____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 14 Number 2

Fall 2004

MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff

A Note from the Editor

Editor Holly Garrett Anthony

Dear 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 both national and international research and commentary. We hope your reading of these articles will be both educational and thought provoking. David Clarke opens this issue with an editorial inviting you to read the research he, Margarita Breed, and Sherry Fraser conducted in the early 1990s and published in this issue. In their research article, they highlight some positive consequences of teaching with the Interactive Mathematics Program (IMP), a problem-based curriculum that has gained attention in recent years. Clarke’s editorial further asks mathematics education researchers to consider research methodology when studying classroom learning. In so doing, he draws on the international comparative research of the Learner’s Perspective Study. Two studies presented in this issue examine the effectiveness of assessment items. Bates and Wiest discuss the impact personalization of word problems can have on students’ performance on mathematics assessments. Contrary to recent research they report no significant difference in students’ performance with personalized and non-personalized problems. Rueda and Sokolowski study the effectiveness of their locally developed mathematics placement test at Merrimack College and show that students who follow the recommendations for course enrollment based on their test scores perform well in those classes. Finally, Cyril Julie, a scholar in South Africa, invites readers to consider the development of democratic competence in students within a newly formed democratic country and the role mathematics might play in that development. He asks whether democratic competence can be realized within Realistic Mathematics Education (RME), a curriculum developed in the Netherlands and recently imported into South Africa. His question is important for consideration and his discussion is stimulating. As I close my final comments as the editor of TME, I encourage readers to support our journal by submitting manuscripts, reviewing articles, or joining our editorial team. TME is growing in recognition, 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 of editors who worked well both together and independently. The publication of TME is a direct result of their time and effort. I appreciate MESA allowing me the opportunity to do this work and I hope that my efforts have been notable. I also extend my thanks to all of the other people who make TME possible: reviewers, authors, peers, and faculty.

Associate Editors Ginger Rhodes Margaret Sloan Erik Tillema Publication Stephen Bismarck Dennis Hembree Advisors Denise S. Mewborn Nicholas Oppong James W. Wilson

MESA Officers 2004-2005 President Zelha Tunç-Pekkan Vice-President Natasha Brewley Secretary Amy J. Hackenberg Treasurer Ginger Rhodes NCTM Representative Angel Abney Undergraduate Representatives Erin Bernstein Erin Cain Jessica Ivey

With Sincere Thanks, Holly Garrett Anthony 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

tme@coe.uga.edu www.coe.uga.edu/tme

About the cover Cover artwork by Tyler M. Ricks. Untitled, 2004. For questions or comments, contact: tricks3@email.byu.edu This piece is the culmination of years of experimentation and study with computer art. Starting with just a simple 3 by 3 grid in an 8th grade art class, the style blossomed into hundreds of different pieces using many different geometrical ideas. The process of hand-drawing each line takes hours of work, but can produce extremely complex mathematical images. The piece featured on the cover was created on the computer program GeoSketchpad™, and is part of a larger series using complex “string frames,” so called because they resemble physical frames on which strings are tightly strung. Other series use complex grids, different geometric 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 Association The University of Georgia

Fall 2004

Volume 14 Number 2

Table of Contents 2

Guest Editorial… Researching Classroom Learning and Learning Classroom Research DAVID CLARKE

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

17 Impact of Personalization of Mathematical Word Problems on Student Performance ERIC T. BATES & LYNDA R. WIEST 27 Mathematics Placement Test: Helping Students Succeed NORMA G. RUEDA & CAROLE SOKOLOWSKI 34 In Focus… Can the Ideal of the Development of Democratic Competence Be Realized Within Realistic Mathematics Education? The Case of South Africa CYRIL JULIE 38 Upcoming Conferences 39 Submissions Information 40 Subscription Form

© 2004 Mathematics Education Student Association. All Rights Reserved

The Mathematics Educator 2004, Vol. 14, No. 2, 2–6

Guest Editorial… Researching Classroom Learning and Learning Classroom Research David Clarke One of the central goals of the mathematics education research community is the identification of classroom practice likely to facilitate student learning of mathematics. In the paper by Clarke, Breed and Fraser in this issue of The Mathematics Educator, the results of an investigation into the outcomes of the Interactive Mathematics Program (IMP) undertaken back in the early 1990s are reported. Why is this important? Because the focus of the analysis was an expanded conception of the outcomes of classroom practice that included both the cognitive and the affective consequences of introducing a problem-based mathematics program. The findings demonstrate that the consequences of a particular curriculum and its associated classroom practices cannot be adequately characterized solely by the mathematical performance of the students. Most importantly, the IMP classrooms studied were most clearly distinguished from conventional classrooms by affective rather than cognitive outcomes. At the time, this was an attempt to embrace a broader vision of valued classroom practice and significant learning outcomes than could be documented in an achievement test. The message of this research has contemporary significance, but in the time since that study was conducted our capacity to investigate classroom practice and to connect it to learning outcomes has increased considerably.

David Clarke is a Professor in the Faculty of Education at the University of Melbourne and Director of the International Centre for Classroom Research. His consistent interests have been Assessment, Learning in Classrooms, and Teacher Professional Development, and he has undertaken research related to all of these areas. Recent publications include the book “Perspectives on Practice and Meaning in Mathematics and Science Classrooms” published by Kluwer Academic Publishers in 2001, and the chapters on Assessment and International Comparative Research in the 1996 and 2003 editions of the “International Handbook of Mathematics Education.” He is currently directing the 14-country Learner's Perspective Study.

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The Participant’s Voice I have argued consistently and persistently (Clarke, 1998, 2001, 2003) that since a classroom takes on different aspects according to how you are positioned within it or in relation to it, our research methodology must be sufficiently sophisticated to accommodate and represent the multiple perspectives of the many participants in complex social settings such as classrooms. Only by seeing classroom situations from the perspectives of all participants can we come to an understanding of the motivations and meanings that underlie their participation. Our capacity to improve classroom learning depends on such understanding. The methodological challenge is how to document and analyze the fundamental differences in how each participant experiences any particular social (classroom) situation. My colleagues, Sverker Lindblad and Fritjof Sahlström (2002), argue that if early researchers had access to the tools for data collection and analysis that are available today, the general view of classroom interaction would be quite different. The most striking of these differences, and a very important one from an education point of view, concerns the role of students in classrooms. Thorsten (2000) has made this point very clearly in relation to the Third International Mathematics and Science Study (TIMSS). What is absent from nearly all the rhetoric and variables of TIMSS pointing to the future needs of the global economy is indeed this human side: the notion 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 the passive, silent student at odds with contemporary learning theory and classroom experience. Research done with technologically more sophisticated approaches 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). Researching Classroom Learning

International Comparative Research Further, classroom researchers have until recently had limited opportunities for engaging in manageable comparative research, where materials from different countries and different periods of time can be accessed and analyzed in feasible ways. At the International Centre for Classroom Research at the University of Melbourne (http://www.edfac.unimelb.edu.au/DSME/ICCR/), contemporary technology makes it possible to carry out comparative analyses of an extensive database that includes three-camera classroom video records of lesson sequences, supplemented by post-lesson videostimulated interviews with students and teachers, scanned samples of written work, and test and questionnaire data, drawn from mathematics classrooms as geographically distant as Sweden and Australia and as culturally distant as Germany and China. Watanabe (2001) quotes White (1987) as writing “we should hold Japan up as a mirror, not as a blueprint.” This powerful and appealing metaphor can serve as a general characterization of one of the major uses of international comparative studies of classroom practice. The agency for the interpretation and adaptation of any documented practice resides with the person looking in the mirror. There is no invocation of absolute best practice – the judgement is a relativist one, and an instructional activity with a high degree of efficacy in Hong Kong may retain little effectiveness when employed in a Swedish classroom, where different cultural values inform and frame the actions of all classroom participants. Most importantly, we are encouraged to study Japanese (or South African or German) classrooms not solely for the purposes of mimicking their practices but for their capacity to support us in our reflection on our own practice. The mutuality of the potential benefit provides further motivation for such research. There is a small but growing body of research that works at developing techniques of documenting classroom interaction in ways that will facilitate highquality analysis of children’s learning. The transfer from single-microphone audio (as in the early studies), via single-camera video (as in many recent studies) to multi-camera and multi-audio (as in the studies at the technological forefront) is not primarily technologydriven, but rather motivated by the recent shifts in education theories on learning, from a view of learning as transfer to a view of learning as constructed in action (see Sfard, 1998, for a discussion). Thus, technological sophistication is a requirement of recent David Clarke

