v14n2

Fall 2004 MATHEMATICS EDUCATION STUDENT ASSOCIATIONTHE UNIVERSITY OF GEORGIA

____ THE_____ MATHEMATICS ___

________ EDUCATOR _____Volume 14 Number 2

Editorial StaffEditorHolly Garrett Anthony

Associate EditorsGinger RhodesMargaret SloanErik Tillema

PublicationStephen BismarckDennis Hembree

AdvisorsDenise S. MewbornNicholas OppongJames W. Wilson

MESA Officers2004-2005PresidentZelha Tunç-Pekkan

Vice-PresidentNatasha Brewley

SecretaryAmy J. Hackenberg

TreasurerGinger Rhodes

NCTMRepresentativeAngel Abney

UndergraduateRepresentativesErin BernsteinErin CainJessica Ivey

A Note from the EditorDear TME readers,

This has been a productive year for the editors of TME. We overhauled and updated our website,broadened our reach nationally and internationally, and doubled our number of reviewers. Collectively,the editors worked with almost 30 authors providing feedback and readying articles for publication.We are now proud to present the final of two issues to be published in 2004. This issue showcases bothnational and international research and commentary. We hope your reading of these articles will beboth educational and thought provoking.

David Clarke opens this issue with an editorial inviting you to read the research he, MargaritaBreed, and Sherry Fraser conducted in the early 1990s and published in this issue. In their researcharticle, they highlight some positive consequences of teaching with the Interactive MathematicsProgram (IMP), a problem-based curriculum that has gained attention in recent years. Clarke’seditorial further asks mathematics education researchers to consider research methodology whenstudying classroom learning. In so doing, he draws on the international comparative research of theLearner’s Perspective Study.

Two studies presented in this issue examine the effectiveness of assessment items. Bates andWiest discuss the impact personalization of word problems can have on students’ performance onmathematics assessments. Contrary to recent research they report no significant difference in students’performance with personalized and non-personalized problems. Rueda and Sokolowski study theeffectiveness of their locally developed mathematics placement test at Merrimack College and showthat students who follow the recommendations for course enrollment based on their test scores performwell in those classes.

Finally, Cyril Julie, a scholar in South Africa, invites readers to consider the development ofdemocratic competence in students within a newly formed democratic country and the rolemathematics might play in that development. He asks whether democratic competence can be realizedwithin Realistic Mathematics Education (RME), a curriculum developed in the Netherlands andrecently imported into South Africa. His question is important for consideration and his discussion isstimulating.

As I close my final comments as the editor of TME, I encourage readers to support our journal bysubmitting manuscripts, reviewing articles, or joining our editorial team. TME is growing inrecognition, and it is through the efforts put forth by all of us that it will continue to thrive.

Serving as the 2004 editor of TME has been truly rewarding. I was privileged to lead a team ofeditors who worked well both together and independently. The publication of TME is a direct result oftheir time and effort. I appreciate MESA allowing me the opportunity to do this work and I hope thatmy efforts have been notable. I also extend my thanks to all of the other people who make TMEpossible: reviewers, authors, peers, and faculty.

With Sincere Thanks,

Holly Garrett Anthony105 Aderhold Hall [email protected] University of Georgia www.coe.uga.edu/tmeAthens, GA 30602-7124

About the coverCover artwork by Tyler M. Ricks. Untitled, 2004.For questions or comments, contact: [email protected] piece is the culmination of years of experimentation and study with computer art. Starting with just a simple 3 by 3 grid in an8th grade art class, the style blossomed into hundreds of different pieces using many different geometrical ideas. The process ofhand-drawing each line takes hours of work, but can produce extremely complex mathematical images. The piece featured on thecover was created on the computer program GeoSketchpad™, and is part of a larger series using complex “string frames,” socalled because they resemble physical frames on which strings are tightly strung. Other series use complex grids, differentgeometric shapes, or skewed frames to create intricate line drawings.

This publication is supported by the College of Education at The University of Georgia.

____________THE___________________________ MATHEMATICS________

______________ EDUCATOR ____________An Official Publication of

The Mathematics Education Student AssociationThe University of Georgia

Fall 2004 Volume 14 Number 2

Table of Contents

2 Guest Editorial… Researching Classroom Learning and Learning ClassroomResearchDAVID CLARKE

7 The Consequences of a Problem-Based Mathematics CurriculumDAVID CLARKE, MARGARITA BREED, & SHERRY FRASER

17 Impact of Personalization of Mathematical Word Problems on StudentPerformanceERIC T. BATES & LYNDA R. WIEST

27 Mathematics Placement Test: Helping Students SucceedNORMA G. RUEDA & CAROLE SOKOLOWSKI

34 In Focus… Can the Ideal of the Development of Democratic Competence BeRealized Within Realistic Mathematics Education? The Case of South AfricaCYRIL JULIE

38 Upcoming Conferences39 Submissions Information40 Subscription Form

© 2004 Mathematics Education Student Association.All Rights Reserved

The Mathematics Educator2004, Vol. 14, No. 2, 2–6

2 Researching Classroom Learning

Guest Editorial…Researching Classroom Learning and Learning Classroom

ResearchDavid Clarke

One of the central goals of the mathematicseducation research community is the identification ofclassroom practice likely to facilitate student learningof mathematics. In the paper by Clarke, Breed andFraser in this issue of The Mathematics Educator, theresults of an investigation into the outcomes of theInteractive Mathematics Program (IMP) undertakenback in the early 1990s are reported. Why is thisimportant? Because the focus of the analysis was anexpanded conception of the outcomes of classroompractice that included both the cognitive and theaffective consequences of introducing a problem-basedmathematics program. The findings demonstrate thatthe consequences of a particular curriculum and itsassociated classroom practices cannot be adequatelycharacterized solely by the mathematical performanceof the students. Most importantly, the IMP classroomsstudied were most clearly distinguished fromconventional classrooms by affective rather thancognitive outcomes. At the time, this was an attempt toembrace a broader vision of valued classroom practiceand significant learning outcomes than could bedocumented in an achievement test. The message ofthis research has contemporary significance, but in thetime since that study was conducted our capacity toinvestigate classroom practice and to connect it tolearning outcomes has increased considerably.

The Participant’s VoiceI have argued consistently and persistently (Clarke,

1998, 2001, 2003) that since a classroom takes ondifferent aspects according to how you are positionedwithin it or in relation to it, our research methodologymust be sufficiently sophisticated to accommodate andrepresent the multiple perspectives of the manyparticipants in complex social settings such asclassrooms. Only by seeing classroom situations fromthe perspectives of all participants can we come to anunderstanding of the motivations and meanings thatunderlie their participation. Our capacity to improveclassroom learning depends on such understanding.The methodological challenge is how to document andanalyze the fundamental differences in how eachparticipant experiences any particular social(classroom) situation. My colleagues, Sverker Lindbladand Fritjof Sahlström (2002), argue that if earlyresearchers had access to the tools for data collectionand analysis that are available today, the general viewof classroom interaction would be quite different.

The most striking of these differences, and a veryimportant one from an education point of view,concerns the role of students in classrooms. Thorsten(2000) has made this point very clearly in relation tothe Third International Mathematics and Science Study(TIMSS).

What is absent from nearly all the rhetoric andvariables of TIMSS pointing to the future needs ofthe global economy is indeed this human side: thenotion that students themselves are agents.(Thorsten, 2000, p. 71)

Single-camera and single-microphone approaches,with a focus on the teacher, embody a view of thepassive, silent student at odds with contemporarylearning theory and classroom experience. Researchdone with technologically more sophisticatedapproaches has described a quite different classroom,where different students are active in different ways,contributing significantly to their own learning (cf.Sahlström & Lindblad, 1998; Clarke, 2001).

David Clarke is a Professor in the Faculty of Education at theUniversity of Melbourne and Director of the InternationalCentre for Classroom Research. His consistent interests havebeen Assessment, Learning in Classrooms, and TeacherProfessional Development, and he has undertaken researchrelated to all of these areas. Recent publications include thebook “Perspectives on Practice and Meaning in Mathematicsand Science Classrooms” published by Kluwer AcademicPublishers in 2001, and the chapters on Assessment andInternational Comparative Research in the 1996 and 2003editions of the “International Handbook of MathematicsEducation.” He is currently directing the 14-country Learner'sPerspective Study.

David Clarke 3

International Comparative ResearchFurther, classroom researchers have until recently

had limited opportunities for engaging in manageablecomparative research, where materials from differentcountries and different periods of time can be accessedand analyzed in feasible ways. At the InternationalCentre for Classroom Research at the University ofMelbourne(http://www.edfac.unimelb.edu.au/DSME/ICCR/),contemporary technology makes it possible to carry outcomparative analyses of an extensive database thatincludes three-camera classroom video records oflesson sequences, supplemented by post-lesson video-stimulated interviews with students and teachers,scanned samples of written work, and test andquestionnaire data, drawn from mathematicsclassrooms as geographically distant as Sweden andAustralia and as culturally distant as Germany andChina.

Watanabe (2001) quotes White (1987) as writing“we should hold Japan up as a mirror, not as ablueprint.” This powerful and appealing metaphor canserve as a general characterization of one of the majoruses of international comparative studies of classroompractice. The agency for the interpretation andadaptation of any documented practice resides with theperson looking in the mirror. There is no invocation ofabsolute best practice – the judgement is a relativistone, and an instructional activity with a high degree ofefficacy in Hong Kong may retain little effectivenesswhen employed in a Swedish classroom, wheredifferent cultural values inform and frame the actionsof all classroom participants. Most importantly, we areencouraged to study Japanese (or South African orGerman) classrooms not solely for the purposes ofmimicking their practices but for their capacity tosupport us in our reflection on our own practice. Themutuality of the potential benefit provides furthermotivation for such research.

There is a small but growing body of research thatworks at developing techniques of documentingclassroom interaction in ways that will facilitate high-quality analysis of children’s learning. The transferfrom single-microphone audio (as in the early studies),via single-camera video (as in many recent studies) tomulti-camera and multi-audio (as in the studies at thetechnological forefront) is not primarily technology-driven, but rather motivated by the recent shifts ineducation theories on learning, from a view of learningas transfer to a view of learning as constructed inaction (see Sfard, 1998, for a discussion). Thus,technological sophistication is a requirement of recent

theory, rather than a matter of sophisticated equipmentfor technologically-minded project coordinators. Thisis an essential point: Educational research, likeresearch in the physical and biological sciences, mustmake optimal use of available technologies inaddressing the major problems of the field. But theprime motivation must be “What are the big questionsand what tools do we need to address these questions?”rather than “What questions can be addressed withavailable tools?” Our research must be fuelled by aneed to answer important questions, not by a need touse new tools. In addition, it is the first question thatwill lead to recognition of the need for new tools andprovide the motivation for their development.

The Learner’s Perspective Study: ComplementaryAccounts

Data collection in the Learner’s Perspective Study(http://www.edfac.unimelb.edu.au/DSME/lps/)involves a three-camera approach (Teacher camera,Student camera, Whole Class camera) that includes theonsite mixing of the Teacher and Student cameraimages into a split-screen video record that is then usedto stimulate participant reconstructive accounts ofclassroom events. So far, these data have beencollected for sequences of at least ten consecutivelessons occurring in the “well-taught” eighth grademathematics classrooms of three teachers in each often participating countries (Australia, Germany, HongKong and mainland China, Israel, Japan, Korea, ThePhilippines, South Africa, Sweden and the USA). Thiscombination of countries gives good representation toEuropean and Asian educational traditions, affluentand less affluent school systems, and mono-culturaland multi-cultural societies. Data collection willcommence next year in the Czech Republic, Englandand Singapore.

Each participating country uses the same researchdesign to collect videotaped classroom data for at leastten consecutive math lessons and post-lesson video-stimulated interviews with at least twenty students ineach of three participating 8th grade classrooms. Thethree mathematics teachers in each country areidentified for their locally-defined ‘teachingcompetence’ and for their situation in demographicallydiverse government schools in major urban settings. Ina major component of the post-lesson studentinterviews, in which a split-screen video record is usedas stimulus for student reconstructions of classroomevents, students are given control of the video replayand asked to identify and comment upon classroomevents of personal importance. Each teacher is


interviewed at least three times using a similarprotocol.

Goffman’s conception of a working consensus as atransient convergence on a locally viable interpretation(Goffman, 1959) is a particularly apt characterizationof the goal of the consensus process operating in manyinterpretive research teams (e.g., Cobb & Bauersfeld,1995; Stigler & Hiebert, 1999). The research in which Ihave been involved (e.g., Clarke, 2001) problematizessuch consensus and attempts to synthesize portrayals ofpractice from ‘complementary accounts’ provided byresearchers and the participants in the research settingrelating to a common body of data (rationale providedin Clarke, 1998).

I would like to assert the inevitable existence ofmultiple reflexivities between theory, research intopractice, and the practice of research. The argument ispredicated on three basic premises:

1. The discourse of the classroom (forexample) acts to position participants in waysthat afford and constrain certain practices.

2. The discourse of educational research actsto position participants in ways that afford andconstrain certain interpretations.

3. The adoption of a theory of learning insocial situations will inevitably find itsreflection in the manner in which thosesituations are researched.These fundamental reflexivities are seldom

acknowledged. Since research activity constitutes aform of learning or knowledge construction, theprocesses by which a research project is conductedshould be in harmony with whatever theory of learningstructures the researcher’s analysis of data.Consistency between methodology and theory shouldbe a matter of purposeful and deliberate design. LorrieShepherd turns this argument delightfully on its headin her paper “Psychometricians’ Beliefs AboutLearning” (Shepard, 1991), where she contends thatthe disputes of the testing community can be explainedin terms of differences in the beliefs about learningheld by the various educational measurementspecialists. In particular, Shepard argues that thebeliefs of many psychometricians derive from animplicit behaviorist learning theory in flagrantcontradiction with evidence from cognitivepsychology. What Shepard does to good effect in herpaper is reverse engineer psychometricians’ learningtheories on the basis of their test instruments. Thefruitfulness of this approach is fully evident in

Shepard’s provocative question, “But what if learningis not linear and is not acquired by assembling bits ofsimpler learning” (Shepard, 1991, p. 7).

In the case of the Learner’s Perspective Study:Research guided by a theory of learning that accordssignificance to both individual subjectivities and to theconstraints of setting and community practice mustconstruct and frame its conclusions (and collect itsdata) accordingly. Such a theory must accommodatecomplementarity rather than require convergence andaccord both subjectivity and agency to individuals notjust to participate in social practice but to shape thatpractice. Research that aims to apply such theoriesmust construct its methodologies accordingly and drawfrom available technologies in ways that afford ratherthan constrain the methodological ambitions of theresearcher.

A Layered VisionInternational comparative classroom research need

not appeal to a separate and distinct research paradigmfrom that enacted in conventional classroom research,although the methodological and theoreticalconsiderations are more complex than research withina single culture. Part of the power of internationalcomparative research lies in its capacity to offer us theopportunity to juxtapose, compare and contrastdocumented practices drawn from settings that simplywould not pertain in our local culture. What form doesteaching competence take when confronted with a classof 60 or more students (as is the case in thePhilippines)? How must we reconceive our notions ofeffective instructional practice to accommodateapparently successful classrooms in which studentsseldom if ever speak to each other (as pertains in someAsian classrooms)? How much more compelling mustour theories of learning become if they can bedemonstrated to accommodate and explain learning insuch disparate settings?

As new theories of learning and social interactiondevelop, research techniques must have the capacity toaccommodate these new theories. All too often it isforgotten that any use of technology in a researchsetting implies the existence of an underlying theory onwhich the type of data, the means of data collection,and the anticipated method of analysis are allpredicated. Of all data sources currently available toresearchers in education, videotape data seems mostamenable to secondary analysis. Further, the potentialof videotape data to sustain secondary analysis carriesan associative potential for the synthesis of thoseanalyses.

David Clarke 5

Multi-site international research projects offeraccess to a layered vision of practice, outcome andtheory development. It may help to illustrate thisstratification with examples from the Learner’sPerspective Study.

