ilea Objectives

There were a number of major themes andobjectives addressed by this unit. The first was

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ifor students to apply their knowledge ofmolecular structure and property relationshipsto construct a model of a protein structure,which they would use to understand the roleof enzymes in chemical reactions. A secondobjective was to integrate the scientific methodinto the students' daily activities by havingthem design and critically analyze theirexperiments. We also wished to provide thestudents with an appreciation of the way inwhich scientific knowledge is acquired andhow it changes as new technologies becomeavailable. Finally, we endeavored to preparepre-service teachers to use a constructivistapproach in their science teaching, anapproach modeled after their own learningexperiences in our course.

Pedagogical Considerations

Before we began the unit, the students wereassigned an article entitled Life Beyond Boiling

(Hive ly, 1993) toengage their interestin protein structureand enzymes. Thearticle describesmicroorganisms thatlive at temperaturesabove the boilingpoint of water anddiscusses aspects ofprotein and enzymestructure that allowlife to exist at suchextremetemperatures. Thearticle wasaccompanied by a setof questions designedto assess the students'prior knowledge andpossiblemisconceptions, todetermine whatfactual information

they had derived from the article, and tochallenge them to use the information learnedin problem-solving and critical thinking.

Through classroom discussion of theengagement article, we provided the students

We ensured that thestudents' ideas and

hypotheses wererespected both by theother students and by

the instructors, bycontinually reinforcing

the idea that anymodel has strengths

and weaknessesregardless

of whether the modelwas constructed by aNobel laureate or abeginning student.

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with the opportunity to examine their priorknowledge. We were able to learn that thestudents had a functional understanding ofenzymes. They knew that enzymes speed upchemical reactions and that they are proteins.We continued to monitor student ideas andbeliefs throughout the unit through theirelectronic-mail journal entries, classroomdiscussion, and cooperative model-building.

The students were provided opportunities toinvent and consider alternate beliefs about theway enzymes function through a series ofexperiments that encouraged them toinvestigate the nature of enzymes and enzyme-catalyzed reactions. Discussion of the way inwhich scientists have modified the concept ofenzyme-substrate interaction over the yearsallowed the students to appreciate the fact thatour understanding of science changes as newinformation is gathered.

Students were encouraged to makeconnections between their classroomexperience and the world around them.Throughout the course students were expectedto communicate these associations both in classand in their electronic-mail journal entries.During this unit the students posed questionsabout the nature of enzymatic contact lenscleaning solutions and digestive enzymes.These questions gave rise to very livelyclassroom discussions.

We ensured that the students' ideas andhypotheses were respected both by the otherstudents and by the instructors, by continuallyreinforcing the idea that any model hasstrengths and weaknesses regardless of whetherthe model was constructed by a Nobel laureateor a beginning student. All student ideas weresubject to discussion, and were frequentlymodified as a result of class discussion. Thiswas not treated as a means of criticizingstudents, but rather as an effective sharing ofideas to establish and extend their level ofunderstanding. All students participated in thismodel-building and each seemed to feelsatisfied that he or she had made a valuablecontribution.

30

Unit Chronoiogy

The unit on enzymes required eight class days.Because constructivist courses are primarilystudent-driven, the time required to coverthese concepts and the precise content coveredwill vary for each class. Included in thefollowing chronology are brief anecdotes thatdescribe the responses of our class to this unit.

The exercises described below involved simple,safe experiments requiring only inexpensiveequipment and supplies available in anundergraduate chemistry or biology laboratory.The complete laboratory exercise is appendedto the end of this paper. Technological supportincluded commercial videotapes ofexperiments and computers for transmittingjournals and preparing reports.

Day 1

To give the students a practical introduction toreaction kinetics, we showed a videotape of anexperiment called "Elephant Toothpaste"(Brown, Wm. C. [Publisher], 1993). Thereaction studied is the breakdown of hydrogenperoxide into water and oxygen:

2H202 (aq) 2H20 (1) + 02 (g)

The evidence of the progress of the reaction isreadily seen by the oxygen bubbles produced.The investigators in the video examined theeffects of temperature, concentration, andcatalysis on the rate of the reaction. Followingthe video, group discussion of the experimentalresults allowed the students to construct anelementary understanding of rates of reactionand the factors affecting reaction rates.

We then gave the class some furtherinformation about this reaction to make itbiologically relevant. We explained thathydrogen peroxide is a toxic by-product ofaerobic respiration, which must be brokendown in the cells of the body so that it doesnot damage our biological molecules. Wepoured some hydrogen peroxide into agraduated cylinder. The students perceivedthat, left to itself, the reaction occurs veryslowly. In fact, their observations led them toconclude that no reaction was occurring andthat spontaneous breakdown of H202 would

BEST COPY AVAILABLE

not be sufficient to protect cells from theharmful effects. We discussed how we couldspeed up this reaction without harming thebiological system. All agreed that if we usedheat it would harm the cells, and that we couldnot control the reactant concentration. Theclass concluded that adding a catalyst would bethe best solution.

We discussed the fact that the catalyst used inthe videotape, potassium iodide, wasinappropriate for biological systems. Thisallowed us to introduce the concept ofenzymes as biological catalysts. Students thenwatched a second videotape in which driedyeast cells (bakers' yeast) were used to"catalyze" the breakdown of hydrogen peroxide(Catalysis. Brown, Wm. C. [Publisher], 1993).They then carried out a modified version of theyeast experiment by pipetting drops ofhydrogen peroxide onto a sterile agar mediumand onto similar plates that had beeninoculated with a variety of bacteria. Noreaction was apparent on the agar; however,the bacterial colonies caused furious bubbling.This reinforced the idea that cells contain anenzyme that dramatically speeds up thisbiologically important reaction.

Day 2

Since the first day of class we had stressedmodel building as a means of understandingcomplex systems. Today the class constructedmolecular models of amino acids as a first steptoward understanding protein structure. Eachstudent constructed a different amino acid.This was easy for them because of theirprevious experience constructing differentkinds of organic molecules. They then modeledpeptide bond formation and constructed atetrapeptide. We drew the reactions on theboard, using a color-coding scheme for thereacting functional groups to help the studentsfocus on the bonds being broken and the bondsbeing formed in the reaction. Again thestudents were able to "perform" the reactionsand summarize them on the board with easebecause of the chemical reactions they hadmodeled earlier in the course.

Using drawings, we then worked through thesecondary and tertiary folding of a peptide

9189

chain. We used the models and diagrams of theamino acids to predict the types of interactionsthat would maintain these folded shapes(Caret, Denniston, & Topping, 1993). Becauseof the solubility experiments that the studentshad previously carried out, they were able tocategorize the amino acids as polar, nonpolar,or charged. They were then able to predictwhich amino acids would be involved in theweak interactions that are responsible forprotein folding: hydrogen bonding,hydrophobic and hydrophilic interactions, ionicbridges, and so on.

Day 3

The students' electronic-mail messages and in-class agitation made it obvious that they wereoverwhelmed by the quantity of informationgenerated in the preceding class period. As aresult, we reviewed the key concepts from theprevious lesson. Having been through theinformation once before, the students had lotsof questions along the way. This interactivediscussion allowed us to clear upmisunderstandings immediately and providethe "missing links" in their model of proteinstructure.

Day 4

We introduced the quaternary level of proteinstructure using hemoglobin as the examplebecause it is a protein familiar to most students(Caret et al., 1993). We then discussed theolder (lock-and-key) and more recent (inducedfit) models of enzyme-substrate interaction.The students were very impressed thatDr. Topping had learned the older model asshort a time ago as when he was in school. Thisseemed to give them a real "feel" for the factthat scientific information is constantlychanging as new information becomesavailable. The incident also impressed upon usthat the experiences of people the students canrelate to are valuable tools in the engagementprocess.

Day 5

The goal for this class period was to have thestudents perform a set of experiments onenzymes, collect the data, and discuss theresults. These experiments are similar in

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content and objective to those used in manyintroductory biology courses for non-majors(Hull, 1995). To adapt them for use in aconstructivist classroom, these laboratorieswere redesigned. Students were required topredict the experimental results. Thesepredictions would be based on the experiencesof the students in the classroom during the firstpart of this unit and on the information theygained from reading the engagement article. Inaddition, the students were required toconstruct their own data tables and, whenapplicable, to prepare a graphic representationof the results. Finally, students were required tointerpret their data, develop a model ofenzyme-catalyzed reactions, and share theirmodels with the class.

We studied the properties of the enzymepolyphenoloxidase (PPO). This enzymecatalyzes the oxidation of catechol to producebenzoquinone and water. For a number ofreasons this is an ideal enzyme to study. First,students have observed this reaction manytimes. When you bite into an apple or cut opena potato, the injured surface darkens. The darkareas are caused by the PPO catalyzedoxidation of catechol to producebenzoquinone, which has been shown toexhibit anti-fungal properties and hence isbeneficial to the injured plant tissues.Benzoquinone is a rust-brown coloredcompound; thus students can easily observethe reaction by the development of this color.Finally, the students can easily prepare theenzyme from a potato or apple (see Appendixfor details).

In the first experiment, positive and negativecontrols were carried out. The positive control,a mixture of enzyme and substrate, allowed thestudents to recognize a meaningful colorchange and understand the chemical andbiological significance of the color change. Thenegative controls, substrate alone or enzymealone, demonstrated that both the enzyme andthe substrate were required for the reaction tooccur. All reactions were carried out at 37°Cand were observed at 5 minute intervals. Thesetubes were saved to compare with the resultsof other experiments.

In the second experiment the studentsinvestigated the biochemical composition ofthis enzyme. We proposed that an enzyme iscomposed of one of three types of biologicalmolecules: starch, protein or lipid. All of theseare polymers that can be destroyed byhydrolysis. Samples of the potato enzyme weretreated with either bacterial protease, whichhydrolyzes protein, amylase which hydrolyzesstarch, or lipase, which hydrolyzes fat. If theenzyme is broken down, it can no longerconvert substrate to end-product and no rust-brown color occurs. Alternatively, if theenzyme is not broken down, it will produce theend-product and the rust-brown color willappear.

In the third experiment, the studentsinvestigated whether PPO requires a metal ioncofactor in order to catalyze the reaction.Phenylthiourea (PTU), which binds verystrongly to divalent cations, was added to thePPO extract to remove any such cations thatmight be acting as cofactors for the enzyme. Byobserving whether the enzyme was still able tocatalyze the formation of benzoquinone,students were able to determine whether ametal ion is required by the enzyme.

In the fourth experiment, the studentsinvestigated the specificity of PPO. Enzymesmay accept only a single substrate into theactive site, in which case they are described asexhibiting absolute specificity. However, someenzymes are able to form complexes withseveral substrates bearing the same functionalgroup and having similar structures. Anenzyme having this property is said to havegroup specificity. The students compared theability of PPO to catalyze the oxidation ofthree different but structurally relatedsubstrates: catechol, phenol, and hydroquinone.Based on product accumulation, students wereto determine whether PPO demonstratesgroup or absolute specificity.

In the fifth experiment, the studentsinvestigated the effects of pH on enzymeactivity. Extremes of pH can affect the structureof an enzyme by disrupting the hydrogen bondsthat link the amino acids between differentportions of the protein strand. This results in a

change in the structure and hence the shape ofthe active site and influences the ability of theenzyme to function. Students estimated thelevel of enzyme activity at three pH levels-2,7, and 14.

In the sixth experiment, the studentsinvestigated the effects of temperature onenzyme function. Increasing the temperaturemay increase the rate of a reaction byincreasing the kinetic energy of the molecules.However, if the temperature becomes too high,the shape of the enzyme active site may bechanged, thereby destroying enzyme activity.Students estimated the level of enzyme activityat 0, 20, 37, and 100°C.

After the students had completed theexperimental work, we began a discussion ofthe purpose of the control experiments.Following a lively discussion involving bothstudents and instructors, the studentsconcluded that both enzyme and substrate arerequired for the reaction to occur. From theirdata, students further concluded that thisenzyme is a protein and that it requires acofactor. We talked about the differencesbetween a cofactor (a metal ion that remainsbound to the enzyme) and a coenzyme (anorganic molecule that participates in theenzyme-catalyzed reaction but is notpermanently associated with the enzymestructure). In response to one of the questionsaccompanying the lab exercise, the classdesigned an experiment to show whether themetal ion was acting as a cofactor or acoenzyme. The students decided that theyneeded to find a way to separate moleculesfrom one another. Because these students haveno molecular tools in their tool kits, wedescribed ways to separate proteins and othercellular components by columnchromatography. We told the students tocontinue working on their experimental designand to consider the laboratory questions for thenext class period.

Day 6

Because the students had shown great interestin separation of cellular macromolecules bycolumn chromatography, we brought somehigh performance liquid chromatography

9391

(HPLC) columns and packings for them toexamine. They were fascinated with the ideathat this fine powder really consisted of tinyglass beads with even tinier pores and channelsthrough them. Because they wanted toexamine the beads in greater detail, webrought out the microscopes and the classexamined them thoroughly and discussed theway in which they were manufactured. Thisallowed us to introduce the history of anancient technology, production of lead shot andminiballs, to explain the methods used tomanufacture some types of chromatographybeads. We then completed our discussion ofthe experiment the students had designed todistinguish between a coenzyme and acofactor. They "argued" back and forth and, inthe end, put together a well-designed, properlycontrolled experiment.

We then continued our discussion of theexperimental results from the previous classperiod, beginning with the enzyme specificityexperiment. We immediately ran into aproblem. The students had recorded different"colors" for their results. Of course, the tubeshad been discarded the period before, so therewas no way to observe the result again. As wediscussed this experiment further, it was alsoclear that one of the experimental substrates,phenol, generated a completely atypical colorreaction. From their description, it sounded asthough the phenol had denatured the enzymes.The students decided that phenol was a poorchoice as an experimental substrate. Thanks toour prior class work on isomers, theyrecognized that another structural isomer ofcatechol was possible, and proposed that itmight work better than phenol. They drew thestructure of the proposed compound on theboard; we determined that it was resorcinoland decided to order some for furtherexperiments. The students also recognizedflaws in their observational skills, ability toaccurately describe results, and record-keepingprocedures. After a rather intense groupdiscussion, the students decided on a moreaccurate, efficient, and reliable method.

We then looked at the experiments designed totest the effect of pH and temperature onenzyme activity. The students prepared graphs

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of their experimental results and concludedthat the enzyme has a very broad range of pHand temperature over which it is active. Wediscussed why this might be the case. The classthen initiated a discussion of digestive enzymesthat must function at extremes of pH.

We concluded the day by deciding to repeatthe control and specificity experiments usingthe substrate and experimental designmodifications suggested by the class. Theseincluded the use of resorcinol instead ofphenol, preparation of enzyme from apples aswell as potatoes, carrying out all reactions induplicate, and trying to improve upon theobservation and data recording skills.

Day 7

The students prepared enzyme from potatoand apple and carried out the experiments asthey had planned during the previous classperiod. The atmosphere in the lab was veryinteresting to observe. The students were veryfamiliar with the procedures. As a result,setting up the reactions required lessconcentration than it had the previous classperiod. They chatted with one another in avery relaxed fashion. When attention to detailwas required, the chatter stopped and theexperiment was attended to. Then the chit-chat started again. This is very reminiscent ofthe atmosphere in a research laboratory. Whenconcentration is required, all is quiet; but atother times there is relaxed conversation.

At the end of the period we examined theresults. Catechol gave the most completereaction; resorcinol resulted in no reaction;hydroquinone showed some darkening. Theclass decided that the reaction was occurring,but at a slower rate because the substrate doesnot fit as well into the active site. We decidedto test this by leaving the reactions in therefrigerator over the weekend.

Day 8

We carried out a discussion of the entireenzyme lab. I told the students that I haddiscussed their experimental modification withDr. J. Hull, editor of the introductory biologylab book (Hull, 1995). The students weredelighted to learn that the experimental

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substrate they had selected would be used inthe next edition of the lab manual. This wasvery important reinforcement of their sensethat they have the ability to contributesignificantly to the sciences.

Since this was the last day of the enzyme unit,we had the students compare theirunderstanding of protein structure and enzymeactivity at that point with their understandingat the time we began the unit. After eachstudent had prepared his or her own writtensummary, the students shared the summarieswith one another. They then came up with aseries of global statements to summarize theirunderstanding of proteins and enzymes. Thisproved to be a very useful assessment tool.

Assessment

Throughout the course we relied on a varietyof types of assessment. In some cases, theassessment was used to assign a grade; morefrequently the assessment was simply a tool toallow us to monitor the level of understandingor to determine the direction the course shouldtake.

Student electronic-mail journals proved to be avery effective means of assessing studentprogress in constructing their models of theconcepts being studied. Because our class wassmall, we frequently began with an informaldiscussion. This allowed us to delve into someof the ideas and questions the studentsexpressed in their journals and to discoverdeveloping misconceptions or faultypreconceptions that students had not yetaddressed. Periodically the students wererequired to summarize their currentunderstanding of a topic and compare this tothe ideas they had at the beginning of the unit.They then shared their summary with theclass, and worked cooperatively to arrive at aclass consensus, which was discussed by all.

Assessment for grading purposes was generallyin the form of a problem-solving exam. Tomake the exams authentic, they reflected theexperience of the students in the class. Manyexam questions focused on critical analysis ofexperiments done in class, requiring thestudent to discuss the limitations of the

experiment and design an improvedexperiment. Others required the students touse the approaches and concepts studied inclass to solve related problems. As the courseproceeded, greater numbers of questionsfocused on the application of the basicprinciples of chemistry, studied in the first halfof the course, to problem-solving in biologicalsystems.

Reflection

One of our concerns was that we would beunable to cover enough content in this coursebecause of the large amount of time requiredby the students to construct their ownunderstanding of each of the principlesinvestigated. To our surprise, the limitednumber of topics and the freedom of thestudents to investigate them allowed thestudents to adopt a"need to know"attitude.As a result, they cameto recognize whatinformation theyneeded to bring tobear on the solutionof a particularproblem and toindependently seekout that information.In addition, theydeveloped anappreciation of theneed to applyinformation from different disciplines to thesolution of a problem. Thus, the students notonly learned a reasonable cross-section ofmaterial, but were able to apply theirknowledge to the solution of problems anddesign of experiments.

Although we tried to model learning scienceby doing science, and thereby hoped toengender an attitude of scientific curiosity, wewere disappointed with two aspects of thestudents' behavior. The first was that theyremained extremely grade conscious. Thesecond was the difficulty with which thestudents adapted to cooperative group work.These two concerns may be, in reality, a single

To our surprise,the limited number oftopics and the freedom

of the students toinvestigate them

allowed the students toadopt a "need toknow" attitude.

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concern. From early days these students havebeen taught to be competitive and that sharingwork is an act of cheating. As we all know,preconceptions are very difficult to overcome.

Summagy

A commonly used set of laboratory exercisesinvestigating the properties of enzymes wasadapted for use in a constructivist classroom.These studies involved student preparation ofan enzyme extract from a potato and use of theextract to investigate the properties of theenzyme. Before each experiment, studentswere required to predict the outcome based onprevious classroom investigation of the role ofenzymes in biochemical reactions. Studentswere further required to carefully analyze theresults of each experiment and present theresults in a meaningful way. Throughout eachexperiment the students were given theopportunity to suggest and conductexperiment design modifications based on their

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analysis of the results. Authentic assessmentwas an ongoing component of the course.

References

Brown, Wm. C. (Publisher). (1993). ElephantToothpaste [Videotape]. Dubuque, IA:Exploring Chemistry Videotapes.