theory, rather than a matter of sophisticated equipment for technologically-minded project coordinators. This is an essential point: Educational research, like research in the physical and biological sciences, must make optimal use of available technologies in addressing the major problems of the field. But the prime motivation must be “What are the big questions and what tools do we need to address these questions?” rather than “What questions can be addressed with available tools?” Our research must be fuelled by a need to answer important questions, not by a need to use new tools. In addition, it is the first question that will lead to recognition of the need for new tools and provide the motivation for their development. The Learner’s Perspective Study: Complementary Accounts 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 the onsite mixing of the Teacher and Student camera images into a split-screen video record that is then used to stimulate participant reconstructive accounts of classroom events. So far, these data have been collected for sequences of at least ten consecutive lessons occurring in the “well-taught” eighth grade mathematics classrooms of three teachers in each of ten participating countries (Australia, Germany, Hong Kong and mainland China, Israel, Japan, Korea, The Philippines, South Africa, Sweden and the USA). This combination of countries gives good representation to European and Asian educational traditions, affluent and less affluent school systems, and mono-cultural and multi-cultural societies. Data collection will commence next year in the Czech Republic, England and Singapore. Each participating country uses the same research design to collect videotaped classroom data for at least ten consecutive math lessons and post-lesson videostimulated interviews with at least twenty students in each of three participating 8th grade classrooms. The three mathematics teachers in each country are identified for their locally-defined ‘teaching competence’ and for their situation in demographically diverse government schools in major urban settings. In a major component of the post-lesson student interviews, in which a split-screen video record is used as stimulus for student reconstructions of classroom events, students are given control of the video replay and asked to identify and comment upon classroom events of personal importance. Each teacher is 3

interviewed at least three times using a similar protocol. Goffman’s conception of a working consensus as a transient convergence on a locally viable interpretation (Goffman, 1959) is a particularly apt characterization of the goal of the consensus process operating in many interpretive research teams (e.g., Cobb & Bauersfeld, 1995; Stigler & Hiebert, 1999). The research in which I have been involved (e.g., Clarke, 2001) problematizes such consensus and attempts to synthesize portrayals of practice from ‘complementary accounts’ provided by researchers and the participants in the research setting relating to a common body of data (rationale provided in Clarke, 1998). I would like to assert the inevitable existence of multiple reflexivities between theory, research into practice, and the practice of research. The argument is predicated on three basic premises:

Shepard’s provocative question, “But what if learning is not linear and is not acquired by assembling bits of simpler learning” (Shepard, 1991, p. 7). In the case of the Learner’s Perspective Study: Research guided by a theory of learning that accords significance to both individual subjectivities and to the constraints of setting and community practice must construct and frame its conclusions (and collect its data) accordingly. Such a theory must accommodate complementarity rather than require convergence and accord both subjectivity and agency to individuals not just to participate in social practice but to shape that practice. Research that aims to apply such theories must construct its methodologies accordingly and draw from available technologies in ways that afford rather than constrain the methodological ambitions of the researcher.

1. The discourse of the classroom (for example) acts to position participants in ways that afford and constrain certain practices.

International comparative classroom research need not appeal to a separate and distinct research paradigm from that enacted in conventional classroom research, although the methodological and theoretical considerations are more complex than research within a single culture. Part of the power of international comparative research lies in its capacity to offer us the opportunity to juxtapose, compare and contrast documented practices drawn from settings that simply would not pertain in our local culture. What form does teaching competence take when confronted with a class of 60 or more students (as is the case in the Philippines)? How must we reconceive our notions of effective instructional practice to accommodate apparently successful classrooms in which students seldom if ever speak to each other (as pertains in some Asian classrooms)? How much more compelling must our theories of learning become if they can be demonstrated to accommodate and explain learning in such disparate settings? As new theories of learning and social interaction develop, research techniques must have the capacity to accommodate these new theories. All too often it is forgotten that any use of technology in a research setting implies the existence of an underlying theory on which the type of data, the means of data collection, and the anticipated method of analysis are all predicated. Of all data sources currently available to researchers in education, videotape data seems most amenable to secondary analysis. Further, the potential of videotape data to sustain secondary analysis carries an associative potential for the synthesis of those analyses.

2. The discourse of educational research acts to position participants in ways that afford and constrain certain interpretations. 3. The adoption of a theory of learning in social situations will inevitably find its reflection in the manner in which those situations are researched. These fundamental reflexivities are seldom acknowledged. Since research activity constitutes a form of learning or knowledge construction, the processes by which a research project is conducted should be in harmony with whatever theory of learning structures the researcher’s analysis of data. Consistency between methodology and theory should be a matter of purposeful and deliberate design. Lorrie Shepherd turns this argument delightfully on its head in her paper “Psychometricians’ Beliefs About Learning” (Shepard, 1991), where she contends that the disputes of the testing community can be explained in terms of differences in the beliefs about learning held by the various educational measurement specialists. In particular, Shepard argues that the beliefs of many psychometricians derive from an implicit behaviorist learning theory in flagrant contradiction with evidence from cognitive psychology. What Shepard does to good effect in her paper is reverse engineer psychometricians’ learning theories on the basis of their test instruments. The fruitfulness of this approach is fully evident in 4

A Layered Vision

Researching Classroom Learning

Multi-site international research projects offer access to a layered vision of practice, outcome and theory development. It may help to illustrate this stratification with examples from the Learner’s Perspective Study. Classroom Practice – Lesson Events At the level of classroom practice, the challenge has been to find a suitable instructional unit to provide the basis for comparative analysis. Demonstration of the inadequacy of “the lesson” to serve this role (at least in the form of nationally characteristic lesson “scripts” or “patterns”) has led to analyses focusing on the “lesson events” from which each lesson is constituted. Lesson events such as “Beginning the Lesson,” “Learning Tasks,” “Guided Development” (Whole class discussion), “Between Desks Instruction,” and “Summing Up” have emerged as internationally recognizable activities, differently and distinctively employed and enacted in classrooms around the world. Patterns of Participation In participating in each of the lesson events identified above, teacher and students position themselves and are positioned within the constraints and affordances offered by the classroom setting and its peculiar practices (peculiar here is used in all possible senses). The consequences of this process of social positioning are characteristic patterns of participation accessible to classroom participants (and co-constructed by them) in ways that reflect each individual’s unique interaction with the classroom setting and community. The Distribution of Responsibility for Knowledge Generation Each classroom affords and constrains access to various patterns of participation. Within the patterns of participation characteristic of a classroom can be found the “distribution of responsibility for knowledge generation” – a much more useful characterization of the classroom than a simplistic dichotomization into teacher-centered and student-centered, and much more revealing of the sociocultural nature of learning. The use of video material supported by post-lesson video-stimulated interviews provides a complex database amenable to analysis at any and all of the three levels indicated above. Complex databases, configured in anticipation of multiple and complementary analyses, offer our best chance to match the complexity of social phenomena with an David Clarke

appropriate sophistication of approach. Advances in technology bring us ever closer to the realization of this vision. The developmental pathway that has led us from early attempts at classroom observation and process-product studies to our present level of sophistication represents an on-going attempt to accommodate the complexity of social situations. Eugene Ionescu is reputed to have said, “Only the ephemeral is of lasting value.” Social interactions are nothing if not ephemeral; and, since it is through social interaction that we experience the world, the understanding of social interactions must underlie any attempts to improve the human condition. Our difficulties in characterizing social interactions for the purpose of theory building in education are compounded by the fluid and transient nature of the phenomena we seek to describe. Attempts to categorize social behavior run the risk of sacrificing the dynamism, contextual-dependence and variation that constitute their essential attributes. This poses a challenge both for methodology and for theory. The ephemeral nature of social interactions is something that must be honored in the methodology but transcended in the analysis. Those of us who have accepted the challenge of researching classroom learning continue to learn how better to undertake classroom research. REFERENCES Clarke, 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 of Mathematics. Clarke, D. J. (Ed.). (2001). Perspectives on practice and meaning in mathematics and science classrooms. Dordrecht, Netherlands: Kluwer Academic Press. Clarke, D. J. (2003). International comparative studies in mathematics education. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (Chapter 5, pp. 145–186). Dordrecht, Netherhlands: Kluwer Academic Publishers. Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale, NJ: Lawrence Erlbaum. 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 classroom research. Invited plenary lecture at the Current Issues in Classroom Research: Practices, Praises and Perspectives Conference, Oslo, Norway.

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

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

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

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

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.

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

Researching Classroom Learning

The Mathematics Educator 2004, Vol. 14, No. 2, 7–16

The Consequences of a Problem-Based Mathematics Curriculum David Clarke, Margarita Breed, & Sherry Fraser Implementation of a problem-based mathematics curriculum, the Interactive Mathematics Program (IMP), at three high schools in California has been associated with more than just differences in student achievement. The outcomes that distinguished students who participated in the IMP program from students who followed a conventional algebra/geometry syllabus were the students’ perceptions of the discipline of mathematics, of mathematical 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 of the set of beliefs, perceptions and performances arising from such a program. Students who have participated in the IMP program appear to be more confident than their peers in conventional classes; to subscribe to a view of mathematics as having arisen to meet the needs of society, rather than as a set of arbitrary rules; to value communication in mathematics learning more highly than students in conventional classes; and to be more likely than their conventionally-taught peers to see a mathematical element in everyday activity. These outcomes occurred while the IMP students maintained performance levels on the mathematics portion of the SAT 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-based curriculum.