Classroom Practice – Lesson Events

At the level of classroom practice, the challengehas been to find a suitable instructional unit to providethe basis for comparative analysis. Demonstration ofthe inadequacy of “the lesson” to serve this role (atleast in the form of nationally characteristic lesson“scripts” or “patterns”) has led to analyses focusing onthe “lesson events” from which each lesson isconstituted. Lesson events such as “Beginning theLesson,” “Learning Tasks,” “Guided Development”(Whole class discussion), “Between DesksInstruction,” and “Summing Up” have emerged asinternationally recognizable activities, differently anddistinctively employed and enacted in classroomsaround the world.Patterns of Participation

In participating in each of the lesson eventsidentified above, teacher and students positionthemselves and are positioned within the constraintsand affordances offered by the classroom setting andits peculiar practices (peculiar here is used in allpossible senses). The consequences of this process ofsocial positioning are characteristic patterns ofparticipation accessible to classroom participants (andco-constructed by them) in ways that reflect eachindividual’s unique interaction with the classroomsetting and community.

The Distribution of Responsibility for KnowledgeGeneration

Each classroom affords and constrains access tovarious patterns of participation. Within the patterns ofparticipation characteristic of a classroom can befound the “distribution of responsibility for knowledgegeneration” – a much more useful characterization ofthe classroom than a simplistic dichotomization intoteacher-centered and student-centered, and much morerevealing of the sociocultural nature of learning.

The use of video material supported by post-lessonvideo-stimulated interviews provides a complexdatabase amenable to analysis at any and all of thethree levels indicated above. Complex databases,configured in anticipation of multiple andcomplementary analyses, offer our best chance tomatch the complexity of social phenomena with an

appropriate sophistication of approach. Advances intechnology bring us ever closer to the realization ofthis vision. The developmental pathway that has led usfrom early attempts at classroom observation andprocess-product studies to our present level ofsophistication represents an on-going attempt toaccommodate the complexity of social situations.

Eugene Ionescu is reputed to have said, “Only theephemeral is of lasting value.” Social interactions arenothing if not ephemeral; and, since it is through socialinteraction that we experience the world, theunderstanding of social interactions must underlie anyattempts to improve the human condition. Ourdifficulties in characterizing social interactions for thepurpose of theory building in education arecompounded by the fluid and transient nature of thephenomena we seek to describe. Attempts to categorizesocial behavior run the risk of sacrificing thedynamism, contextual-dependence and variation thatconstitute their essential attributes. This poses achallenge both for methodology and for theory. Theephemeral nature of social interactions is somethingthat must be honored in the methodology buttranscended in the analysis. Those of us who haveaccepted the challenge of researching classroomlearning continue to learn how better to undertakeclassroom research.

REFERENCESClarke, D. J. (1998). Studying the classroom negotiation of meaning:

Complementary accounts methodology. In A. Teppo (Ed.),Qualitative research methods in mathematics education (Chapter 7,pp. 98–111). Journal for Research in Mathematics Education,Monograph No. 9. Reston, VA: National Council of Teachers ofMathematics.

Clarke, D. J. (Ed.). (2001). Perspectives on practice and meaning inmathematics and science classrooms. Dordrecht, Netherlands: KluwerAcademic Press.

Clarke, D. J. (2003). International comparative studies in mathematicseducation. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, &F. K. S. Leung (Eds.), Second international handbook of mathematicseducation (Chapter 5, pp. 145–186). Dordrecht, Netherhlands: KluwerAcademic Publishers.

Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematicalmeaning: Interaction in classroom cultures. Hillsdale, NJ: LawrenceErlbaum.

Goffman, E. (1959). The presentation of self in everyday. New York:Doubleday. [Cited in Krummheuer (1995)]

Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb &H. Bauersfeld (Eds.), The emergence of mathematical meaning:Interaction in classroom cultures (Chapter 7, pp. 229–269). Hillsdale,NJ: Lawrence Erlbaum.

Lindblad, S., & Sahlström, F. (2002, May). From teaching to interaction:On recent changes in the perspectives and approaches to classroomresearch. Invited plenary lecture at the Current Issues in ClassroomResearch: Practices, Praises and Perspectives Conference, Oslo,Norway.


Sahlström, F., & Lindblad, S. (1998). Subtexts in the science classroom –an exploration of the social construction of science lessons and schoolcareers. Learning and Instruction, 8(3), 195–214.

Sfard, A. (1998). On two metaphors for learning and the danger of choosingjust one. Educational Researcher, 27(2), 3–14.

Shepard, L. A. (1991). Psychometrician’s beliefs about learning.Educational Researcher, 20(6), 2–16.

Stigler, J., & Hiebert, J. (1999). The teaching gap. New York: Free Press.

Thorsten, M. (2000). Once upon a TIMSS: American and Japanesenarrations of the Third International Mathematics and Science Study.Education and Society, 18(3), 45–76.

Watanabe, T. (2001). Content and organization of teacher’s manuals: Ananalysis of Japanese elementary mathematics teacher’s manuals.School Science and Mathematics, 101(4), 194–201.

White, M. (1987). The Japanese educational challenge: A commitment tochildren. New York: The Free Press. [Cited in Watanabe (2001)]


Clarke, Breed, & Fraser 7

The Consequences of a Problem-Based MathematicsCurriculum

David Clarke, Margarita Breed, & Sherry Fraser

Implementation of a problem-based mathematics curriculum, the Interactive Mathematics Program (IMP), atthree high schools in California has been associated with more than just differences in student achievement. Theoutcomes that distinguished students who participated in the IMP program from students who followed aconventional algebra/geometry syllabus were the students’ perceptions of the discipline of mathematics, ofmathematical activity and the origins of mathematical ideas, of the mathematical nature of everyday activities,and of school mathematics and themselves as mathematicians. A coherent and consistent picture has emerged ofthe set of beliefs, perceptions and performances arising from such a program. Students who have participated inthe IMP program appear to be more confident than their peers in conventional classes; to subscribe to a view ofmathematics as having arisen to meet the needs of society, rather than as a set of arbitrary rules; to valuecommunication in mathematics learning more highly than students in conventional classes; and to be morelikely than their conventionally-taught peers to see a mathematical element in everyday activity. Theseoutcomes occurred while the IMP students maintained performance levels on the mathematics portion of theSAT at or above those of their peers in conventional classes. If student achievement outcomes are comparable,the mathematics education community must decide whether it values these consequences of a problem-basedcurriculum.

Among the debates engaging the energies of themathematics education community, one of the moreenergetic has concerned the role of problem solving inmathematics instruction. This debate has encompassedissues from what constitutes a problem to whetherproblem solving should be the medium or the messageof the mathematics curriculum (cf., Clarke &McDonough, 1989; Lawson, 1990; Owen & Sweller,1989; Schoenfeld, 1985). Claims and counter-claims

have been made regarding the advisability and thefeasibility of basing a mathematics syllabus on non-routine mathematics tasks. Attempts to evaluate thesuccess of such curricula have typically employedachievement tests to distinguish student outcomes.The authors of this study felt that a problem-basedcurriculum would be characterized moreappropriately by the belief systems which theinstructional program engendered in participatingstudents than by the students’ achievement onconventional mathematical tasks.

It is students’ belief systems that are likely toinfluence the students’ subsequent participation inthe study of mathematics, to structure theirconsequent learning of mathematics, and to guideand facilitate the application of mathematical skillsto everyday contexts. If it could be demonstratedthat student achievement on conventionalmathematics tasks was enhanced by a problem-based program, and if student performance on non-routine problem-solving tasks was heightened bysuch a program, the ultimate value of the instructionwould depend still on whether the student choosesto continue to study mathematics, develops a set ofbeliefs which supports and empowers furtherlearning, and sees any relevance in the skillsacquired in class for situations encountered in theworld beyond the classroom. Conventionalinstruction does little to address such concerns, andresearch has commonly ignored such outcomes.

David Clarke is a Professor in the Faculty of Education at theUniversity of Melbourne and Director of the InternationalCentre for Classroom Research. He is currently directing the14-country Learner's Perspective Study.Margarita Breed is currently a fulltime Ph. D. student fundedunder the APAI (Australian Postgraduate Award Industry)Scheme for the Scaffolding Numeracy in the Middle YearsResearch Project at RMIT University. Her background hasbeen in primary teaching and as a Middle Years NumeracyLeader for Eastern Metropolitan Region and she is particularlydedicated to students in the Middle Years. Whilst completingher Master of Education (Research) she was ResearchAssistant for the Mathematics Teaching and Learning Centre atthe Australian Catholic University.Sherry Fraser is currently the Director of the IMPlementationCenter for the Interactive Mathematics Program. She continuesto be interested in providing both students and teachers accessto rich secondary mathematics materials.AcknowledgementsThe cooperation of the teachers whose pupils participated inthis study is gratefully acknowledged. The comments of BarryMcCrae, Kevin Olssen, Diane Resek and Peter Sullivan onearly drafts of this paper are also gratefully acknowledged.

8 Problem-Based Mathematics Curriculum

The evaluation of teaching experiments currentlyin progress must address these other consequences ofinstruction. In discussing their work on “one-on-oneconstructivist teaching,” Cobb, Wood, and Yackel(1990) drew attention to non-conventional learningoutcomes. This instructional approach providesopportunities for the children to constructmathematical knowledge not found in traditionalclassrooms. The difficulty for researchers evaluatinginnovative classroom practices is that many of theconventional research tools are insensitive to thebehaviors and the knowledge that distinguishes suchinstruction. This concern is also relevant where thegoals of the program are affective as well as cognitive.Since studies such as that of Erlwanger (1975) drewattention to the significance of a student’s beliefsystem regarding mathematics and mathematicalbehavior, research into effective teaching practice hashad an obligation to address student belief outcomes.This obligation is linked to the recognition of“cognition as socially situated activity” (Lave, 1988,p.43). While the subject of student beliefs has beendiscussed usefully in a variety of forums (for instance,Clarke, 1986; Cobb, 1986), research studies have stillto accept a responsibility to address student belief andperception outcomes routinely in the evaluation ofinstructional programs. The study reported here is oneattempt to do so.

The Instructional ProgramIn 1989, the California Postsecondary Education

Commission (CPEC) released a request for proposalsthat would drastically revamp the Algebra I-Geometry-Algebra II sequence. The curriculum envisioned in theguidelines would set “problem solving, reasoning andcommunication as major goals; include such areas asstatistics and discrete mathematics; and makeimportant use of technology” (CPEC, 1989, p. 4). TheInteractive Mathematics Project (IMP) CurriculumDevelopment Program obtained funding to developand field test three years of problem-basedmathematics that would satisfy six of the University ofCalifornia requirements for high school mathematics.

Program GoalsThe goals of IMP were to:• broaden who learns mathematics, by making the

learning of core mathematics accessible to groupspreviously underrepresented in collegemathematics classes;

• expand what mathematics was learned, consistentwith the recommendations of the Curriculum and

Evaluation Standards (NCTM, 1989),emphasizing problem solving and thecommunication of mathematical ideas;

• change mathematics instruction, by requiringstudents to be active learners and investigators,by integrating the study of mathematicaldomains, such as algebra, geometry andstatistics, with each other and with areas ofapplication, and by making use of currenttechnology;

• change how teachers perceive their roles, byemphasizing the role of the teacher as guide andmodel learner and by changing dominant modesof classroom communication from teacherexplanation to student interaction;

• change how mathematics learning is assessed,by assessing students’ use of mathematicalknowledge to solve complex problems, and bydiversifying assessment strategies to includestudent portfolios, self-assessment, teacherobservations, oral presentations, and groupprojects, as well as written homework and tests.

Pupil SelectionMethods of selection of pupils for participation

in the IMP classes varied. The principal criterionwas student self-nomination. One high schoolcollected information on student performance,instructional preferences, and academic history andthen selected “60% of the group who would havebeen placed in Algebra and 40% from those below.”It was the opinion of the various schooladministrations that the academic standing of thesample of IMP students arising from the variousselection criteria was certainly no higher than thatof the students in conventional Algebra classes. Infact, in the case of the high school just mentioned,the overall academic standing of studentscommencing IMP was almost certainly lower thanthat of commencing Algebra students.

Teacher SelectionTeachers were also self-nominated.

The IMP MaterialsThe IMP materials consist of modular units,

each requiring approximately five weeks ofinstructional time. These units employ historical,literary, scientific and other contexts to provide athematic coherence to the pupils’ exploration ofmathematics. For instance, in one unit the Edgar


Allan Poe short story The Pit and the Pendulum is usedto facilitate student investigation of variation,measurement uncertainty, normal distribution,graphing, mathematical modeling, and non-linearfunctions. The instructional sequence of each unitaddresses mathematical concepts and skills andmathematical problem solving in a context thatprovides both the rationale for the skills being acquiredand a means of integrating newly acquired knowledgewithin a coherent structure.

An IMP ClassroomClass size averaged around 32 students. Classroom

activities were typified by group work, writing, andoral presentations. Graphing calculators were availableat all times. The characteristics of IMP and Algebraclassrooms, as perceived by the pupils, weredocumented in the course of this study, and aredetailed in the results presented later in this paper.

Assessment PracticesPriority was given in IMP classrooms to a diversity

of assessment strategies, consistent with the programgoals. For example, in one IMP class, grades werecalculated from student performance on homework(30%), classwork and class participation (30%),problems of the week (30%), and unit assessments(10%). It appeared that most assessing of Algebrastudents was through weekly quizzes and chapter tests.

MethodSubjects

The subjects of this study were 182 students atthree Californian high schools participating in the IMPprogram outlined above. In addition, matching datawere collected on 74 Algebra 2 students and 143Algebra 4 students from the same schools. Data on anadditional 52 Algebra 2 students were collected from afourth high school to provide a comparable sample ofstudents at the same level as the IMP pupils.

Procedures and MeasuresDuring June, towards the end of the academic year,

all students completed two questionnaires. The studentquestionnaire was constructed in large part bycombining items developed and tested in a study ofstudent mathematics journal use and a further study ofstudent self-assessment.

The Mathematics Belief questionnaire examinedstudent perceptions of their mathematical competence,and student beliefs about mathematical activity and theorigins of mathematical ideas. Students were asked to

report their perceptions of those valued activities,which, in their opinion, assisted their learning ofmathematics, in addition to their perceptions ofwhat constituted typical classroom activities inmathematics and their attitudes towardsmathematics. The Mathematics World questionnairerequired students to identify the extent to whichspecific everyday activities were mathematical. Atthe time of administration of the questionnaires,IMP students had completed almost one year in theprogram.

In addition, the next fall, the MathematicsScholastic Aptitude Test (SAT) was administered tothe school populations, facilitating comparison ofthe mathematics performance of IMP students withtheir peers in conventional classes.

Mathematics belief. The mathematics beliefquestionnaire was adapted from an instrumentemployed to measure the student belief outcomes ofan innovative program employing student journals(Clarke, Stephens & Waywood, 1992; Clarke,Waywood, & Stephens, 1994). Every item wasvalidated through interviews with students. Minorchanges in phrasing were made for administration inAmerican schools. Some sample items were:1. If I had to give myself a score out of 10 to

show, honestly, how good I think I am at math,the score I would give myself would be…

3. The ideas of mathematics:

A. Have always been true and will always betrue.

Agree Disagree

D. Developed as people needed them ineveryday life.

Agree Disagree

F. Are most clearly explained using numbers.Agree Disagree

5. When I am doing mathematics at school, I amlikely to be:

A. TalkingAlways Often Sometimes Seldom Never

C. Writing wordsAlways Often Sometimes Seldom Never

F. Working with a friendAlways Often Sometimes Seldom Never

I. Listening to other studentsAlways Often Sometimes Seldom Never

K. Working from a textbookAlways Often Sometimes Seldom Never


7. An adaptation of the IMPACT instrument (Clarke,1987) was included as item 7, including such sub-items as:

Write down one new problem that you can now do.How could math classes be improved?

Student attitudes towards mathematics classes weremeasured explicitly through the sub-item:

How do you feel in math classes at the moment?(circle the words which apply to you.)

A. Interested B. Relaxed C. WorriedD. Successful E. Confused F. CleverG. Happy H. Bored I. RushedJ. (Write one word of your own) _____________

The response alternatives provided in this sub-itemarose from extensive interviewing of high schoolstudents in the course of a study of studentmathematical behavior at the point of transition fromprimary school (elementary school) mathematics tohigh school mathematics (Clarke, 1985, 1992). TheIMPACT instrument, from which the sub-item wasdrawn, was extensively field-tested with 753 grade 7students over a period of one year (Clarke, 1987).