Brown, Wm. C. (Publisher). (1993). Catalysis[Videotape]. Dubuque, IA: ExploringChemistry Videotapes.

Caret, R.L., Denniston, K.J., & Topping, J.J.(1993). Principles and Applications of Inorganic,Organic, and Biological Chemistry. Dubuque,IA: Wm. C. Brown.

Hive ly, W. (1993, May). Life Beyond Boiling.Discover, 87-91.

Hull, J.C. (Ed.). (1995). Properties of enzymes.In Contemporary General Biology Lab Book (5thed.). Dubuque, IA: Kendall/Hunt.

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Appendix

PROPERTIES OF ENZYMES

but II111 anctOcon

Objectives

The purpose of this exercise is to examine biological reactions with respect to the characteristics ofthe enzymes which control them. You will examine how enzymes are affected by various factorsincluding substrate availability and type, inhibitors, temperature, cofactors, and pH. In addition,youwill draw conclusions from a series of observations. Upon completion of this exercise, you should beable to:

O describe the effects of pH, temperature, substrate availability, and cofactors on enzymaticreactions;

O draw conclusions from observed experimental results; and

O develop hypotheses to propose the cause of observed results.

Materials and Equipment

O Each of the following solutions should be placed in wash bottles and kept in an ice bath: potatoextract containing polyphenoloxidase (PPO), 1% catechol, 1% resorcinol, 1% hydroquinone, 1%-bacterial protease, 1% amylase, 1% lipase, pH 2.0 buffer, pH 7.0 buffer, pH 14.0 buffer.

O The following should be available at each table: test tube rack with 18 test tubes, glass markingpencils, wash bottle containing distilled water, 15 cm ruler, phenylthiourea crystals (PTU).

O The following should be available for the entire class use: test tube brushes, carboy filled withdistilled water, electric blender, 1000-ml plastic pitcher, cheese cloth, white potato or apple,waterbath with capacity for 120 test tubes, potato peeler, paring knife, 600-ml beaker ice bath,hot plate with 600-m1 beaker.

Preparations

Immediately prior to class, you should prepare the potato extract as follows: Peel and slice a whitepotato and place into a blender with 700 ml distilled water. Homogenize for two minutes. Strainhomogenate through cheese cloth. The liquid portion contains the enzyme, polyphenoloxidase(PPO). This extract should be divided into several wash bottles and placed into an ice bath. Thispotato extract contains polyphenoloxidase as well as numerous other enzymes and materials whichwe will not be measuring. In addition there is some catechol which occurs in the potato which willserve as a naturally occurring substrate for the reaction we will be studying.

You should work in cooperative groups. Before beginning, all test tubes should be rinsed withdistilled water. Using a glass marking pencil, divide each tube into three 1 cm units beginning at theinside bottom of the tube. To save time, substances will be "squirted" into the tubes in 1 cm units.All of the following experiments may be set up and run at the same time.

Key Concepts

Develop an understanding of each of the following concepts: catalyst, coenzyme, cofactor, enzyme,hydrolysis, and substrate.

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General Introductory Questions

What happens to the white flesh of an apple, banana, or potato when you cut it open and exposeit to the air?

O Why do you suppose that this change occurs?

O Do you think this is a physical or a chemical change?

Have you ever prepared a fruit salad and added an ingredient that stopped this reaction fromoccurring?

Background

Most chemical reactions which occur in living cells are catalyzed by enzymes. Without thesenaturally occurring biocatalysts, the rate of physiological reactions at physiological temperatureswould be so slow that life as we know it could not exist.

We shall study the properties of one particular enzyme, polyphenoloxidase (PPO). This enzymecatalyzes the oxidation of catechol to produce benzoquinone and water.

OH

Catechol

OH

+ 1/2 02Polyphenoloxidaser>

Benzoquinone

+ H2O

This reaction is one which you have observed many times. Many plants contain catechol and PPO intheir tissues. When these tissues are damaged (e.g., when you bite into an apple), the injured surfacedarkens. The dark areas contain polymers of benzoquinone. Benzoquinone has been shown toexhibit anti-fungal properties and hence is beneficial to the injured plant tissues.

This reaction has practical application for the food processing industry. Fruits and vegetables areprocessed in reduced oxygen conditions, such as in a SO2 atmosphere, until the enzyme is brokendown by heat processing. Therefore, processed foods do not show the unappetizing appearancecaused by the polyphenoloxidase reaction.

Benzoquinone is a rust-brown colored compound, so its presence can be easily detectedqualitatively. This property allows us to detect that the reaction has occurred by the development ofa rust-brown color.

Experimerrot 2.: Rehrence Reaction

Obtain three tubes and mark off three 1 cm increments on each. Each tube must be labeled with anappropriate symbol. Place 1 cm of potato extract (w/PPO) + 1 cm of catechol + 1 cm of distilledwater in the first tube, 1 cm of potato extract (w/PPO) + 2 cm of distilled H2O into the secondtube, and 1 cm of catechol (1% solution) + 2 cm of distilled H2O into the third tube.

Shake all tubes and place in a waterbath at 37°C for 10 minutes. Save the tubes to use forcomparison with the results in the other experiments.

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Prediction:

Which of the tubes in Experiment 1 do you think will show the color change that indicates that thereaction has occurred? "Why?

Observations and Interpretation of Results:

O Construct a data table. At 0, 5 and 10 minute intervals note and record the color of the solutionin each tube.

O What is the brown-colored substance in Tube A-P

O From this experiment, what materials are necessary for the reaction to occur?

O Why were the tubes placed in a 37°C waterbath?

Experiment 2: The Chemical Comi.eition of Polyphenoioxidase

In this experiment you will test the kind of chemical substance which makes up an enzyme such aspolyphenoloxidase. To perform this test we will propose that an enzyme is composed of one ofthree types of chemicals: starch, protein or lipid (fat). All of these are polymers which can behydrolyzed into their component parts. A hydrolysis reaction is one in which the larger molecule isbroken down enzymatically by inserting water into the molecule, breaking it down into smallermolecules. The basis of this experiment is that if we treat the extract with a material known tobreak down a specific type of substance, we can deduce the chemical nature of polyphenoloxidasefrom the results. If the enzyme is broken down, it can no longer convert substrate-to end productand no rust-brown color occurs. Alternatively, if the enzyme is not broken down, it will produce theend product and the rust-brown color will appear. We will use the following hydrolyzing enzymes:Bacterial protease hydrolyzes protein; Amylase hydrolyzes starch; Lipase hydrolyzes fat.

Mark four tubes into 3 intervals of 1 cm and label them. To each of the four tubes add 1 cm ofpotato extract (w/PPO). Then add the following: 1 cm of distilled water to the first tube; 1 cmbacterial protease solution to the second tube; 1 cm of amylase solution to the third tube; and 1 cmof lipase solution to the fourth tube.

Shake all tubes thoroughly and place them in a waterbath at 37°C. After 45 minutes, add 1 cm ofcatechol to each tube. Shake. Return the tubes to the waterbath for 10 additional minutes.

In the tubes in which polyphenoloxidase was not digested by the hydrolytic enzyme, benzoquinoneand a rust-brown color will be formed when catechol is added. This would be a negative result. Inthe tube in which PPO was hydrolyzed, no benzoquinone will be formed and hence no colorchange will occur when catechol is added.

Predictions:

O If the enzyme polyphenoloxidase is a lipid, which hydrolytic enzyme will destroy it?

O If the enzyme PPO is a starch, which hydrolytic enzyme will destroy it?

O If PPO is a protein, which hydrolytic enzyme will destroy it?

How will you tell if the PPO has been destroyed?

Observations and Interpretation of Results:

O Construct a data table and record your results.

In which tube(s) was benzoquinone formed?

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o In which tube(s) was benzoquinone not formed?

o What is the purpose of Tube B-1?

® From the observations in Experiment 2, what can you conclude about the chemical structure ofpolyphenoloxidase?

Experiment 3: Cofactors

Some enzymes require the presence of other molecules or ions to perform their function. Thesenon-enzyme substances are called cofactors. In this experiment we will use phenylthiourea (PTU),which binds very strongly to divalent cations (ions with a plus-two charge such as copper,manganese, magnesium, ferrous iron, etc.), to remove any cation from the enzyme and, therefore,determine if a cofactor is needed for the catechol-PPO reaction.

Label two tubes and mark two intervals of 1 cm on each. To each tube (C-1 and C-2) add 1 cm ofpotato extract. Add a few crystals of PTU to the second tube. Shake both tubes thoroughly andfrequently for 5 minutes. Add 1 cm of catechol solution to each tube and place them in a waterbathat 37°C for 10 minutes.

Predictions:

® If PPO requires a divalent cation (a positively charged ion with a +2 charge) to remain active,what effect do you think the PTU will have?

O What do you think is the function of a metal ion in the function of an enzyme?

Observations and Interpretation of Results:

O Construct a data table and record your observations.

What is the importance of tube C-1?

O What conclusion can you make concerning the necessity of a cofactor for polyphenoloxidase tofunction?

o Some cofactors are prosthetic, which means they form an integral part of the enzyme, whileother cofactors are free and must simply be present in the medium. If the cofactor for PPO wascopper, suggest a treatment you might perform to test if the PPO cofactor is prosthetic or free.

Experiment 4: Enzyme Specificity

The current theory of enzyme activity states that an enzyme catalyzes a reaction by forming anenzyme-substrate complex. Formation of this complex is dependent upon the substrate fitting intothe enzyme at a location called the "active site." Some enzymes are able to form complexes withseveral substrates which are of similar structure. An enzyme having this property is said to have"group specificity." Other enzymes are known to react with only one substrate and are said toexhibit "absolute specificity." The following experiment compares the ability of polyphenoloxidaseto combine with three different but structurally related substrates: catechol, resorcinol, andhydroquinone.

Label three test tubes and add 1 cm of potato extract (w/PPO) to each. Add 1 cm of catechol to thefirst tube; add 1 cm of resorcinol to the second tube; and add 1 cm of hydroquinone to the thirdtube.

Shake the tubes gently. Place in waterbath at 37°C for 10 minutes.

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0 0

Catechol

OH

OH

OH

OH

Resorcinol Hydroquinone

Predictions:

O If PPO were group specific, which substrates do you predict would be most likely to serve as analternate substrate?

O Explain your reasoning.

Observations and Interpretation of Results:

O Construct a data table and record any color change.

O With which substrate does polyphenoloxidase react best? Least?

O Does polyphenoloxidase exhibit absolute specificity, group specificity or something in between?Explain your reasoning.

Expealment 5: Effects © pH

Because the shape of the active site has important influences on the ability of an enzyme to form anenzyme-substrate complex, the pH of a solution often influences the ability of an enzyme tofunction. The hydrogen ion concentration in a solution is measured by the pH. Hydrogen ions (H+),or alternatively hydroxyl ions (OH-), affect the secondary (and to a much less extent the tertiaryand quaternary) structure of enzymes by disrupting the hydrogen bonds which link the amino acidsbetween different portions of the protein strand. This results in a change in the secondary structureand hence the shape of the active site.

Mark three tubes with three intervals of 1 cm. Label these tubes. Into each of the tubes add thefollowing: 1 cm pH 2.0 buffer + 1 cm potato extract (w/PPO) to the first tube; 1 cm pH 7.0 buffer +1 cm potato extract (w/PPO) to the second tube; and 1 cm pH 14.0 buffer + 1 cm potato extract(w/PPO) to the third tube.

Shake. Add 1 cm of catechol to each tube and shake again. Place in the 37°C waterbath for 10minutes.

Prediction:

O The pH in the interior of a cell is near 7 (neutral). Knowing this, at which pH do you think theenzyme will function optimally?

O Explain your reasoning.

Observations and Interpretation of Results:

O Construct a data table and record your results. Rank the tubes from lightest to darkest as 1 to 3.

O Prepare a graph representing your results.

O Does pH affect polyphenoloxidase?

O Does pH slow the reaction or stop it altogether?

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Experriment 6: Effect of Temperature

Temperature affects enzymatic reactions in several ways. During the reaction, temperature affectsthe kinetic energy of the molecules, which in turn affects the frequency of collisions betweensubstrate and enzyme as well as the energy of activation for the reaction. In addition, enzymes aredirectly affected by temperature. If temperatures are not appropriate, the shape of the enzyme'sactive site may be changed. In some cases this change is permanent, and the enzyme is referred to asdenatured. Denatured proteins usually appear as a white precipitate.

Mark four tubes with two intervals of 1 cm and label. Into each tube add 1 cm of potato extract.Place one tube into the following conditions for 5 minutes: ice bath (0°C); room temperature(20°C); 37°C water bath; boiling water (100°C).

After the 5 minutes add 1 cm of catechol to each tube and place back into the conditions above.Observe after an additional 5 and 10 minutes.

Predictions:

O Most living systems that we know about function best in the temperature range of 20-40°C.Knowing this, at what temperature do you think the enzyme will function optimally?

O From reading the paper on life above the boiling point of water, why do you think that the potatoenzyme is most active at the temperature you have predicted?

Observations and Interpretation of Results:

O Construct a data table and record your observations.

O At which temperature did the most benzoquinone form? At which temperature did the leastbenzoquinone form?

O Did any temperature treatment denature the enzyme?

Coneud5ng questions

Why do cooks sprinkle lemon juice on cut bananas which are used for decorations on top ofcream pies? What is the mechanism of action of the lemon juice?

O Explain the effects of increasing temperature, shown in the graph below, on enzyme activity.

EnzymeActivity

Temperature

O Propose a mechanism for the effects of pH on enzyme activity.

O What determines with which substrate an enzyme will react?

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102

THE AIR WE BREATHE: IS DILUTION THESOLUTION TO POLLUTION?

Thomas C. O'HaverDepartment of Chemistry and Biochemistry

University of Maryland College Park

After participating in the Maryland..Collaborative for Teacher Preparation for a

year, I redesigned Chemistry 121, anintroductory chemistry course for non-sciencemajors, and taught it in the fall of 1994. Priorto joining MCTP, I had been a fairly traditionalfaculty member in a large, research-orientedchemistry department. I taught a wide range ofcourses, mostly for chemistry majors, and didresearch in analytical chemistry with mygraduate students. I had always been interestedin teachingover the years I had developedseveral new courses and had engaged in someeducational software development projects.However, I had never taught a non-sciencemajor course before my involvement in MCTP.Only the lecture portion of the course (CHEM121) had been taught before; the laboratoryportion (CHEM 122) had never had sufficientenrollment to be offered.

What's Diffevent About This Course?

My redesigned CHEM 121/122 differed fromprevious courses in the following ways:

Course content. I "covered" less material than Iprobably would have, but I spent more timethinking about which concepts and topics werereally important. We did use a textbook"Chemistry in Context", which had just beenpublished by the American Chemical Societyspecifically for non-science major courses(Schwartz et al., 1994). I liked this textbook'scase-study approach, in which chemicalprinciples are imbedded within, and arisenaturally from, a consideration of majorsocietal issues such as global warming,

destruction of the ozone layer, energy, humannutrition, and health. I based some of thecourse activities on that text, but I alsodeveloped a number of hands-on activities ofmy own. It should be no surprise that we werenot able to cover the entire book.

Course structure. I scheduled three two-hourclass meetings per week rather than the usualthree one-hour lectures and three-hour labschedule. This structure allowed for greaterflexibilitydepending on the topic and thestudents' progress, sometimes we would meetfor several consecutive class periods in thelaboratory, the computer lab or the regularclassroom.

Teachlng methods. I did not lecture; rather, mostclass time was devoted to small-groupcooperative activities. Students worked withinformation sources, data, observations, hands-on manipulatives, computer-based activities, orlaboratory experiments, interspersed withsmall-group and whole-class discussion.

Use of technology. I had previously usedcomputers in my upper-division and graduate-level teaching, but not in the first-year courses.In the redesigned CHEM 121, I introducedspreadsheets early on as an approach tocomplex, multi-step, quantitative problemsolving and graphical data display. I alsointroduced software for organic chemistrynomenclature and molecular modeling to helpdevelop the students' ability to visualize andwork with bonding and three-dimensionalstructures. When I taught the course again in1996, I integrated the use of the World WideWeb into two of the experiments.

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Assessment. Grades were based primarily onwritten assignments, experiments, and exams.The exams used an essay and word problemformat rather than multiple-choice. In the lastformal activity of the course, I asked studentgroups to list the most important concepts andskills they developed in the course, and tocompose several test questionsincluding one"performance assessment"to assess mastery ofthose skills and concepts. I promised them thatI'd use the best questions on the finalexamination.

Onstructor's Come Boma°. For the first time, I kepta course journal that recorded my daily classactivities, difficulties, student reactions, andthings that did and did not work. I composed myjournal on a word processor and distributed itvia e-mail to the MCTP listsery for scrutiny andcomment by the whole Collaborative. Thefeedback from my colleagues in theCollaborative made me feel less alone in thisenterprise and provided many helpful tips.

Themes and Big Ideas

Most of the instructional units and activities inthis course were centered around two mainthemes or "big ideas":

1. The observable macroscopic behavior ofsubstances is determined by their unseenmicroscopic structure and behavior.

Activities that supported this theme wereentitled:

o Temperature and Molecular Motion

Explaining Cold Boiling and Other PuzzlingObservations

O Counting Bonds and Calories: A MolecularView of Reaction Energy

Chemical Reactions and Energy

® Does a Gas Have Mass?

O pH Balanced: The Meaning of pH

2. Atoms and molecules are far too small to beseen, handled, weighed, and countedindividually, so scientists have devised a numberof models and methodologies that overcome this"inconvenient fact."

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Activities that supported this theme wereentitled:

O Counting by Weighing: Bead JewelryConstruction as a Metaphor for ChemicalStoichiometry

O Visualizing Molecules with Computers

O The Structure of Simple CarbonCompounds

O Drugs and Molecular Symmetry

O Molecular Modeling with "Ball-and-Stick"Model Kits

O The Air We Breathe: Is Dilution theSolution to Pollution? (described in thispaper)

Additional units were not relatedspecifically to either of these themes; theseinclude: "Problem Solving withSpreadsheets"(also described in this paper),"Chemical Information Scavenger Hunt onthe Internet," and "A Penny for YourThoughts."

The student handouts for all of theseactivities are available on the World WideWeb; go to http://www.wam.umd.edui-tohand click on Chem 121: IntroductoryChemistry Course for Non-Science Majors. Theexams, quizzes, and instructor's daily courselog are also included on this Web site.

Caesars Last Breath: INodeIing theUnImaginable

In developing the unit, "The Air We Breathe:Is Dilution the Solution to Pollution?," I wasguided by the AAAS Benchmarks for ScienceLiteracy (AAAS, 1993). In particular, I foundguidance in the benchmarks for "PhysicalSetting" ("The Earth," p. 66; "The Structureof Matter," p. 75), "Habits of Mind"("Computation and Estimation," p. 288;"Manipulation and Observation," p. 292), andothers related to ratio and proportion, scale,solid geometry, and multi-step problemsolving. The benchmarks aligned well withthe text's first chapter on air pollution,which was the springboard for this unit. (SeeAppendix A for student handout.)

10 4

In this unit, the students interpret publishedair pollution data, compare local vehicleemission standards to national air pollutionstandards (using actual Maryland Vehicle TestReports that I provided), and attempt to draw ascale model of the Earth and its atmosphere. Intrying to draw the scale models, students find, totheir surprise, that the task is virtuallyimpossible, because most of the atmosphere liesin a layer that is only 1/cooth of the radius of theEarth. After attempting this, they have a bettergrasp of why the drawings of the Earth and itsatmosphere in their textbook are not to scale.