Among the debates engaging the energies of the mathematics education community, one of the more energetic has concerned the role of problem solving in mathematics instruction. This debate has encompassed issues from what constitutes a problem to whether problem solving should be the medium or the message of the mathematics curriculum (cf., Clarke & McDonough, 1989; Lawson, 1990; Owen & Sweller, 1989; Schoenfeld, 1985). Claims and counter-claims David Clarke is a Professor in the Faculty of Education at the University of Melbourne and Director of the International Centre for Classroom Research. He is currently directing the 14-country Learner's Perspective Study. Margarita Breed is currently a fulltime Ph. D. student funded under the APAI (Australian Postgraduate Award Industry) Scheme for the Scaffolding Numeracy in the Middle Years Research Project at RMIT University. Her background has been in primary teaching and as a Middle Years Numeracy Leader for Eastern Metropolitan Region and she is particularly dedicated to students in the Middle Years. Whilst completing her Master of Education (Research) she was Research Assistant for the Mathematics Teaching and Learning Centre at the Australian Catholic University. Sherry Fraser is currently the Director of the IMPlementation Center for the Interactive Mathematics Program. She continues to be interested in providing both students and teachers access to rich secondary mathematics materials. Acknowledgements The cooperation of the teachers whose pupils participated in this study is gratefully acknowledged. The comments of Barry McCrae, Kevin Olssen, Diane Resek and Peter Sullivan on early drafts of this paper are also gratefully acknowledged. Clarke, Breed, & Fraser

have been made regarding the advisability and the feasibility of basing a mathematics syllabus on nonroutine mathematics tasks. Attempts to evaluate the success of such curricula have typically employed achievement tests to distinguish student outcomes. The authors of this study felt that a problem-based curriculum would be characterized more appropriately by the belief systems which the instructional program engendered in participating students than by the students’ achievement on conventional mathematical tasks. It is students’ belief systems that are likely to influence the students’ subsequent participation in the study of mathematics, to structure their consequent learning of mathematics, and to guide and facilitate the application of mathematical skills to everyday contexts. If it could be demonstrated that student achievement on conventional mathematics tasks was enhanced by a problembased program, and if student performance on nonroutine problem-solving tasks was heightened by such a program, the ultimate value of the instruction would depend still on whether the student chooses to continue to study mathematics, develops a set of beliefs which supports and empowers further learning, and sees any relevance in the skills acquired in class for situations encountered in the world beyond the classroom. Conventional instruction does little to address such concerns, and research has commonly ignored such outcomes. 7

The evaluation of teaching experiments currently in progress must address these other consequences of instruction. In discussing their work on “one-on-one constructivist teaching,” Cobb, Wood, and Yackel (1990) drew attention to non-conventional learning outcomes. This instructional approach provides opportunities for the children to construct mathematical knowledge not found in traditional classrooms. The difficulty for researchers evaluating innovative classroom practices is that many of the conventional research tools are insensitive to the behaviors and the knowledge that distinguishes such instruction. This concern is also relevant where the goals of the program are affective as well as cognitive. Since studies such as that of Erlwanger (1975) drew attention to the significance of a student’s belief system regarding mathematics and mathematical behavior, research into effective teaching practice has had 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 been discussed usefully in a variety of forums (for instance, Clarke, 1986; Cobb, 1986), research studies have still to accept a responsibility to address student belief and perception outcomes routinely in the evaluation of instructional programs. The study reported here is one attempt to do so. The Instructional Program In 1989, the California Postsecondary Education Commission (CPEC) released a request for proposals that would drastically revamp the Algebra I-GeometryAlgebra II sequence. The curriculum envisioned in the guidelines would set “problem solving, reasoning and communication as major goals; include such areas as statistics and discrete mathematics; and make important use of technology” (CPEC, 1989, p. 4). The Interactive Mathematics Project (IMP) Curriculum Development Program obtained funding to develop and field test three years of problem-based mathematics that would satisfy six of the University of California requirements for high school mathematics. Program Goals The goals of IMP were to: •

broaden who learns mathematics, by making the learning of core mathematics accessible to groups previously underrepresented in college mathematics classes;

•

expand what mathematics was learned, consistent with the recommendations of the Curriculum and

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Evaluation Standards (NCTM, 1989), emphasizing problem solving and the communication of mathematical ideas; •

change mathematics instruction, by requiring students to be active learners and investigators, by integrating the study of mathematical domains, such as algebra, geometry and statistics, with each other and with areas of application, and by making use of current technology;

•

change how teachers perceive their roles, by emphasizing the role of the teacher as guide and model learner and by changing dominant modes of classroom communication from teacher explanation to student interaction;

•

change how mathematics learning is assessed, by assessing students’ use of mathematical knowledge to solve complex problems, and by diversifying assessment strategies to include student portfolios, self-assessment, teacher observations, oral presentations, and group projects, as well as written homework and tests.

Pupil Selection Methods of selection of pupils for participation in the IMP classes varied. The principal criterion was student self-nomination. One high school collected information on student performance, instructional preferences, and academic history and then selected “60% of the group who would have been placed in Algebra and 40% from those below.” It was the opinion of the various school administrations that the academic standing of the sample of IMP students arising from the various selection criteria was certainly no higher than that of the students in conventional Algebra classes. In fact, in the case of the high school just mentioned, the overall academic standing of students commencing IMP was almost certainly lower than that of commencing Algebra students. Teacher Selection Teachers were also self-nominated. The IMP Materials The IMP materials consist of modular units, each requiring approximately five weeks of instructional time. These units employ historical, literary, scientific and other contexts to provide a thematic coherence to the pupils’ exploration of mathematics. For instance, in one unit the Edgar Problem-Based Mathematics Curriculum

Allan Poe short story The Pit and the Pendulum is used to facilitate student investigation of variation, measurement uncertainty, normal distribution, graphing, mathematical modeling, and non-linear functions. The instructional sequence of each unit addresses mathematical concepts and skills and mathematical problem solving in a context that provides both the rationale for the skills being acquired and a means of integrating newly acquired knowledge within a coherent structure. An IMP Classroom Class size averaged around 32 students. Classroom activities were typified by group work, writing, and oral presentations. Graphing calculators were available at all times. The characteristics of IMP and Algebra classrooms, as perceived by the pupils, were documented in the course of this study, and are detailed in the results presented later in this paper. Assessment Practices Priority was given in IMP classrooms to a diversity of assessment strategies, consistent with the program goals. For example, in one IMP class, grades were calculated 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 Algebra students was through weekly quizzes and chapter tests. Method

report their perceptions of those valued activities, which, in their opinion, assisted their learning of mathematics, in addition to their perceptions of what constituted typical classroom activities in mathematics and their attitudes towards mathematics. The Mathematics World questionnaire required students to identify the extent to which specific everyday activities were mathematical. At the time of administration of the questionnaires, IMP students had completed almost one year in the program. In addition, the next fall, the Mathematics Scholastic Aptitude Test (SAT) was administered to the school populations, facilitating comparison of the mathematics performance of IMP students with their peers in conventional classes. Mathematics belief. The mathematics belief questionnaire was adapted from an instrument employed to measure the student belief outcomes of an innovative program employing student journals (Clarke, Stephens & Waywood, 1992; Clarke, Waywood, & Stephens, 1994). Every item was validated through interviews with students. Minor changes in phrasing were made for administration in American 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 beâ€Ś

3.

The ideas of mathematics: A. Have always been true and will always be true. Agree Disagree

Subjects The subjects of this study were 182 students at three Californian high schools participating in the IMP program outlined above. In addition, matching data were collected on 74 Algebra 2 students and 143 Algebra 4 students from the same schools. Data on an additional 52 Algebra 2 students were collected from a fourth high school to provide a comparable sample of students at the same level as the IMP pupils. Procedures and Measures During June, towards the end of the academic year, all students completed two questionnaires. The student questionnaire was constructed in large part by combining items developed and tested in a study of student mathematics journal use and a further study of student self-assessment. The Mathematics Belief questionnaire examined student perceptions of their mathematical competence, and student beliefs about mathematical activity and the origins of mathematical ideas. Students were asked to Clarke, Breed, & Fraser

D. Developed as people needed them in everyday life. Agree Disagree F. Are most clearly explained using numbers. Agree Disagree 5.

When I am doing mathematics at school, I am likely to be: A. Talking Always Often Sometimes Seldom Never C. Writing words Always Often Sometimes Seldom Never F. Working with a friend Always Often Sometimes Seldom Never I. Listening to other students Always Often Sometimes Seldom Never K. Working from a textbook Always Often Sometimes Seldom Never 9

7.

An adaptation of the IMPACT instrument (Clarke, 1987) was included as item 7, including such subitems as: Write down one new problem that you can now do. How could math classes be improved?