Mathematics world. The mathematics worldquestionnaire was adapted for American administrationfrom an instrument employed in a study of communityperceptions of mathematical activity (Clarke &Wallbridge, 1989; Wallbridge, 1992). In thisquestionnaire, students were asked to indicate whetherthey thought specific everyday activities were highlymathematical, quite mathematical, slightlymathematical, barely mathematical, or notmathematical. The activities listed included:4. Cooking a meal using a recipe7. Playing a musical instrument9. Buying clothing at a sale

A complete listing of all questionnaire items isavailable in Clarke, Wallbridge, and Fraser (1992).

ResultsThe results that follow make reference to three

groups of students to whom questionnaires wereadministered:i. 180 IMP students – mean age 15.3 yearsii. 126 Algebra 2 students – mean age 15.4 yearsiii. 137 Algebra 4 students – mean age 16.9 years

Comparing the Algebra 2 and Algebra 4 SamplesIn all, 48 student measures were generated through

the two questionnaires. Algebra 2 and Algebra 4samples (n = 126 and n = 137, respectively) did not

differ significantly on any of the 48 measuresexcept the use of worksheets, for which the Algebra4 students recorded an even lower incidence thandid the Algebra 2 students, and the importanceaccorded to the teacher's explanations: Algebra 4students attached lower importance to these than didthe Algebra 2 students. It seems reasonable tosummarize these findings by observing that, withrespect to the beliefs documented here,conventionally-taught students adhere to a specificset of beliefs with a high level of stability over time.These beliefs and the associated perceptions ofclassroom practice were sufficiently distinct fromthose held by IMP students to clarify thecharacteristics of both class types. Results are givenas comparisons between IMP and Algebra 2students since these represent the most similarsample populations.

In each table where comparisons are madebetween groups the corresponding p value is given.Differences between groups which achievedstatistical significance are asterisked.

Student Mathematics Achievement on ConventionalTests

Where comparison was possible between IMPand Algebra students at the same school, mean SATscores for IMP classes were higher than mean SATscores for traditional Algebra/Geometry classes.Pair-wise comparison of group means (t test) wasused to identify any statistically significantdifference under a conventional null hypothesisassumption. At one high school the difference inperformance was statistically significant. Theseresults are documented in Table 1.Table 1SAT Scores for Algebra and IMP Students on Two SchoolSites

School Classtype

Mean SATscore SD p value

Algebra(n = 83) 420.48 82.96

AIMP

(n = 74) 443.37 77.21.0372*

Algebra(n = 86) 367.56 57.02

BIMP

(n = 67) 373.88 60.95.1003


Student Perceptions of Their Mathematics CompetenceIMP students were significantly more likely to rate

themselves highly on how good they were atmathematics than were Algebra 2 students (Table 2).Sample Item:

If I had to give myself a score out of 10 to show,honestly, how good I think I am at math, the score Iwould give myself would be:

Two comments should be made concerning thishigher self-rating by IMP students. First, SAT scoresindicated that where comparison was possible IMPstudents tended to be more capable at conventionalmathematics tasks than were their peers in Algebraclasses, which suggests that these self-ratings had somebasis in fact. Second, the difference in self-ratings canalso be interpreted as a difference in confidence. Wewould suggest that heightened self-confidence inmathematics is likely to lead to increased participationin further mathematics, and a greater likelihood that thestudent will make use of the mathematical skillsacquired. Both are desirable outcomes.Table 2Self-rating Scores for IMP and Algebra 2 Students

Class type Mean SD p value

Algebra 2(n = 125) 6.86 1.2

IMP(n = 173) 7.5 1.38

.0012*

Student Attitude Toward Mathematics ClassesIMP students were significantly more likely to feel

positive about mathematics classes (Table 3).Sample item:

How do you feel in math classes at the moment? (circlethe words which apply to you.)

A. Interested B. Relaxed C. WorriedD. Successful E. Confused F. CleverG. Happy H. Bored I. RushedJ. (Write one word of your own) _____________

A student attitude index was calculated by scoring eachpositive response +1 and each negative response –1,and summing for each student.

Table 3Student Attitude Index for IMP and Algebra 2 Students

Class type Mean SD p value

Algebra 2(n = 126) –.52 1.85

IMP(n = 174) .97 2.14

.0001*

Student Perceptions of Mathematical ActivityThe distinguishing characteristic between the

problem solving students and the Algebra 2 studentswas the degree to which they perceivedmathematics to be a mental activity (Table 4).Sample Item:

Mathematics is something I do (circle one ormore):

A. Every day as a natural part of livingB. Mostly at schoolC. With a pencil and paperD. Mostly in my headE. With numbers

Table 4Student Perceptions of Mathematical Activity

Responsealternatives Class type Proportion

(%)Every day as a naturalpart of living

Algebra 2IMP

4952

Mostly at school Algebra 2IMP

6364

With a pencil and paper Algebra 2IMP

4142

Mostly in my head Algebra 2IMP

2739

With numbers Algebra 2IMP

5146

Table 4 is significant in the context of this paperin that it was only in these perceptions ofmathematical activity that the IMP and Algebrastudents responded in a similar fashion. The markeddifferences in beliefs and perceptions reported bythe two groups, which constitute the essentialfindings of this study, are only evident in Table 4 inthe significantly greater inclination for IMP studentsto report mathematics as being a mental activity.

Student Perceptions of Mathematical IdeasIMP students were more likely to agree that

mathematical ideas could be clearly explained using


every day words that anyone could understand, thanwere Algebra 2 students. IMP students were also lesslikely to view the ideas of mathematics as ones that canonly be explained using numbers and language specificto mathematics. The IMP students were more likely toview mathematics as having developed in response topeople’s needs. The IMP students were also less likelythan the Algebra 2 students to view mathematics ashaving been invented by mathematicians or to hold thatthe ideas of mathematics have always and will alwaysbe true. Figure 1 and Table 5 document thesedifferences.Sample Item:

The ideas of mathematics

A. Have always been true and will always be true.Agree Disagree

B. Were invented by mathematicians.Agree Disagree

C. Were discovered by mathematicians.Agree Disagree

D. Were developed as people needed them in dailylife.

Agree Disagree

E. Have very little to do with the real world.Agree Disagree

F. Are most clearly explained using numbers.Agree Disagree

G. Can only be explained using mathematicallanguage and special terms.

Agree Disagree

H. Can be explained in everyday words thatanyone can understand.

Agree Disagree

In summary: IMP students were more likely tohold a socially-oriented view of the origins andcharacter of mathematical ideas rather than a Platonistbelief in the existence of mathematical absolutesawaiting discovery.

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

Figure 1. Students’ perceptions of the ideas ofmathematics.

Table 5Students’ Perceptions of the Ideas of Mathematics

Sub-items ClassType Mean

Have always been true and willalways be true.

Algebra 2IMP

.02–.28

Were invented by mathematicians. Algebra 2IMP

–.13–.36

Were discovered bymathematicians.

Algebra 2IMP

.18

.01Developed as people needed themin daily life.

Algebra 2IMP

.57

.77Have very little to do with the realworld.

Algebra 2IMP

–.72–.82

Are most clearly explained usingnumbers.

Algebra 2IMP

.26–.02

Can only be explained usingmathematical language and specialterms.

Algebra 2IMP

–.41–.69

Can be explained in everydaywords that anyone can understand.

Algebra 2IMP

.21

.63

Always Discovered Unreal Special True Terms Invented Developed Numbers Everyday

Words

Algebra students = light bars, IMP students = dark bars;a positive mean value indicates agreement;

a negative mean value indicates disagreement


Student Perceptions of School MathematicsThe IMP students were significantly more likely to

agree that writing was important in helping them tounderstand mathematics. The IMP students were alsomore likely to see value in talking to other studentsthan were the Algebra 2 students. The IMP studentswere significantly less likely than the Algebra 2students to view drill and practice as the best way tolearn mathematics.Sample Item:

Circle the alternative which best describes how trueyou think each statement is (SA = Strongly Agree,A = Agree, D = Disagree, and SD = StronglyDisagree):

1. Explaining ideas clearly is an important part ofmathematics. SA A D SD

2. Mathematics does not require a person to usevery many words.

SA A D SD

3. Writing is an important way for me to sort outmy ideas in mathematics.

SA A D SD

4. Talking to other students about the mathematicswe are doing helps me to understand.

SA A D SD

5. Drill and practice is the best way to learnmathematics.

SA A D SD

The distinguishing characteristics between theIMP students and the Algebra 2 students were:

• the importance attached by IMP students to writingin mathematics (p = .04*)

• the degree to which IMP students perceived talkingto other students as useful in helping them tounderstand mathematics (p = .0005*)

• the relative importance attached to drill andpractice by the Algebra 2 students (p = .0001*)

Student Perceptions of Mathematical Activity at SchoolThe greatest degree of difference between IMP

students and the Algebra 2 students was evident intheir perceptions of mathematical activity at school.Table 6 illustrates the differences in studentperceptions of their mathematics classrooms. In thesestatistics, the differences between the two class typesare most clearly illustrated. Key differences betweenIMP and Algebra 2 classes can be summarized asfollows:

• IMP students were significantly more likely tobe writing words and drawing diagrams, andless likely to be writing numbers.

• IMP students were significantly more likely tobe working with a friend or with a group, andless likely to be working on their own.

• While there was no difference between IMP andAlgebra 2 classes in the relative frequency oflistening to the teacher, IMP students weresignificantly more likely to be listening to otherstudents than were students in Algebra 2classes.

• IMP students were significantly more likely tobe working from a worksheet and less likely tobe copying from the board or working from atextbook.Students were asked to respond on a four-point

scale to the cue “When doing mathematics atschool, I am likely to be...” The mean values inTable 6 should be read as students’ perceptions ofthe relative frequency (on a 5-point scale) withwhich they engaged in each of the listed activities.Table 6Mean Relative Frequency of Student Engagement

Sub-items Classtype Mean SD p

value

Talking Algebra 2IMP

2.252.59

1.0.88 .002*

Writing numbers Algebra 2IMP

3.132.66

.86

.83 .0001*

Writing words Algebra 2IMP

1.702.68

.93

.94 .0001*

Drawing diagrams Algebra 2IMP

1.852.70

.82

.83 .0001*

Working on myown

Algebra 2IMP

2.591.91

.90

.87 .0001*

Working with afriend

Algebra 2IMP

2.102.69

.90

.80 .0001*

Working with agroup

Algebra 2IMP

1.963.17

1.0.80 .0001*

Listening to theteacher

Algebra 2IMP

2.802.75

.98

.96 .63

Listening to otherstudents

Algebra 2IMP

2.192.79

.96

.83 .0001*

Copying from theboard

Algebra 2IMP

2.401.91

1.021.04 .0001*

Working from atextbook

Algebra 2IMP

3.150.25

.93

.70 .0001*

Working from aworksheet

Algebra 2IMP

2.053.32

1.06.97 .0001*


Student Perceptions of the Relative Importance ofCourse Components

IMP students placed more value on working withothers than d id Algebra 2 s tudents(p = .0001*). By contrast, Algebra 2 students valuedthe teacher’s explanations (p = .0005), and thetextbook (p = .0001) more than did IMP students.

Student Perceptions of Mathematics in EverydayActivity

IMP students were significantly more likely toidentify a mathematical component in everydayactivities than were Algebra 2 students. This result isevident in Table 7.Table 7Mean Math World Index for Algebra 2 and IMP Students(Incomplete responses from some students led to a slightlysmaller sample size for both groups.)

Class type Mean SD p valueAlgebra 2(n = 113) 19.292 5.591

IMP(n = 172) 21.477 5.598

.0014*

In particular, the IMP students were more likely toview as mathematical:• using a calculator to work out interest paid on a

housing loan over 20 years (p = .003*)• planning a family’s two week holiday (p = .006*)• chopping down a pine tree (p = .007*)• buying clothing at a sale (p = .03*)• painting the house (p = .0001*)Gender Differences

Comparison was made in this study of the attitudesto mathematics of boys and girls in IMP and Algebraclasses, and of the boys’ and girls’ self-ratings of theirmathematics competence. These results are shown inTable 8.

Girls in both class types were less likely than boysto rate highly their own mathematical competence.However this difference was only statisticallysignificant for students in Algebra classes. Both boysand girls in IMP classes had similar positive attitudestowards mathematics. In Algebra classes, both maleand female students felt negatively towardsmathematics, however boys’ attitudes were lessnegative than those of girls. On the basis of thesefindings, it appears that the IMP program was ofparticular value to female students. The statistical

significance of the direct comparison of Algebra 2girls with IMP girls is quite clear from Table 8,where the difference in mean attitude and self-ratingfor the two groups of girls is even more strikingthan in the comparison of the Algebra and IMPcohorts reported in Tables 2 and 3.Table 8Gender Comparison of Self-ratings and AttitudeMeasures for Algebra 2 and IMP Classes

Class type &Measure Gender Mean SD p value

Male(n = 58) 7.333 1.875Algebra 2

Self-rating Female(n = 67) 6.433 1.994

.0101*

Male(n = 77) 7.636 1.297

IMPSelf-rating Female

(n = 96) 7.271 1.410.081

Male(n = 58) –0.345 1.821

Algebra 2Attitude Female

(n = 68) –0.676 1.872.3176

Male(n = 78) 0.756 2.021

IMPAttitude Female

(n = 96) 1.146 2.234.2344

ConclusionsFor the purpose of drawing conclusions from

the findings reported here, the inclusion of theAlgebra 4 sample in the study encourages theextrapolation of conclusions from comparisons ofclass types at a specific grade level to more generalconclusions comparing problem-based andconventional instruction for high schoolmathematics classes.

The conclusions that follow, however, relatespecifically to the study sample.

The Students as Learners1. IMP students rated themselves as significantly

more mathematically able than did the Algebrastudents.

2. IMP students held a significantly more positiveattitude towards their mathematics classes thandid the Algebra students.

3. On school sites where comparison was possible,IMP students averaged higher SAT scores thandid pupils of conventional classes.


4. IMP classes appeared to have less negativeoutcomes for girls than did conventional Algebraclasses.

Student Perceptions of Mathematics5. IMP students were significantly more likely to

perceive mathematics as a mental activity.6. IMP students held beliefs consistent with a view of

mathematics as arising from individual and societalneed; while Algebra students were more likely toview mathematical ideas as having an independent,absolute and unvarying existence.

7. The IMP students were significantly more likely toperceive mathematics as having applications indaily use.

8. IMP students were significantly more likely thanAlgebra students to believe that mathematical ideascan be expressed “in everyday words that anyonecan understand.”

Instructional Alternatives

9. IMP students attached significantly more value tointeractive learning situations; whereas Algebrastudents valued “the teacher’s explanations” and“the textbook.”

10. IMP students valued writing and talking to otherstudents as assisting their learning. Algebrastudents were significantly more likely to value“drill and practice.”

11. (a) It is possible to identify a coherent andconsistent set of classroom practices which can beassociated with conventional instruction (cf.Clarke, 1984).

11 (b) It is similarly possible to identify a set ofclassroom practices which identify, in the students’view, the characteristics of the IMP classroom.

11 (c) The characteristics of these two instructionalmodels are sufficiently distinct to represent clearalternatives.

In conclusion, the classroom practices of the IMPprogram, as reported by the students, placed greateremphasis on a variety of modes of communication andon facilitating student-student interaction than was thecase with conventional instruction. By contrast,conventional instruction was perceived as solitary,text-driven, and typically expressed through specialterms and numbers.

To what aspect of the IMP experience might weattribute the student beliefs documented in this study?The small-group, interactive classroom and the

problem-based mathematics curriculum representtwo key characteristics of the InteractiveMathematics Project. Whether such belief systemswould arise in interactive classrooms lacking aproblem-based emphasis or in more conventionallytaught, problem-based classrooms is a matter forfurther research.

Certainly the IMP program has providedstudents with significantly different experiencesfrom those found in conventional mathematicsclasses, and these experiences appear to have led todemonstrably different beliefs about mathematicalactivity, mathematics learning, school mathematics,and the mathematics evident in everyday activity.The findings of other studies suggest that studentswhose instruction has included experience withopen-ended tasks can be expected to perform moresuccessfully on both conventional and non-routinetasks than students lacking that experience (forinstance, Sweller, Mawer & Ward, 1983). Incombination, this research suggests that a problem-based curriculum is capable of developingtraditional mathematical skills at least assuccessfully as conventional instruction, whilesimultaneously developing non-traditionalmathematical skills and engendering measurablydifferent belief systems in participating students.The nature of these different beliefs has formed thebasis of this study.