This paper describes the last activity in thisunit. It relates to a question raised in the textabout whether the atmosphere is large enoughthat toxic pollutant emissions would simply be"diluted away to virtually nothing" by mixingwith the atmosphere. The textbook asks thisrhetorical question: "Is it possible that we arebreathing air that was once breathed by famoushistorical people: Julius Caesar's last breath, forexample?" The textbook did riot actuallyanswer this question, but it greatly interestedthe students and generated a good deal ofdiscussion in class.

I decided to challenge the class to consider arelated but simpler question: "If I took one liter(about one deep breath) of an inert gas andreleased it into the outside air, then waiteduntil it completely mixed with the entireatmosphere of the Earth, what would be theaverage number of molecules of that substancethat I would breathe in each breath?" All of thestudents felt intuitively that the volume of theatmosphere must certainly be so large that asingle liter of air would be "negligible" incomparison, so that we are not likely to bebreathing air once breathed by Julius Caesarmuch less the air in his last breath.

I myself had no idea initially how this mightplay out numerically, but I was confident that,if nothing else, the process of answering thisquestion would draw together aspects ofscience, math, and technology in a rewardingmanner. In essence we were all curious aboutthis questionand curiosity is one of theimportant driving forces in science andmathematics that is often neglected.

Because an actual physical experiment isclearly out of the question, we sought todevelop a mathematical model that would giveus at least an approximate answer to thisquestion. This computation has threechallenging aspects: First, it is a multi-stepcomputation; second, it involves some roughorder-of-magnitude estimations of severalquantities; and third, it involves very largenumbers.

I asked the students to map out a strategy forestimation. During one class period, thefollowing questions arose for discussion: Whatinformation would we need? What is thevolume of one breath? How many moleculesof gas are in a given volume of air (e.g., oneliter)? How could weestimate the totalvolume of theatmosphere in liters?What assumptions arereasonable? Can wemodel the Earth as asphere and theatmosphere as aspherical shell? Whatis the volume of aspherical shell (thedifference betweenvolumes of twospheres)? What is the formula for the volumeof a sphere? How can we estimate thethickness of the atmosphere? How can we takeinto account the fact that the atmospheregradually "thins out" with increasing altituderather than ending abruptly?

Some of this information could be looked up,such as the radius of the Earth, the number ofmolecules in a liter under normal atmosphericconditions, and plots of air pressure versusaltitude. But it soon became clear that thecomputations themselves would be challengingand the students would need more guidance.Nevertheless, they were very interested in anactual numerical answer.

In essence we wereall curious about this

questionandcuriosity is one of the

important drivingforces in science andmathematics that is

often neglected.

IA Spreadsheet Solution

Accordingly, we tackled the actual numericalcomputation during the unit we started in the

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next class period, which happened to be,"Problem Solving with Spreadsheets" (seeAppendix B for student handout). The use ofspreadsheets is specifically recommended inthe AAAS benchmarks (AAAS, 1993, p. 291).As luck would have it, I had scheduled thecomputer lab for this exact time in thesemester, which enabled the students to pursuethe "pollution dilution" question while it wasfresh in their minds.

It turned out that most of the students hadused spreadsheets before, but none hadconstructed their own spreadsheets. During thistwo-day spreadsheet unit, the students

developed this skill byconstructing andtesting a series ofsmall spreadsheets tosolve practicalnumerical problems,starting with verysimple unitconversion problemsand progressinggradually to more

complex constructions involving volume andconcentration. We decided that theculminating activity would be to develop aspreadsheet to solve the "pollution dilution"problem stated above.

This calculation really came down to acomparison of two unimaginably largenumbers: the volume of the Earth'satmosphere (in liters) versus the number ofmolecules of gas in one liter. From our earlierdiscussions, students realized that the latter is a"standard" chemistry textbook number andcould be looked up or calculated; it happens tobe 2 x 1022 at normal atmospheric pressureand temperature. They also knew that thevolume of the atmosphere could be estimatedby subtracting the volume of the Earth fromthe volume of the Earth plus its atmosphere.

Estimating the volume of the Earth from itsradius (6300 km) was not difficult for most ofthe students. But to derive the combinedvolume of the Earth and its atmosphere, theyneeded to have a value for the thickness of theatmosphere. That was the hard part, because

. . . we were allamazed at the result,which seemed counter-intuitive to everyone,

including myself

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the atmosphere thins out gradually withincreasing altitude. I suggested that treating theatmosphere as a homogeneous layer with adiscrete cut-off height would be an acceptableapproximation for our purposes. I encouragedthem to choose their own value for thethickness, based on an inspection of a graph ofair pressure versus altitude found in theirtextbook (Schwartz et al., 1994, p. 6). Moststudents chose values around 4 to 8 km, wherethe air pressure is about half of its ground-levelvalue. Different final answers would beexpected, therefore, depending on the choiceof this value.

As the students worked their computationsusing the spreadsheets, a few got bogged downin the early stages and some were unable tocomplete the final solution. Several of them,however, not only finished, but arrived atanswers that substantially agreed. Four studentsdetermined that, on the average, there wouldbe about 8 molecules of the original gas perliter of atmosphere, so the probability ofbreathing at least one molecule of the gas perbreath is excellent. (They estimated thevolume of the atmosphere to be 2.5 x 1021liters, and found that the ratio of 2 x 1022molecules to 2.5 x 1021 liters is 8 molecules/liter.) Thus, we could literally be breathingsome of the molecules in Caesar's last breath!During a discussion of the results on the nextday of class, we were all amazed at the result,which seemed counter-intuitive to everyone,including myself

These spreadsheet activities were challengingassignments for these students, many of whomstruggled with logic, basic math, ratio andproportion, and unit conversions. It was toomuch for one student, who objected to the useof mathematics in a chemistry course. Anotherstudent confided that she never thought shewould ever be able to "program" a computer tosolve such "hard" problems. I pointed out thatit was really she who had solved thoseproblems, using the computer as a tool.

Later, in class, I asked the students if they feltthat the use of a spreadsheet was helpful,harmful, or made no difference in working outcomplex multi-step "statement problems" such

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ias the atmospheric dilution problem. Studentsunanimously agreed that the use of aspreadsheet was helpful. They volunteeredwith the following three advantages: Use of aspreadsheet reduces arithmetic errors; allowsthe variables to be changed to see what effectthey have; and makes the problems clearer byvirtue of the spatial layout of the numbers onthe page, with cells and columns clearlylabeled. I felt that this last aspect wasparticularly interesting and was not one that Ihad recognized explicitly.

Course Changes and Student Perfforonance

This course differed substantially from theusual freshman chemistry course (taken mainlyby science majors) that I had taught in thepast, in that:

O This class was much smaller, with only 10students as opposed to perhaps 200 in atypical general chemistry lecture.

Less material was "covered," and the topicsthat were covered were differenttypicallymore practical and less theoretical.

There was less emphasis on solving thestandard chemistry word problems that areconsidered by most chemistry faculty to bethe "meat" of general chemistry. Moreemphasis was placed on conceptualunderstanding rather than numericalcomputation of correct answers.

O The instructional approaches were muchmore varied, and there was much more classparticipation and discussion than in thestandard lecture-discussion-laboratorymodel.

As to the effect of those changes on thecontent knowledge of the students, I found,based on an analysis of student answers on thefinal examination, that the students did verywell on questions that were essentiallyconceptual and spacial/visualization oriented.For example, all students answered correctly aperformance assessment question on the finalexam that asked them to use kits to constructmolecular models that met certain structuraland symmetry requirements.

Students also did very well on complex, non-quantitative, cause-and-effect relationships;they scored an aggregate 98% in explaininghow the unabated large-scale use ofchlorofluorocarbons in refrigeration units andaerosol can propellants would be expected toresult in an increase in the incidence of skincancer. They performed adequately onquestions that involved visualizing quantitativerelationships; they scored an aggregate 84% ona question that required them to explain whyequal weights of two different compoundsmight have different numbers of molecules.

In contrast, they did less well on complex,multi-step word problems. This is to beexpected, as even science majors typically havedifficulty with those types of problems, evenwhen they get lots of practice learningalgorithmic solution methods. However, thekinds of knowledge and skills that the MCTPstudents have demonstrated are much morealigned with the AAAS benchmarks andtherefore more likely to serve them well intheir future teaching jobs.

Student Reflections

At various points throughout the course, Iasked the students to reflect on their learningby having them write responses to questionssuch as: "Describe something that you havelearned in this course that was somewhat of asurprise to you, that is, something that youdidn't expect," and "Describe one scientificconcept or idea that you have learned in thiscourse that changed one of your previously-held concepts or ideas about science orchemistry." Several students mentioned theenvironmental chemistry topics that we hadcovered, including the "pollution dilution"problem and the spreadsheet tool that allowedus to obtain an estimated answer:

fit] surprised me to know that there are somany liters in our atmosphere yet we exhaleso many molecules that our breath can fillevery liter of air with 8 molecules!

000000

I . . . didn't know that if only one liter of atoxic substance was released into the whole

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atmosphere, it was likely that I would bebreathing it. . . . I always thought that 1 literwas nothing compared to the atmosphere.

000000

II was surprised by] the use of computers inrelationship to chemistry, specifically, thespreadsheets. I always thought those were for

business and statistics.

. . . it has beenrewarding to watchstudents workingtogether, actually

talking about scienceand chemistry ratherthan about grades,

points, right and wronganswers, requirements,

course policy, andthe like.

In some of theircomments, evidence ofconceptual change isclear:

I had previously hadthe idea that anelectron was similarto a moon orbiting aplanet. This imagerestricted me fromunderstanding, in avisual way (which isnecessary for me tounderstand mostthings), the wayatoms are said toshare electrons. The

fact that electrons can be thought of as cloudsor shells helps me tremendously in visualizingchemical bonds.

000000

I always thought that environmentalists werepeople who loved nature and had a lot of timeon their hands. Now I see why they madesuch a big deal about the destruction of theozone. I also found it fascinating how thelayer of ozone is created.

Some of the students' comments were moregeneral, reflecting the overall approach and thechoice of topics:

I was kind of surprised when this coursestarted off with environmental chemistryissues. (But I was happily surprised.)

000000

. . . this class relates chemistry to real life,which makes it far more interesting to thelayperson.

000000

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I [learned that] chemistry is useful or evenfits in with everyday life such as about thechemical spills, gasoline, pollution, and howit affects us, the ozone layer and how it isbeing destroyed.

Some of the students' comments admittedtheir surprise that doing science activitiesmight actually be fun:

What I DID expect was to learn aboutbonding, atoms, elements, and typicalchemistry stuff, but what I DIDN'T expectwas to learn about them in fun ways.

000000

I didn't think the course would be so muchfun! Chemistry is generally not my forte.

Onstnoctoes Reflections

While the easiest aspect of teaching this newcourse was its interesting content, the mostchallenging aspect was the time and effort ittook to develop new instructional materials forthat content, and to set up a laboratoryprogram from "scratch." In the "regular" generalchemistry course, the laboratory program isdesigned by a separate faculty laboratorydirector and implemented by a staff of severalstockroom employees who order equipmentand chemicals, stock the laboratories, and soon. None of these services was available to me.

In a standard chemistry class, however, I wouldnever have had the luxury of following thestudents' curiosity, and my own, into unknownterritory. The new format gave me thatopportunity, and it was a very satisfyingexperience. In addition, it has been rewardingto watch students working together, actuallytalking about science and chemistry rather thanabout grades, points, right and wrong answers,requirements, course policy, and the like.

In my regular chemistry classes, I have alwaysmade extensive use of hands-on laboratoryexperiments, which are generally done in smallgroups because of equipment limitations.Having seen the benefits of cooperativelearning in this new course, however, I nowhave another justification for small-group workin the labs of my standard courses. Also, I nowattempt to make my laboratory activities a

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little more "discovery" oriented whereverpossible and to de-emphasize the "verification"aspects of experiments.

My MCTP experience has prompted me toundertake a little educational "research" of myown. For my lecture-only classes, I created a setof theory-based interactive computersimulations that I use for homeworkassignments. These simulations give studentshands-on, "experiment-like" activities,complete with goal-driven questions and open-ended explorations. I have evaluated thestudent responses to those simulations andhave now compiled a considerable amount ofevidence over the last several years that thesekinds of simulations drive students to thinkmore deeply, and to uncover and confrontmisunderstandings.

Another area of personal change is that I amnow more willing to allow students to workthings out for themselves, especially when theyare working in groups, even if they seem to bestruggling. In the past, I would have stepped insooner to "help." Now I will keep my distanceand let them thrash it out. I am even beginningto answer questions with other questionsrather than with answers.

Regerences

American Association for the Advancement ofScience. (1993). Benchmarks for ScienceLiteracy. New York: Oxford University Press.

Schwartz, A.T., Bunce, D.M., Silberman, R.G.,Stanitski, C.E, Stratton, W.J., and Zipp, A.P.(1994). Chemistry in Context: Applying Chemistryto Society. Dubuque, IA: Wm. C. Brown.

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Appendix A

CHAPTER 1: THE AIR WE BREATHE

This is not a quiz, but rather a class exercise. The papers will be collected and graded. You may talkto your classmates, but you must write your own individual answers to each question.

1. List the air pollutants that are discussed in Chapter 1 (give the names or chemical formulas).

2. Of the air pollutants listed on page 8 of the textbook, which do you think is the mosthazardous?

3. On the basis of the data on page 15:

a. Would you say that air pollution has been getting better or worse since 1975?

b. Which air pollutant has had the largest change since 1975?

c. What do you think is the largest source of this pollutant?

d. Why do you think this pollutant has decreased so much since 1975?

e. Why were lead compounds (e.g. tetraethyl lead) added to most gasoline that was sold before1975?

f. Why do you think that the gasoline companies were able to "get away" with adding leadasubstance that has been known to be toxic for many years? In other words, why was there nota public outcry from the very beginning?

4. Some copies of recent Maryland Vehicle Emission Test Reports are being circulated around theclass.

a. Look at these and list the pollutants that are measured here (HC = hydrocarbons).

b. Convert the test reading for carbon monoxide (CO) on the test ticket from percent (PCT =percent) to PPM (parts per million).

c. How does the test reading for CO compare to the permissible limit for CO listed on page 8 ofthe textbook?

d. How can the car pass the test when the CO emission is so much greater than the permissiblelimit?

5. Look at the textbook drawings of the Earth and its atmosphere on pages 54 and 64 of thetextbook. Using the data in the textbook, draw a scale model of the Earth and the atmosphereon the back of this paper, representing the Earth as a large circle and the atmosphere as anenclosing concentric circle. Rulers and compasses are available for your use. Based only on yourexperience, what can you say about the accuracy of the scale of the drawings on pages 54 and64?

6. Consider the following hypothetical experiment. Suppose you were to take one liter of a toxicbut otherwise stable gas and release it into the outside air and wait until it is completely mixedwith the entire atmosphere of the Earth. What would be the concentration of the substanceexpressed in ppm by volume?

a. Without actually performing this calculation, what other piece of information would you haveto know to obtain the solution of this problem?

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0

b. What is the probability that one molecule of that substance would be found in one breath ofair? How many molecules of that substance would I breathe in each breath of air for the restof my life? How can the concentration expressed in ppm seem so low but the number ofmolecules in a breath seem so large? Do they not express the same concentration? On thebasis of this result, comment on the reasonableness of achieving truly zero levels of toxicmolecules such as carbon monoxide.

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1 1 I

Appendix B-Part 1

OVERVIEW OF BASIC SPREADSHEET OPERATIONS

Claris Works is an integrated, multi-purpose program that contains an easy-to-use spreadsheet.

A. To launch the Claris Works spreadsheet program, click on the Claris Works button on the mainmenu screen, then click on the Spreadsheet button.

B. To enter a label or a number into a cell, click on the cell, type, and press the enter key.

C. To move to another cell, either click on the new cell or use the cursor (arrow) keys to move.

D. To edit a cell, click on it, make the changes in the "entry box" at the top of the window, thenpress the enter key.

E. To enter an equation into a cell, click on the cell, type an = sign followed by the desiredequation, and press the enter key. When typing equations, use * for multiplication, / for division,+ and for addition and subtraction, A for exponents (e.g. A3 means to raise to the third power).The values of other cells are referred to by location (Al, B12, etc.). Use pi( ) for the value of ILExample: if the radius of a sphere is contained in cell B7, then the equation for the volume ofthat sphere is "=(%)*pi( )*B7A3." Don't forget to press enter when you are finished entering orediting an equation.

E You may optionally change the way a number is displayed in a cell by double-clicking on it. Thisbrings up the dialog box shown in part here. Click on desired buttons and then click on the OKbutton to return to the spreadsheet.

-Number® Generei

CurrencyPercentScientificPilied

CommasNegatiues M 0

Precision

You may also use the Format pull-down menu to change the Font, Size, Style, Text Color, andAlignment of the contents of a cell, just like in a word processor.

G. To save a spreadsheet on your floppy disk, select Save from the File pull-down menu, insert yourfloppy disk into the disk drive, type a file name, and press return.

H. To print a spreadsheet, select Print . . . from the File pull-down menu and press return.

I. To get more help, select ClarisWorks Help from the menu.

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Appendix B-Part 2

SPREADSHEET CONSTRUCTION EXERCISE

In each of the following spreadsheet layouts, <INPUT> marks the cells into which you are to typethe numeric inputs (variables) and the blank cells are calculated cells that contain the equationsreferring to the input cells. Write your cell equations in the blank cells on this paper.

A. Construct a simple spreadsheet that converts distances entered in kilometers into meters andcentimeters. 1 kilometer = 1000 meters; 1 meter = 100 centimeters. Suggested layout:

<INPUT> kilometers (km)

meters (m)

centimeters (cm)

Use your spreadsheet to compute the radius of the Earth (6300 km) in meters:and in centimeters:

B. Construct a spreadsheet that performs the calculations needed to draw a scale drawing of theEarth and its atmosphere, given the actual radius of the Earth (6300 km), the thickness of theatmosphere (6 km estimate), and the radius in inches of the Earth in the scale drawing.Suggested layout:

Radius Thickness

<INPUT> <INPUT> kilometers

<INPUT> inches

Use your spreadsheet to compute the thickness, in inches, of the atmosphere in a scale drawingthat has a radius of 5 inches:

C. Construct a spreadsheet that performs calculations related to car travel, taking the first fouritems in the table below as givens and computing the last three.

Distance, miles <INPUT>

Speed, miles/hour <INPUT>

Mileage, miles/gallon <INPUT>

Price of gas, $/gallon <INPUT>

Time required, hours

Fuel used, gallons

Cost of trip, dollars

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If you drove 1000 miles at a steady 55 miles/hour, in a car that gets 20 miles/gallon, when gascosts $1.20/gallon, how long would it take? How many gallons of gas would youuse? And how much would you spend on gas?

D. Construct a spreadsheet equation that computes the number of molecules of an air pollutant perliter, given the concentration of the pollutant in parts per million (ppm). Any gas contains a totalof 2 x 1022 molecules per liter at atmospheric pressure. Given that the permissible concentrationof sulfur dioxide according to the 1991 EPA standards is 0.03 ppm, how many molecules ofsulfur dioxide would there be in one liter of air at this concentration?(Hint: 1,000,000 ppm = 100% = 2 x 1022 molecules per liter.)