Student attitudes towards mathematics classes were measured 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. Worried D. Successful E. Confused F. Clever G. Happy H. Bored I. Rushed J. (Write one word of your own) _____________

The response alternatives provided in this sub-item arose from extensive interviewing of high school students in the course of a study of student mathematical behavior at the point of transition from primary school (elementary school) mathematics to high school mathematics (Clarke, 1985, 1992). The IMPACT instrument, from which the sub-item was drawn, was extensively field-tested with 753 grade 7 students over a period of one year (Clarke, 1987). Mathematics world. The mathematics world questionnaire was adapted for American administration from an instrument employed in a study of community perceptions of mathematical activity (Clarke & Wallbridge, 1989; Wallbridge, 1992). In this questionnaire, students were asked to indicate whether they thought specific everyday activities were highly mathematical, quite mathematical, slightly mathematical, barely mathematical, or not mathematical. The activities listed included: 4. 7. 9.

Cooking a meal using a recipe Playing a musical instrument Buying clothing at a sale

A complete listing of all questionnaire items is available in Clarke, Wallbridge, and Fraser (1992). Results The results that follow make reference to three groups of students to whom questionnaires were administered: i. 180 IMP students – mean age 15.3 years ii. 126 Algebra 2 students – mean age 15.4 years iii. 137 Algebra 4 students – mean age 16.9 years

Comparing the Algebra 2 and Algebra 4 Samples

differ significantly on any of the 48 measures except the use of worksheets, for which the Algebra 4 students recorded an even lower incidence than did the Algebra 2 students, and the importance accorded to the teacher's explanations: Algebra 4 students attached lower importance to these than did the Algebra 2 students. It seems reasonable to summarize these findings by observing that, with respect to the beliefs documented here, conventionally-taught students adhere to a specific set of beliefs with a high level of stability over time. These beliefs and the associated perceptions of classroom practice were sufficiently distinct from those held by IMP students to clarify the characteristics of both class types. Results are given as comparisons between IMP and Algebra 2 students since these represent the most similar sample populations. In each table where comparisons are made between groups the corresponding p value is given. Differences between groups which achieved statistical significance are asterisked. Student Mathematics Achievement on Conventional Tests Where comparison was possible between IMP and Algebra students at the same school, mean SAT scores for IMP classes were higher than mean SAT scores for traditional Algebra/Geometry classes. Pair-wise comparison of group means (t test) was used to identify any statistically significant difference under a conventional null hypothesis assumption. At one high school the difference in performance was statistically significant. These results are documented in Table 1. Table 1 SAT Scores for Algebra and IMP Students on Two School Sites School

A

B

Class type

Mean SAT score

SD

Algebra (n = 83)

420.48

82.96

IMP (n = 74)

443.37

77.21

Algebra (n = 86)

367.56

57.02

IMP (n = 67)

373.88

60.95

p value

.0372*

.1003

In all, 48 student measures were generated through the two questionnaires. Algebra 2 and Algebra 4 samples (n = 126 and n = 137, respectively) did not 10

Problem-Based Mathematics Curriculum

Student Perceptions of Their Mathematics Competence IMP students were significantly more likely to rate themselves highly on how good they were at mathematics 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 I would give myself would be:

Two comments should be made concerning this higher self-rating by IMP students. First, SAT scores indicated that where comparison was possible IMP students tended to be more capable at conventional mathematics tasks than were their peers in Algebra classes, which suggests that these self-ratings had some basis in fact. Second, the difference in self-ratings can also be interpreted as a difference in confidence. We would suggest that heightened self-confidence in mathematics is likely to lead to increased participation in further mathematics, and a greater likelihood that the student will make use of the mathematical skills acquired. Both are desirable outcomes. Table 2 Self-rating Scores for IMP and Algebra 2 Students Class type

Mean

SD

Algebra 2 (n = 125)

6.86

1.2

IMP (n = 173)

7.5

1.38

p value

Table 3 Student Attitude Index for IMP and Algebra 2 Students Class type

Mean

SD

Algebra 2 (n = 126)

â€“.52

1.85

IMP (n = 174)

.97

Student Attitude Toward Mathematics Classes IMP students were significantly more likely to feel positive about mathematics classes (Table 3).

A. Interested B. Relaxed C. Worried D. Successful E. Confused F. Clever G. Happy H. Bored I. Rushed J. (Write one word of your own) _____________

A student attitude index was calculated by scoring each positive response +1 and each negative response â€“1, and summing for each student.

2.14

The distinguishing characteristic between the problem solving students and the Algebra 2 students was the degree to which they perceived mathematics to be a mental activity (Table 4). Sample Item: Mathematics is something I do (circle one or more): A. Every day as a natural part of living B. Mostly at school C. With a pencil and paper D. Mostly in my head E. With numbers Table 4 Student Perceptions of Mathematical Activity Class type

Proportion (%)

Every day as a natural part of living

Algebra 2 IMP

49 52

Mostly at school

Algebra 2 IMP

63 64

With a pencil and paper

Algebra 2 IMP

41 42

Mostly in my head

Algebra 2 IMP

27 39

With numbers

Algebra 2 IMP

51 46

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

.0001*

Student Perceptions of Mathematical Activity

Response alternatives .0012*

p value

Table 4 is significant in the context of this paper in that it was only in these perceptions of mathematical activity that the IMP and Algebra students responded in a similar fashion. The marked differences in beliefs and perceptions reported by the two groups, which constitute the essential findings of this study, are only evident in Table 4 in the significantly greater inclination for IMP students to report mathematics as being a mental activity. Student Perceptions of Mathematical Ideas IMP students were more likely to agree that mathematical ideas could be clearly explained using

Clarke, Breed, & Fraser

11

every day words that anyone could understand, than were Algebra 2 students. IMP students were also less likely to view the ideas of mathematics as ones that can only be explained using numbers and language specific to mathematics. The IMP students were more likely to view mathematics as having developed in response to people’s needs. The IMP students were also less likely than the Algebra 2 students to view mathematics as having been invented by mathematicians or to hold that the ideas of mathematics have always and will always be true. Figure 1 and Table 5 document these differences. 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 daily life. 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 mathematical language and special terms. Agree Disagree H. Can be explained in everyday words that anyone can understand. Agree Disagree

In summary: IMP students were more likely to hold a socially-oriented view of the origins and character of mathematical ideas rather than a Platonist belief in the existence of mathematical absolutes awaiting discovery.

12

1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80 -1.00

Always True

Discovered Invented

Unreal

Developed

Special Terms Numbers Everyday Words

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

Figure 1. Students’ perceptions of the ideas of mathematics. Table 5 Students’ Perceptions of the Ideas of Mathematics Class Type

Sub-items

Mean

Have always been true and will always be true.

Algebra 2 IMP

.02 –.28

Were invented by mathematicians.

Algebra 2 IMP

–.13 –.36

Were discovered by mathematicians.

Algebra 2 IMP

.18 .01

Developed as people needed them in daily life.

Algebra 2 IMP

.57 .77

Have very little to do with the real world.

Algebra 2 IMP

–.72 –.82

Are most clearly explained using numbers.

Algebra 2 IMP

.26 –.02

Can only be explained using mathematical language and special terms.

Algebra 2 IMP

–.41 –.69

Can be explained in everyday words that anyone can understand.

Algebra 2 IMP

.21 .63

Problem-Based Mathematics Curriculum

Student Perceptions of School Mathematics The IMP students were significantly more likely to agree that writing was important in helping them to understand mathematics. The IMP students were also more likely to see value in talking to other students than were the Algebra 2 students. The IMP students were significantly less likely than the Algebra 2 students to view drill and practice as the best way to learn mathematics.

•

•

•

Sample Item: Circle the alternative which best describes how true you think each statement is (SA = Strongly Agree, A = Agree, D = Disagree, and SD = Strongly Disagree): 1. Explaining ideas clearly is an important part of mathematics. SA A D SD 2. Mathematics does not require a person to use very many words. SA A D SD 3. Writing is an important way for me to sort out my ideas in mathematics. SA A D SD 4. Talking to other students about the mathematics we are doing helps me to understand. SA A D SD 5. Drill and practice is the best way to learn mathematics. SA A D SD

The distinguishing characteristics between the IMP students and the Algebra 2 students were: • •

•

the importance attached by IMP students to writing in mathematics (p = .04*) the degree to which IMP students perceived talking to other students as useful in helping them to understand mathematics (p = .0005*) the relative importance attached to drill and practice by the Algebra 2 students (p = .0001*)

Student Perceptions of Mathematical Activity at School The greatest degree of difference between IMP students and the Algebra 2 students was evident in their perceptions of mathematical activity at school. Table 6 illustrates the differences in student perceptions of their mathematics classrooms. In these statistics, the differences between the two class types are most clearly illustrated. Key differences between IMP and Algebra 2 classes can be summarized as follows: Clarke, Breed, & Fraser

•

IMP students were significantly more likely to be writing words and drawing diagrams, and less likely to be writing numbers. IMP students were significantly more likely to be working with a friend or with a group, and less likely to be working on their own. While there was no difference between IMP and Algebra 2 classes in the relative frequency of listening to the teacher, IMP students were significantly more likely to be listening to other students than were students in Algebra 2 classes. IMP students were significantly more likely to be working from a worksheet and less likely to be copying from the board or working from a textbook.