REFERENCES

Baird, J. R., & Mitchell, J. (Eds.). (1986). Improving the qualityof teaching and learning: An Australian case study – thePEEL project. Melbourne: Monash University Printery.

Bishop, A. J. (1985). The social construction of meaning – asignificant development for mathematics education. Forthe Learning of Mathematics, 5(1), 24–28.

California Post-Secondary Education Commission (CPEC).(1989). Request for program proposals under theEducation for Economic Security Act, Federal Public Law98-377 Fourth Year, San Francisco: California Post-Secondary Education Commission and California StateDepartment of Education.

Clarke, D. J. (1984). Secondary mathematics teaching: Towardsa critical appraisal of current practice. Vinculum, 21(4),16–21.

Clarke, D. J. (1985). The impact of secondary schooling andsecondary mathematics on student mathematicalbehaviour. Educational Studies in Mathematics, 16(3),231–251.

Clarke, D. J. (1986). Conceptions of mathematical competence.Research in Mathematics Education in Australia, 2,17–23.

Clarke, D. J. (1987). The interactive monitoring of children’slearning of mathematics. For the Learning ofMathematics, 7(1), 2–6.


Clarke, D. J. (1992). The role of assessment in determiningmathematical performance. In G. Leder (Ed.), Assessementand learning of mathematics (Chapter 7, pp. 145–168).Hawthorn: Australian Council for Educational Research.

Clarke, D. J., & McDonough, A. (1989). The problems of theproblem solving classroom. The Australian MathematicsTeacher, 45(2), 20–24.

Clarke, D. J., & Sullivan, P. (1990). Is a question the best answer?The Australian Mathematics Teacher, 46(3), 30–33.

Clarke, D. J., Stephens, W. M., & Waywood, A. (1992).Communication and the learning of mathematics. In T. A.Romberg (Ed.), Mathematics assessment and evaluation:Imperatives for mathematics educators (Chapter 10). Albany,New York: The State University of New York (SUNY) Press.

Clarke, D. J., & Wallbridge, M. (1989). How many mathematicsare there? In B. Doig (Ed.), Everyone counts. Parkville:Mathematical Association of Victoria.

Clarke, D. J., Wallbridge, M., & Fraser, S. (1992). The otherconsequences of a problem-based mathematics curriculum.Research Report No. 3. Mathematics Teaching and LearningCentre, Australian Catholic University (Victoria). Oakleigh:Mathematics Teaching and Learning Centre.

Clarke, D. J., Waywood, A., & Stephens, W. M. (1994). Probingthe structure of mathematical writing. Educational Studies inMathematics, 25(3), 235–250.

Cobb, P. (1986). Contexts, goals, beliefs, and learningmathematics. For the Learning of Mathematics, 6(2), 2–9.

Cobb, P., Wood, T., & Yackel, E. (1990). Classrooms as learningenvironments for teachers and researchers. In R. B. Davis, C.A. Maher, & N. Noddings (Eds.), Constructivist views on the

teaching and learning of mathematics (pp. 125–146).Journal for Research in Mathematics Education,Monograph No. 4. Reston, VA: National Council ofTeachers of Mathematics.

Erlwanger, S. H. (1975). Case studies of children’s conceptionsof mathematics – part 1. Journal of Children’sMathematical Behaviour, 1, 157–283.

Lave, J. (1988). Cognition in practice. Cambridge: CambridgeUniversity Press.

Lawson, M. J. (1990). The case for instruction in the use ofgeneral problem-solving strategies in mathematicsteaching: A comment on Owen and Sweller. Journal forResearch in Mathematics Education, 21(5), 403–410.

Meece, J. L., Parsons, J. E., Kaczala, C. M., Goff, S. B., &Futtennan, R. (1982). Sex differences in mathachievement: Toward a model of academic choice.Psychological Bulletin, 91(2), 324–348.

Owen, E., & Sweller, 1. (1989). Should problem-solving beused as a learning device in mathematics? Journal forResearch in Mathematics Education, 20, 322–328.

Schoenfeld, A. (1985). Mathematical problem-solving.Orlando, FL: Academic Press.

Sullivan, P., & Clarke, D. J. (1991). Catering to all abilitiesthrough the use of “good” questions. Arithmetic Teacher,39(2), 14–21.

Wallbridge, M. (1992). Community perceptions ofmathematical activity. Unpublished master’s thesis,Australian Catholic University (Victoria) – Christ Campus,Oakleigh, Victoria, Australia.


Bates & Wiest 17

Impact of Personalization of Mathematical Word Problems onStudent Performance

Eric T. Bates & Lynda R. Wiest

This research investigated the impact of personalizing mathematical word problems using individual studentinterests on student problem-solving performance. Ten word problems were selected randomly from amathematics textbook to create a series of two assessments. Both assessments contained problems exactly asthey appeared in the textbook and problems that were personalized using student interests based on student-completed interest inventories. Fourth-grade students’ scores on the non-personalized and personalized problemswere compared to investigate potential achievement differences. The scores were then disaggregated to examinethe impact of reading ability and problem type on the treatment outcomes. The results showed no significantincrease in student achievement when the personalization treatment was used regardless of student readingability or word problem type (t = –.10, p = .46)..

“Problem solving is the cornerstone of schoolmathematics. Unless students can solve problems, thefacts, concepts, and procedures they know are of littleuse” (National Council of Teachers of Mathematics,2000, p. 181). Students can learn mathematicalprocedures, but without real-world applications, theseskills are rendered meaningless and are forgottenreadily. In the school curriculum, word problems allowone means by which students can work towarddeveloping problem-solving skil ls withincontextualized settings that do not require applicationof rote procedures. However, research has shown thatstudents have difficulty solving word problems (Hart,1996).

At least three reasons have been proposed for whystudents have little success solving word problems:limited experience with word problems (Bailey, 2002),lack of motivation to solve word problems (Hart,1996), and irrelevance of word problems to students’lives (Ensign, 1997). These factors should be addressedin an effort to improve student performance on wordproblems, a fundamental component of mathematicseducation. Personalizing word problems—replacingselected information with students’ personalinformation—can address the latter two, motivationand relevance, which may in turn lead to the first,greater experience with word problems.

The purpose of this study was to investigate theimpact of personalizing word problems on fourth-gradestudents’ problem-solving performance. Results of thisresearch, conducted at Copper Flats ElementarySchool1 in Northern Nevada, were disaggregated toexamine how reading ability and problem type mightinfluence scores in solving personalized versus non-personalized problems.

Review of Related LiteratureThe Role of Word Problems

Conventional word problems, despite theirartificial nature, are likely to “stick around” in schoolmathematics (D. Brummett, personal communication,February 29, 1996; Sowder, 1995; J. Stephens,personal communication, February 29, 1996). Thismay be due to their strong grounding in tradition, theirpotential for fostering mathematical thinking, their easeof use (e.g., conciseness and practicality within theconfines of school walls), and a lack of abundant andpragmatic alternatives. Word problems may, in fact,serve several important functions in the mathematicsclassroom: They provide questions that challengestudents to apply mathematical thinking to varioussituations, and they may be an efficient means ofrelating this thinking to the real world. Practicallyspeaking, word problems are either readily available inmathematics texts or can be written in a short period oftime, which makes them useful to time-consciousteachers (Fairbairn, 1993).

Personalization and Student InterestThe idea of individualizing instruction certainly is

not new. Almost a quarter of a century ago, Horak(1981) stated, “Meeting the educational needs of the

Eric T. Bates is a fourth-grade teacher at SunnycrestElementary School in Lake Stevens, Washington. He holds aMaster’s degree in elementary education with a concentrationin technology in mathematics education.Lynda R. Wiest is an Associate Professor of Education at theUniversity of Nevada, Reno. Her professional interests includeK–8 mathematics education, educational equity, and teachereducation.

18 Personalization of Mathematical Word Problems

individual student has long been a concern ofprofessional educators” (p. 249). Personalizinginstruction to student experiences and interests is oneway to individualize instruction that may be importantfor mathematics learning (Ensign, 1997). In particular,it can enhance interest and motivation, which arecritically important factors in teaching and learning.

Mathematical word problems have been targetedfor personalization. Students “don’t care how manyapples Bob gave to Suzy. They’re much moreinterested in things like music, video games, movies,trading cards, money, and friends” (Bailey, 2002, p.61). Giordano (1990) adds, “student fascination withproblems can be enhanced when names, locations, andevents are changed to personal referents” (p. 25). It isimportant that word problems appeal to students inorder to generate interest in and motivation for solvinga problem (Fairbairn, 1993; Hart, 1996) However, inpractice, classroom mathematics rarely links tostudents’ life experiences (Ensign, 1997).

Research on Personalized Word ProblemsNumerous studies have investigated the impact of

personalizing problems—inserting individual students’names and/or information from their backgroundexperiences into the problems they solve—on studentinterest/motivation and problem-solving success.Personalized problems have been computer-generatedin some cases. Most of these studies found positiveeffects on the three major variables investigated—interest, understanding, and achievement (Anand &Ross, 1987; d’Ailly, Simpson, & MacKinnon, 1997;Davis-Dorsey, Ross, & Morrison, 1991; Hart, 1996;Ku & Sullivan, 2002; López & Sullivan, 1991, 1992;Ross & Anand, 1987; Ross, McCormick, & Krisak,1985; Ross, McCormick, Krisak, & Anand, 1985).

Several researchers and educators creditpersonalization of word problems with positivelyinfluencing student affect, such as interest andmotivation. Hart (1996) notes, “Most students areenergized by these problems and are motivated to workon them” (p. 505). Davis-Dorsey et al. (1991) saypersonalization fosters and maintains attentiveness toproblems, and Jones (1983) claims that personalizedproblems invest students in wanting to solve themcorrectly.

López and Sullivan’s (1992) research foundindividual personalization (tailoring problems toindividual rather than whole-class interests) to beparticularly effective in fostering positive attitudestoward word problems. However, Ku and Sullivan’s(2002) study involving 136 fourth-grade Taiwanese

students and their teachers also found grouppersonalization to have a positive impact. Bothstudents and teachers using personalized problemsshowed better attitudes toward the program than thoseusing non-personalized word problems. Ku andSullivan argue that familiarity (reduced cognitive load)and interest are the major factors that lead to greatersuccess solving personalized versus non-personalizedproblems.

Another major area where personalization of wordproblems has yielded favorable results is studentunderstanding. Davis-Dorsey et al. (1991) saypersonalization supports development of meaningfulmental representations of problems and theirconnections to existing schemata, and that it createsstrong encoding that aids retrieval of knowledge.Personalized word problems may be more meaningfulin general and make contexts more concrete and morefamiliar (López & Sullivan, 1992). Familiar people andsituations in personalized problems can aidunderstanding (Davis-Dorsey et al., 1991; López &Sullivan, 1992).

In their research, d’Ailly et al. (1997) employed atype of personalization known as self-referencing. Avariety of problems were taken from a standardmathematics text and some of the character nameswere replaced with the word you . One hundredstudents in grades three, four, and five were asked tosolve the problems within a mix of self-referencing andnon-self-referencing problems. The researchers found,“When a you word was involved in the problem,children asked for fewer repeats for the problems, andcould solve the problems in a shorter amount of timeand with a higher accuracy” (p. 566).

As noted, d’Ailly et al.’s (1997) study found thatpersonalized word problems (specifically, those usingself-referencing) positively impacted studentachievement—the third main area where word problempersonalization can benefit students. Numerous otherresearchers have attained similar results in this area,although some findings demonstrate positive effects insome cases but not others, as some of the followingstudies show.

For their study, Ku and Sullivan (2002)personalized problems using the most popularitems—as determined by a completed interestsurvey—for students as a whole. Students attainedhigher problem-solving scores on personalizedproblems both on the pretest and on the posttest (i.e.,before and after instruction). The 53-minute interiminstruction and review used either personalized or non-personalized problems. Students who worked with

Bates & Wiest 19

personalized problems performed better on bothpersonalized and non-personalized problems than thosewho received non-personalized instruction, suggestingthat transfer of learning had occurred from thepersonalized to non-personalized problems.

Davis-Dorsey et al. (1991) studied the effects ofpersonalizing standard textbook word problems on 68second-grade students and 59 fifth-grade students.Prior to the treatment, all of the students completed abiographical questionnaire that was later used todevelop the personalized problems. Personalizationproved to be highly beneficial to the fifth graders, but itdid not positively impact the second-grade students’test scores.

Wright and Wright (1986) researched the use ofpersonalized word problems with 99 fourth-gradestudents. They examined both the processes used tosolve the problems and the accuracy of the answers.Interestingly, the researchers found that a correctprocess was chosen more often when the problemswere personalized, but correct and incorrect answerswere given equally on personalized and non-personalized problems.

López and Sullivan (1992) found significantdifferences favoring personalization on problem-solving scores for two-step but not for one-stepproblems, although the seventh graders in their studyalso scored higher on the latter in comparison withnon-personalized problems. The researchers saypersonalization may be particularly important for moredemanding (e.g., unfamiliar or mathematicallycomplex) cognitive tasks. They found personalizationto be effective on a group basis—personalizingproblems using dominant interests of a group ofstudents—as well as on an individualbasis—personalizing problems for each student usingindividual interests—in relation to students’ problem-solving scores.

Most evidence indicates that personalizing wordproblems can be an effective technique in teaching andunderstanding mathematical word problems.Nevertheless, some research data suggest caution inassuming that personalization of word problemsalways yields positive results. As noted, López andSullivan (1992) found significant differences favoringpersonalization for two-step but not one-step problems,and the Wright and Wright (1986) study showed nosignificant improvement in student achievement onpersonalized word problems, even though studentsmore often chose appropriate solution strategies forpersonalized problems. Ross, Anand, and Morrison(1988) raise other issues for consideration. They

suggest that the effectiveness of the personalizedtreatment may wear off over time. The researchersexpress concern that the higher scores on personalizedtests could be due, in part, to the novelty of thepersonalization and that the novelty might dissipate ifthe treatment were used often. They also point out thatpreparing individualized materials could limit its use inthe classroom due to time constraints. Finally, in theirresearch with 11-year-olds, Renninger, Ewen, andLasher (2002) found that personalized contexts basedon individual interests can have a differential effect onstudents. For the most part, these contexts encouragestudents to connect with the meaning of problems. Thisleads some students to consider a task more carefullyto be sure they understand it. However, it leads othersto assume falsely that they have answered a problemcorrectly, which hinders a “healthy skepticism” thatencourages problem solvers to check their work aftercompleting problems.

More research is needed to address how problemtype interacts with word problem personalization,where personalization has its greatest impact—studentattitude, understanding, or achievement, grade levelsand types of students that are most responsive topersonalization, the long-term effects ofpersonalization, and the potentially differential impactof individual versus group personalization. Ku andSullivan (2002) also call for future research on tappingtechnology’s potential for creating personalizedproblems and on investigating the implications of usingpersonalized problems for assessment.

Research PurposeThe purpose of this exploratory study was to

investigate if the predominantly positive researchresults concerning personalization of mathematicalword problems would apply to elementary schoolstudents regardless of reading ability or word problemtype. The intent was to contribute to the body ofknowledge about the impact of personalizing wordproblems and to extend previous explorations byconsidering particular student subgroups and problemtypes (simple translation and process, discussed underInstrumentation). If the benefits of personalizationwere to outweigh the time constraints of planning andpreparing for this type of activity, the use ofpersonalized mathematical word problems could be aneffective tool for elementary teachers working withstudents who struggle to understand and solve wordproblems.


Research MethodSample

Participants in this study were fourth-gradestudents at Copper Flats Elementary School. CopperFlats is a small desert community in rural NorthernNevada. The school receives federal Title 1 money,reflecting the fact that Copper Flats Elementary servesstudents from a lower-income community. The schoolhouses four fourth-grade classrooms. Students whoreturned parental consent forms in all four of theseclassrooms were selected for the present study.

Ninety-seven parental consent forms weredistributed. Of those, 42 were returned in time for thestudy. Therefore, the sample included 42 students— 22boys and 20 girls. Students not participating in thestudy worked on classroom assignments given by theirregular teacher, while the participants completed theassessments. All participants in this project werepresent for the two data-collection sessions. By readingability, 20 participants ranked high, 8 ranked medium,and 14 ranked low.