E. Construct a spreadsheet that computes the number of liters (1 liter = 1000 cm3) of air in arectangular room of a given height, width, and length in feet (1 inch = 2.54 cm). Compute thevolume of a 12' x 10' x 8' room: Suggested layout:

1.

Length Width Height Volume,,,I1

Feet <INPUT> <INPUT> <INPUT> 'id

Inches r ,, .,I ',I

Centimeters Cubic cm

11 1 ; 11 Liters

F. Construct a spreadsheet that computes the volume in liters of the atmosphere of the Earth,assuming that the Earth is a sphere with a radius of 6300 km and that the atmosphere is 5 kmthick. Suggested layout:

Radius ofthe Earth

Thickness ofatmosphere

Radius of Earth+ atmosphere

Volume ofatmosphere

kilometers <INPUT> <INPUT>

metersii

centimeters i qf,

volume, cm3E' 4' ," T 11 ,1, ' ' 1, I

volume, liters,1 1.

Suppose that you used a value of 10 km for the thickness of the atmosphere. How would thisaffect the calculated volume of the atmosphere?

G. Expand the above spreadsheet to solve the following problem: Suppose that one liter of a stablegas X is released into the outside air and completely mixed with the entire atmosphere of theEarth. What would be the concentration in molecules per liter of X in the atmosphere aftermixing? (Any gas contains a total of 2 x 1022 molecules per liter at atmospheric pressure.)

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4

CONSTRUCTING THE STREAM COMMUNITY:THE ECOLOGY OF FLOWING WATERS

Joanne Settel and Tom N. HooeDepartment of Natural and Physical Sciences

Baltimore City Community CollegeBaltimore, MD 21215

"An individual can and cannot step into the same river twice"(Heraclitus)

n restructuring our introductory biologyAcourse, we wanted to set a stage for ourstudents that was not based upon dogma, sterilecontent, or rigid process, but rather on thechanging nature of systems so well exemplifiedby the "stream." As implied by the quotation byHeraclitus above, nothing stays the same; thepassage of time brings change. The streamcommunity, a prime setting for observing theeffects of time on a system, became a symbol ofchange for our course as well as its centraltheme. During the course we changed bothprocesses and pedagogy to meet the needs ofstudents at various points in time. We were notfixed to a set schedule and felt the need to befree to adapt to any learning situation.

Context and Goals

We first taught the MCTP course "Principles ofBiology" in the fall semester of 1995. Our coursewas an "Honors Section" with 24 students whowere screened to be a part of the honorsprogram; three of them were preparing tobecome teachers.

Prior to our MCTP section, this foundationcourse had been taught in a traditional lecture/laboratory format. Most of the labs in thetraditional section were hands-on, but notconstructivist-based. Students were givenbackground information and explicit instructionson how to perform procedures; this wasfollowed by discussions of results. Students wereasked to do very little connecting and integrating

of knowledge they had gained. Content wasmore important than process.

Our major goal for the course was torestructure the way in which students (andinstructors) assemble, learn and evaluatebiological principles. Based on our study ofconstructivist principles, we drastically changedthe course design. We agreed that instead ofjust learning biological principles in an isolatedformat, students might better master theconcepts if they could tag biological principlesonto real life experiences. We also decided tovary from traditional lecture/lab courses byensuring that students observed phenomenaand engaged in activities before formallystudying biological principles. By reversing theorder of exposure for students, we hoped thatthe laboratory would become more than just a"lookseeI told you so" experience. (In factit did; it actually became a place for discovery.)Here is an overview of the changes we made:

o The course followed a central theme orstrand, "The Stream Community."

O The course was team-taught, with bothprofessors present in the classroom most ofthe time.

O The biological principles examined in thecourse were integrated with underlyingmathematics and physics concepts.

O The course was inquiry-based: Introductionof concepts often began with observing

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phenomena, experimenting, and collectingdata, rather than through lectures.

O Technology was integrated into courseactivities, with students using computerspreadsheets, graphing software, andcomputer statistical evaluation.

O Preconceptions and misconceptions ofstudents were identified and sometimesbecame the starting point for classroomactivities.

O Questioning and question development wascentral to student-teacher interactions.Great care was taken to develop questioningtechniques.

O Cooperative learning was supported fromthe beginning of the course and wasrecognized by both students and instructorsas a powerful learning tool.

O Literature and writing reinforced thelearning of biological concepts. Relevantresearch articles were assigned for reading,and both professors and students keptjournals.

O Words like predicting, hypothesizing,analyzing, and classifying became a part ofour daily routine. Students sometimeswondered if the instructors were ever goingto answer one of their questions directly.

O The instructors had to "buy in" to the ideathat "less is more" from a learning/instructional standpoint. Nevertheless, manyof the principles that would be found in anintroductory biology text were included.

In adopting the less is more principle, wedevised three main themes for the course:Energy in Biological Systems, Evolution, andthe Stream Community. Concepts related to

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the first theme, energy in biological systems,were constructed and reinforced through avariety of activities on surface area-to-volumeratios, photosynthesis, metabolism, andorganism adaptations to thermal loss. For ourevolution theme, we engaged students in plantgenetics and population dynamics activities,and added a zoo field trip, after which studentswere able to construct models for vertebratelimb evolution. This case report focuses on ourthird and central theme, that of "The StreamCommunity."

ExpOoring the Stream: Mew Depths ofUnderstaroding

In keeping with our goal to help students tagprinciples of biology onto real life experiences,we began the course with a field trip to astream in southern Pennsylvania. There, ourgroup of inner-city students got what was, formost of them, their first chance to observenumerous parameters of a stream ecosystem.They collected data on both living and non-living stream features, which became the focusfor later units on the following topics:

O water and its properties;

o pH and the effects of acid rain and minewaste on ecosystems;

© classification and the use of simpledichotomous keys;

o photosynthesis and primary production inthe stream; and

O construction of food webs and pyramids ofbiomass.

The student handout to follow outlines ourexpectations for this activity.

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Student Handout

CONSTRUCTING THE STREAM COMMUNITY:THE ECOLOGY OF FLOWING WATERS

Ontroduction

The ever-changing nature of streams makes them interesting communities to study. Because thewater is flowing, the condition of a stream one minute is not exactly the same as in another minute.

Streams are divided into two major sectionsriffles and poolswhich usually alternate down astream. The nature of each is determined by flow rate. Water flow also plays a major role in thekinds of adaptations made by the aquatic life found in the stream.

The stream is also a unique community due to the nature of its energy flow. Where does all theenergy come from to support such an abundance of life? In most ecosystems the driving energycomes from within the system, from photosynthesis by plants. By contrast, the stream communitygains much of its energy from external plant material that falls into the stream.

A stream's measurable physical parameterssuch as pH, flow rate, and dissolved oxygenare of_great importance in determining its health, that is, its ability to support plant and animal life. Formost organisms in the stream to thrive, these parameters must stay within very narrow ranges. Inaddition, the adverse effects of pollution often work against the health of a stream.

In this unit you will measure physical factors and observe stream organisms prior to your formalinvestigation of this community. You will be asked to arrange the organisms in the community byassembling a food web and a pyramid of energy. You will be further asked to predict the effect ofacid rain, one changing parameter, on the community. This activity will be accomplished early in thesemester and references will be made to it throughout the remainder of the course. Math andchemistry concepts, along with classification skills, will be integrated into this unit.

In today's study we will measure and observe a stream, looking in particular at the organisms there.We will also try to determine if the stream is healthy or polluted. So let's get started observing ourstream. Remember to wear some real old shoes or sneakers because to do our work we will have towade in some very rocky, shallow water.

The Riffles

The riffles of a stream are waters that move very rapidly (50 cm/second or faster), have a highoxygen concentration (at least 10mg/L) and a good pH value (above 7), and contain organisms likecaddisflies, mayflies and stone flies. Trout and other stream fishes are also found in riffles.

With your instructors' help you should:

1. Measure the speed of the stream using the meter stick, watch and cork. Record your data.

2. Using a kick seine or by picking up rocks, collect as many aquatic insects as possible and identifythem using the attached keys. Record your data.

3. Using the pH kit, measure the pH of your riffles. Record your data.

4. Using the thermometer, measure the temperature of your riffles. Record your data.

5. Using the dissolved oxygen kit, measure the dissolved oxygen level of the riffles. Record your data.

U5

n7

The Pooh

The pool is much quieter than the riffles. Water in pools moves more slowly, is cloudier, has loweroxygen levels, and contains a much different group of organisms. Some of the organisms you willfind in pools will be trout, bass, crayfish, leeches and plankton.

With your instructors' help you should:

1. Measure the speed of the stream using the meter stick, watch and cork. Record your data.

2. Help your leader use the drag seine to collect and identify as many organisms as possible. Recordyour data.

3. Use the pH kit to measure the pH of your pool. Record your data.

4. Use the thermometer to measure the temperature of your pool. Record your data.

5. Use the dissolved oxygen kit to measure the dissolved oxygen level of the riffles. Record your data.

We believed that the stream experience wouldbe an interesting one for our students, but bothof us were surprised by the amount ofexcitement and energy generated by this fieldtrip. Even before we got to the site, thestudents were showing signs of bothexpectation and trepidation. Many of thestudents had not ventured far from BaltimoreCity previously and were at first hesitant towade into the stream. After some bold studentsproceeded, all eventually got caught up in theexcitement and were soon collecting data andorganisms. The site itself was a quiet, woodedarea reached through winding dirt roads, andstudents remarked later of the beauty of thecountryside.

Comments excerpted from student and facultyjournals illustrate how the stream environmentwas initially a little intimidating for an urbanstudent, and how it also served to facilitate the"bonding" of classmates.

The thing that I am beginning to enjoyabout our class is that we are learning whileat the same time being occupied with funand interesting activities. . . . We went to astream near Professor Hooe's house inPennsylvania today. I never could havethought I would have enjoyed myself somuch. We split into groups and collecteddata on velocity, temperature, oxygen andpH and we sampled specimens from the

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riffles and pools. I think that our classbonded on this trip. . . . Although at timesduring the course I have felt discouragedbecause what's expected of us was a littleambiguous, I am very happy that I am inthe program.

(David, 9/21/95)

000000

Well, we are back from the stream fieldactivity and we had no casualties. Weassembled at 8:00 a.m. and were back by12:20 p.m. Quite a task since we drove allthe way to Pennsylvania, did the stream inabout an hour and a half and were back forother classes and the honors reception in theafternoon. The field experience went welland they made a great number ofobservations of biotic and physicalcomponents of the stream. The students werehesitant at first to immerse themselves intothe stream but after a few minutes they werebusy exploring the community.

(Professor Hooe, 9/21/95)

000000

Most of the students were quick to waderight into the stream and several exclaimedwith delight as they experienced a stream forthe first time. Several students were verytentative and held back until fellow students

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chided them to join in. As the studentsbegan to feel more comfortable in the streamthey began to venture further, collectingorganisms from the different riffles and poolsand taking pride in the number and varietyof organisms found.

When we left the stream for the drive backto school, the students were still talkingabout the experience. They were activelygenerating ideas about how to determine ifthe stream was healthy. We were delightedby the experience and plan to use examplesfrom the stream throughout the rest of thecourse.

(Professor Settel, 9/21/95)

Reflection on the Course as a Whoie

We have spent many hours assessing the courseand specific units. Below we share some of ourfeelings both of success and of possible failure.

We tried a new class formata three-hourperiod, twice a week. At the onset we wereafraid that the period would be too long, butwe have found it to be just the opposite. Itallowed us to vary the activities and tocomplete some activities that demanded thetime. Students surprised us by spending morethan the three-hour period engaged in classactivities. Having the time, then planning andusing it wisely, was essential.

We realized early on that we were not going tobe able to cover all of the content that we hadoriginally planned. Modifying our courseoutline, mid-stream, allowed us to select whatwe could reasonably do and still give thestudents experiences that we felt would bevaluable in their construction of the concepts.We came to the realization that simply"covering" more material does not improvestudent learning. Instead, we adopted the "lessis more" concept, which is hard for manyprofessors to accept. But we believe that whenassessments show that students really aren'tlearning effectively when you just "cover" theinformation, then you must search for newways to facilitate student understanding.

One part of the teaching-learning process thatstill poses the greatest problem for us is the

students' ability to apply constructed conceptsto new problems or situations. For example, inour units on "Percent Composition of LivingCells" and "Surface-Area-to-Volume Ratios andEnergetics," students were able to manipulatethe math to solve the biological problems. Butwhen we asked them to apply these skills andknowledge to a new situation, only about one-third of the class could reasonably do so. Weplan in the next semester to do some initialassessment ofdevelopmental level,math proficiency andbiological conceptionof students. Weespecially want tomonitor conceptaccuracy, conceptdevelopment andmisconception ofstudents throughmore extensivewriting exercises inthe course.

An area of studentconcern that surfacedearly on was theirfrustration with thefact that we did not answer all of theirquestions, and when we did, it was usually withanother question. They eventually caught on tothis inquiry-based approach, and they saw thatwe were trying to lead them to an answerinstead of giving them the answer. They alsobegan to use one another as resources, whichreally supported collaborative learning in thecourse.

We came tothe realization that

simply "covering" morematerial does notimprove student

learning. Instead, weadopted the "less is

more" concept, which ishard for many

professors to accept.

We plan to change several other aspects of thecourse for the second offering, based uponobservations that we made from the firstcourse:

O We are changing how we use the journal as adevice to give us student feedback about thecourse. We will probably be a little morestructured in what we ask students todocument in their journals.

O We are adding more structure andorganization to the course to allow us tobetter manage it. This does not mean that we

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are going away from the constructivistapproach.

O We are going to devise new techniques toengage students with their preconceptions,misconceptions, and final conception of thetopics covered. For example, we plan to havethe students write their preconceptions andrevisit them periodically, so that theyconfront the old concepts with the new.

® Team instructors are going to meet prior toevery class to review strategy and pedagogy.

O We are going to respond to student requestsfor some closure at the end of units andsome sort of a "big picture" review.

o We are changing the grading policy toinclude quizzes spaced at shorter intervals inthe course.

Some final thoughts: We are committed tomaking student learning and assessment thedriving forces for what we do as professors.Indeed, we will not spew information atstudents; we will continue to search for new

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and different ways to facilitate student learning.This will mean continuing to change pedagogymid-stream to find alternative ways to assiststudents in their learning.

References

Eddy, S. & Hodson, A.C. (1970). TaxonomicKeys To the Common Animals of the NorthCentral States. Minneapolis, MN: BurgessPublishing.

Maryland Save Our Streams. (1975). Key tostream dwelling macro-invertebrates. (exerptedfrom Aquatic and Stream Sampling Kit).Severn, MD: Author.

Odum, E.P. (1971). Fundamentals of Ecology.Philadelphia: W. B. Saunders.

Pennsylvania Game Commission. (1969).Pennsylvania Fishes. [Brochure]. Harrisburg, PA:Author.

Smith, R. L. (1990). Ecology and Field Biology(4th. ed.). New York: Harper and Row.

Appendix

COURSE OBJECTIVES ANDASSESSMENT METHODS

Objedives

The student will be able to:

O construct concepts and biological ideas using knowledge gained from experimentation,observations, readings and classroom discussions;

O define selected biological nomenclature at the comprehension level of understanding in each ofthe topic areas of the course;

O construct and perform experiments in biology using standard laboratory apparatus and employingskills of hypothesizing, data collection, evaluation and drawing conclusions;

construct an ecological community and its interactions using data from the stream study andinformation presented throughout the course; and

O maintain and organize d_ ata in a laboratory notebook, evaluate data by computer, and keep acourse journal.

gssessment Methods

O Laboratory Notebook and Journal (20%). This book and journal will be reviewed every two weeksand at the end of the course for form, completeness, and accuracy.

(3 Three Major Exams (20°k each). These exams will evaluate the student in the areas of knowledge(recall of facts), terminology comprehension (putting definitions in the students' own words),analysis and synthesis (analyzing data and putting together elements of a biological concept), andapplication of knowledge to new situations.

O Project and Presentation (20%). This aspect of evaluation will attempt to involve the student withlibrary research on a topic of ethical or technological significance to biology and will measure theability of the student to research, analyze and critically evaluate a topic.

O Quizzes were added to the course about half-way through at the request of students who felt thatthey needed to assess their progress at shorter intervals of time.

ADDITIONAL INFORMATION FOR INSTRUCTORS

Student Preconceptions

O A stream community might not have many living things in it (since they are not easily seen).

O All the energy that drives the stream community comes from within the stream.

O Stream life might be the same along its length.

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Student Prerequisite Knowledge

O The student will have an understanding of water chemistry.

O The student will have an understanding of pH.

O The student will be able to measure using the metric system.

Student Learning Outcomes

The student will be able to:

O measure flow rate, pH, dissolved oxygen and temperature of two selected regions of a stream;

O observe and classify stream organisms using appropriate biological keys;

O predict the effect of flow rate on living and physical factors of the stream;

O construct a non-quantitative pyramid of energy and food web for the stream community; and

O predict and conduct research on the effects of acid rain on the stream community.

Related Activities

O Both before and after the field trip, we held laboratory activities in which we guided studentsthrough the process of identification of stream organisms using taxonomic keys. At the stream,we did not indicate any detail of any parameter of organism found, but instead asked questionsthat led the students to further research and investigation.

O After the stream study, students were asked to make a list based on class discussion of observablefeatures of the stream, and to predict the effects of flow rate on living and non-living aspects ofthe stream.

Students were asked to predict the effect of acid rain as one changing parameter on thecommunity. This activity was accomplished early in the semester, and references were made to itthroughout the course.

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THE CHALLENGE OF TEACHING BIOLOGY 100:CAN I REALLY PROMOTE ACTIVE LEARNING

IN A LARGE LECTURE?

Phillip G. SokoloveDepartment of Biological Sciences

University of Maryland Baltimore CountyBaltimore, MD 21250

A Then I returned to full-time teaching afternine years in academic administration, I

spent a number of months reexamining myteaching practice. I realized that the students inmy classes read a lot, and memorized a lot, butdidn't always understand a lot. Often what Isaid in lecture was simply not sufficient toconnect similar concepts across seeminglydifferent topics. Many students expectedlectures to follow the text closely, and did noteasily absorb explanations and examples drawnfrom different chapters or from non-textsources. A typical student response at the endof the semester was that my lectures had oftenbeen disorganized and difficult to follow.

In the fall semester of 1995, I decided to see ifany of the theory and practice associated withconstructivist teaching could be used toadvantage in a large lecture environment. I sawexamples and read descriptions of how MCTPstudents were learning actively in laboratorysettings and smaller classes, but I wonderedwhether active learning could be encouraged ina large lecture. I decided to experiment withmy teaching practice in a large, introductorybiology course (Biology 100). I asked to begiven half of the class (approximately 240students) as a single section for which I wouldbe solely responsible. I felt strongly that theemphasis on a large body of factual knowledgedid not provide students with the sense ofwonder and excitement that scientists normallyexperience. I resolved to place a greateremphasis on questions, particularly those posed

by students, and to use whatever meanspossible to challenge students to take greaterresponsibility for their own learning.