Students were asked to respond on a four-point scale to the cue “When doing mathematics at school, I am likely to be...” The mean values in Table 6 should be read as students’ perceptions of the relative frequency (on a 5-point scale) with which they engaged in each of the listed activities. Table 6 Mean Relative Frequency of Student Engagement Class type

Mean

SD

p value

Talking

Algebra 2 IMP

2.25 2.59

1.0 .88

.002*

Writing numbers

Algebra 2 IMP

3.13 2.66

.86 .83

.0001*

Writing words

Algebra 2 IMP

1.70 2.68

.93 .94

.0001*

Drawing diagrams

Algebra 2 IMP

1.85 2.70

.82 .83

.0001*

Working on my own

Algebra 2 IMP

2.59 1.91

.90 .87

.0001*

Working with a friend

Algebra 2 IMP

2.10 2.69

.90 .80

.0001*

Working with a group

Algebra 2 IMP

1.96 3.17

1.0 .80

.0001*

Listening to the teacher

Algebra 2 IMP

2.80 2.75

.98 .96

.63

Listening to other students

Algebra 2 IMP

2.19 2.79

.96 .83

.0001*

Copying from the board

Algebra 2 IMP

2.40 1.91

1.02 1.04

.0001*

Working from a textbook

Algebra 2 IMP

3.15 0.25

.93 .70

.0001*

Working from a worksheet

Algebra 2 IMP

2.05 3.32

1.06 .97

.0001*

Sub-items

13

Student Perceptions of the Relative Importance of Course Components IMP students placed more value on working with others than did Algebra 2 students (p = .0001*). By contrast, Algebra 2 students valued the teacher’s explanations (p = .0005), and the textbook (p = .0001) more than did IMP students. Student Perceptions of Mathematics in Everyday Activity IMP students were significantly more likely to identify a mathematical component in everyday activities than were Algebra 2 students. This result is evident in Table 7. Table 7 Mean Math World Index for Algebra 2 and IMP Students (Incomplete responses from some students led to a slightly smaller sample size for both groups.) Class type

Mean

SD

Algebra 2 (n = 113)

19.292

5.591

IMP (n = 172)

• • • •

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 attitudes to mathematics of boys and girls in IMP and Algebra classes, and of the boys’ and girls’ self-ratings of their mathematics competence. These results are shown in Table 8. Girls in both class types were less likely than boys to rate highly their own mathematical competence. However this difference was only statistically significant for students in Algebra classes. Both boys and girls in IMP classes had similar positive attitudes towards mathematics. In Algebra classes, both male and female students felt negatively towards mathematics, however boys’ attitudes were less negative than those of girls. On the basis of these findings, it appears that the IMP program was of particular value to female students. The statistical 14

Class type & Measure Algebra 2 Self-rating

IMP Self-rating

Algebra 2 Attitude

5.598

In particular, the IMP students were more likely to view as mathematical: •

Table 8 Gender Comparison of Self-ratings and Attitude Measures for Algebra 2 and IMP Classes

p value .0014*

21.477

significance of the direct comparison of Algebra 2 girls with IMP girls is quite clear from Table 8, where the difference in mean attitude and self-rating for the two groups of girls is even more striking than in the comparison of the Algebra and IMP cohorts reported in Tables 2 and 3.

IMP Attitude

Gender

Mean

SD

Male (n = 58)

7.333

1.875

Female (n = 67)

6.433

1.994

Male (n = 77)

7.636

1.297

Female (n = 96)

7.271

1.410

Male (n = 58)

–0.345

1.821

Female (n = 68)

–0.676

1.872

Male (n = 78)

0.756

2.021

Female (n = 96)

1.146

p value

.0101*

.081

.3176

.2344 2.234

Conclusions For the purpose of drawing conclusions from the findings reported here, the inclusion of the Algebra 4 sample in the study encourages the extrapolation of conclusions from comparisons of class types at a specific grade level to more general conclusions comparing problem-based and conventional instruction for high school mathematics classes. The conclusions that follow, however, relate specifically to the study sample. The Students as Learners 1. IMP students rated themselves as significantly more mathematically able than did the Algebra students. 2. IMP students held a significantly more positive attitude towards their mathematics classes than did the Algebra students. 3. On school sites where comparison was possible, IMP students averaged higher SAT scores than did pupils of conventional classes. Problem-Based Mathematics Curriculum

4. IMP classes appeared to have less negative outcomes for girls than did conventional Algebra classes. Student Perceptions of Mathematics 5. 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 societal need; while Algebra students were more likely to view mathematical ideas as having an independent, absolute and unvarying existence. 7. The IMP students were significantly more likely to perceive mathematics as having applications in daily use. 8. IMP students were significantly more likely than Algebra students to believe that mathematical ideas can be expressed “in everyday words that anyone can understand.” Instructional Alternatives 9. IMP students attached significantly more value to interactive learning situations; whereas Algebra students valued “the teacher’s explanations” and “the textbook.” 10. IMP students valued writing and talking to other students as assisting their learning. Algebra students were significantly more likely to value “drill and practice.” 11. (a) It is possible to identify a coherent and consistent set of classroom practices which can be associated with conventional instruction (cf. Clarke, 1984). 11 (b) It is similarly possible to identify a set of classroom practices which identify, in the students’ view, the characteristics of the IMP classroom. 11 (c) The characteristics of these two instructional models are sufficiently distinct to represent clear alternatives. In conclusion, the classroom practices of the IMP program, as reported by the students, placed greater emphasis on a variety of modes of communication and on facilitating student-student interaction than was the case with conventional instruction. By contrast, conventional instruction was perceived as solitary, text-driven, and typically expressed through special terms and numbers. To what aspect of the IMP experience might we attribute the student beliefs documented in this study? The small-group, interactive classroom and the Clarke, Breed, & Fraser

problem-based mathematics curriculum represent two key characteristics of the Interactive Mathematics Project. Whether such belief systems would arise in interactive classrooms lacking a problem-based emphasis or in more conventionally taught, problem-based classrooms is a matter for further research. Certainly the IMP program has provided students with significantly different experiences from those found in conventional mathematics classes, and these experiences appear to have led to demonstrably different beliefs about mathematical activity, mathematics learning, school mathematics, and the mathematics evident in everyday activity. The findings of other studies suggest that students whose instruction has included experience with open-ended tasks can be expected to perform more successfully on both conventional and non-routine tasks than students lacking that experience (for instance, Sweller, Mawer & Ward, 1983). In combination, this research suggests that a problembased curriculum is capable of developing traditional mathematical skills at least as successfully as conventional instruction, while simultaneously developing non-traditional mathematical skills and engendering measurably different belief systems in participating students. The nature of these different beliefs has formed the basis of this study. REFERENCES Baird, J. R., & Mitchell, J. (Eds.). (1986). Improving the quality of teaching and learning: An Australian case study – the PEEL project. Melbourne: Monash University Printery. Bishop, A. J. (1985). The social construction of meaning – a significant development for mathematics education. For the Learning of Mathematics, 5(1), 24–28. California Post-Secondary Education Commission (CPEC). (1989). Request for program proposals under the Education for Economic Security Act, Federal Public Law 98-377 Fourth Year, San Francisco: California PostSecondary Education Commission and California State Department of Education. Clarke, D. J. (1984). Secondary mathematics teaching: Towards a critical appraisal of current practice. Vinculum, 21(4), 16–21. Clarke, D. J. (1985). The impact of secondary schooling and secondary mathematics on student mathematical behaviour. 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’s learning of mathematics. For the Learning of Mathematics, 7(1), 2–6.

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

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

Clarke, D. J., & McDonough, A. (1989). The problems of the problem solving classroom. The Australian Mathematics Teacher, 45(2), 20–24.

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

Problem-Based Mathematics Curriculum

The Mathematics Educator 2004, Vol. 14, No. 2, 17–26

Impact of Personalization of Mathematical Word Problems on Student Performance Eric T. Bates & Lynda R. Wiest This research investigated the impact of personalizing mathematical word problems using individual student interests on student problem-solving performance. Ten word problems were selected randomly from a mathematics textbook to create a series of two assessments. Both assessments contained problems exactly as they appeared in the textbook and problems that were personalized using student interests based on studentcompleted interest inventories. Fourth-grade students’ scores on the non-personalized and personalized problems were compared to investigate potential achievement differences. The scores were then disaggregated to examine the impact of reading ability and problem type on the treatment outcomes. The results showed no significant increase in student achievement when the personalization treatment was used regardless of student reading ability or word problem type (t = –.10, p = .46). .