Research DesignThis study was a quantitative analysis of the effects

of personalizing word problems on fourth-gradestudents’ achievement in solving the problems. In thefall of 2002, participants completed an interestinventory to provide individual information forpersonalizing the assessments. One week later,participants were administered an instrumentcontaining 10 word problems to solve. On thatassessment, 5 problems were personalized and 5 werenot. Two weeks later, participants were given a similar10-item instrument. On this second and finalassessment, parallel versions of the 5 problems that hadbeen non-personalized on the initial instrument werepersonalized, and vice versa. Therefore, allparticipants—across the two test administrations inwhich they took part—answered 20 problems, 10personalized problems and 10 similar problems thatwere not personalized. The two-week period betweenthe two tests provided necessary time to reduce threatsto validity due to repeated testing of participants onsimilar test items (Parsons & Brown, 2002).

During each test administration, each participantwas given an instrument and a blank sheet of paper onwhich to solve the problems. Participants were allowed15 minutes to complete each assessment. Allparticipants finished within the allotted time.

Teacher-reported scores on the DevelopmentalReading Assessment (DRA) (Beaver, 2001)

established participants’ reading level for the purposeof comparing achievement on personalized versus non-personalized problems in relation to reading ability.DRA levels 30 to 38 are considered to be third-gradereading ability, DRA level 40 is fourth-grade readingability, and DRA level 44 is fifth-grade reading ability.For this study, participants with DRA scores higherthan 40 were considered high readers, or above gradelevel. Participants with DRA scores at 40 wereconsidered m e d i u m readers, or at grade level.Participants with DRA scores below 40 wereconsidered low readers, or below grade level.

This research was not designed to include aqualitative component. However, student commentswere recorded as field notes on the few occasionswhere students made relevant, unsolicited remarks.

InstrumentationTen problems were randomly selected from

Mathematics: The Path to Math Success (Fennell et al.,1999), the third-grade mathematics text, for use indeveloping the assessment instruments (seeAppendices B and C). This text was chosen because itwas the text used for teaching third-grade mathematicsat Copper Flats Elementary School; therefore, theparticipants were familiar with the format of theproblems. The problems were selected by scanningevery third page of the text that contained wordproblems. Of the word problems selected from thosepages, five of each of the two problem types describedbelow were drawn from a basket and incorporated intothe assessments.

The problems selected for the assessments weredifferentiated by problem type. Five of the problemsselected for the assessments were simple translationproblems and five were process problems (L. R. Wiest,personal communication, August 27, 2002). Simpletranslation problems can be solved using a one-stepmathematical algorithm. An example of a simpletranslation problem is: “There are 7 seats in each of 6vans. How many seats are there in all?” (Fennell, et al.,1999, p. 360). Process problems typically are notsolved through direct application of an algorithm.Another strategy is generally sought and chosen, suchas working backward, drawing a picture or diagram, orusing guess-and-check. A sample process problem is:“Jen is older than Arnie. Paul is older than Jen. Who isthe oldest?” (p. 313).

An interest inventory (see Appendix A) wascreated to determine selected participant preferences.Inventory items included students’ name, favorite toy,favorite store, something to buy at that store, names of

Bates & Wiest 21

friends, something they like to make, name of a game,and favorite type of vehicle. Each inventory was usedto personalize the original textbook word problems.

Two assessments were developed from the wordproblems taken from the mathematics text. Items fromthe interest inventory replaced the original characters,objects, and situations in order to personalize theproblems for each individual student on five of the tenproblems on each of the two instruments. On oneassessment, the odd-numbered problems werepersonalized (see Appendix B). On the other, the even-numbered problems were personalized (see AppendixC). Participants randomly received one instrument onthe first administration and the other instrument on thesecond administration two weeks later. This method ofalternating personalized and non-personalized items onthe assessments was shown to be an effectivetechnique used in other research on this topic (Davis-Dorsey et al., 1991; d’Ailly, Simpson, & MacKinnon,1997).

Data AnalysisA paired-samples t test (available online at

http://faculty.vassar.edu/lowry/tu.html) was performedto compare the number of correct answers onpersonalized versus non-personalized problems. Meanscores and standard deviations were calculated andsignificance was tested at the .05 level using a one-tailed test. This analysis included 42 pairs of scores.

An additional paired-samples t test was performedto compare the number of correct responses onpersonalized versus non-personalized itemsdisaggregated by participants’ predetermined readinglevels. Again, mean scores and standard deviationswere calculated and significance was tested for at the.05 level using a one-tailed test. There were 20 pairs ofscores at the high level, 8 pairs of scores at the mediumlevel, and 14 pairs of scores at the low level.

Two final paired-samples t tests were performed tocompare the number of correct responses onpersonalized versus non-personalized test itemsdisaggregated by problem type. Each assessmentcontained five simple translation problems and fiveprocess problems. On the first of these two paired-samples t tests, correct responses on the personalizedsimple translation problems were compared to correctresponses on the non-personalized simple translationproblems. On the second of the two paired-samples ttests, correct responses on the personalized processproblems were compared to correct responses on thenon-personalized process problems. Through thesestatistical methods, mean scores and standard

deviations were calculated with significance tested atthe .05 level using a one-tailed test. Both of theseanalyses included 42 pairs of scores.

ResultsMean scores for the number of items answered

correctly out of ten showed a difference of .03 pointsbetween the personalized and non-personalizedproblems (see Table 1). This difference was notstatistically significant (t = –.10, p = .46).Table 1Paired-Samples t Test for Personalized and Non-Personalized Problems

Context n Mean SD t pPersonalized 42 5.26 2.07

Non-personalized 42 5.29 2.28 –.10 ,46

Table 2 provides mean scores for the number ofproblems answered correctly out of ten, separated bystudent reading level. The high-reader scores for non-personalized problems were .10 points higher than forpersonalized problems, a nonsignificant difference (t =–.26, p = .39). The medium-reader scores for non-personalized and personalized problems differed by .50points, also favoring non-personalized problems. Apaired-samples t-test indicated that this difference wasnot significant (t = –1.08, p = .15). The low-readerscores were .35 points higher for personalizedproblems than for non-personalized problems. Thiswas the only group who attained better scores onpersonalized problems, although the scores were notsignificantly higher (t = –.84, p = .20).Table 2Paired-Samples t Test for Personalized and Non-Personalized Scores by Reading Ability

Personalized Non-Personalized

ReadingGroup n Mean SD Mean SD t p

High 20 5.90 1.77 6.00 2.15 –.26 .39Medium 8 5.50 2.07 6.00 2.14 –1.08 .15

Low 14 4.21 2.19 3.85 1.96 –.84 .20

Mean scores for the number of problems answeredcorrectly out of 5 were separated by problem type (seeTable 3). Scores for simple translation problem meanswere 0.16 points higher for non-personalized than forpersonalized problems. A paired-samples t-test showedthat these differences were not significant (t = –.84, p =.20). Process problem means showed a difference of0.1 points between the personalized and non-


personalized problems, favoring the former. Again, thisdifference was not significant (t = .45, p = .32).Table 3Paired-Samples t Test for Personalized and Non-Personalized Scores by Problem Type

Personalized Non-Personalized

ProblemType n Mean SD Mean SD t p

SimpleTrans. 42 2.41 1.56 2.57 1.74 –.84 .20

Process 42 2.86 0.98 2.76 1.12 .45 .32

DiscussionThe results of this study suggest that students are

no more successful answering word problems when theword problems are personalized and reflect their areasof interest than when the problems are taken verbatimfrom a mathematics textbook. Only in the subgroup oflow-reading-level students and the subcategory ofprocess problems did the personalized problem scoresimprove slightly, although statistically significantdifferences were not found in either case. The meanscores in each other subgroup and subcategory weresomewhat lower on the personalized versions of theword problems than on the non-personalized versions.

These research results point to a differentconclusion than many previous studies on this topic.However, given the rather substantial amount ofprevious research weighted toward positive effects ofpersonalizing word problems and the reasons explainedbelow, it is still quite possible that personalized wordproblems can be a beneficial part of schoolmathematics programs. Several factors may havecaused the lack of positive findings in this study. First,the personalized problems may not have adequatelyaddressed the three aforementioned reasons studentsfail at mathematical word problems. Second, the age ofthe students may have been a factor in the treatment’slack of success. Third, this study looked only atcomparisons of personalized and non-personalizedproblems on assessments. No attempt was made tointroduce personalization as an instructional practice.

The three reasons offered earlier for why studentsfail at solving mathematical word problems werelimited student experience with word problems(Bailey, 2002), lack of motivation to solve the wordproblems (Hart, 1996), and irrelevance of wordproblems to students’ lives (Ensign, 1997). The formatof this study could not—and did not intend to—have

much impact on student experience with wordproblems. By simply taking two 10-problemassessments, student exposure to word problems wasnot greatly increased. Increased motivation wasnoticed, however, when students saw their names orfavorite things included in a problem. On severaloccasions while completing the assessments, studentsmade comments such as, “Hey, this has my name,” or“These problems are fun ones.” This acknowledgementand the smiles that followed were taken as signs ofincreased student motivation. It was anticipated that byutilizing student names and other referents to studentlives, relevance would be increased. This may havebeen the case to an extent, but just seeing their namesand favorite things may not have given the problemsenough personal context to encourage correct answers.In effect, the ability of this study to address the threemajor reasons students fail at solving word problemswas not substantial or sustained enough to helpdistinguish performance on the two problem contexts.Personalized problems per se might not beadvantageous unless they are an integral part of alarger instructional effort.

The young age of the students may also havecontributed to the results of the present study. Thesestudents fall at the lower end of the grade levelspreviously researched on this topic. Most studies thatfound positive results for personalized problems tookplace at upper elementary or middle grades (Anand &Ross, 1987; d’Ailly et al., 1997; Davis-Dorsey et al.,1991; Hart, 1996; Ku & Sullivan, 2002; López &Sullivan, 1991, 1992; Renninger et al., 2002; Ross &Anand, 1987; Wright & Wright, 1986). Only twostudies included younger grades—second andthird—among the older grades they investigated(Davis-Dorsey et al., 1991; d’Ailly, et al., 1997). Thepresent study dealt exclusively with fourth-gradestudents and found no relationship betweenpersonalization and student scores. Perhaps fourthgrade is somewhat early for the personalizationtreatment to be effective. Interest in problem contextsmay become more important across the many years inwhich students encounter school word problems. Inrelation to their study involving the impact of wordproblem context, Parker and Lepper (1992) state that itis “clear that the need for techniques to enhancestudent interest in traditional educational materials mayactually increase with age” (p. 632). Advancing gradelevels also deal with increasingly difficult mathematicsproblems, the complexity of which may allow for afactor such as personalization to influence studentperformance. As noted earlier, López and Sullivan

Bates & Wiest 23

(1992) found personalization to have a positive impacton two-step but not one-step problems, leading them toconclude that personalization may be particularlyimportant for more difficult problems. Use of third-grade problems in this study may also have reduced thecognitive demands of this research task, creating lesssensitivity to or discrimination among problemvariations.

Several previous studies found a significantincrease in correct answers on mathematical wordproblems when students were taught with thepersonalized format (e.g., Anand & Ross, 1987; López& Sullivan, 1992). After the instructional period,participants in these studies were assessed usingstandard word problems. The present study sought todiscover the effects of the personalized format onstudent achievement on the test items themselveswithout prior instruction using these types of problems.Perhaps these two approaches yield different results.Students may need time to adjust to the new problemcontexts.

One benefit that did appear in using this treatmentwas student excitement. Similar to the Ross,McCormick, and Krisak study (1985), manyparticipants were visibly and audibly excited todiscover the personalized problems. In informaldiscussions after the test administrations, participantsreported that they really liked reading about themselvesand their friends. They enjoyed seeing familiar storesand games they like to play in this testing situation.This affirms Hart’s (1996) reference to thepersonalized treatment that “students are energized bythese problems” (p. 505). It must be recalled, however,that interest in problems can be detrimental to somestudents, who may incorrectly assume that they haveattained correct answers (Renninger et al., 2002). Also,too much interest in a problem context can distractsome students, particularly girls (Boaler, 1994; Parker& Lepper, 1992). If these potential negative effectstook place in this study, they might have countered andthus masked potentially positive effects in the overallresults.

Limitations of the StudyThe two major limitations of this study were the

sample size and the somewhat simplistic nature of theresearch design. The sample size was reduced due tothe small number of parental consent forms that weresigned and returned so that students could participatein the study. Ninety-seven consent forms weredistributed, but only 42 (43%) were signed andreturned in time for the first test administration. (Time

constraints prohibited a second distribution of consentforms, which might have raised the return rate.) Thisgreatly reduced the sample size, thus limiting thepower of the data used to determine the effectivenessof the treatment.

This study was also limited by its lack ofcomplexity. Merely assessing student performancebased on two test administrations was restrictive. Itonly gave a look into the results of those two tests. Itwould be interesting to discover how students mightperform on word problems when they were taught withthe personalized format. Time and other constraints didnot allow for this additional research component.Analysis of solution strategies might have yieldedfurther information. It is also difficult to know whatlong-term impact the motivational aspects of theseproblems may have.

Implications for Further ResearchThis study, in conjunction with the professional

literature discussed earlier, yields at least three majorimplications for future research.

• The potential of personalization of word problemsas an instructional method should be studied.

• Alternative technologies should be explored todecrease the time-intensive nature of preparingindividualized word problems.

• Longitudinal research should be conducted on theimpact of personalizing mathematical wordproblems.As a teaching strategy, personalization of

mathematical word problems has been shown toincrease student achievement, particularly in the upperelementary and middle grades (e.g., Anand & Ross,1987; López & Sullivan, 1992). While this study didnot find such results for assessment problems, whichmay be due to the mitigating factors discussed earlier,it did find some anecdotal evidence that supportedother research findings of increased interest in theseproblems. This might be an important underpinning ofmathematics learning. Personalization as aninstructional strategy could be implemented at variousgrade levels and studied to assess its effectiveness forstudents of those ages. Rather than comparing testitems only, as the present study did, students could betaught with the personalization treatment and assessedon standard textbook word problems to determine thelevel of transferability of any possible positive effects.This instructional method may increase studentmotivation and interest when learning how to solveproblems in mathematics, thereby increasing their


comprehension of the material and increasing theirscores on textbook and other assessments.

In order to employ personalization as a teachingstrategy—based on the assumption that it may yieldpositive results in affect, understanding, and/orachievement—alternative methods of personalizingword problems would be needed to decrease theamount of time researchers and educators spendcreating personalized problems. One such methodmight utilize the Internet. The capability of the Internetto deliver individualized materials immediately andsimultaneously to a large population of studentsremains untapped. A researcher could develop a website that allows students to complete an interestinventory online and then submit the inventory to theserver. The server would then apply that information toan existing word problem template document, updatingthe characters and other referents to individualize theproblems for each student. This process would takeonly seconds and would eliminate researchers’ (andlater teachers’) time investment in personalizingindividual worksheets and tests. Students could theneither print the problems or complete them on thecomputer screen. The preparation time would begreatly reduced and the number of participants couldbe increased significantly. This technique would allowresearchers to create countless individualized wordproblems for student instruction, practice, andassessment. Such research should include attention towhat types of problems lend themselves well to thistype of problem generation. In the research reported inthis paper, for example, problems using names werethe easiest to personalize, with difficulty increasingwhere gender-specific pronouns were included. Theprocess problems seemed to require greater attentionthan the simple translation problems in preparingpersonalized problems, mirroring the greatermathematical complexity of the former compared withthe latter.

The present study and similar earlier studiesdiscussed here have been shorter than three months induration. Long-term effects of the personalizingstrategy have not been determined at any educationallevel. Researchers might look at the use ofpersonalized word problems in a classroom over thecourse of a school year and its relationship to wordproblem achievement on standardized tests.

Closing ThoughtsPersonalization of mathematical word problems

may not be an efficacious approach in fourth-gradeclassrooms due to the age of the students and the

simplistic nature of the word problems the students arerequired to complete. This should not, however,discount other research on the personalization ofeducational materials. Other researchers have shownpersonalization to be an effective method in teachingolder students to solve mathematical word problems.

Excitement and interest tend to be rare whenstudents are working on word problems. Fairbairn(1993) suggested that the terms story problems andword problems can invoke uncomfortable memoriesfor many people. This may be due to the fact that wordproblems can be boring and tedious to solve.Unfortunately, student motivation is difficult toquantify. In the present study, as well as in others, theexcitement level of individual students visibly andaudibly rose when personalized problems werepresented. At the very least, personalization could beused as an instructional strategy to break the monotonyof word problems containing unknown people, dealingwith unfamiliar situations, asking uninspiringquestions.