The summer before I started my experiment, Iwas lucky enough to meet Dr. Susan Blunck,who was being recruited as a science educationspecialist at UMBC. Her area of expertise isconstructivist education. When I asked if shewould collaborate with me in the MCTPprogram, she enthusiastically agreed. Moreover,she agreed to participate actively as a colleagueand on-site observer in the classroom in thefall. At MCTP meetings that summer, Dr.Blunck and I gathered ideas and informationfrom other MCTP faculty, and worked togetherto develop a set of somewhat radical changes inthe way Biology 100 was finally taught thatfall. Our goal was to employ a multiplicity ofapproaches to see if we could promote active(or participatory) learning in a large lectureenvironment.

he Course and Grading

Biology 100 ("Concepts of Biology") is a 4-credit, one-semester course designed primarilyfor beginning biology and biochemistry majors.It is traditionally offered by two, oroccasionally three, tenure-track facultymembers who take turns lecturing three timesa week in 50-minute sessions to classes rangingfrom 240 to over 400 students. Students alsomeet in weekly, small-class discussion sectionsof 25 to 50 students led by graduate teaching

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assistants. The corresponding laboratory(Biology 100L) is taken as a separate courseeither simultaneously or in a subsequentsemester. Students other than those majoringin biology or biochemistry (or allied healthstudents for whom Biology 100 and 100L arealso required), can enroll in the lecture courseto meet a general education requirement inscience without taking the associatedlaboratory course.

Traditionally, students in Biology 100 received agrade based entirely on their performance onfour multiple choice exams consisting of 35-50 questions: the top scores earned on threeout of four hourly exams, plus their score onthe comprehensive final exam. In our new,active learning course, multiple choice examswere also administered, but these counted forless than a third of the course grade. Studentswere given three hourly midterm exams of tenquestions each and a comprehensive final oftwenty-five questions. All exams were openbook. The top two midterm scores plus thescore earned on the final exam determined 30percent of a student's grade. Students were toldthat the remaining 70 percent of their finalgrade would be determined by theirperformance on written, take-homeassignments (30 percent); by their participationboth in the large class and in smaller, weeklydiscussion sections led by graduate teachingassistants (35%); and by their scores on oraland written quizzes administered during thediscussion sections (5%).

Approaches Used to Promote Active Learning

We employed five specific approaches that weanticipated would make the large-lectureenvironment more personal, student centeredand interactive. These were: (1) name badges;(2) cooperative learning groups; (3) large-group discussion; (4) in-class written responses;and (5) written, take-home assignments. Anexpanded description of each is given below:

1. Name badges. Large lecture halls and classesof more than 100 students seem impersonalespecially to freshmen. To help personalizeBiology 100, we prepared individual, pin-on,name badges for each student at the beginning

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of the semester from the official enrollmentlist. I explained that each student should viewhim/herself as a member of a professionalcommunity and that, like scientists attending aprofessional symposium, they should weartheir name badges whenever they were in class.It was noted that wearing name badges was anon-verbal means of participating.

One of the most immediate effects of thebadges was that both the students and I couldidentify people in the class by name. Whenstudents raised their hand, they could berecognized by name without the need for aseating chart and assigned seating. This helpedgreatly to personalize the large lectureenvironment and encouraged more studentquestions and responses from a wider array ofstudents than is typical in a large lecturecourse. Badges also helped me to learnstudents' names so that by the end of thesemester I could recognize more than half ofthe students on sight even outside of theclassroom.

Many students initially resisted wearing theirname badges in class, but by the end of thesemester most had become accustomed to thepractice. Surprisingly few comments aboutname badges were made on studentevaluations, and almost all were positive. Onestudent wrote, "Name tags were a good idea. Itis good to know that at least one instructor isinterested in learning the names of his/herstudents!"

2. Cooperative kerning groups. Research suggeststhat students learn higher order thinking skillswhen they can discuss their ideas and activelyengage in dialog (Johnson, Johnson & Smith,1991). Therefore, randomly selectedcooperative learning groups, or teams, of up tofour students were used to foster dialogue inBiology 100. In a class session at the start ofthe semester, four sets of cards numbered 1through 70 were randomly distributed. One ofthe numbered sets was colored, and the 70students who received these colored cardswere asked to hold them up. Then all studentswhose numbers matched were asked to movetogether so they could easily converse. Groupswere instructed to turn in the colored cards

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listing the members' names and one sentencedescribing something unusual or uncommonthat they all had in common. Finally, they wereinvited (but not required) to agree upon ateam name.

Once formed, learning groups remainedremarkably stable throughout the semester.Few students asked to switch teams. Whengiven the opportunity at mid-semester to formnew teams, the class overwhelmingly chose notto change partners. Team members were askedto sit near each other whenever they came toclass, but sometimes individuals waited until Igave directions to start a small group activitybefore moving to sit with their other teammembers.

Team activities were varied throughout thesemester as was the format of the response. Allteam activities took place during class andrequired either an oral or written response. Insome cases I posed a short-answer questionrelated to the topic at hand, and teams wereasked to come to a consensus which would bereported to the class by a team spokesperson.At other times students were shown results ofan experiment related to the day's topic, andthen were asked to work together to develop awritten team response that described andinterpreted the results displayed. On still otheroccasions the class was presented with aconcept question, and students were asked toprovide individual written responses after ashort period of team discussion.

For example, after a unit on cell respiration, theclass might be asked to answer the following:"Do plant cells have mitochondria? Explainwhy or why not." Or, following a discussion ofphotosynthesis, I might ask: "How do plantsthat lose their leaves stay alive during thewinter?" Or, as part of a unit on naturalselection, students might be asked to respondto this question: "If an allele is recessive and isharmful, can selection eliminate it entirelyfrom the population? Explain your reasoning."

Consistent with others who have usedcooperative learning methods, we found thatstudents enjoyed working together in smallgroups and reported that group work improvedtheir understanding of the material. One

student wrote on an end-of-semesterevaluation, "I love being able to do group workbecause I am able to discuss my answers withother people." Another noted that as a non-native speaker of English she sometimes haddifficulty with concepts such as osmosis, butcould always turn to her group for help. Whenasked to assess their own contribution to groupdiscussions, almost all students rated theirparticipation as moderate to high. In contrast,when asked to evaluate the contribution of theother team members, students were typicallyquite candid and tended to be morediscriminating in their assessments.

3. Large group discussion. Students wereencouraged to ask questions in class, to reply toquestions posed by me or by other students,and to respond to others' replies. In order toassure that students could hear each otherspeak, my wirelessmicrophone waspassed to any studentwho had a question orwho wanted torespond to a questionor statement. Inprevious classes, whenI called on studentsfor recitation, theywere often reluctantto speak into themicrophone I washolding. However, inthe experimentalsection, suchreluctance was muchless evident, in part,because I only calledon students whoasked to be recognized. Perhaps moreimportant was the fact that in this class I gavethe students the microphone to hold duringtheir responses. Thus, students easily andnaturally obtained control of the means ofcommunication with the entire class, and Icould assume the role of listener rather thanlecturer.

Students were initially reluctant to speak outin a large lecture hall. Many who felt this wayalso resented the heavy weighting of

Consistent withothers who have usedcooperative learning

methods, we found thatstudents enjoyed

working together insmall groups and

reported that groupwork improved their

understanding ofthe material.

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participation in the final course grade, sincethey assumed that speaking out in the largeclass was the only (or major) method ofparticipation. I tried to correct this impressionby reminding students they could alsoparticipate in the smaller, weekly discussionsections and by emphasizing that there wereother, non-verbal means of participation. Atabout mid-semester, students who had becomeconcerned about their participation gradecomplained that although they raised theirhands in class, they were being recognizedrarely or not at all.

I responded by providing a sign-up sheet at theend of class for all students who had raisedtheir hand during classwhether or not theyhad been recognized. Not only did the numberof complaints decline, but students did notabuse this unmonitored means of assessingtheir participation. About the same number ofstudents signed the participation sheet as raisedtheir hands in classtypically 20 to 30and,aside from those who were already known tous as "high-frequency responders," names onthe list differed from one class period to thenext. I plan to continue this practice when thecourse is repeated.

On the course evaluation, students were asked,"What was the best part of the course andwhy?" Many responded that they appreciatedthe open atmosphere of the class and likedbeing able to hear different points of view. Onestudent wrote, "The course involved lots of giveand take between Dr. Sokolove and the pupilsduring the lecture. This made the class moredynamic. Also it provided an atmospherewhere questions and a variety of perspectiveswere encouraged." Another appreciated " . . .

the open discussions during which everyonehad an opportunity to participate and sharetheir ideas." A third commented, "I liked thefact that the students were able to give somuch input. It made the class much moreinteresting. It was better than listening tolectures every day."

On the other hand, a few felt that thediscussions in the large lecture were toodisorganized and that there were too manypeople in class for participation. One student

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wrote, "The interaction takes away from theclass. Straight lecture would be better. And[the instructor] should go over more chapters[in the textbook] ." Another commentedthoughtfully, "Although the discussions duringlecture were great, we need to perhaps setaside [time for] a structured lecture in order tocover more information . . . "

4. Duo - class written responses. Some educationspecialists have emphasized the need to getwritten feedback from students no matter thesize of the class. In Biology 100 students wererequired to bring to every class session alaboratory research notebook (published byJones & Bartlett; Sudbury, MA) in which theycould write their responses and comments. Thenotebook had numbered, duplicate pages ofcarbonless paper that allowed students to retaina permanent copy of everything they wrote. Anotebook page might be turned in to askquestions about the day's material, or to reply toa short-answer question. Other examples ofhow the notebook was used include respondingto informal survey questions (such as "Whatwould you suggest to improve this course?"),and writing one-minute papers relating onemain point that they learned in class that dayalong with a question that they had about thatday's topic.

The advantage of the notebook was that itoffered a vehicle for shy students tocommunicate with the instructor withouthaving to raise their hands and speak out, or toremain after class to ask questions, or to makean appointment to meet with the instructor.As the semester progressed, the notebook wasused more and more frequently to facilitatenon-verbal, in-class student input. Whenstudents wrote down their questions, responses,and comments, they were generally thoughtfuland explicit in expressing themselves.

Reading and responding to one or moreexcellent questions from students at thebeginning of class was sometimes used toprovide positive feedback on student questionsand to exemplify what I considered to be a goodquestion. Students' written responses were alsohelpful in identifying misconceptions,misunderstandings, or confusion about the

material so that these could be addressed in thesubsequent class period.

5. EIVritien, take-home assignments. There werethree take-home assignments during thesemester, each of which focused on anapplication-oriented question. For example:"You are a track star scheduled to compete in a1,500 meter race next week. Your coach tellsyou to 'load up' on carbohydrates the daybefore the track meet so that your body willhave plenty of energy in reserve for the race. Isthe coach correct in her advice? Why or why not?Cite evidence to support your position."

For each assignment, students were asked toprovide a three-part response of up to five typedpages: Part I asked students to describe whatresearch sources they had consulted and explainwhy these had been chosen, and also to list anyresources that were not consulted and to explainwhy these had not been used. Part II asked for aresponse to the question that had been posedwith appropriate citations. Part III askedstudents to pose two questions of their own thatresulted from the research they had done.

All papers were read and graded either by me orby graduate teaching assistants using a rubricbased upon the following grading approach. Foreach section of the paper that met a minimumlevel of acceptability, the student was guaranteeda "good" grade of B-(e.g., 8 out of 10 possiblepoints). There were no correct or incorrectresponses. To be minimally acceptable, a paperhad to demonstrate an honest effort by thestudent, and each part of the response had to becomplete (all directions followed) and readable(clearly organized with few spelling or syntaxerrors). Additional points could be earned if astudent displayed extra effort and thoughtfulnessin his/her response; points could be lost if astudent failed to follow fully the directionsprovidedfor example, noting which researchsources were consulted (books in the library, myhigh school track coach, etc.), but failing toexplain why these were used and others (theWorld Wide Web, sports or nutrition journals,articles in the popular press, etc.) were not.

Take-home assignments served to initiatediscussion about the types of informationsources that are available and about the

reliability of the information they provide.Students typically reached differentconclusions about the answer to the questionposed in the assignment. These were aired inclass discussion to illustrate the importance ofscientific discourse and to allow students togain experience in defending the interpretationof their research findings. Take-home questionswere assigned that could be readily related toone or more biologically important concepts ortopics. In the case of "carbo-loading," suchtopics might include digestion, cellularrespiration and muscle energetics. Studentscould thereby learn key facts and concepts inan applied contextwith guidance fromthe instructor, ratherthan feel they neededto memorize themfrom lecture notes orthe textbook onlybecause "they mightbe on the next exam."

The response ofstudents to take-homeassignments wasgenerally positive,although one studentadvised us to " . . . doaway with those sillyresearch papers." Mostfelt that theassignments provideda chance to engage inself-directed learning and to demonstrate whatthey could do as opposed to what they could notdo. One student commented that, "The takehome assignments were a help in theunderstanding of certain topics." Anotherwrote, "The take home questions were a goodopportunity for students to demonstrateknowledge without being in an examsituation." A third noted, "I learned the mostfrom the take-home assignments, researchingand discovering information on my own. I feelthere should be twice as many take-homeassignments as there was [sic]! They are greatpractice [for writing] and they are very helpfulin learning on our own! I feel it is important tolearn on your ownit really sinks in that way."

Students could. . . learn key factsand concepts in an

applied context withguidance from the

instructor, rather thanfeel they needed to

memorize them fromlecture notes or the

textbook only because"they might be on the

next exam."

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What Nave We Learned?

"He makes us think." This is how one studentin fall 1995 introductory biology responded tothe following question on the courseevaluation questionnaire: "What personalqualities did this instructor have whichhindered his/her teaching?" Another studentanswered, "He doesn't answer questionsdirectly and lets students find it forthemselves." A third replied, "He openly statedhe didn't know. Made opinions about himskeptical about his abilities [as an instructor]."

Dr. Blunck and I have learned a number oflessons that we hope will allow us to improveour efforts to promote active learning inintroductory biology. First, we recognize thatstudents do, indeed, have different ways oflearning, and we will continue to providechoices both within class and between classesfor those with different learning styles. Withinthe class, we have used multiple teaching/learning approaches and will continue to do so.But we will emphasize at the outset that thereare many ways for students to succeed besidessimply doing well on exams or speaking up inlecture.

Based on our findings, we will refine someapproaches, modify others, and eliminate stillothers (see examples of planned changes under"Work in Progress" below). We may also addnew approaches such as "concept tests" thatshow much promise (Mazur, 1995). We willalso respect those students who feel they learnbest in a more familiar lecture course. Whenthe course is repeated, such students will beadvised that they can enroll in a differentsection of Biology 100 that is offered by thedepartment using a more traditional approach.

Second, we have confirmed that cooperativelearning is as fruitful in professionaldevelopment as it is in the classroom. Thecollaboration between Dr. Blunck and me was, Ithink, essential for effective modification of myteaching. It has often been said that studentevaluations are of limited use in assessing thequality of teaching. Nevertheless, there are fewinstances at the college level where othermethods of assessment, such as peer review, areregularly and conscientiously employed. Even

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fewer are cases in which teaching practice isassessed by an experienced field evaluator on anongoing basis. The model of continuousimprovement in teaching requires continualfeedback. In reforming Biology 100, we havefound that sharing the experience of aprofessional science education specialist (Dr.Blunck) and a scientist instructor (me) hasallowed us to meet common educational goalsthat could not be achieved by relying on studentassessments alone. In the case of Biology 100,regular, constructive input from Dr. Blunck wasinstrumental in promoting rapid and effectivechanges in my teaching practice.

Third, we have found it possible to implementa variety of active learning approaches in alarge-lecture environment, but we cannot yetstipulate which of these has been more (orless) effective in promoting understanding ofbasic concepts. From the perspective ofstudents, it appears that cooperative learninggroups were the most helpful of the methodsemployed. Virtually all students whocommented on small group discussion claimedthat it significantly aided their understanding,and no student commented negatively aboutteam learning. However, an approach such aslarge group discussion that elicited positiveresponses from many students, was viewed byothers as a distraction. While a large numberindicated that the best part of the course wasclass interaction and having input from thestudents in the class during lectures, onestudent commented, "Sometimes class seemedlike a talk show." A significant fraction of theclass felt that there should have been morestructure, less class participation, and more textbook related material on exams]. Clearly notall agreed with the student who wrote, "I lovedthis new teaching technique. I learned more inthis class than I have in any science course.Everybody could benefit from this change ifthey don't resist it."

Work in Progress

Dr. Blunck and I regard our efforts as a "workin progress" and not as a finished product.Students who completed the semester did notnecessarily appreciate being asked toparticipate in their own learning and a number

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continued to prefer traditional ways of learningbiology. Some of the changes we plan to makewhen the course is repeated will be in responseto student input, while others will be new.Among the changes we plan are the following:

Students will be given more explicitinformation about what they can do toparticipate during class so that speaking outin front of the whole class is not viewed asthe sole means of participating.

Additional wireless microphones will bepurchased so that the instructor does nothave to race around the lecture hall in orderto hand the microphone to a student.

More effort will be made to providestructure by providing a short list of themajor concept or concepts we hope to covereach week together with a list of therelevant text chapters.

Cooperative learning groups will beemployed more frequently during class usinga wider range of activities.

Peer evaluation of written take-homeassignments will be used in conjunction witha grading rubric that will be discussed (andpossibly modified) before assignments aremade.

Students will be given the option ofpurchasing a higher level textbook in lieu ofthe "required" text, but those who choosethis option will need to assumeresponsibility for determining what sectionsof that text relate to the concept list.

What About Content?

Virtually all of the teachers who have tried topromote active learning in their classrooms,whether in grades K-12 or at the post-secondary level, have had to struggle with thefact that active learning approaches take moretime and consequently reduce the timeavailable in class for "coverage" of material. I,too, have had to struggle with the coverageissue in Biology 100. However, I recognized atthe outset that even in a traditional lecturecourse, it is impossible to cover within a singlesemester all the material contained in even the

least challenging introductory textbook. Thus,it is always the case that regardless of whoteaches the course (or, perhaps, depending onwho teaches), some topics will be covered andsome not. In our case, I tried to reduce thenumber of topics to a core of essentialconcepts. However, it will take many iterationsof teaching Biology 100 before I amcomfortable with the choices. No doubt eventhese choices will change with time andexperienceand after many discussions withother faculty members in my department.

National groups such as the AmericanAssociation for the Advancement of Science andthe National Research Council have beenengaged in developing new guidelines for scienceeducation. Aconsistent findingexpressed in theseefforts is that studentslearn best whenexposed toenvironments thatencourage reflection,self-directed andinquiry-based learning,and higher order skillsthat go beyondmemorizing items ofinformation (AAAS,1993; NRC, 1996; seealso, Yager, 1993).Content issues are notignored. Indeed,almost half of theNRC's National Science Education Standards isdevoted to content standards. However, theemphasis in these content standards is shifted inimportant ways. One shift is from expectingstudents to know scientific facts and informationto expecting them to understand scientificconcepts and develop abilities of inquiry.Another major shift is from covering manyscience topics to studying a few fundamentalscience concepts (Standards, p.113).

Still, there are many who continue to wonder,are there any "objective" data regarding studentlearning when these standards are applied inthe college classroom? Can students reallylearn "enough" even though one covers less

I recognized atthe outset that even ina traditional lecture

course, it is impossibleto cover within a

single semester all thematerial contained

in even the leastchallengingintroductory

textbook.

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material? One way Dr. Blunck and I attemptedto determine whether students in our student-centered, active-learning version of Biology 100had been "short changed" was to examine theirperformance on the same final exam questionsgiven to students in the traditional lecturesection taught that semester. Whether weexamined average performance on the entireset of questions or looked at the percentage ofthe class that answered each question correctly,we found that students in the active learningsection performed as well as or better than thestudents in the traditional lecture section. Thefollowing student comment captures oursubjective impression of how well studentslearned: "I liked the new format. I tried to takethis class a year ago and was confussed [sic]and bored senseless. I picked up a lot this timeand I enjoyed it!"