“Problem solving is the cornerstone of school mathematics. Unless students can solve problems, the facts, concepts, and procedures they know are of little use” (National Council of Teachers of Mathematics, 2000, p. 181). Students can learn mathematical procedures, but without real-world applications, these skills are rendered meaningless and are forgotten readily. In the school curriculum, word problems allow one means by which students can work toward developing problem-solving skills within contextualized settings that do not require application of rote procedures. However, research has shown that students have difficulty solving word problems (Hart, 1996). At least three reasons have been proposed for why students 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 addressed in an effort to improve student performance on word problems, a fundamental component of mathematics education. Personalizing word problems—replacing selected information with students’ personal information—can address the latter two, motivation and relevance, which may in turn lead to the first, greater experience with word problems. Eric T. Bates is a fourth-grade teacher at Sunnycrest Elementary School in Lake Stevens, Washington. He holds a Master’s degree in elementary education with a concentration in technology in mathematics education. Lynda R. Wiest is an Associate Professor of Education at the University of Nevada, Reno. Her professional interests include K–8 mathematics education, educational equity, and teacher education.

Bates & Wiest

The purpose of this study was to investigate the impact of personalizing word problems on fourth-grade students’ problem-solving performance. Results of this research, conducted at Copper Flats Elementary School1 in Northern Nevada, were disaggregated to examine how reading ability and problem type might influence scores in solving personalized versus nonpersonalized problems. Review of Related Literature The Role of Word Problems Conventional word problems, despite their artificial nature, are likely to “stick around” in school mathematics (D. Brummett, personal communication, February 29, 1996; Sowder, 1995; J. Stephens, personal communication, February 29, 1996). This may be due to their strong grounding in tradition, their potential for fostering mathematical thinking, their ease of use (e.g., conciseness and practicality within the confines of school walls), and a lack of abundant and pragmatic alternatives. Word problems may, in fact, serve several important functions in the mathematics classroom: They provide questions that challenge students to apply mathematical thinking to various situations, and they may be an efficient means of relating this thinking to the real world. Practically speaking, word problems are either readily available in mathematics texts or can be written in a short period of time, which makes them useful to time-conscious teachers (Fairbairn, 1993). Personalization and Student Interest The idea of individualizing instruction certainly is not new. Almost a quarter of a century ago, Horak (1981) stated, “Meeting the educational needs of the 17

individual student has long been a concern of professional educators” (p. 249). Personalizing instruction to student experiences and interests is one way to individualize instruction that may be important for mathematics learning (Ensign, 1997). In particular, it can enhance interest and motivation, which are critically important factors in teaching and learning. Mathematical word problems have been targeted for personalization. Students “don’t care how many apples Bob gave to Suzy. They’re much more interested in things like music, video games, movies, trading cards, money, and friends” (Bailey, 2002, p. 61). Giordano (1990) adds, “student fascination with problems can be enhanced when names, locations, and events are changed to personal referents” (p. 25). It is important that word problems appeal to students in order to generate interest in and motivation for solving a problem (Fairbairn, 1993; Hart, 1996) However, in practice, classroom mathematics rarely links to students’ life experiences (Ensign, 1997). Research on Personalized Word Problems Numerous studies have investigated the impact of personalizing problems—inserting individual students’ names and/or information from their background experiences into the problems they solve—on student interest/motivation and problem-solving success. Personalized problems have been computer-generated in some cases. Most of these studies found positive effects 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 credit personalization of word problems with positively influencing student affect, such as interest and motivation. Hart (1996) notes, “Most students are energized by these problems and are motivated to work on them” (p. 505). Davis-Dorsey et al. (1991) say personalization fosters and maintains attentiveness to problems, and Jones (1983) claims that personalized problems invest students in wanting to solve them correctly. López and Sullivan’s (1992) research found individual personalization (tailoring problems to individual rather than whole-class interests) to be particularly effective in fostering positive attitudes toward word problems. However, Ku and Sullivan’s (2002) study involving 136 fourth-grade Taiwanese 18

students and their teachers also found group personalization to have a positive impact. Both students and teachers using personalized problems showed better attitudes toward the program than those using non-personalized word problems. Ku and Sullivan argue that familiarity (reduced cognitive load) and interest are the major factors that lead to greater success solving personalized versus non-personalized problems. Another major area where personalization of word problems has yielded favorable results is student understanding. Davis-Dorsey et al. (1991) say personalization supports development of meaningful mental representations of problems and their connections to existing schemata, and that it creates strong encoding that aids retrieval of knowledge. Personalized word problems may be more meaningful in general and make contexts more concrete and more familiar (López & Sullivan, 1992). Familiar people and situations in personalized problems can aid understanding (Davis-Dorsey et al., 1991; López & Sullivan, 1992). In their research, d’Ailly et al. (1997) employed a type of personalization known as self-referencing. A variety of problems were taken from a standard mathematics text and some of the character names were replaced with the word you. One hundred students in grades three, four, and five were asked to solve the problems within a mix of self-referencing and non-self-referencing problems. The researchers found, “When a you word was involved in the problem, children asked for fewer repeats for the problems, and could solve the problems in a shorter amount of time and with a higher accuracy” (p. 566). As noted, d’Ailly et al.’s (1997) study found that personalized word problems (specifically, those using self-referencing) positively impacted student achievement—the third main area where word problem personalization can benefit students. Numerous other researchers have attained similar results in this area, although some findings demonstrate positive effects in some cases but not others, as some of the following studies show. For their study, Ku and Sullivan (2002) personalized problems using the most popular items—as determined by a completed interest survey—for students as a whole. Students attained higher problem-solving scores on personalized problems both on the pretest and on the posttest (i.e., before and after instruction). The 53-minute interim instruction and review used either personalized or nonpersonalized problems. Students who worked with Personalization of Mathematical Word Problems

personalized problems performed better on both personalized and non-personalized problems than those who received non-personalized instruction, suggesting that transfer of learning had occurred from the personalized to non-personalized problems. Davis-Dorsey et al. (1991) studied the effects of personalizing standard textbook word problems on 68 second-grade students and 59 fifth-grade students. Prior to the treatment, all of the students completed a biographical questionnaire that was later used to develop the personalized problems. Personalization proved to be highly beneficial to the fifth graders, but it did not positively impact the second-grade students’ test scores. Wright and Wright (1986) researched the use of personalized word problems with 99 fourth-grade students. They examined both the processes used to solve the problems and the accuracy of the answers. Interestingly, the researchers found that a correct process was chosen more often when the problems were personalized, but correct and incorrect answers were given equally on personalized and nonpersonalized problems. López and Sullivan (1992) found significant differences favoring personalization on problemsolving scores for two-step but not for one-step problems, although the seventh graders in their study also scored higher on the latter in comparison with non-personalized problems. The researchers say personalization may be particularly important for more demanding (e.g., unfamiliar or mathematically complex) cognitive tasks. They found personalization to be effective on a group basis—personalizing problems using dominant interests of a group of students—as well as on an individual basis—personalizing problems for each student using individual interests—in relation to students’ problemsolving scores. Most evidence indicates that personalizing word problems can be an effective technique in teaching and understanding mathematical word problems. Nevertheless, some research data suggest caution in assuming that personalization of word problems always yields positive results. As noted, López and Sullivan (1992) found significant differences favoring personalization for two-step but not one-step problems, and the Wright and Wright (1986) study showed no significant improvement in student achievement on personalized word problems, even though students more often chose appropriate solution strategies for personalized problems. Ross, Anand, and Morrison (1988) raise other issues for consideration. They Bates & Wiest

suggest that the effectiveness of the personalized treatment may wear off over time. The researchers express concern that the higher scores on personalized tests could be due, in part, to the novelty of the personalization and that the novelty might dissipate if the treatment were used often. They also point out that preparing individualized materials could limit its use in the classroom due to time constraints. Finally, in their research with 11-year-olds, Renninger, Ewen, and Lasher (2002) found that personalized contexts based on individual interests can have a differential effect on students. For the most part, these contexts encourage students to connect with the meaning of problems. This leads some students to consider a task more carefully to be sure they understand it. However, it leads others to assume falsely that they have answered a problem correctly, which hinders a “healthy skepticism” that encourages problem solvers to check their work after completing problems. More research is needed to address how problem type interacts with word problem personalization, where personalization has its greatest impact—student attitude, understanding, or achievement, grade levels and types of students that are most responsive to personalization, the long-term effects of personalization, and the potentially differential impact of individual versus group personalization. Ku and Sullivan (2002) also call for future research on tapping technology’s potential for creating personalized problems and on investigating the implications of using personalized problems for assessment. Research Purpose The purpose of this exploratory study was to investigate if the predominantly positive research results concerning personalization of mathematical word problems would apply to elementary school students regardless of reading ability or word problem type. The intent was to contribute to the body of knowledge about the impact of personalizing word problems and to extend previous explorations by considering particular student subgroups and problem types (simple translation and process, discussed under Instrumentation). If the benefits of personalization were to outweigh the time constraints of planning and preparing for this type of activity, the use of personalized mathematical word problems could be an effective tool for elementary teachers working with students who struggle to understand and solve word problems.