REFERENCES

Anand, P. D., & Ross, S. M. (1987). Using computer-assistedinstruction to personalize arithmetic materials for elementaryschool students. Journal of Educational Psychology, 79(1),72–78.

Bailey, T. (2002). Taking the problems out of word problems.Teaching PreK–8, 32(4), 60–61.

Beaver, J. (2001). Developmental Reading Assessment. UpperArlington, OH: Celebration Press.

Boaler, J. (1994). When do girls prefer football to fashion? Ananalysis of female underachievement in relation to ‘realistic’mathematics contexts. British Educational Research Journal,20, 551–564.

d’Ailly, H. H., & Simpson, J. (1997). Where should ‘you’ go in amath compare problem? Journal of Educational Psychology,89(3), 567–567.

Davis-Dorsey, J., Ross, S. M., & Morrison, G. R. (1991). The roleof rewording and context personalization in the solving ofmathematical word problems. Journal of EducationalPsychology, 83(1), 61–68.

Ensign, J. (1997, March). Linking life experiences to classroommath. Paper presented at the Annual Meeting of the AmericanEducational Research Association, Chicago, IL.

Fairbairn, D. M. (1993). Creating story problems. ArithmeticTeacher, 41(3), 140–142.

Fennell, F., Ferinni-Mundy, J., Ginsburg, H. P., Greenes, C.,Murphy, S., Tate, W., et al. (1999). Mathematics: The path tomath success. Parsippany, NJ: Silver Burdett Ginn.

Giordano, G. (1990). Strategies that help learning-disabled studentssolve verbal mathematical problems. Preventing SchoolFailure, 35(1), 24–28.

Hart, J. M. (1996). The effect of personalized word problems.Teaching Children Mathematics, 2(8), 504–505.

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Horak, V. M. (1981). A meta-analysis of research findings onindividualized instruction in mathematics. Journal ofEducational Research, 74(4), 249–253.

Jones, B. M. (1983). Put your students in the picture for betterproblem solving. Arithmetic Teacher, 30(8), 30–33.

Ku, H. Y., & Sullivan, H. J. (2002). Student performance andattitudes using personalized mathematics instruction.Educational Technology Research and Development, 50(1),21–34.

López, C. L., & Sullivan, H. J. (1991). Effects of personalized mathinstruction for Hispanic students. Contemporary EducationalPsychology, 16, 95–100.

López, C. L., & Sullivan, H. J. (1992). Effect of personalization ofinstructional context on the achievement and attitudes ofHispanic students. Educational Technology Research &Development, 40(4), 5–13.

National Council of Teachers of Mathematics. (2001). Principlesand standards for school mathematics. Reston, VA: Author.

Parker, L. E., & Lepper, M. R. (1992). Effects of fantasy contextson children’s learning and motivation: Making learning morefun. Journal of Personality and Social Psychology, 62,625–633.

Parsons, R. D., & Brown, K. S. (2002). Teacher as reflectivepractitioner and action researcher. Belmont, CA: Wadsworth.

Renninger, K. A., Ewen, L., & Lasher, A. K. (2002). Individualinterest as context in expository text and mathematical wordproblems. Learning and Instruction, 12, 467–491.

Ross, S. M., & Anand, P. G. (1987). A computer-based strategy forpersonalizing verbal problems in teaching mathematics.Educational Communication and Technology, 35(3), 151–162.

Ross, S. M., Anand, P. G., & Morrison, G. R. (1988). Personalizingmath problems: A modern technology approach to an old idea.Educational Technology, 28(5), 20–25.

Ross, S. M., McCormick, D., & Krisak, N. (1985). Adapting thethematic context of mathematical problems to students’interests: Individual versus group-based strategies, Journal ofEducational Research, 79(1), 245–252.

Ross, S. M., McCormick, D., Krisak, N., & Anand, P. (1985).Personalizing context in teaching mathematical concepts:Teacher-managed and computer-assisted models. EducationalCommunication and Technology, 33(3), 169–178.

Sowder, L. (1995). Addressing the story-problem problem. In J. T.Sowder & B. P. Schappelle (Eds.), Providing a foundation forteaching mathematics in the middle grades (pp. 121–142).Albany: NY, State University of New York Press.

Wright, J. P., & Wright, C. D. (1986). Personalized verbalproblems: An application of the language experienceapproach. Journal of Educational Research, 79(6), 358–362.

1 Pseudonym.

Appendix A: Interest InventoryFavorite Toy ____________________Name of a Store You Shop At ____________________Something You Would Like to Buy at That Store ___________Name Three Friends ________ ________ ________Name a School Supply ____________________Something You Like to Make ____________________A Game You Like to Play With One Partner ________________Name a Type of Vehicle ____________________

Appendix B: Sample Assessment(odd numbered problems personalized)

1. Four students are collecting empty soda cans. Josh has morethan Jon but fewer than Warren. Robby has the same numberas Josh. Who has the greatest number of cans so far?

2. Tom has a ball. He passes it to Wally, and Wally passes it toAnne. Anne passes it back to Tom. If they continue in thisorder, who will catch the ball on the 10th throw?

3. Suppose 30 bottles of glue are shared equally among 6 classes.How many bottles of glue would each class get?

4. It’s the grand opening of Futura Florists! Every day for 8 daysthey give away 50 roses. How many roses in all do they giveaway?

5. Josh read 67 pages of a book. Jon read 32 pages. How manymore pages did Josh read than Jon?

6. Jordan, Nina, Amy, and Gia are practicing for a dance. Theytake turns dancing in pairs. If each girl practices one dancewith each of the other girls, how many dances do they practicein all?

7. A toy maker can put together 1 Gameboy™ every 6 minutes.How many Gameboys™ can he put together in 60 minutes?

8. There are 7 seats in each of 6 vans. How many seats are therein all?

9. Josh is older than Jon. Warren is older than Josh. Who is theoldest of the three?

10. Paula made first-aid kits to sell at the fair. She made 1 kit onMonday, 2 kits on Tuesday, 3 kits on Wednesday, and so on,until Saturday. How many kits did Paula make on Saturday?


Appendix C: Sample Assessment(even numbered problems personalized)

1. Four students are collecting empty soda cans. Meg has morethan Jo but fewer than Sid. Bart has the same number as Meg.Who has the greatest number of cans so far?

2. Josh has a ball. Josh passes it to Jon, and Jon passes it toRobby. Robby passes it back to Josh. If they continue in thisorder, who will catch the ball on the 10th throw?

3. Suppose 30 musical instruments are shared equally among 6classes. How many instruments would each class get?

4. It’s the grand opening of Winco! Every day for 8 days theygive away 50 chocolates. How many chocolates in all do theygive away?

5. Wendy read 67 pages of a book. Ellie read 32 pages. Howmany more pages did Wendy read than Ellie?

6. Josh, Jon, Robby, and Warren are playing Battleship™. Theytake turns playing Battleship™ in pairs. If each kid plays onegame of Battleship™ with each of the other kids, how manygames do they play in all?

7. A toy maker can put together 1 toy robot every 6 minutes.How many toy robots can he put together in 60 minutes?

8. There are 7 seats in each of 6 Toyotas™. How many seats arethere in all?

9. Jen is older than Arnie. Paul is older than Jen. Who is theoldest of the three?

10. Josh made dented cars to sell at the fair. Josh made 1 onMonday, 2 on Tuesday, 3 on Wednesday, and so on, untilSaturday. How many did Josh make on Saturday?


Rueda & Sokolowski 27

Mathematics Placement Test: Helping Students SucceedNorma G. Rueda & Carole Sokolowski

A study was conducted at Merrimack College in Massachusetts to compare the grades of students who took therecommended course as determined by their mathematics placement exam score and those who did not followthis recommendation. The goal was to decide whether the mathematics placement exam used at MerrimackCollege was effective in placing students in the appropriate mathematics class. During five years, first-yearstudents who took a mathematics course in the fall semester were categorized into four groups: those who tookthe recommended course, those who took an easier course than recommended, those who took a course moredifficult than recommended, and those who did not take the placement test. Chi-square tests showed astatistically significant relationship between course grade (getting a C– or higher grade) and placement advice.The results indicate that students who take the recommended course or an easier one do much better than thosewho take a higher-level course or do not take the placement exam. With achievement in coursework as themeasure of success, we concluded that the placement test is an effective tool for making recommendations tostudents about which courses they should take.

There is a widespread recognition of the need forappropriate placement in the mathematics courses forundergraduate freshmen. Many colleges anduniversities around the nation have used theMathematical Association of America (MAA)Placement Test; others have designed their own examsor used a combination of placement exams and othermeasurements, such as ACT or SAT mathematicsscores and high school GPAs. Since the MAA hasdiscontinued its placement test program, theresponsibility has been put on individual institutions todevelop their own placement exam. The purpose of ourstudy was to determine the effectiveness of MerrimackCollege’s placement test by examining the connectionbetween students enrolling in recommended coursesand their success in those courses.

Literature ReviewIn this section, we investigate some of the specific

methods reported in the literature for placingundergraduates in their first mathematics course. Wepoint to some of the assumptions in these reports andsuggest some of the drawbacks to the method ofplacement.

Cederberg (1999) reported on the three placementtests administered at St. Olaf College in Minnesota.

She explained that the placement recommendationswere based on a large number of regression equationsthat required considerable expertise in developmentand periodic redefinition. The placement test alsorequired the coordination of numerous categories ofstudent data used in the equations. Approximately 85%of students who enrolled in a calculus course based onthe recommendations from the placement test at St.Olaf College received a grade of at least C–.

Cohen, Friedlander, Kelemen-Lohnas and Elmore(1989) recommended a placement procedure that wasless technically sophisticated than St. Olaf’s, but stillrequired considerable background data about students.They recommended multiple criteria methods, whichincluded a placement test customized to an institution’scurriculum. They started with sixty variables, andfound the eight best predictors: high school graduationstatus, number of hours employed, units planned, age,high school grade point average, mathematicsplacement test score, reading placement test score, andEnglish placement test score. Cohen et al.’s study wasbased on (a) thousands of surveys completed bystudents and faculty members at eight Californiacommunity colleges, (b) a comparison between studentscores on assessment tests and grades in differentcourses, and (c) the relationship between studentcharacteristics and their grades.

Krawczyk and Toubassi (1999) described a simplerplacement procedure used by the University ofArizona. The University of Arizona used twoplacement tests adapted from the 1993 CaliforniaMathematics Diagnostic Testing Project (seehttp://mdtp.ucsd.edu/). Students chose which test theyfelt was most appropriate for their ability and choice of

Norma G. Rueda is a professor in the Department ofMathematics at Merrimack College, North Andover, MA. Herresearch interests include mathematical programming andapplied statistics.Carole Sokolowski is an Assistant Professor in the Departmentof Mathematics at Merrimack College, North Andover, MA.Her research area is undergraduate mathematics education.

28 Mathematics Placement Test

major. One test covered intermediate algebra skillsand placed students in one of three levels of algebra ora liberal arts mathematics course. The second testcovered college algebra and trigonometry skills andplaced students in courses from finite mathematicsthrough Calculus I. In the fall of 1996, their dataindicated that approximately 17% of freshmen placedin College Algebra through Calculus I failed orwithdrew from their respective courses, compared witha 50% attrition rate in the early 1980’s before themandatory testing and placement. Apart from a test,they also considered other factors, such as high schoolGPA.

A number of studies have investigated the use ofstandardized tests, such as ACT and SAT. Bridgemanand Wendler (1989) found that the mathematics SATscore was a relatively poor predictor of gradescompared to placement exams. Their results werebased on grades from freshman mathematics courses atten colleges. Odell and Schumacher (1995) showedthat a placement test used in conjunction withmathematics SAT scores could be a better predictorthan SAT scores alone. Their conclusion was based ondata from a private business college in Rhode Island.Callahan (1993) studied the criteria followed at CotteyCollege in Missouri to place students in the appropriatelevel course, as well as their placement success rates.As with the studies mentioned above, Cottey Collegeused several variables to achieve their results – theMAA Placement Test, ACT and SAT mathematicsscores, and years of high school mathematics taken.Each of these studies assumed that the success rateswere based on students following the placementadvice. Mercer (1995), on the other hand, conducted astudy to compare pass rates in a college-levelmathematics class for mathematically unpreparedstudents who enrolled in developmental courses andthose who did not. The results of this study showed astatistically significant relationship between passingthe course and following placement advice.

BackgroundMerrimack College, located in North Andover,

Massachusetts, is a small four-year Catholic collegeoffering programs in the liberal arts, business, thesciences, and engineering. Among the college’sdistribution requirements, students must complete threemathematics or science courses, with no more than twocourses from the same department. Most of thestudents take at least one mathematics course. Mostliberal arts majors usually choose Basic Statistics,Finite Mathematics, or Discrete Mathematics to satisfy

the mathematics/science requirement. During datacollection for this study, business administrationmajors were required to take Applied College Algebra,Calculus for Business, and one other mathematicscourse. Students majoring in science or engineeringgenerally were required to take more mathematics; forexample, engineering students were required to takethree calculus courses and one course in differentialequations. They also took Precalculus if they did notplace out of this course on their placement exam.

Since some incoming freshmen are not prepared totake a college-level mathematics course, a non-creditdevelopmental mathematics course, Math I, has beenoffered at Merrimack College since the fall of 1994.Before 1994 we administered a mathematics diagnosticexam to incoming students, but were unable toaccommodate students who were not ready to take acollege-level mathematics course. Instead, theyenrolled in a mathematics course at a higher level thantheir exam score indicated they should. There was ahigh failure rate among these students.

Because students at Merrimack College often havedifficulty in other courses if they have not completedMath I, proper mathematics placement has becomeimportant to all of our departments. For example, theChemistry Department now requires the students whoplace into Math I to complete the course before theytake several of their chemistry courses. Krawczyk andToubassi (1999) have found similar results at theUniversity of Arizona in which all chemistry studentsand 90% of biology students—whose placement testscores indicated they should be placed in intermediatealgebra or lower—received grades below C– in theirchemistry or biology courses.

This interest in correct mathematics placementextends beyond the chemistry department as evidencedby the many questions from the business and liberalarts faculty, as well as science and engineering facultyat the meetings in preparation for orientation advising.A major reason for our concern about student successis that successful students are more likely to remain intheir studies. A high level of student retention is notonly academically and socially desirable at a school,but it makes sense economically.

We do not place students according to their priorhigh school GPA or whether they have taken calculus.We do not assume that these factors indicate whetheror not they need algebra. In fact, many students placedinto Math I have had four years of high schoolmathematics, including precalculus (and occasionallycalculus), but according to our placement test they donot appear to understand the basic concepts of algebra.


Mathematics Placement TestAll incoming freshmen at Merrimack College are

expected to take a mathematics placement testdeveloped by members of the MathematicsDepartment. There are two versions of the exam, onefor students who will major in Business Administrationor Liberal Arts, and one for students in Science andEngineering. The version for Business Administrationand Liberal Arts consists of two parts, elementary andintermediate algebra. The version for Science andEngineering students contains a third part that testsstudents’ understanding of functions and graphs.Students are instructed not to use calculators. From1994 until 1999 the placement exam was taken atMerrimack College in June during orientation or at thebeginning of the academic year. Since 2000 the examhas been mailed to students at home. A Scantron formwith the students’ answers is mailed back toMerrimack College and graded. Students are informedthat using outside resources or calculators may result intheir being placed in a course for which they areunprepared and may result in their failing orwithdrawing from the course. There is not a differencebetween these mail-in results and the previousmonitored exam results with respect to the percentagesof students who place into the various mathematicscourses. Thus, we believe that most students heed ourwarnings. The recommendations are available tostudents and advisors during June orientation. SeeAppendix A for some problems similar to those givenin the mathematics placement test.