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Refevences:

American Association for the Advancement ofScience. (1993). Project 2061, Benchmarks forScience Literacy. New York: Oxford UniversityPress.

Johnson, D.W., Johnson, R.T. & Smith, K.A.(1991). Active Learning: Cooperation in theCollege Classroom. Edina, Minnesota:Interaction Book Company.

Mazur, E. (1995). Concept Tests in Physics.Englewood Cliffs, New Jersey: Prentice-Hall.

National Research Council. (1996). NationalScience Education Standards. Washington, DC:National Academy Press.

Yager, R. (ed.) (1993) What Research Says tothe Science Teacher, Volume 7: The Science,Technology, Society Movement. Washington,DC: National Science Teachers Association.

A SHIFT IN MATHEMATICS TEACHING:GUIDING STUDENTS TO BECOME

INDEPENDENT LEARNERS

Richard C. Weimer, Karen Parks, and John JonesMathematics Department

Frostburg State UniversityFrostburg, MD 21532

n the fall semester of 1994, we began testing..a new approach to teaching special sectionsof Math 207, "Fundamental Concepts ofMathematics II." This is the secondmathematics course required for our earlychildhood/elementary education majors. Twoof usDrs. Weimer and Parkstaught sectionsof Math 207 that fall, and Dr. Jones added hissection a year later.

In the past we all had used a standard textbookfor Math 207. We spent the major portion oftime teaching geometric concepts (usuallyabout five chapters) and a much smallerportion of time (about one chapter each) onprobability, statistics, and computers (usingBASIC and LOGO). Each of us felt that thestudents in the traditional classes had not beenfully involved in the learning of mathematics.We believed that the students had been passivelearners at best and non-learners at worst.Therefore, our major goal for the "new" coursewas to get the students actively involved inconstructing their own understanding ofmathematics.

Facilitators of Learning

The two of us who pioneered the coursedecided to take a constructivist approach toteaching our sections. We decided to do verylittle lecturing; instead, we would serveprimarily as facilitators of learning. During thesummer of 1994, in conjunction with theMCTP program, we developed new coursematerials to support this approach. (Later,

when Dr. Jones observed our classes in action,he decided to join forces with us.)

When our first two Math 207 sections gotunder way, we took a new approach toresponding to student questions. We answeredquestions directly when appropriate, but moreoften we "answered a question with aquestion." We tried to aim the students' inquiryin ways that would enable them to find theirown answers. Sometimes we would do a mini-lecture at the beginning of a project to set thestage, review, or provide backgroundinformation. At other times we would beprompted to give a mini-lecture during theproject for clarification when several groupsseemed to be struggling with the sameconcept.

Cooperative learning became our standardmode of operation. Throughout the semester,students worked together to complete twelveclass projects designed to actively engage themin learning the course content. Working ingroups of four, they constructed their ownknowledge using a hands-on, laboratoryapproach to learning. To demonstrate that theyhad learned the material involved in anactivity, they were required to write a reportexplaining the major mathematical ideasdeveloped within the context of the activity..

The students worked in pairs to produce thereports, and we required them to switchpartners after completing two projects. Thisallowed the students to experience differentlearning styles. (One student commented that

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her partner approached problems by drawingpictures or diagrams while she approachedproblems algebraically. She said she liked thevisual approach and decided to try to use itmore often.)

The "new" course materials we developedincluded much of the original mathematicscontent but in a radically different format.Each of the twelve projects included questionsthat asked students to investigate concepts bycollecting and analyzing data. Students werethen required to justify and explain theiranswers. All class time was devoted to theprojects, each of which took approximatelyone week to complete. In addition, in order tocomplete the projects, groups needed to meetoutside of class to answer questions as well aswrite and type the final reports (one per

group). There were notests; instead, thegroup project reportsformed the basis forassessment. Studentswere also involved inanalyzing theirthought processes bysubmitting e-mailjournals at least onceeach week.

In short, themajority of my

students were not beingreached by the

traditional methodsfrom which I had

learned so well as astudent in a differentera. For me, the time

was ripe fora change!

Time for a Change

While our first twosections were in fullswing, Dr. Jones oftenstopped by the classesand couldn't help but

notice a major difference in the level of studentinterest and involvement. Ultimately, hedecided to join our efforts, adding a newsection a year later. Below, in his words, is adescription of his evolution.

I have been teaching mathematics contentcourses for prospective elementary schoolteachers for over 25 years using somewhattraditional methods of instruction. AlthoughI have experienced a moderate degree ofsuccess in teaching Math 207 in the past, inrecent years I had been haunted with somenagging reservations about how these future

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educators are being prepared to teachmathematics.

My typical format consisted of lecture forthree weeks, interspersed with encourageddiscussion (often unsuccessful), which wasfollowed by the students' cramming for thetraditional timed examination. Thisconventional approach seemed to becounterproductive with respect to motivatingstudents to truly learn mathematics.

Furthermore, while a small minority of mystudents did respond with genuine interestin learning mathematics, the overallatmosphere in my classroom bordered on"siesta time." In short, the majority of mystudents were not being reached by thetraditional methods from which I hadlearned so well as a student in a differentera. For me, the time was ripe for a change!

I became interested in the MCTP projectafter my colleagues shared some of their newexperiences with me. The constructivistapproach to teaching sounded promising,especially on those occasions when I wouldwalk by their classrooms and watchinterested students working actively ingroups. I noted that these students wereactually communicating with one anotherabout mathematics and they appeared toenjoy the process (a far cry from my owntypically apathetic spectator-type students).

When my colleagues first approached meabout joining this constructivist experiment,I was a bit apprehensive. I was concernedabout the considerable content that mightnot be covered, about this new role of theteacher as a facilitator, about turning overmore of the responsibility of learning to thestudents themselves, about forfeiting periodictimed examinations, and so on. Yet, I knewall too well from the past the consequencesof maintaining the status quo. So, in spite ofthese lingering reservations, I decided to"jump in and get my feet wet."

This move proved to be "a shot in the arm"for my teaching. My entire focus in theclassroom became dramatically changedwith a stronger emphasis on student

learning. In the past, I would focus mythinking on the issue, "How can I (as ateacher) best explain a particular topic?"Using the constructivist approach, I foundmyself asking, "How can I best provide anopportunity for significant student learningwhere discovery is encouraged andunderstanding is promoted?" I began toacquire a new appreciation of what issometimes referred to as the art of teaching.My energies were redirected towardquestions such as, "How much of a hintshould be shared so that the student is notrobbed of the discovery experience, and yet isnot so completely discouraged with thefeeling of batting his or her head against thewall?"

Although our individual circumstances differsomewhat, much of Dr. Jones' story can beconsidered "our" story, in that all three of uswent through similar shifts in our thinking andin our approaches to teaching.

Circumference vs. Diameter: A Sampie Activity

To follow is a description of the first activitythat each of us used in our own sections ofMath 207. In "Circumference vs. Diameter,"the students were asked at the outset to makea conjecture as to how the height of anordinary tennis ball can compares with itscircumference (see the Appendix for theactivity). This counterintuitive experienceseemed to hook the students from thebeginning. Most students guess that the heightis greater than the circumference, or thosestudents who think that this is a trick questionwill say that the height and the circumferenceof the can are the same. Rarely do studentsguess correctly.

This question, of course, is a good attention-getter for the first activity. When studentsmeasured the can and found that their answerwas incorrect, they discussed this findingwithin their groups. They then proceeded tothe next phases of the activity to see whatfurther challenges lay ahead. While it is truethat our students had been informed about "pi"in their past studies, in this first activity theydeveloped a first-hand understanding of how

any circle's diameter and circumference relateto each other. Indeed, as in future activities, thestudents were able to come up with theformulas themselves. Admittedly, this processtook longer than our previous traditionalapproaches, but once our students wereequipped with a stronger understanding ofcircles, they were able to effectively handle, ontheir own, deeper thought processes andapplications of the topic.

Change is not easy. Although the studentsfound the projects interesting, they dislikedwriting out explanations. They would muchprefer to give a one-word or simple numericalanswer to a problem than to explain how andwhy they came up with an answer. More thanone student has lamented, "I hate the wordEXPLAIN." But since they are pre-serviceteachers, they knowthat explaining is animportant skill forthem to develop.Many admit to havinghad poor experienceswith mathematics inelementary schoolthemselves andtherefore know thevalue of a teacher'sunderstanding andanswering questionsclearly and precisely.

The difficult part forall of us was notgiving students theanswers directly orshowing them the easy way to solve a problem.We had to learn to be patient with theirlearning and to ask leading, yet judiciousquestions. This was an adjustment both for usand for the students. Students often look totheir instructors for validation, approval, andan indication that they are on the right trackwhen solving a problem. We wanted our pre-service teachers to become independentlearners and to be able to critically evaluatetheir conclusions.

The following student comments, compiledfrom each of our classes, were written

The difficult partfor all of us was notgiving students theanswers directly or

showing them the easyway to solve a

problem. We had tolearn to be patient with

their learning and toask leading, yet

judicious questions.

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immediately following completion of the firstactivity:

I never realized exactly how circumferenceand diameter related, in all past classes, Ihad always just been told, diameter times piequals circumference. Without thinkingabout why this formula was true I neverreally understood it. After seeing it, hearingit, touching it, and actively using it, I nowunderstand it.

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I think what challenges me most is takingthe math from the equation type form andapplying it to a problem in real life. I haveno problem computing things once I'mshown how to do that type of problem, but Ireally don't know how to use it any otherway. I guess what I'm trying to say is thatthere is more to using math than what Irecall being taught.

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I like working hands-on. Sometimesnumbers and formulas seem so abstract, butwhen you have something like the tennisball can that you can actually touch andfeel it makes things easier to understand.

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I am realizing that I learn a lot more when Iwork problems out for myself than havingyou [the instructor] tell me the answer eventhough it would be ten times faster

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I like the way you format the project in theform that questions lead to one another Thequestions were written in a manner that youhad to think because the answer was notthere in the question. The project also ishelping me become more comfortable withword problems. The questions werechallenging but not overwhelming.

Learning Together

Having the students work in cooperativegroups appealed to all three of us. We believedthat the students would bring various skills andknowledge about mathematics to the groups

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and that by working together on a project theywould be more inclined to participate. We alsohoped that by discussing mathematics witheach other they would learn to rely uponthemselves to decide if an answer made sense.For most of the students, working incooperative groups was a new experience. Thefollowing are some student thoughts abouttheir first experiences with partners:

I'm finding that I really like working with apartner. When you are unsure of a questionyou always have someone there to help out.I also have found that having a partner, italso helped my thinking process. At onepoint during our discussion on the project Iasked my partner a question. She wasunsure of the answer also. But, to mysurprise simply by verbalizing the questionout loud I was able to find the answermyself.

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C00000

Working with a partner helps me to remainfocused because you know that anotherperson is depending on you to help get thework done. This is great practice in the areaof responsibility.

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In terms of working with a partner, I foundit to be very challenging. My partner and Igo about things in very different ways. Forexample, when we got stuck on a problem Iwanted to skip it and move on, where shewanted to search through the book until shefound an answer

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After doing this project I have discoveredthat I am capable of resolving a problem.However I find it difficult some of the timeto explain why and how I did what I did tosolve the problem. I found this out when Iworked with my partner. I had to read theproblem several times then answer it theway I thought might be right, then try tohelp her also understand the problem after Idid. This was good practice for me so I canbecome the best teacher possible. I think thatwe should have the chance to work with

everybody in the class so that we have thechance to learn and help each other better

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Using cooperative learning is a very goodinstructional tool for a mathematics class. Ithelps me to be reassured and it builds myconfidence by having someone to work with.The first activity was very thoughtprovoking, and required my partner and I tocooperate and share the work, whileallowing each of us to learn from the other'sexperience.

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I have come to like working with differentpartners. It is more challenging to learnabout each person's learning style, as well asvarious patterns of thinking.

Activity-Driven Ongennity

Throughout the course we were sometimesdiscouraged with students' lack of standardknowledge (i.e. the Pythagorean Theorem, howto solve simple equations, or the fact that1 mile is equivalent to 5,280 feet), but in thiskind of course the students' thought processesare revealed and there is no place for them tohide. On the other hand, we were oftendelightfully surprised at many students'ingenuity in attacking problems.

For instance, we asked students to determinethe maximum volume of a box that can beconstructed from an ordinary sheet of paper bycutting squares of the same size from eachcorner. Students were to construct various sizesof open boxes and look for patterns among thedata. One student quietly folded her papertwice and cut out one square from a corner,which magically multiplied in front of herpartner's eyes to four squares when the paperwas unfolded and a box was formed.

In retrospect, we have no doubt that betterlearning did occur within this "activity-centered" classroom than within our past"teacher-centered" courses. Although there mayhave been some sacrifice in the amount ofcontent covered, the entire process involvingstudent discoveries, peer interactions,

individual journaling, oral communication, andwritten reporting provided a valuablemultidimensional learning experience for ourstudents.

The Male of Journa0s

As the semester progressed, we learned alongwith the students. During each period, we hadthe opportunity to interact with students asthey interacted with their partners andsometimes within larger groups. It wasespecially gratifying to witness the thoughtprocesses utilized by students as theycommunicated with one another. We were alsoimpressed with the weekly student journalentries describing their learning experiences,after which we attempted to offer appropriatewritten feedback. Throughout the semesterthese journal writingsdisplayed a variety offeelings aboutmathematics, butoverall they reflectedthat significantlearning wasoccurring. A sampleof these studentponderings includesthe following quotes:

This lesson is puttogether well and forces us to THINK aboutwhy things happen. What we find in theactivities are often counterintuitive, and thisforces us to think through our answers alittle more carefully.

. . . we wereoften delightfully

surprised at manystudents' ingenuity

in attackingproblems.

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I am beginning to lose some of the anxiety Ifirst had about venturing into the unknown.The labor involved in our pursuit is wherethe real learning occurs and this seems farmore valuable to me as a future educatorthan merely obtaining the actual "answers"to this assignment.

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Class is more fun, but it's a lot more work.

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C00000

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I find that when I am forced to write aboutthe mathematics we are doing that I gain abetter understanding of the concepts we arelearning.

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I am beginning to think of my future as anelementary school teacher. Already I havedecided one goal of my career will be toteach math as a fun, interesting, andvaluable subject that anyone can learn. Ireally want to "breed" a generation ofyoungsters who like math and do not fear it.

On occasion, students did have legitimateconcerns, which they were able to share withus through their journaling. One problem inparticular that surfaced in all of our classes isthat of partner-shared responsibility. Onestudent writes:

I was a little disappointed with my partnerduring the second project we did together. Ifelt she really didn't do her share of thework. The last part of the project was themost challenging and I was left to do italone. In certain cases like this one, whenyour partner doesn't pull his/her share ofthe weight in helping with the project, itdoesn't seem quite fair that they receive anequal grade.

Although we have not completely resolved thisissue, we feel that it is important for studentsto be able to privately describe their learningfrustrations by way of their journals, and for usto be able to give suggestions on how theymight go about trying to solve their problems.We also point out that these problems exist notjust in our current classroom, but that they willface problems of motivation, responsibility, andfairness in grading in their future teachingassignments as well. The journaling process alsoprovides a way of showing the students that wereally are listening to them and that we careabout them as learners.

At the end of the semester, one studentsummarized her experience as follows:

I think that I really have learned a lot thissemester. I think that going into this class, Ifelt as if I really wouldn't LEARN anything

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at all but I found that I did. . . . I learnednew ways of approaching [the concepts] andhow to work together with people to get tounderstanding those concepts.

I also discovered how much more I learnedfrom the report writing rather than thestandard tests and quizzes that the student Itutor has. She still doesn't truly understandthe concepts that she has learned becauseher way of explaining them is to take a test,whereas ours involves an understanding inorder to complete an explanation. Thesereports have also taught me to think before Iwrite and to EXPLAIN myself more clearly.I may understand what I am trying to writebut the reader (or my future student) maynot.

Working with a partner was also helpful.Although in the beginning, I wasapprehensive about working with someonenew every other week, I have found thatworking with a partner helps a lot becausewe both may be thinking of two differentways of doing something. We can combineour ideas to make something better, ormaybe just to solve a problem.

And most of all I liked the journal writing. Ithink that it gave us another way ofcommunication and it helped a lot with thesolving of the problems in our reports.Sometimes when I would write and ask aquestion, I would end up solving it in theway that I asked you. I think it made us feelcloser to you as a teacher, giving us a linkthat most students and teachers don't havein their classroom.

Basically, what I am trying to say is thatthis is probably the most worthwhile class Ihave had here. I was beginning to hate mathin general until now. I am now lookingforward to taking other math classes for myconcentration.

Looking Back

In reflecting on our experiences, we all admitthey were most rewarding. The most difficultpart during the semester was learning not "tospill the beans" too quickly. In the traditional

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classroom a "good" teacher will always answerquestions clearly and succinctly for his/herstudents. We had to learn how not to answerquestions directly, but to gently guide ourstudents to find their own answers withoutcausing them so much frustration that theywould have given up on the problem.

There was a major adjustment period for ourstudents at the beginning of each semester.Most of the students, even the good students,were dependent upon us to tell them if theywere doing the problem correctly. So eventhough the students worked in groups, theystill wanted us to tell them if they were doingthe activities correctly. So we had to make surewe visited with all groups and that one or twogroups could not be allowed to dominate ourtime.

We found that if we did our jobs correctly, ourstudents would ask us different kinds ofquestions, and fewer questions towards the endof the semester. They also learned to knowwhen they were doing the problems correctly.

As mentioned earlier, we still have not solvedthe problem of all students contributingequally to each and every part of the projects.When we were able to identify students whowere not contributing, we paired themtogether for the next project. That at leastforced one of them (if not both) to become

more involved. However, we do believe thatby using the activity-centered materials andrequiring written reports, many more studentswere truly involved in learning mathematicsand in learning moremathematicsthan inour traditional classes.In a traditionalteacher-centered classit is much easier for astudent to hidebecause the teacheroften does themajority of the work.

Finally, it was awonderful experienceto stand back andobserve studentsmaking sense ofmathematics forthemselves. Therewere many occasionswhen we could see"the light bulb turning on." In the"Circumference vs. Diameter" activity, manystudents were surprised to learn for themselveshow pi was related to the circumference anddiameter of a circle and how pi was thenrelated to the slope of a line on the graph. Itwas great to hear a student say, "Hey, that'scool!" in a mathematics class.

Clearly, whenthe students began to

find their own answersthey were pleased with

themselves, whichincreased theirconfidence and

encouraged them tocontinue to find more

answers ontheir own.

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Appendix

CIRCUMFERENCE vs. DIAMETER ACTIVITY

1. How does the height of the can compare to the circumference of the can? What is yourconjecture? Explain your reasoning.

2. Test your conjecture by measuring, and describe your process and result. How does this resultcompare with your original conjecture? Explain.

3. Choose five cans of different sizes and complete Table 1. For each can, measure the diameter andcircumference three times.

TABLE 1

Can Diameter Circumference

Trial 1 Trial 2 Trial 3 Average Trial 1 Trial 2 Trial 3 Average

1

2

3

4

5

4. Why is it a good idea to measure the diameter and circumference three times and then take anaverage?

5. Plot the diameter, D, on the horizontal axis and the circumference, C, on the vertical axis. Doesit look like the best fit curve should be a straight line?