19

Research Method Sample Participants in this study were fourth-grade students at Copper Flats Elementary School. Copper Flats is a small desert community in rural Northern Nevada. The school receives federal Title 1 money, reflecting the fact that Copper Flats Elementary serves students from a lower-income community. The school houses four fourth-grade classrooms. Students who returned parental consent forms in all four of these classrooms were selected for the present study. Ninety-seven parental consent forms were distributed. Of those, 42 were returned in time for the study. Therefore, the sample included 42 students— 22 boys and 20 girls. Students not participating in the study worked on classroom assignments given by their regular teacher, while the participants completed the assessments. All participants in this project were present for the two data-collection sessions. By reading ability, 20 participants ranked high, 8 ranked medium, and 14 ranked low. Research Design This study was a quantitative analysis of the effects of personalizing word problems on fourth-grade students’ achievement in solving the problems. In the fall of 2002, participants completed an interest inventory to provide individual information for personalizing the assessments. One week later, participants were administered an instrument containing 10 word problems to solve. On that assessment, 5 problems were personalized and 5 were not. Two weeks later, participants were given a similar 10-item instrument. On this second and final assessment, parallel versions of the 5 problems that had been non-personalized on the initial instrument were personalized, and vice versa. Therefore, all participants—across the two test administrations in which they took part—answered 20 problems, 10 personalized problems and 10 similar problems that were not personalized. The two-week period between the two tests provided necessary time to reduce threats to validity due to repeated testing of participants on similar test items (Parsons & Brown, 2002). During each test administration, each participant was given an instrument and a blank sheet of paper on which to solve the problems. Participants were allowed 15 minutes to complete each assessment. All participants finished within the allotted time. Teacher-reported scores on the Developmental Reading Assessment (DRA) (Beaver, 2001) 20

established participants’ reading level for the purpose of comparing achievement on personalized versus nonpersonalized problems in relation to reading ability. DRA levels 30 to 38 are considered to be third-grade reading ability, DRA level 40 is fourth-grade reading ability, and DRA level 44 is fifth-grade reading ability. For this study, participants with DRA scores higher than 40 were considered high readers, or above grade level. Participants with DRA scores at 40 were considered m e d i u m readers, or at grade level. Participants with DRA scores below 40 were considered low readers, or below grade level. This research was not designed to include a qualitative component. However, student comments were recorded as field notes on the few occasions where students made relevant, unsolicited remarks. Instrumentation Ten problems were randomly selected from Mathematics: The Path to Math Success (Fennell et al., 1999), the third-grade mathematics text, for use in developing the assessment instruments (see Appendices B and C). This text was chosen because it was the text used for teaching third-grade mathematics at Copper Flats Elementary School; therefore, the participants were familiar with the format of the problems. The problems were selected by scanning every third page of the text that contained word problems. Of the word problems selected from those pages, five of each of the two problem types described below were drawn from a basket and incorporated into the assessments. The problems selected for the assessments were differentiated by problem type. Five of the problems selected for the assessments were simple translation problems and five were process problems (L. R. Wiest, personal communication, August 27, 2002). Simple translation problems can be solved using a one-step mathematical algorithm. An example of a simple translation problem is: “There are 7 seats in each of 6 vans. How many seats are there in all?” (Fennell, et al., 1999, p. 360). Process problems typically are not solved through direct application of an algorithm. Another strategy is generally sought and chosen, such as working backward, drawing a picture or diagram, or using guess-and-check. A sample process problem is: “Jen is older than Arnie. Paul is older than Jen. Who is the oldest?” (p. 313). An interest inventory (see Appendix A) was created to determine selected participant preferences. Inventory items included students’ name, favorite toy, favorite store, something to buy at that store, names of Personalization of Mathematical Word Problems

friends, something they like to make, name of a game, and favorite type of vehicle. Each inventory was used to personalize the original textbook word problems. Two assessments were developed from the word problems taken from the mathematics text. Items from the interest inventory replaced the original characters, objects, and situations in order to personalize the problems for each individual student on five of the ten problems on each of the two instruments. On one assessment, the odd-numbered problems were personalized (see Appendix B). On the other, the evennumbered problems were personalized (see Appendix C). Participants randomly received one instrument on the first administration and the other instrument on the second administration two weeks later. This method of alternating personalized and non-personalized items on the assessments was shown to be an effective technique used in other research on this topic (DavisDorsey et al., 1991; d’Ailly, Simpson, & MacKinnon, 1997). Data Analysis A paired-samples t test (available online at http://faculty.vassar.edu/lowry/tu.html) was performed to compare the number of correct answers on personalized versus non-personalized problems. Mean scores and standard deviations were calculated and significance was tested at the .05 level using a onetailed test. This analysis included 42 pairs of scores. An additional paired-samples t test was performed to compare the number of correct responses on personalized versus non-personalized items disaggregated by participants’ predetermined reading levels. Again, mean scores and standard deviations were calculated and significance was tested for at the .05 level using a one-tailed test. There were 20 pairs of scores at the high level, 8 pairs of scores at the medium level, and 14 pairs of scores at the low level. Two final paired-samples t tests were performed to compare the number of correct responses on personalized versus non-personalized test items disaggregated by problem type. Each assessment contained five simple translation problems and five process problems. On the first of these two pairedsamples t tests, correct responses on the personalized simple translation problems were compared to correct responses on the non-personalized simple translation problems. On the second of the two paired-samples t tests, correct responses on the personalized process problems were compared to correct responses on the non-personalized process problems. Through these statistical methods, mean scores and standard Bates & Wiest

deviations were calculated with significance tested at the .05 level using a one-tailed test. Both of these analyses included 42 pairs of scores. Results Mean scores for the number of items answered correctly out of ten showed a difference of .03 points between the personalized and non-personalized problems (see Table 1). This difference was not statistically significant (t = –.10, p = .46). Table 1 Paired-Samples t Test for Personalized and NonPersonalized Problems Context

n

Mean

SD

t

p

Personalized Non-personalized

42 42

5.26 5.29

2.07 2.28

–.10

,46

Table 2 provides mean scores for the number of problems answered correctly out of ten, separated by student reading level. The high-reader scores for nonpersonalized problems were .10 points higher than for personalized problems, a nonsignificant difference (t = –.26, p = .39). The medium-reader scores for nonpersonalized and personalized problems differed by .50 points, also favoring non-personalized problems. A paired-samples t-test indicated that this difference was not significant (t = –1.08, p = .15). The low-reader scores were .35 points higher for personalized problems than for non-personalized problems. This was the only group who attained better scores on personalized problems, although the scores were not significantly higher (t = –.84, p = .20). Table 2 Paired-Samples t Test for Personalized and NonPersonalized Scores by Reading Ability Personalized

NonPersonalized

Reading Group

n

Mean

SD

Mean

SD

t

p

High Medium Low

20 8 14

5.90 5.50 4.21

1.77 2.07 2.19

6.00 6.00 3.85

2.15 2.14 1.96

–.26 –1.08 –.84

.39 .15 .20

Mean scores for the number of problems answered correctly out of 5 were separated by problem type (see Table 3). Scores for simple translation problem means were 0.16 points higher for non-personalized than for personalized problems. A paired-samples t-test showed that these differences were not significant (t = –.84, p = .20). Process problem means showed a difference of 0.1 points between the personalized and non21

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

NonPersonalized

Problem Type

n

Mean

SD

Mean

SD

t

p

Simple Trans.

42

2.41

1.56

2.57

1.74

–.84

.20

Process

42

2.86

0.98

2.76

1.12

.45

.32

Discussion The results of this study suggest that students are no more successful answering word problems when the word problems are personalized and reflect their areas of interest than when the problems are taken verbatim from a mathematics textbook. Only in the subgroup of low-reading-level students and the subcategory of process problems did the personalized problem scores improve slightly, although statistically significant differences were not found in either case. The mean scores in each other subgroup and subcategory were somewhat lower on the personalized versions of the word problems than on the non-personalized versions. These research results point to a different conclusion than many previous studies on this topic. However, given the rather substantial amount of previous research weighted toward positive effects of personalizing word problems and the reasons explained below, it is still quite possible that personalized word problems can be a beneficial part of school mathematics programs. Several factors may have caused the lack of positive findings in this study. First, the personalized problems may not have adequately addressed the three aforementioned reasons students fail at mathematical word problems. Second, the age of the students may have been a factor in the treatment’s lack of success. Third, this study looked only at comparisons of personalized and non-personalized problems on assessments. No attempt was made to introduce personalization as an instructional practice. The three reasons offered earlier for why students fail at solving mathematical word problems were limited student experience with word problems (Bailey, 2002), lack of motivation to solve the word problems (Hart, 1996), and irrelevance of word problems to students’ lives (Ensign, 1997). The format of this study could not—and did not intend to—have 22