Part I of the placement test consists of seventeenquestions on elementary algebra. If a student does notanswer at least fifty percent of these questionscorrectly, then that student must take Math I. Forstudents who score above fifty percent on Part I, liberalarts majors may take any mathematics elective course;business administration majors are placed into acollege algebra or a business calculus course,depending on their overall score; and science andengineering students may be placed into a collegealgebra, Precalculus or Calculus I course, depending ontheir total score. The specific recommendationsresulting from the mathematics placement exam are asfollows:Science and Engineering:

Score below 9 in Part I ⇒ Math IFor those who score above 8 in Part I:• Score 20 or lower in Parts I–III ⇒ Applied College

Algebra• Score between 21 and 34 in Parts I–III ⇒ Precalculus• Score of 35 or higher in Parts I–III ⇒ Calculus I

Business:Score below 9 in Part I ⇒ Math IFor those who score above 8 in Part I:• Score below 30 in Parts I–II ⇒ Applied College Algebra• Score of 30 or higher in Parts I–II ⇒ Calculus for

BusinessLiberal Arts:

Score below 9 in Part I ⇒ Math IAny other score ⇒ Enroll in a mathematics elective course

Data AnalysisWe have performed statistical analyses since 1997

to study whether there was a relationship between thescore on the placement test and how well first yearstudents did on the first mathematics course taken atMerrimack College. In order to determine thisrelationship, we first examined the correlation betweentotal score on the placement test and students’ grades.A preliminary study with n = 372 showed that thecorrelation between the grade earned in a mathematicscourse and the total score on the placement test was0.466. For the same study, the correlation betweengrades earned and SAT scores was 0.334. A multipleregression model to estimate the final grade based onthe SAT score and the placement exam score gave theequation: Final Grade = 0.80 + 0.00122(SAT) +0.0641(Placement Exam). In addition, a t test for eachof the variables in the model indicated that the SAThad a t value of 1.01 (p = 0.314) and the PlacementExam had a t-value of 6.58 (p < 0.0005). Based on the ttest values and the p-values, we removed the SATvariable from the model and concluded that theplacement exam was a better predictor of studentsuccess.

Although we first used t-tests to compare the totalscore on the placement test with SAT scores aspredictors of students’ final grades, we ultimatelydecided against them as a means to further examine theeffectiveness of our placement test as a placement toolfor two reasons. First, there was not a true linearrelationship between the total score on the placementtest and placement. It was more like a step-function,with a range of scores in each part of the test beingconsidered for placement. Second, there was arelatively weak relationship between total score on theplacement test and final grades because students areplaced into so many different levels of courses. Forexample, a student may have a very low placement testscore, be properly placed into an elementary algebraclass, and earn a high grade in that course. Thus, asimple correlation between total score on theplacement test and grades earned was consideredinappropriate to determine the effectiveness of the

30 Mathematics Placement Test

placement test, and therefore we decided to categorizethe data.

Each first-year student was categorized accordingto the level of mathematics course taken: (1) the coursewas easier than that recommended by the placementtest score; (2) the course was the recommended course;(3) the course was more difficult than thatrecommended (higher-level); or (4) the placement testwas not taken. Although the test was required, somestudents—usually transfer students—were allowed totake a mathematics course based on their mathematicsgrade(s) at a previous institution. This policy has notworked well and is being changed to require allincoming students to complete the placement test.Within these categories, students were countedaccording to whether they (a) did well (received agrade of C– or better) or (b) did poorly (received agrade below C–). Chi-square tests were performed todetermine whether there was a relationship between thelevel of the course taken and the grade received in theclass. Given the eight possible categories previouslydescribed in this study, the chi-square test indicatedwhether the percentage of students, say, who did wellin each category of course level taken was significantlydifferent from that in any of the other categories. Theuse of the chi-square test assumes that a randomsample, representative of the population, was taken. Inthis study, we used the entire population of freshmenstudents who took mathematics in their entering fallsemester at our college for the years from 1997 to 2001(n = 1710). The null hypothesis stated that there was noassociation between the two variables. The alternativehypothesis stated that the grade depended on whether

or not the student followed the placement test resultrecommendation.

ResultsWe wanted to know whether there was an

association between the level of the course taken andthe grade earned. Table 1 shows the number andpercentage of students who did well in the class (C– orhigher) and the number and percentage of students whodid poorly (D+ or lower, or withdrew from the class)from 1997 through 2001. It was generally accepted thatgrades of D and F were unsatisfactory, as evidenced bythe fact that almost all comparable studies used cut-offgrades of C or C–. Those that used the C cut-off oftenhad a minimum grade requirement of C for a student tomove on to a subsequent course. Our department doesnot have such a requirement, and thus the choice of C–for this study was somewhat arbitrary. We felt that ourprofessors might be less likely to slightly inflate thegrade of C– to C than would those at schools with theminimum requirement. As described above, studentswere classified according to the level of the coursetaken: easier than the one recommended, the onerecommended, or a higher-level course than the onerecommended. A fourth category was used in order toinclude students who did not take the placement exam,but took a mathematics course.

The data were analyzed using the chi-square test(see Table 2). We found that there was a relationshipbetween the two variables for first year students whotook a mathematics course in the fall of 1997 [χ2 (3, n= 369) = 24.66, p < 0.0005]. The same conclusion heldfor the data corresponding to the fall of 1998

Table 1Number and Percentage of Students Each Year Disaggregated by Grade and Course Category

1997 1998 1999 2000 2001Course < D+ > C– < D+ > C– < D+ > C– < D+ > C– < D+ > C–

6 23 2 17 1 12 5 21 2 8Easier

21% 79% 11% 89% 8% 92% 19% 81% 20% 80%

47 168 57 168 55 220 57 205 50 200Recom.

22% 78% 25% 75% 20% 80% 22% 78% 20% 80%

30 30 33 34 6 16 4 12 7 15Higher-Level 50% 50% 49% 51% 27% 73% 25% 75% 32% 68%

28 37 22 28 4 18 12 16 18 16No Exam

43% 57% 44% 56% 18% 82% 43% 57% 53% 47%

n 111 258 114 247 66 266 78 254 77 239


Table 2Contingency Table and Chi-Square Test (Expected counts are printed below observed counts; shaded cells indicateexpected counts less than 5.)

1997 1998 1999 2000 2001Course < D+ > C– < D+ > C– < D+ > C– < D+ > C– < D+ > C–

6 23 2 17 1 12 5 21 2 8Easier

8.72 20.28 6.00 13.00 2.58 10.42 6.11 19.89 2.44 7.56

47 168 57 168 55 220 57 205 50 200Recom.

64.67 150.33 71.05 153.95 54.67 220.33 61.55 200.45 60.92 189.08

30 30 33 34 6 16 4 12 7 15Higher-Level 18.05 41.95 21.16 45.84 4.37 17.63 3.76 12.24 5.36 16.64

28 37 22 28 4 18 12 16 18 16No Exam

19.55 45.45 15.79 34.21 4.37 17.63 6.58 21.42 8.28 25.72

n 111 258 114 247 66 266 78 254 77 239

Chi-Squaretest 24.66 21.22 N/A 6.56 18.42

p value <0.0005 <0.0005 N/A 0.087 <0.0005

[χ2 (3, n = 361) = 21.22, p < 0.0005]. We did notperform a chi-square test for the data corresponding to1999 because there were 3 cells with expected countsless than 5 (see shaded cells in Table 2) and Moore(2001) does not recommend the use of the chi-squaretest when more than 20% of the expected counts areless than 5. The relationship was not significant for thefall of 2000 at an alpha level of 0.05 [χ2 (3, n = 332) =6.56, p = 0.087] when taking p < 0.05 to be statisticallysignificant. The relationship was again significant forthe data corresponding to the fall of 2001 [χ2 (3, n =316) = 18.42, p < 0.0005]. In sum, the percentage ofstudents who did well in their first undergraduatemathematics course was higher for those students whofollowed the advice or took an easier course than theone recommended based on their placement test score.

The chi-square test only showed evidence of someassociation between the variables. We then looked atthe tables to determine the nature of the relationship orassociation (Moore 2001). Table 3 shows the numberof students who took the recommended course, aneasier one, a higher-level course, as well as the numberof students who did not take the placement exam, andthe mean and median grades on a 4.0 scale for eachgroup from 1997 to 2001. The followingcorrespondence between letter-grades and number-grades was used at Merrimack:

A 4.0 B 3.0 C 2.0 D 1.0A– 3.7 B– 2.7 C– 1.7 D– 0.7B+ 3.3 C+ 2.3 D+ 1.3 F 0.0

In addition, students who withdrew from a course werecounted and were assigned a number grade of 0 for thisstudy.

Discussion and ConclusionFrom Table 1 we saw that students who took therecommended course or an easier one did much betterthan those who took a higher-level course or did nottake the placement exam. The same conclusion can bedrawn from Table 3, with the exception of the year2000. In 2000 there was no significant differenceamong the average grades received by those studentswho took the recommended course and those who tooka higher-level course. An explanation for this may bethat the proportion of students who took a higher-levelcourse dropped from 19% in 1998 to 7% in 1999, andto 5% in 2000. The few students who took a higher-level course seemed to know that they would be able tosucceed. In addition, the percentage of students placedinto our developmental course, Math I, has beendecreasing. Twenty five percent of the 521 who tookthe mathematics placement exam in 1998 wererecommended to take Math I. Twenty percent of the532 who took the mathematics placement exam in1999 were in that category. Those figures went downto 14% out of 525 students in 2000 and 16% out of 517in 2001. One of the reasons for this decrease was thatstudents were allowed to retake the placement examand to place out of our developmental mathematicscourse. Even though that possibility was available tostudents before, more effort has been made in the last

32 Mathematics Placement

Table 3Mean and Median Grade (on a 4.0 Scale) by Year and Course Category

1997 1998 1999 2000 2001

n n n n n

Course (%)

Mea

n

Med

ian

(%)

Mea

n

Med

ian

(%)

Mea

n

Med

ian

(%)

Mea

n

Med

ian

(%)

Mea

n

Med

ian

29 2.8 3.0 19 2.7 2.7 13 3.1 3.3 26 2.4 2.5 10 2.4 2.7Easier

8% 5% 4% 8% 3%

215 2.4 2.7 225 2.4 3.0 275 2.5 2.7 262 2.5 2.7 250 2.6 3.0Recom.

58% 62% 83% 79% 79%

60 1.7 1.5 67 1.6 1.7 22 2.2 2.0 16 2.4 2.5 22 1.9 1.7Higher-

Level 16% 19% 7% 5% 7%

65 1.8 2.0 50 1.7 2.0 22 2.4 2.2 28 1.8 2.2 34 1.6 1.2No Exam

18% 14% 7% 8% 11%

n 369 361 332 332 316

few years to avoid improperly placing students in anon-credit course (Math I).

With the exception of 1999, there is no significantdifference, using z tests, between the proportions ofstudents who did well or poorly among those studentswho did not take the mathematics placement exam.

It is not surprising that students who enrolled in therecommended course or an easier course performedbetter than did students who enrolled in a higher-levelcourse than the one recommended. What is importanthere is that approximately 80% of these students whotook the recommended or easier course succeeded witha grade of C– or higher.

We have found the mathematics placement examto be a useful tool to place students in the appropriatemathematics course, and we have been successful inconvincing most of our students to follow our advicewith respect to which courses to take. A challenge forus has been persuading students to take Math I, thenon-credit class, when they are not ready for a collegelevel course. While there is no perfect placementmethod, we have found that our test is better than SATscores in placing students into the appropriate course.In addition, our multiple-choice test is easier toadminister than methods used at other schoolsmentioned in this paper.

From our experience, a well-designed in-houseplacement test geared towards our curriculum is asimple and powerful tool for placing incoming studentsin an appropriate mathematics course. Keeping

adequate records and analyzing them with regard to theplacement test’s effectiveness, as we have done in thisstudy, is a key component in maintaining the validityand reliability of the test itself. A number of years ago,the placement test score was viewed as the basis for a“recommended” mathematics course for each student,to be followed or not, as the student chose. Today, theentire Merrimack community appears to view the testscore with increased respect because of the resultspresented in this paper. These ongoing statisticalvalidations of the connections between properplacement and successful achievement have served tolegitimize the placement test as part of a larger effort toincrease retention on our campus.

REFERENCES

Bridgeman, B., & Wendler, C. (1989). Prediction of grades incollege mathematics courses as a component of the placementvalidity of SAT-mathematics scores. (College Board ReportNo. 89-9). New York, NY: College Entrance ExaminationBoard.

Callahan, S. (1993, November). Mathematics placement at CotteyCollege. Paper presented at the Annual Conference of theAmerican Mathematical Association of Two-Year Colleges,Boston. (ERIC Document Reproduction Service No. ED373813)

Cederberg, J. N. (1999). Administering a placement test: St. OlafCollege. In B. Gold, S. Keith, & W. Marion (Eds.),Assessment practices in undergraduate mathematics (pp.178−180). Washington, DC: Mathematics Association ofAmerica.


Cohen, E., Friedlander, J., Kelemen-Lohnas, E., & Elmore, R.(1989). Approaches to predicting student success: Findingsand recommendations from a study of California CommunityColleges. Santa Barbara, CA: Chancellor’s Office of theCalifornia Community Colleges. (ERIC DocumentReproduction Service No. ED 310808)

Krawczyk, D., & Toubassi, E. (1999). A mathematics placementand advising program. In B. Gold, S. Keith, & W. Marion(Eds.), Assessment practices in undergraduate mathematics(pp. 181−183). Washington, DC: Mathematics Association ofAmerica.

Mercer, B. (1995). A comparison of students who followedmathematics advisement recommendations and students whodid not at Rochester Community College. Practicum preparedfor Nova Southeastern University, Ft. Lauderdale, FL. (ERICDocument Reproduction Service No. ED 400014)

Moore, D. (2001). Statistics: Concepts and controversies (5th ed.).New York: W. H. Freeman and Company.

Odell, P., & Schumacher, P. (1995). Placement in first-year collegemathematics courses using scores on SAT-math and a test ofalgebra skills. PRIMUS, 5, 61−72.

Appendix A: Sample ProblemsThe following problems are similar to the ones given on the actual exam, but the format is different. These sample problems are free response.The actual exam has a multiple choice format, in which several answers are provided to each problem, and only one of them is correct.

1. Without a calculator, evaluate:

a. |5 – 9| b.

€

−83 c. 02.

42.3 d. 12

5

9

8−

2. Simplify the following:

a. x

xx

3

312 2 + b. 3

3

)2(

2

x

x

−

− c. 23

2/184

)3(

)25(

x

yx − d. 3

4

2 −+

+ aa

a

e.

€

5x +18 − 4(x + 7) f. xxx log)2log()1log( 2 ++−−

3. Solve the following for x:

a. 5=− bax b. | 12 +x | = 5 c. 152 += xx

d. 644

1=

x e.

74

52

=+

=+

yx

yx f. 62 <+ xx

4. Solve, then simplify the radical: xx 422 −=−

5. Find an equation for the line through the points (–1, –2) and(1, 4). Give the slope, m, and the y-intercept, b.

The following problems are representative of the additional sectionof the Placement Test for Science and Engineering majors.

6. Let

€

f (x) =1x −1

. Find the domain of f.

7. Find the zeros of the function .1

32)(

+

−=x

xxf

8. Find the inverse function, ),(1 xf − if .2)( 3 += xxf

9. Which of the following points is not on the graph of 12 −= xey ?

(0, 1−e ), (1, 0), (2, 3e ), (3, 8e )

10. Convert 135° to radians.

11. Simplify in terms of =− θθ 2cos1:sin ?

12. Which of the following is greatest? sin °30 , sin °45 , sin °90 ,sin °180


34 Democratic Competence

In Focus…Can the Ideal of the Development of Democratic Competence

Be Realized Within Realistic Mathematics Education? The Caseof South Africa

Cyril Julie

As is the case in any country there is a constantsearch to improve the mathematics offerings presentedto school-goers in South Africa. The activitysurrounding this search was intensified after theattainment of democracy. The primary aim of thissearch was to establish a mathematics curriculum thatwould result in productive learning and the mastery ofthe goals set for the curriculum. These goals arepredetermined and are embedded within the country’sideological intent of its school educational endeavors.Explicitly it is stated that school education should be amechanism to contribute towards the development of“a participating citizen in a developing democracy[who has] a critical stance with regard to mathematicalarguments presented in the media and other platforms”(South African Department of Education, 2003, p. 9).This goal was already proffered during the struggle forliberation against apartheid and encapsulated in thealternative school mathematics program during thelatter periods of that struggle. The alternativemathematics curricular movement found expression inPeople’s Mathematics (PM). People’s Mathematicswas an independent development in South Africaduring a particular historical moment but sharedcommonalities with the varieties of CriticalMathematics Education (Skovsmose, 1994;Frankenstein, 1989). It differed from other varieties inthat it adopted the stance of critique but alsoemphasized action against those practices which inhibithuman possibility.