6. Should the best fit curve go through the origin? Explain.

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7. Draw in the best fitting curve.

8. What does the best-fitting curve represent, and how can it be used?

Comprehension Questions:

Use your graph to answer the next three questions. For each, show your work using dotted lines onyour graph.

9. a. If a circle has a diameter of 3 cm, what is its circumference?b. Did you use interpolation or extrapolation?

10. a. If a circle has a diameter of 7 cm, what is its circumference?b. Did you use interpolation or extrapolation?

11. a. If a circle has a circumference of 75 cm, what is its diameter?b. Did you use interpolation or extrapolation?

12. From your curve find a formula relating variables C and D. Explain your reasoning, includingany constants used. What do you think the exact value of the constant should be?

13. For the three circles on the attached page, measure the circumference and diameter and add thisinformation to Table 1. Plot a new graph with all eight points. Does the best-fitting curvechange? Explain.

For the next five problems, use your formula relating C and D with the exact constant to answerthe questions.

14. What is the diameter of a circle if the circumference is 117 cm?

15. A tire on a bicycle has an inside diameter of 45 cm and an outside diameter of 52 cm.

a. What is the inner circumference?

b. What is the outer circumference?

c. How far will the tire roll in two complete turns?

d. If the distance from your home to school is 4000 meters, how many revolutions will thewheel make?

16. It takes 25 adults, arms outstretched, to surround a Redwood tree. Approximate the tree'sradius. Explain how you arrived at this approximation.

17. Which would have the larger circumference, a heavy copper cylinder that is 5 cm in diameter ora light plastic cylinder that is 5 cm in diameter? Explain.

18. The diameter of a large soup can is 4.5 inches. Its height is 6 inches.

a. What shape did the label have before it was put on the can?

b. What are the dimensions of the label if it covers the can?

19. Return to the tennis ball problem. Without using measurements, provide somemathematicaljustification for your conjecture.

20. a. Construct a circle.

b. Inscribe a square in your circle.

c. Circumscribe a square about the circle.

d. How do the perimeters of the two squares compare? Explain your reasoning.

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21. Refer to the large circle below. Determine the length of the arc between any two successivesmall marks. Explain what you did.

Critical Thought Questions:

22. Is the circumference part of the circle?

23. Is the diameter part of the circle?

24. Is the center of a circle part of the circle?

25. Define a circle, its circumference, diameter, radius, and center.

26. a. Define pi.

b. Is pi equal to 22/7? Explain.

27. Can a straight line intersect a circle in one point, that point being the center of the circle?Explain.

05

SO

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95

SO

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35

20

30

23

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41

A TALE OF TWO PROFESSORS:FACULTY PROFILES

he mathematics and science professors whojoined the MCTP program came aboard

through different routes. Some had beenactively seeking new teaching methods toboost their students' level of interest and classparticipation. When they heard about theMCTP program, they signed up readily. Othershad resigned themselves to student disinterestas "part of the package" of teaching collegemathematics or science. They wound upjoining the program, however, after a colleagueconvinced them to try something new.

The following interviews convey the journeysof two professors who came to MCTP fromopposite ends of the spectrum. First is the storyof chemistry professor Pallassana Krishnan ofCoppin State College, who had not consideredaltering his methods of instruction until acolleague persuaded him to give this new

program a try. Second is a profile of physicalsciences professor Rich Cerkovnik of AnneArundel Community College, who knew hewanted to change his teaching strategies beforeMCTP came into being.

Regardless of their paths of entry, each of theseprofessors took from the program many newideas for creating active learning environmentsfor their students. They also gained supportfrom a statewide network of colleagues as theyventured into new ways of teaching. And bothagree that what they're doing in the classroomnow is definitely better than what they weredoing before.

These stories, written by Maureen Gardner,MCTP Faculty Research Assistant, wereoriginally published in the MCTP Quarterly,the project newsletter.

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CHEMISTRY PROFESSOR TRANSFORMSENERGY HIS OWN

Profile of Professor Pallassana Krishnan, Coppin State College

He was not looking for a new way to teach.After more than two decades as a

chemistry professor at Coppin State College,Dr. Pallassana Krishnan was resigned to the lackof student participation in his traditionallectures and labs. Although it bothered him, hesaid, "you get adapted; you say this is life . . .

it's a job you have to do.'"

It's all different now. Since two summers ago,when a colleague at Coppin persuaded him togive MCTP a try, Dr. Krishnan and his studentshave been energized bymajor changes in hiscourse. Gone is thelecture room, theclassroom format, thetable behind which heused to sit, separatefrom the students. Nowprofessor and studentsinteract, side-by-side inthe lab, throughouteach class.

Instead of listeningpassively, Dr.Krishnan's studentsnow learn chemistry bydoing things, like flash-freezing items in liquidnitrogen or heating them in a solar oven. Theactivities drive students to ask question afterquestion, which Dr. Krishnan tosses back tothem to figure out. "You know he's not goingto tell us the answer," said one student toanother during a recent class. Textbooks areused for reference purposes only.

The transformation wasn't quick, and itcertainly wasn't easy. The way Dr. Krishnan

sees it, however, it's now a part of him, andthere is no turning back. Here are some of histhoughts on how this all happened:

On the first attempt. The first time I taught anMCTP class it was a disaster. I was trying tochange from one mode of teaching to another,and after lecturing for 24 years, I wasn't able todo that properly. If the professor is confused,the students sense that so easily, and thatcauses a lot of disturbance in the students. Ittook a whole semester for me to adjust.

On why he didn't "bag it"when times got tough."Bagging it" flewthrough my mindseveral times, but bagit and do what? If yougo back to the old way,that isn't any better;it's equally bad.

On trying again. Thenext semester I triedagain. Fortunately thatwas a smaller classonly 12 studentsandthat helped a lot. ThenI saw that maybe thisis the way to go after

all. It takes awhile. That time was good and thethird time, this semester, is good. . . . If youwant it to work it will work.

On resisting the urge to answer student questions.This is the hardest part for me, becausestudents expect an answer immediately. Notanswering comes naturally now. . . . There is aquestion, and you just open it up and let thestudents kick it around. Eventually the answerevolves from the discussion; if it doesn't then

Dr. Krishnan, now in the role of the "guide on the side" asopposed to the "sage on the stage," as his studentsexperiment with liquid nitrogen in his physical science class.

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Now We're Really Cookin'

We hear a lot of talk about motivatingstudents, said Dr. Krishnan, but MCTPteaching offers motivation for professors, too.He offers a prime example: It began during a

Dr. Krishnan (foreground, left) talks with students aboutthe solar-powered oven on which they will be doing

1 research this summer.

you give them a small hint and then it keepson. It takes a little bit of time, but then thislearning takes place, which is exciting.

On content coverage and meaning. Others may thinkthat in teaching this way we don't cover enoughmaterial as we do otherwise. I don't think so; Ithink we cover a whole lot. Once the students'interest is sparked, they are going to read, andthey are going to find out. Then how you sparkthat interest becomes the question. And if youdo not spark that interest, no matter what youdo, it has no meaning at all.

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4 3

class discussion about papers the studentswere writing on energy topics, when acouple of his students from Kenya broughtup the use of solar energy in developingcountries. Dr. Krishnan said, "Wait, I mightbe able to help you with that." He had readthe work of Dr. Dan Kammen at Princeton,who is developing solar ovens for people inKenya and other African nations. "I sent himan e-mail and he responded immediatelyand suggested that we meet," Dr. Krishnansaid.

During their meeting at Princeton, Dr.Kammen said that some undergrads therewould be doing research on the oven, suchas measuring temperature differentials.Noting that both good students and goodsunshine were available at Coppin, Dr.Krishnan asked whether his class could helpcollect the data. "Professor Kammen thoughtit was fantastic," said Dr. Krishnan. "Hejumped on it very quickly, and carried thissolar oven all the way out to my car."

Back at Coppin, "the students were reallythrilled," said Dr. Krishnan. "Six of themhave volunteered to come in this summer totake the measurements." He adds, "Thesethings I would not have done five years ago.That's the difference MCTP has made."

On how you know you've got 'em hooked. Duringthe second semester, one class was scheduledfor Friday afternoons, from 4-5 p.m. Typicallyone does not get too many students for aphysical science class held at that time, but Ihad almost full attendance in every class. Thatmeans they were understanding it. Once youunderstand it, you like it.

On an innovative way to introduce other professorsto PAM-style teaching. I invited a newly hiredchemistry professor, who teaches anothersection of the same course, to just come sit inmy class. I didn't tell him anything about

BEST COPY AVAILABLE

illiCTP student Amanda Rice of Coppin State College dips anobject into liquid nitrogen as Dr. Krishnan looks on.

constructivism or about MCTP; I just askedhim to observe and make some comments. Hecame in and watched. Right after the class hejumped up and said "I am amazed at the way

the students participate in your classit istotally different! They all do the work and youdo the least amount of work." I said that'sinteresting to know, that's good to know. Hehad some very positive thoughts about thatclass and I felt good about it. Now we canprobably talk to him about MCTP.

On the importance of institutional backing. If itwere not for MCTP, I don't think that this kindof change could have taken place. If the MCTPprogram was not here when I started teachingthis way, I would not have survived. Peoplewould have jumped on me very quickly, saying"Oh the professor has gone senile after all theseyears." The education system is a bigmachineyou just don't change things easilyor quickly. So MCTP was the backing; I knewat least that somebody was behind me to helpme out.

Co resolve. Now that we have started this newway of teaching, I think it can go on. Even ifsomeone were_to question it, I still would notgo back to the lecture again. I guarantee youthat I would not do itthat's all there is to it.The students are learning and they're excited;that's all I need. Having made this change, it'ssomething you feel that's inside. You will notchange back once you really are convinced. Itmay not be perfect yet, but it's certainly betterthan what I was doing before.

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STUDENT QUESTIONS HOLiI THE ANSWER

Profile of Professor Rich Cerkovnik, Anne Arundel Community College

Wondering how to reform a collegephysical science course? When Assistant

Professor Rich Cerkovnikat Anne ArundelCommunity College wasnot satisfied with hisstudents' level of interest inhis physical science course,he began searching for newways to teach. The searchled him to MCTP, and to arealization that valuingstudent questionsto thepoint of building a coursearound themwouldprovide him with ananswer to his own inquiry.The following interviewgives a little insight intohow this came about.

Now would you describe yourteaching style or philosophybefore you entered the itOCTPprogram?

I'd have to say it wasn'tstraight lecture, becausewithin the first month ofteaching I decided thelecture wasn't doing it. So Ijust tried different methodsfor the next few yearsgroup work, differentassessments and so on. Iwas adjusting a little bit at atime because I wasreluctant to changedramatically. I was reallysearching for some new methods but I didn'thave the big picture.

What led you to the conclusion that straight lecturewasn't working?

As Prof. Cerkovnik looks on, MCC students JohnHardy and Jama Carey rollerblade their waythrough a demonstration of Newton's third law ofmotion, that is, "to every action there is an equalbut opposite reaction." Inspired by classdiscussions, the students themselves suggestedusing rollerblades to help them understand thisconcept as well as Newton's other laws ofmotion.

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Students just weren'tgetting it. They didn't care.Exams and quizzes andtests weren't comingthrough; and in addition tothat, the students justweren't excited and there'sso much in science to beexcited about. The reason Igot into physics teaching isbecause I had so manyuneventful physics classes,and I always thought Icould do it better. But thenof course I was teachingthe way I was taught, andit just wasn't working.

Now did joining itfiCIP helpyou?

By interacting with all theMCTP participants I couldgather ideas and makejudgments as to whatwould work for me. Igained a focus for my lab,which I needed. Then lastsemester I just changed myentire course and dideverythinggroup work,questions, demonstrations,microcomputer-based labs,and was in general muchmore student-centered.

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How is your course student-centered?

Although I provide an overall direction for thecourse, generally the class works off questionsposed by students. In the lecture, for example,we do a lot of demonstrations that are drivenby the students' questions about what theythink will happenand why. I then setup activities that leadthem where theywant to go in termsof scientificunderstanding, thatbuild to the pointthat the studentsmake connectionsamong conceptsbased on their ownobservations andanalyses. Before wedo demonstrations Ihave them writetheir ideas down,then they discussthem and offer theiropinions. We then dothe demonstration,observe it, and goback into the "why's" and whether or not weneed to change our models. I also incorporatedstudent questions into their journals so that Iget at what they want to learn.

Do you require students to pose questions?

At the beginning of the semester I requiredeach student to generate a list of 15-20questions. I had them read some material inthe later chapters of the text that deal withastronomyplanetary motion, star formation,galaxiesthings they're interested in. Theywere to pull out ideas, concepts that they didnot understand, that they felt they needed toknow in order to understand what was goingon in those chapters. The students typed theirquestions into the lab computers, then Icompiled them and organized demonstrationsand activities to address them throughout thesemester. I even used some of them as the basisfor their final exams.

What tells you that the changes you made in yourclass are working?

One is that I get a larger variety of studentsasking questions and answering questions. Also,when I give them time to talk with a neighborabout a question, they appear to be talking

about what's going

Prof. Cerkovnik goes for a spin to help students understandrotational inertia and its effect on rotational motion. Before thedemonstration, he asked students to predict how and whydifferent arm positions would affect rotational inertia and in turnhow that would affect the motion.

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on. Some of the besttimes are when Iwalk into theclassroom, and Iknow it's workingwhen I ask if thereare any questions,and that's all I haveto do. Before I knowit the period's over,and we've set updemonstrations,possibly ones I'veanticipated orperhaps ones we puttogether on the spurof the moment. Ourscience division hasa new combined laband lecture room,with easy access todemonstration

equipment and student materials, which hasbeen a great help.

You mean you walk in and ask "Are there anyquestions?" and that drives the day?

Yes. Generally I'll ask "Are there anyquestions?" and I'll wait as long as 2-5 minutes,because they need a certain amount ofadjustment to this active learning. Sometimesstudents will come up with questions from thestudent-generated list. If I get no questions,then I ask a student to tell me what we did inthe last class. This usually brings up somequestions that give direction for the day's class.

How do you manage to keep the course on trackwhile giving the students so much input into the day-to-day activities?

There is a healthy tension between my wantingto use every demonstration and activity my

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technician and I have planned and wanting thestudents to have a significant impact upon thedirection of the course. It's important to methat each student feel that he or she hasownership of this course, but of course I haveownership of the course, too. Sometimes thismeans that we start with a student questionand while we are investigating it, through aseries of gentle nudges, I extend it and connectit to other concepts that I feel are important.Other times, I bypass a planned demonstrationor activity because it no longer fits thedirection the class is currently taking.

With this student-centered approach, do you knowweek-by-week where you're going?

Not week by week, but within a 3-week periodwe'll get to where I think we want to be, andwe'll generally do it in association with thestudent questions. For my two sections of thecourse, the demonstrations I use in each classmay be different because of what gets asked. Ifeel it's much more meaningful when they seetheir professor responding to their questions.

On teaching this new way, what's been the mostsurprising aspect? The most difficadt?

The most surprising aspect has been thestudents' being able to drive a class. The hardesthas been getting the students to do that. I didthis by making very sure that within the first two

weeks, any time that a student mentioned aquestion in a journal I addressed it within thenext class period and indicated that it came froma journal. Or if a student asked a question and Ididn't have the demonstration materials availableat that time, we did it the next class period. Thisreinforced their questions and helped themrealize that what they were saying was feedingback into the course and coming back out. I thensaw an increase in the questioning and anincrease in the journal writing in terms of thingsthey were confused about. They would ask if wecould do something about their questions, sothey were taking more responsibility for thecourse. It was hard for me in that it took a lot ofwriting back to the students in their journals andspending time putting together stuff that Ihadn't anticipated, to meet what they needed atthat time.

What's been '.1 e easiest aspect of teaching this way?The ntost rewarding?

It's been coming in and saying 'Are there anyquestions?" and having students ask wonderfulquestions that give the class direction. Thatmakes things easy and fun. This is a rewardingstructureonce you've got the students movingin this direction, it's such a wonderful experienceto come in and ask what they want to talk abouttoday. It's really rewarding to not have to forcepeople to follow you on the board.

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Appendix I

REFERENCES FOR READING

he references provided below were.11 compiled by Genevieve Knight and John

Layman, MCTP Co-Principal Investigators, aspart of a Collaborative-wide effort to define aframework of guiding principles for MCTPcourses.

Mathematics References (Genevieve Knight)

Artzt, A.F. & Newman, C.M. (1990). How toUse Cooperative Learning in the MathematicsClassroom. Reston, VA: National Council ofTeachers of Mathematics.

Case, R. & Bereiter, C. (1984). Frombehaviorism to cognitive development.Instructional Science, 13, 141-158.

Cobb, P. & Steffe, L.P. (1983). Theconstructivist researcher as teacher and modelbuilder. Journal for Research in MathematicsEducation, 14, 83-94:

Confrey, J. (1990). What constructivism impliesfor teaching. (Journal For Research inMathematics Education, Monograph No. 4).Reston, Va.: National Council of Teachers ofMathematics.

Davidson, N. (1990). Cooperative Learning inMathematics: A Handbook for Teachers.Reading, MA: Addison-Wesley.

Davis, R.B. (1984). Learning Mathematics: TheCognitive Science Approach to MathematicsEducation. Norwood, NJ: Ablex.

Davis, R.B., Maher, C.A. & Noddings, N.(1990). Constructivist views on the teachingand learning of mathematics. Journal forResearch in Mathematics Education, MonographNo. 4.

Feldt, C. C. (1993, May). Becoming a teacherof mathematics: A constructive, interactiveprocess. The Mathematics Teacher, 86 (5),400-403.

Gadanidis, G. (1994, February).Deconstructing constructivism. TheMathematics Teacher, 87 (2), 91-96.

Goldin, G. A. (1990). Epistemology,constructivism, and discovery. (Journal ForResearch in Mathematics Education,Monograph No. 4). Reston, VA: NationalCouncil of Teachers of Mathematics.

Graves, N. & Graves, T. (Eds.). (1993).Cooperative learning and mathematics [SpecialIssue]. Cooperative Learning, 14(1). SantaCruz, CA: International Association for theStudy of Cooperation in Education.

Harmin, M. (1994). Inspiring Active Learning: AHandbook for Teachers. Alexandria, VA:Association for Supervision and CurriculumDevelopment.

Hiebert, J. (Ed.). (1986). Conceptual andProcedural Knowledge: The Case of Mathematics.Hillsdale, NJ: Lawrence Erlbaum Associates.

National Council of Teachers of Mathematics.(1989). Curriculum and Evaluation StandardsFor School Mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics.(1991). Professional Standards for TeachingMathematics. Reston, VA: NCTM.

Noddings, N. (1990). Constructivism inmathematics education. (Journal For Researchin Mathematics Education, Monograph No. 4).Reston, VA: National Council of Teachers ofMathematics.

Ross, R. & Kurtz, R. (1993, January). Makingmanipulatives work: A strategy for success.Arithmetic Teacher, 40 (5), 254-257.

Simon, M.A. (1995). Reconstructingmathematics pedagogy from a constructivistperspective. Journal for Research in MathematicsEducation, 26, 114-145.

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Sowell, E.J. (1989). Effects of manipulativematerials in mathematics instruction. Journalfor Research in Mathematics Education 20,498-505.

Steffe, L. P. (1990). On the knowledge ofmathematics teachers. (Journal for Research inMathematics Education, Monograph No 4).Reston, VA: National Council of Teachers ofMathematics.