The broad umbrella goals of People’s Mathematicswere “political, intellectual and mathematical

empowerment” (Julie, 1993, p. 31). It is with thesegoals in mind that Realistic Mathematics Education(RME) was found a viable approach to schoolmathematics with which the People’s Mathematicsmovement could “live.” The particular characteristicsof RME that PM found useful were:(a) It has a Lakatosian research program nature

(Gravemeijer, 1988). Being such a researchprogram there was some certainty of sustainabilitydue to Lakatos’s notion of strong researchprograms fulfilling their predictions.

(b) The retention of the integrity of mathematicsthrough RME’s vertical and horizontalmathematization (Streefland, 1990).

(c) The centrality of applications and modeling inRME (De Lange, 1987).

(d) The seamless integration of the history ofmathematics and educational contexts from extra-mathematical domains (De Lange, 1987).

(e) Mathematics curriculum development that iscontinuous and not of a once-off tinkering nature.

(f) A curriculum development research methodologythat is classroom-based and action-oriented(Gravemeijer, 1994) with an accompanyingreporting strategy of research findings that isunderstandable to practitioners.

In terms the umbrella goals enunciated above inRME fulfilled the mathematical and intellectual ideals((b) and (d)) of PM but not the political. This does notimply that contexts of an overt political nature are notincluded in the RME program. For instance, in thefollowing activity (De Lange & Verhage, 1992) aroundnational budgets and military expenditure, the overtpolitical dimensions are clearly discernable.

In a certain country, the national defense budget is$30 million for 1980. The total budget for that yearwas $500 million. The following year the defensebudget is $35 million, while the total budget is$605 million. Inflation during the period coveredby the two budgets amounted to 10 per cent.

Cyril Julie obtained his doctorate in Mathematics Educationand Computer-Based Education from the University of Illinoisat Urbana-Champaign. He teaches courses at undergraduateand graduate level in Mathematics Education at the Universityof the Western Cape. He is currently involved in a majorproject related to the development of research capacitybuilding in Mathematics Education in Sub-Saharan Africa. Hismajor research centers around the application and modeling ofschool mathematics and the relevance of school mathematicsfor students in grades 8 to 10.

Cyril Julie 35

A. You are invited to give a lecture for a pacifistsociety. You intend to explain that the defensebudget decreased over this period. Explain howyou would do this.

B. You are invited to lecture to a military academy.You intend to explain that the defense budgetincreased over this period. Explain how you woulddo this.

What is not clear from these and other similaractivities is how these activities are to be used inclassrooms or how and whether there are follow-upactivities that take the intent of the activities beyondthe purely mathematical. The political empowermentideal within People’s Mathematics implied amovement beyond this purely mathematical treatmentof issues of political import. This begs the question ofwhere does the political reside in mathematicalactivity?

The political find expression in at least three areasof mathematical activity. They are all within the arenaof the applications and modeling of mathematics. Thefirst of these is akin to the activity of De Lange andVerhage above. What is added to this is opportunity forovert reflection on those issues that relate to inequalityand discrimination on the basis of race, sex, socialclass and economic developmental level of countries.This is aptly illustrated by the following activity(Frankenstein, 1989, p. 140):

Review the comparisons made in the followingthree tables and write briefly about the connectionsamong the data in each table and any conclusionsand any questions you have about the giveninformation. (Note: Only one table is given here.)

Median incomes of full-time workers by occupation(persons fourteen and above)

Major occupation group 1976 income ($)Women Men

Professional and technical workers 11,081 16,296Non-farm managers & administrators 10,177 17,249Sales workers 6,350 14,432Clerical workers 8,138 12,716Operatives (including transport) 6,696 11,747Service workers (except private household) 5,969 10,117

A second area where the political is overtly presentis during the model construction process. When amathematical model is constructed, interpretations andtranslations take place. The given reality situation fromoutside of mathematics is stripped down to make itamenable for mathematical treatment, and the resultingmathematical model is more a mathematicalrepresentation of a stripped-down version of the

situation. In essence, there are three domains involvedin mathematical model making. These are the extra-mathematical reality, the consensus-generated realitydomain, and the intra-mathematical domains. Thecharacteristics of these domains are summarized inFigure 1.

Extra-Mathematical

Reality Domain

Consensus-Generated

Reality Domain

Intra-Mathematical

Domain

Issues of atechnical,physical,financial, social,political,environmental,and so forthnature are atstake.Issues arecomplex andunder a variety ofinfluencingfactors.

Issues arestripped of someof the influencingfactors.Consensus isreached based onpurposes andinterests.

Mathematicalprocedures andideas aredeveloped andused.Mathematicalconclusions arereached.

Figure 1. The translation of reality issues through differentdomains.

It is during this process of translation that issues ofinterests, ideological preferences and power are atstake and contestations manifest themselves. Thesecontestations occur prior to the subjucation of the issuefor which a mathematical representation is to beconstructed to mathematical treatment. They occurbetween the domain of the real and that ofmathematics. The resolution of conflicts,interpretational variances, and interests render adifferent reality. This reality is realized throughconsensus and hence the postulation of a consensus-generated reality domain as outcome of deliberationson differences, interests, and intentions. It is within theconsensus-generated reality domain that ideologicalintentions are explicitly revealed. A resultingmathematical model is always a product of itsconsensus-generated domain and thus different andnon-equivalent models might result for the samephenomenon.

Lastly, the political rears its head during the finalphase of the modeling process when the adequacy-of-fit of the model to the reality situation is considered.Here the issue is whether the derived mathematicalconclusions should be accepted or not. Consider, forexample, the mathematics of voting. In this instance a

36 Democratic Competence

dictatorship is defined as “[a member] in [a voting set]A is called a dictator in A if and only if {x} is aminimal coalition” with a minimal coalition “a subsetof K…if and only if K is a winning coalition and noproper subset of K a winning coalition” (Steiner, 1968,p. 189, 184). From this definition, if the results of ageneral election in a country is such that a politicalparty has a majority such that they need to form nocoalition with any other party to carry an issue, thenthat political party, and by implication, the governmentof the day, is a dictatorship, at least mathematically.For example, for the South African 2004 election, theAfrican National Congress (ANC) commanded that69.75% of the parliamentary seats and a two-thirdsmajority was needed to carry any decision. Thus,according to the mathematical definition of adictatorship, South Africa is under the dictatorship ofthe ANC. However, the lived experiences in SouthAfrica are such that this is not the case. Here then it isclear that mathematical conclusions and livedexperiences might at times be in conflict and for allintents and purposes it is sometimes wiser to be guidedby lived experiences rather than by the dictates ofmathematical conclusions.

One of the features that stand out in modern schoolmathematics curricula in terms of the goals that areoffered is the development of democratic competence.This competence is an individual’s (and a collective’s)ability to make sound judgments about those issueswhich structure and steer the affairs and practices ofhumankind. The judgments are about theappropriateness or not of the development andimplementation of the mechanisms that guide theseaffairs and practices. These mechanisms areprofoundly driven by worldviews on issues such asrace, gender, and class differentials and generally thekind of world that is envisioned. During the modelconstruction process issues of this political naturecome into play and hence the need for democraticcompetence as stated. Further, as enunciated above,part of this competence is a considered skepticismtowards being convinced through mathematicalargumentation. Democratic competence is normallycaptured in the definition of Mathematical Literacy as,for example, given below for the Programme forInternational Student Assessment (Organization forEconomic Co-Operation and Development, 2003, p.24):

Mathematical literacy is the individual’s capacityto identify and understand the role thatmathematics plays in the world, to make well-founded judgments and to use and engage with

mathematics in ways that meet the needs of thatindividual’s life as a constructive, concerned, andreflective citizen.

A question that can be asked is whether the goal ofthe development of democratic competence can berealized within the Realistic Mathematics Educationframework. What was indicated above is that RMEfalls short as a paradigm in this regard. It does allowfor reflection on issues of a political nature but remainsat an awareness and conscientization level. It does notallow for a consensus generation phase in model-construction. Neither does it explicitly allow for thequestioning of mathematical conclusions in relation tolived and other experiences. It is suggested that RMEneeds to be broadened to incorporate at least thesethree issues in order to contribute more to the goal ofdevelopment of democratic competence. In doing sothere is a need to move beyond awareness andconscientization. This “moving beyond” is what Ellis,a leading South African and internationally recognizedcosmologist, suggested about 20 years ago. He stated:

…the only true basis of freedom is a realistic visionof the alternative possibilities for change before us.Mathematical studies can sometimes help us inunderstanding what these alternative possibilitiesare. But such an understanding is quite valuelessunless it affects our actions. An understanding ofthe causes of any social wrong, which does notlead to some corrective action to right that wrong,is meaningless. (Ellis, 1974, p. 17)

Democratic competence is thus about anindividual’s (or collective’s) capacity to interact withmechanisms which affect their lives and those ofsociety and to act where such mechanisms are to thedetriment of humankind. Where these mechanismshave a mathematical base, or where they can beexplained and understood through mathematicalmeans, necessitates that schooling in mathematics becalled upon to provide the spaces for such interactionsand actions.

REFERENCESde Lange, J. (1987). Mathematics, insight and meaning. Utrecht

University: Utrecht, Netherlands: OW & OC.de Lange, J., & Verhage, H. (1992). Data visualization.

Pleasantville, New York: Sunburst.Ellis, G. (1974). On understanding the world and the universe.

Professorial Inaugural Lecture, University of Cape Town,Cape Town, South Africa.

Frankenstein, M. (1989). Relearning mathematics: A different thirdR – Radical maths. London: Free Association Books.

Gravemeijer, K. (1988). Een Realistisch research programma. InK. Gravemeijer & K. Koster (Eds.), Onderzoek, ontwikkeling

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en ontwikkelingsonderzoek (pp. 106–117). Utrecht University,Utrecht, Netherlands: OW & OC.

Gravemeijer, K. (1994). Educational development and developmentresearch in mathematics education, Journal for Research inMathematics Education. 25(5), 443–471.

Julie, C. (1993). People’s mathematics and the applications ofmathematics. In J. de Lange, C. Keitel, I. Huntley, & M. Niss(Eds.), Innovation in maths education by modelling andapplications (pp. 31–40). Chichester: Ellis Horwood.

Organisation for Economic Co-operation and Development(OECD). (2003). The PISA 2003 assessmentframework—Mathematics, reading, science and problemsolving knowledge and skills. Retrieved November 23, 2004from http://www.pisa.oecd.org/

Skovsmose, O. (1994). Towards a philosophy of criticalmathematics education. Dordrecht, Netherlands: KluwerAcademic Publishers.

Steiner, H. G. (1968). Examples of exercises in mathematization insecondary school level. Educational Studies in Mathematics,1, 181–201

South African Department of Education. (2003). The nationalcurriculum statement: Mathematical literacy. Pretoria, SouthAfrica: Government Printers.

Streefland, L. (1990). Fractions in realistic mathematics education:A paradigm of developmental research. Dordrecht,Netherlands: Kluwer Academic Publishers.

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CONFERENCES 2005…

MAA-AMS Atlanta, GA January 5 – 8Joint Meeting of the Mathematical Association of Americaand the American Mathematical Societyhttp://www.ams.org

AAMT 2005 Sydney, Australia January 17 – 20Australian Association of Mathematics Teachershttp://www.aamt.edu.au/mmv

AMTE Dallas, TX January 27 – 29Association of Mathematics Teacher Educatorshttp://amte.sdsu.edu/conf_info_2005.shtml

RCML Little Rock, AK February 24 – 26Research Council on Mathematics Learninghttp://www.unlv.edu/RCML

Mα Coventry, UK March 30 – April 2The Mathematical Associationhttp://m-a.org.uk/resources/conferences/

NCTM Anaheim, CA April 6 – 9National Council of Teachers of Mathematicshttp://www.nctm.org/meetings/index.htm#annual

AERA Montreal, Canada April 11 – 15American Educational Research Associationhttp://www.aera.net/meeting/

AMESA Kimberley, June 27 – 30Eleventh Annual National Congress South Africahttp://academic.sun.ac.za/mathed/AMESA/Index.htm

PME-29 Melbourne, Australia July 10 – 15International Group for the Psychology of Mathematics Educationhttp://staff.edfac.unimelb.edu.au/~chick/PME29/

JSM of the ASA Minneapolis, MN August 7 – 11Joint Statistical Meetings of the American Statistical Associationhttp://www.amstat.org/meetings/jsm/2005/

GCTM Rock Eagle, GA TBAGCTM Annual Conferencehttp://www.gctm.org/georgia_mathematics_conference.htm

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The Mathematics Educator (ISSN 1062-9017) is a student-produced journal published semiannually bythe Mathematics Education Student Association (MESA) at The University of Georgia. The journal promotes theinterchange of ideas among the mathematics education community locally, nationally, and internationally, andpresents a variety of viewpoints on a broad spectrum of issues related to mathematics education. TME also provides avenue for the encouragement and development of leaders and editors in mathematics education. The MathematicsEducator is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on MathematicalEducation).

The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and otherprofessionals in mathematics education including:

• reports of research (including experiments, case studies, surveys, philosophical studies, and historical studies),curriculum projects, or classroom experiences;

• commentaries on issues pertaining to research, classroom experiences, or public policies in mathematicseducation;

• literature reviews;• theoretical analyses;• critiques of general articles, research reports, books, or software;• mathematical problems (framed in theories of teaching and learning; classroom activities);• translations of articles previously published in other languages;• abstracts of or entire articles that have been published in journals or proceedings that may not be easily

available.

The Mathematics Educator strives to provide a forum for collaboration of mathematics educators with varying levelsof professional experience. The work presented should be well conceptualized; should be theoretically grounded; andshould promote the interchange of stimulating, exploratory, and innovative ideas among learners, teachers, andresearchers.

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(including references and footnotes). An abstract should be included and references should be listed at the end ofthe manuscript. The manuscript, abstract, and references should conform to the Publication Manual of theAmerican Psychological Association, Fifth Edition (APA 5th).

• An electronic copy is required. (A hard copy should be available upon request.) The electronic copy may be inWord, Rich Text, or PDF format. The electronic copy should be submitted via an email attachment [email protected]. Author name, work address, telephone number, fax, and email address must appear on thecover sheet. The editors of The Mathematics Educator use a blind review process therefore no author identificationshould appear on the manuscript after the cover sheet. Also note on the cover sheet if the manuscript is based ondissertation research, a funded project, or a paper presented at a professional meeting.

• Pictures, tables, and figures should be camera ready and in a format compatible with Word 95 or later. Originalfigures, tables, and graphs should appear embedded in the document and conform to APA 5th - both in electronicand hard copy forms.

To Become a Reviewer:Contact the Editor at the postal or email address below. Please indicate if you have special interests in reviewingarticles that address certain topics such as curriculum change, student learning, teacher education, or technology.

Postal Address: Electronic address:The Mathematics Educator [email protected] Aderhold HallThe University of GeorgiaAthens, GA 30602-7124

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The Mathematics Education Student Association is an official affiliate of theNational Council of Teachers of Mathematics. MESA is an integral part ofThe University of Georgia’s mathematics education community and isdedicated to serving all students. Membership is open to all UGA students,as well as other members of the mathematics education community.

Visit MESA online at http://www.coe.uga.edu/mesa

TME SubscriptionsTME is published both online and in print form. The current issue as well as back issues are available online athttp://www.coe.uga.edu/tme. A paid subscription is required to receive the printed version of The MathematicsEducator. Subscribe now for Volume 15 Issues 1 & 2, to be published in the spring and fall of 2005.

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In this Issue,

Guest Editorial… Researching Classroom Learning and Learning Classroom ResearchDAVID CLARKE

The Consequences of a Problem-Based Mathematics CurriculumDAVID CLARKE, MARGARITA BREED, & SHERRY FRASER

Impact of Personalization of Mathematical Word Problems on Student PerformanceERIC T. BATES & LYNDA R. WIEST

Mathematics Placement Test: Helping Students SucceedNORMA G. RUEDA & CAROLE SOKOLOWSKI

In Focus… Can the Ideal of the Development of Democratic Competence Be RealizedWithin Realistic Mathematics Education? The Case of South AfricaCYRIL JULIE

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