Science References (John Layman)

American Institute of Physics. (1990).Operation physicsheat. Operation Physics.One Physics Ellipse, College Park, MD. AIP.

Andersson, B. (1979). Some aspects ofchildren's understanding of boiling point. In WF. Archenhold , R.H. Driver, A. Orton, & C.Wood-Roginsion (Eds.), Cognitive DevelopmentResearch in Science and Mathematics. Leeds:University of Leeds Printing Service.

Arons, A. B. (1990). A Guide To IntroductoryPhysics Teaching. (1st ed.) New York: JohnWiley & Sons, Inc.

Arons, A. B. (1993, May). Guiding insight andinquiry in the introductory physics laboratory.The Physics Teacher, 31, 278-282.

Driver, R. & Bell, B. (1986, March). Students'thinking and the learning of science: Aconstructivist view. School Science Review, 67(240), 443-456.

Driver, R. (1992). Research in physics learning:Theoretical issues and empirical studies. In R.Duit, F. Goldberg, & H. Niedderer (Eds.),International Workshop on Research in PhysicsLearning: Theoretical Issues and EmpiricalStudies. (pp. 403-409) Kiel: University of Kiel,IPN/Institute for Science Education.

Garfalo, F. & LoPresti, V. (1993, May).Evolution of an integrated college freshmancurriculum. Journal of Chemical Education, 70(5), 352-359.

Greeno, J. G. (1992). Mathematical andscientific thinking in classrooms and othersituations. Enhancing Thinking Skills in theSciences and Mathematics. (pp. 39-61).Hillsdale, NJ: Laurence Erlbaum Associates.

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Gunstone, R. F. (1992). Constructivism andmetacognition: Theoretical issues andclassroom studies. In R. Duit, F. Goldberg, & H.Niedderer ( Eds.), Research in Physics Learning:Theoretical Issues and Empirical Studies. (pp.129-140). Kiel: University of Kiel,IPN/Institute for Science Education.

Halpern, D. (1992). A cognitive approach toimproving thinking skills in the sciences andmathematics. Enhancing Thinking Skills in theSciences and Mathematics. (pp. 1-14). Hillsdale,NJ: Laurence Erlbaum Associates.

Hewson, P. & Hewson, M. (1992). The statusof students' conceptions. In R. Duit, F.Goldberg, & H. Niedderer (Eds.), Research inPhysics Learning: Theoretical Issues andEmpirical Studies, (pp. 59-73). Kiel: Universityof Kiel, IPN/Institute for Science Education.

Hewson, P. W. & Thorley, N. R. (1989). Theconditions of conceptual change in theclassroom. International Journal of ScienceEducation, 11 [ Special Issue], 541-553.

McDermott, L. C.& Shaffer, P. S. (1992).Research as a guide for curriculumdevelopment: An example from introductoryelectricity. Part II: Design of instructionalstrategies. American Journal of Physics 60(11),1003-1013.

Mestre, J. (1987, Summer). Why shouldmathematics and science teachers be interestedin cognitive research findings? AcademicConnections, 3-11.

National Research Council. (1996). NationalScience Education Standards. Washington, DC:National Academy Press.

Needham, R. & Hill, P. (1987, June). Teachingstrategies for developing understanding in science.Children's Learning in Science Project, Centerfor Studies in Science and MathematicsEducation. Leeds: University of Leeds.

Pfundt, H. & Duit, R. (Eds.). (1991).Bibliography: Students' Alternative Frameworksand Science Education. Kiel: University of Kiel,IPN/Institute for Science Education.

Posner, G. J., Strike, K. A., Hewson, P. W. &Gertzog, W. A. (1982). Accommodation of a

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scientific conception: Toward a theory ofconceptual change. Science Education, 66,211-227.

Resnick, L. B. (1987). Education and Learningto Think. Washington, DC: National AcademyPress.

Scott, P. H., Asoko, H. M. & Driver, R.H.(1992). Teaching for conceptual change: Areview of strategies. In R. Duit, E Goldberg, &H. Niedderer (Eds.), Research in PhysicsLearning: Theoretical Issues and EmpiricalStudies. (pp. 310-329). Kiel: University of Kiel,IPN/Institute for Science Education.

Scott, P. H., Dyson, T. & Gater, S. (1987, June).A constructivist view of learning and teaching inscience. Children's Learning in Science Project,Center for Studies in Science and MathematicsEducation. Leeds: University of Leeds.

Tiberghien, A. (Ed.). (1980). Modes andconditions of learning. An example: Thelearning of some aspects of the concepts ofheat. Cognitive Development Research in Scienceand Mathematics. Leeds: University of LeedsPrinting Service.

Tobias, S. (1990). They're Not Dumb, They'reDifferent. Tucson, AZ: Research Corporation.

Watson, B. & Konicek, R. (1990, May). Teachingfor conceptual change: Confronting children'sexperience. Phi Beta Kappan, 681-685

General References

Brandt, R. (1992, April). On research onteaching: A conversation with Lee Shulman.Educational Leadership, 14-19.

Brooks, J. G. & Brooks, M. G. (1993). In Searchof Understanding: The Case for ConstructivistClassrooms. Alexandria, VA: Association forSupervision and Curriculum Development.

Elmore, R. E (1992, April). Why restructuringalone won't improve teaching. EducationalLeadership, 44-48.

Leinhardt, G. (1992, April). What research onlearning tells us about teaching. EducationalLeadership, 20-25.

Rosenshine, B. & Meister, C. (1992;April). Theuse of scaffolds for teaching higher-level cognitivestrategies. Educational Leadership, 26-33.

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Appendix II

HIGE LIG -TS OF NATIONALRECOMMENDATIONS

he excerpts below emanate from recent,national efforts to delineate standards for

mathematics and science education, and toestablish guidelines for school reform based onlearner-centered principles. These wereselected, annotated, and distributed to allMCTP participants as part of a collaborative-wide effort, led by Genevieve Knight and JohnLayman, to define "guiding principles" forMCTP courses. (Note: When distributed in1995, the materials from the National ResearchCouncil and the American PsychologicalAssociation were in draft form. The excerptsbelow are from the final products.)

Included are excerpts from:

O The National Council of Teachers ofMathematics (NCTM): ProfessionalStandards for Teaching Mathematics andCurriculum and Evaluation Standards forSchool Mathematics

o The National Research Council (NRC):National Science Education Standards

o The American Psychological Association(APA): Learner-Centered PsychologicalPrinciples: Guidelines for School Redesign andReform

Highlights fromThe National Council of

Teachers of Mathematics (NCTM)Professional Standards for

Teaching Mathematics (1991) andCurriculum and Evaluation Standards for

School Mathematics (1989)

The Professional Standards for TeachingMathematics rest on two assumptions:

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O Teachers are key figures in changing theways in which mathematics is taught andlearned in schools.

O Such changes require that teachers have long-term support and adequate resources.

(NCTM, 1991, p. 2)

The NCTM teaching standards explicitlydescribe how the environment of mathematicsclassrooms must change to support activelearning. There must be major shifts to:

O Classrooms as mathematical communitiesaway from classrooms as simply a collection ofindividuals;

O Logic and mathematical evidence asverificationaway from the teacher as thesole authority for right answers;

o Mathematical reasoningaway from simplymemorizing procedures;

O Conjecturing, inventing, and problemsolvingaway from an emphasis on mechanicanswer-finding;

o Connecting mathematics, its ideas, and itsapplicationsaway from treating mathematicsas a body of isolated concepts and procedures.

(NCTM, 1991, p. 3)

The development of teachers of mathematics is alife-long activity; standards for professionaldevelopment rest on assumptions including thefollowing:

O Teachers are influenced by the teaching theysee and experience.

® Learning to teach is a process of integration oftheory, research and practice.

® The education of teachers of mathematics is

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4

an ongoing process; their growth requirescommitment to professional development.

O There are level-specific needs for theeducation of teachers of mathematics; thelearning needs of elementary, middle, andsecondary school students must beaddressed when considering teachers'knowledge of students and teaching.

(NCTM, 1991, p. 124-125)

The following quote is particularly relevant toMCTP's philosophy and goals:

Prospective teachers must be taught in amanner similar to how they are to teachby exploring, conjecturing, communicating,reasoning and so forth. . . . all teachers needan understanding of both the historicaldevelopment and current applications ofmathematics. Furthermore, they should befamiliar with the power of technology.

(NCTM, 1989, p.253)

Highlights fromThe National Research Council (NRC)

National Science EducationStandards (1996)

The goals for school science that underlie theNational Science Education Standards are toeducate students who are able to:

O Experience the richness and excitement ofknowing about and understanding thenatural world;

O Use appropriate scientific processes andprinciples in making personal decisions;

O Engage intelligently in public discourse anddebate about matters of scientific andtechnological concern; and

O Increase their economic productivitythrough the use of knowledge,understanding, and skills of thescientifically literate person in theircareers.

(NRC, 1996, p. 13)

Two of the guiding principles behind theNSES have been central to MCTP:

O Science is for all students.

O Learning science is an active process.

(NRC, 1996, p. 19)

The standards for science teaching are groundedin the following assumptions:

O What students learn is greatly influenced byhow they are taught.

O The actions of teachers are deeply influencedby their perceptions of science as an enterpriseand as a subject to taught and learned.

O Student understanding is actively constructedthrough individual and social processes.

o The actions of teachers are deeply influencedby their understanding of and relationshipswith their students.

(NRC, 1996, p. 28)

To guide and facilitate learning, teachers:

O Focus and support inquiries while interactingwith students.

O Orchestrate discourse among students aboutscientific ideas.

O Challenge students to accept and shareresponsibility for their own learning.

O Recognize and respond to student diversityand encourage all students to participate fullyin science learning.

o Encourage and model the skills of scientificinquiry, as well as curiosity, openness to newideas and data, and skepticism thatcharacterizes science.

(NRC, 1996, p. 32)

To engage in ongoing assessment of theirteaching and of student learning, teachers:

O Use multiple methods and systematicallygather data about student understanding andability.

O Analyze assessment data to guide teaching.

O Use student data, observations of teaching,and interactions with colleagues to reflect onand improve teaching practice.

(NRC, 1996, p. 37-38)

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4

To provide students with time, space, and theresources needed for learning science, teachers:

Structure the time available so that studentsare able to engage in extendedinvestigations.

Create a setting that is flexible andsupportive of science inquiry.

Ensure a safe working environment.

Make science tools, materials, printresources, media, and technologicalresources accessible to students.

Identify and use resources outside theschool.

Engage students in designing the learningenvironment.

(NRC, 1996, p. 43)

To develop communities of science learnersthat reflect the intellectual rigor of scientificinquiry and the attitudes and social valuesconducive to science learning, teachers:

Display and demand respect for the ideasand skills of all students.

Enable students to have a significant voice indecisions about the content and context oftheir work and require students to takeresponsibility for the learning of allmembers of the community.

Nurture collaboration among students.

Facilitate ongoing formal and informaldiscussion based on a shared understandingof rules of scientific discourse.

(NRC, 1996, p. 46)

Highlights fromThe American Psychological Association (APA)

Learner-Centered Psychological Principles:Guidelines for School Redesign and Reform (1995)

The following 14 principles are intended todeal holistically with learners in the context ofreal-world learning situations. Thus, they arebest understood as an organized set ofprinciples; no principle should be viewed inisolation. These principles are intended to

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apply to all learnersfrom children, toteachers, to administrators, to parents, and tocommunity members involved in oureducational system (APA, 1995).

Cognitive and Nletacognitive Factors

Nature of the learning process. Thelearning of complex subject matter is mosteffective when it is an intentional processof constructing meaning from informationand experience.

© Goals of the learning process. Thesuccessful learner, over time and withsupport and instructional guidance, cancreate meaningful, coherentrepresentations of knowledge.

© Construction of knowledge. The successfullearner can link new information withexisting knowledge in meaningful ways.

© Strategic thinking. The successful learnercan create and use a repertoire of thinkingand reasoning strategies to achievecomplex learning goals.

O Thinking about thinking. Higher orderstrategies for selecting and monitoringmental operations facilitate creative andcritical thinking.

Context of learning. Learning is influencedby environmental factors, includingculture, technology, and instructionalpractices.

Motivational and Affective Factors

Motivational and emotional influences onlearning. What and how much is learned isinfluenced by the learner's motivation.Motivation to learn, in turn, is influencedby the individual's emotional states, beliefs,interests and goals, and habits of thinking.

Intrinsic motivation to learn. The learner'screativity, higher order thinking, andnatural curiosity all contribute tomotivation to learn. Intrinsic motivation isstimulated by tasks of optimal novelty anddifficulty, relevant to personal interests,and providing for personal choice andcontrol.

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O Effects of motivation on effort. Acquisition ofcomplex knowledge and skills requiresextended learner effort and guided practice.Without learners' motivation to learn, thewillingness to exert this effort is unlikelywithout coercion.

Developmental and Social]

Developmental influences on learning. Asindividuals develop, there are differentopportunities and constraints for learning.Learning is most effective when differentialdevelopment within and across physical,intellectual, emotional, and social domains istaken into account.

Social influences on learning. Learning isinfluenced by social interactions,interpersonal relations, and communicationwith others.

individual Differences

O Individual differences in learning. Learnershave different strategies, approaches, andcapabilities for learning that are a functionof prior experience and heredity.

Learning and diversity. Learning is mosteffective when differences in learners'linguistic, cultural, and social backgroundsare taken into account.

0 Standards and assessment. Settingappropriately high and challenging standardsand assessing the learner as well as learningprogressincluding diagnostic, process, andoutcome assessmentare integral parts ofthe learning process.

References

American Psychological Association. (1995).Learner-Centered Psychological Principles:Guidelines for School Redesign and Reform. [On-line]. Available: http://www.apa.org/ed/lcp.html[1997, Dec 18].

National Council of Teachers of Mathematics.(1989). Curriculum and Evaluation Standardsfor School Mathematics. Reston, VA: NCTM.Also available: http://www.enc.org/reform/journals/ENC2280/280dtocl.htm [1997,Dec 18].

National Council of Teachers of Mathematics.(1991). Professional Standards for TeachingMathematics. Reston, VA: NCTM.

National Research Council. (1996). NationalScience Education Standards. Washington, DC:National Academy Press. Also available: http://www.nap.edu/readingroom/books/nses/ [1997,Dec 18].

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Appendix III

MCTP ADVISORS AND PARTICIPANTS

National Visiting Committee

Edward Ernst, ChairUniversity of South CarolinaHenry HeikkinenUniversity of Northern ColoradoFlossie Is lerWashington, D.C. Public Schools (Retired)

Glenda LappanMichigan State UniversityMartha McLarenGettysburg PA Public SchoolsJames RutherfordAmerican Association for the Advancement of Science

National Science Foundation

James LightboumeCurtis SearsTerry Woodin

Anne Arundel Community College

Richard Cerkovnik (Physical Sciences)John Wisthoff (Mathematics)

Baltimore City Community College

James Coleman (Mathematics)Thomas Hooe (Biological Sciences)Donald Hoster (Chemistry)Joanne Settel (Biological Sciences)Felice Shore (Mathematics)Jack Taylor (Physics)Joachim Bullacher (Math/Engineering/

Computer Science)

Bowie State University

Karen Benbury (Mathematics)Rebecca Berg (Mathematics)Claudette Burge (Mathematics)Elaine Davis (Natural Sciences)Joan Langdon (Math/Computer Science)Mohammed Moharerrzaded (Natural Sciences)Marlene Reagin (Education)Sadanand Srivastava (Math/Physical Sciences)

Catonsville Community College

Catherine Alban (Mathematics)Marie Skane (Mathematics)

Coppin State College

Glenn Dorsey (Mathematics)Delores Harvey (Education)Genevieve Knight (Math/Computer Science)Pallassana Krishnan (Science)Gilbert Ogonji (Science)Mary Owens (Science)

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Faculty

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Program OfficerConsultantProgram Officer

Frostburg State University

Renny Azzi (Mathematics Education)Marcia Cushall (Math/Science Education)Joseph Hoffman (Physics)Joan Lindgren (Education)Karen Parks (Mathematics)Robert Riley (Biology)Francis Tam (Physics)Richard Weimer (Mathematics)Kenneth Witmer (Education).

Morgan State University

Roselyn Hammond (Biology)

Prince George's Community College

Gladys Whitehead (Mathematics)

Salisbury State University

Donald Cathcart (Mathematics)Sharon Clark (Science/Education)Phillip Creighton (Science)Mark Holland (Biology)Thomas Horseman (Math/Computer Science)C. Richard McKenzie (Physics)David Parker (Math/Computer Science)Geraldine Rossi (Education)Elichia Venso (Environmental Health/Biology)Gail Welsh (Physics)

Towson University

Virginia Anderson (Science Education)Lawrence Boucher (Math & Natural Sciences)Rachel Burks (Geology)Katherine Denniston (Biology)R. Michael Krach (Mathematics Education)

Towson University (continued)

Loretta Molitor (Physics/Center for Math/Science Education)

Joseph Topping (Chemistry)Leon Ukens (Physics)Tad Watanabe (Mathematics Education)John Wessner (Physics)Maureen Yarnevich (Mathematics)Jay Zimmerman (Mathematics)

University of Maryland, Baltimore County

Susan Blunck (Science Education)Barbara Kinach (Mathematics)Ivan Kramer (Physics)Thomas Seidman (Mathematics)Alan Sherman (Computer Science)Phillip Sokolove (Biological Sciences)

University of Maryland Eastern Shore

Eddie Boyd (Mathematics)James Doyle (Biology)Gurbax Singh (Science)

University of Maryland, College Park

William Adams (Mathematics)Joseph Auslander (Mathematics)Kenneth Berg (Mathematics)James Fey (Mathematics)Douglas Gill (Zoology)Anna Graeber (Mathematics)David Lay (Mathematics)John Layman (Science)J. Randy McGinnis (Science Education)Jerome Motta (Zoology)Thomas O'Haver (Science)

Institutional Associates

Wayne Bell, Center for Environmental ScienceLynn Dierking, Science Learning, Inc.Alan Place, MD Biotechnology InstituteJordan Warnick, University of Maryland, Baltimore-School of Medicine

Public School Colleagues

Beth Abraham Baltimore County Margaret Miller Baltimore CitySally Bell Baltimore City Evelyn Nicks Prince George's Co.Karen Burgess Baltimore City Mary O'Haver Montgomery CountyPenny Caldwell Wicomico County Jean Regin Baltimore CityEleanor Ennis Wicomico County Mary Thurlow Baltimore CountyBarbara Faltz-Jackson Baltimore City Marlene Weimer Allegany CountySharon Hoffman Prince George's Co. Steve Wilson Allegany CountyAudrey Melton Baltimore City Colleen Zaloga Allegany County

Susan BoyerKaren LangfordDonna AyresMaureen GardnerSusan HanleyMary Ann HuntleyKaren KingSteven KramerCarolyn ParkerAmy Roth-McDuffieSusan SchultzStephanie StockmanWilliam Sullivan

Evaluators

Gilbert AustinLois Williams

Staff

Executive DirectorAssociate DirectorProgram Management SpecialistFaculty Research AssistantGraduate Assistant, Science EducationGraduate Assistant, Mathematics EducationGraduate Assistant, Mathematics EducationGraduate Assistant, Mathematics EducationGraduate Assistant, Science EducationGraduate Assistant, Mathematics EducationGraduate Assistant, Mathematics EducationGraduate Assistant, Science EducationGraduate Assistant, Mathematics

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