much impact on student experience with word problems. By simply taking two 10-problem assessments, student exposure to word problems was not greatly increased. Increased motivation was noticed, however, when students saw their names or favorite things included in a problem. On several occasions while completing the assessments, students made comments such as, “Hey, this has my name,” or “These problems are fun ones.” This acknowledgement and the smiles that followed were taken as signs of increased student motivation. It was anticipated that by utilizing student names and other referents to student lives, relevance would be increased. This may have been the case to an extent, but just seeing their names and favorite things may not have given the problems enough personal context to encourage correct answers. In effect, the ability of this study to address the three major reasons students fail at solving word problems was not substantial or sustained enough to help distinguish performance on the two problem contexts. Personalized problems per se might not be advantageous unless they are an integral part of a larger instructional effort. The young age of the students may also have contributed to the results of the present study. These students fall at the lower end of the grade levels previously researched on this topic. Most studies that found positive results for personalized problems took place 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 two studies included younger grades—second and third—among the older grades they investigated (Davis-Dorsey et al., 1991; d’Ailly, et al., 1997). The present study dealt exclusively with fourth-grade students and found no relationship between personalization and student scores. Perhaps fourth grade is somewhat early for the personalization treatment to be effective. Interest in problem contexts may become more important across the many years in which students encounter school word problems. In relation to their study involving the impact of word problem context, Parker and Lepper (1992) state that it is “clear that the need for techniques to enhance student interest in traditional educational materials may actually increase with age” (p. 632). Advancing grade levels also deal with increasingly difficult mathematics problems, the complexity of which may allow for a factor such as personalization to influence student performance. As noted earlier, López and Sullivan Personalization of Mathematical Word Problems

(1992) found personalization to have a positive impact on two-step but not one-step problems, leading them to conclude that personalization may be particularly important for more difficult problems. Use of thirdgrade problems in this study may also have reduced the cognitive demands of this research task, creating less sensitivity to or discrimination among problem variations. Several previous studies found a significant increase in correct answers on mathematical word problems when students were taught with the personalized format (e.g., Anand & Ross, 1987; López & Sullivan, 1992). After the instructional period, participants in these studies were assessed using standard word problems. The present study sought to discover the effects of the personalized format on student achievement on the test items themselves without prior instruction using these types of problems. Perhaps these two approaches yield different results. Students may need time to adjust to the new problem contexts. One benefit that did appear in using this treatment was student excitement. Similar to the Ross, McCormick, and Krisak study (1985), many participants were visibly and audibly excited to discover the personalized problems. In informal discussions after the test administrations, participants reported that they really liked reading about themselves and their friends. They enjoyed seeing familiar stores and games they like to play in this testing situation. This affirms Hart’s (1996) reference to the personalized treatment that “students are energized by these problems” (p. 505). It must be recalled, however, that interest in problems can be detrimental to some students, who may incorrectly assume that they have attained correct answers (Renninger et al., 2002). Also, too much interest in a problem context can distract some students, particularly girls (Boaler, 1994; Parker & Lepper, 1992). If these potential negative effects took place in this study, they might have countered and thus masked potentially positive effects in the overall results. Limitations of the Study The two major limitations of this study were the sample size and the somewhat simplistic nature of the research design. The sample size was reduced due to the small number of parental consent forms that were signed and returned so that students could participate in the study. Ninety-seven consent forms were distributed, but only 42 (43%) were signed and returned in time for the first test administration. (Time Bates & Wiest

constraints prohibited a second distribution of consent forms, which might have raised the return rate.) This greatly reduced the sample size, thus limiting the power of the data used to determine the effectiveness of the treatment. This study was also limited by its lack of complexity. Merely assessing student performance based on two test administrations was restrictive. It only gave a look into the results of those two tests. It would be interesting to discover how students might perform on word problems when they were taught with the personalized format. Time and other constraints did not allow for this additional research component. Analysis of solution strategies might have yielded further information. It is also difficult to know what long-term impact the motivational aspects of these problems may have. Implications for Further Research This study, in conjunction with the professional literature discussed earlier, yields at least three major implications for future research. • •

•

The potential of personalization of word problems as an instructional method should be studied. Alternative technologies should be explored to decrease the time-intensive nature of preparing individualized word problems. Longitudinal research should be conducted on the impact of personalizing mathematical word problems.

As a teaching strategy, personalization of mathematical word problems has been shown to increase student achievement, particularly in the upper elementary and middle grades (e.g., Anand & Ross, 1987; López & Sullivan, 1992). While this study did not find such results for assessment problems, which may be due to the mitigating factors discussed earlier, it did find some anecdotal evidence that supported other research findings of increased interest in these problems. This might be an important underpinning of mathematics learning. Personalization as an instructional strategy could be implemented at various grade levels and studied to assess its effectiveness for students of those ages. Rather than comparing test items only, as the present study did, students could be taught with the personalization treatment and assessed on standard textbook word problems to determine the level of transferability of any possible positive effects. This instructional method may increase student motivation and interest when learning how to solve problems in mathematics, thereby increasing their 23

comprehension of the material and increasing their scores on textbook and other assessments. In order to employ personalization as a teaching strategy—based on the assumption that it may yield positive results in affect, understanding, and/or achievement—alternative methods of personalizing word problems would be needed to decrease the amount of time researchers and educators spend creating personalized problems. One such method might utilize the Internet. The capability of the Internet to deliver individualized materials immediately and simultaneously to a large population of students remains untapped. A researcher could develop a web site that allows students to complete an interest inventory online and then submit the inventory to the server. The server would then apply that information to an existing word problem template document, updating the characters and other referents to individualize the problems for each student. This process would take only seconds and would eliminate researchers’ (and later teachers’) time investment in personalizing individual worksheets and tests. Students could then either print the problems or complete them on the computer screen. The preparation time would be greatly reduced and the number of participants could be increased significantly. This technique would allow researchers to create countless individualized word problems for student instruction, practice, and assessment. Such research should include attention to what types of problems lend themselves well to this type of problem generation. In the research reported in this paper, for example, problems using names were the easiest to personalize, with difficulty increasing where gender-specific pronouns were included. The process problems seemed to require greater attention than the simple translation problems in preparing personalized problems, mirroring the greater mathematical complexity of the former compared with the latter. The present study and similar earlier studies discussed here have been shorter than three months in duration. Long-term effects of the personalizing strategy have not been determined at any educational level. Researchers might look at the use of personalized word problems in a classroom over the course of a school year and its relationship to word problem achievement on standardized tests. Closing Thoughts Personalization of mathematical word problems may not be an efficacious approach in fourth-grade classrooms due to the age of the students and the 24

simplistic nature of the word problems the students are required to complete. This should not, however, discount other research on the personalization of educational materials. Other researchers have shown personalization to be an effective method in teaching older students to solve mathematical word problems. Excitement and interest tend to be rare when students are working on word problems. Fairbairn (1993) suggested that the terms story problems and word problems can invoke uncomfortable memories for many people. This may be due to the fact that word problems can be boring and tedious to solve. Unfortunately, student motivation is difficult to quantify. In the present study, as well as in others, the excitement level of individual students visibly and audibly rose when personalized problems were presented. At the very least, personalization could be used as an instructional strategy to break the monotony of word problems containing unknown people, dealing with unfamiliar situations, asking uninspiring questions. REFERENCES Anand, P. D., & Ross, S. M. (1987). Using computer-assisted instruction to personalize arithmetic materials for elementary school 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. Upper Arlington, OH: Celebration Press. Boaler, J. (1994). When do girls prefer football to fashion? An analysis 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 a math compare problem? Journal of Educational Psychology, 89(3), 567–567. Davis-Dorsey, J., Ross, S. M., & Morrison, G. R. (1991). The role of rewording and context personalization in the solving of mathematical word problems. Journal of Educational Psychology, 83(1), 61–68. Ensign, J. (1997, March). Linking life experiences to classroom math. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL. Fairbairn, D. M. (1993). Creating story problems. Arithmetic Teacher, 41(3), 140–142. Fennell, F., Ferinni-Mundy, J., Ginsburg, H. P., Greenes, C., Murphy, S., Tate, W., et al. (1999). Mathematics: The path to math success. Parsippany, NJ: Silver Burdett Ginn. Giordano, G. (1990). Strategies that help learning-disabled students solve verbal mathematical problems. Preventing School Failure, 35(1), 24–28. Hart, J. M. (1996). The effect of personalized word problems. Teaching Children Mathematics, 2(8), 504–505.

Personalization of Mathematical Word Problems

Horak, V. M. (1981). A meta-analysis of research findings on individualized instruction in mathematics. Journal of Educational Research, 74(4), 249–253. Jones, B. M. (1983). Put your students in the picture for better problem solving. Arithmetic Teacher, 30(8), 30–33. Ku, H. Y., & Sullivan, H. J. (2002). Student performance and attitudes using personalized mathematics instruction. Educational Technology Research and Development, 50(1), 21–34. López, C. L., & Sullivan, H. J. (1991). Effects of personalized math instruction for Hispanic students. Contemporary Educational Psychology, 16, 95–100.

Appendix A: Interest Inventory Favorite 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

________________

____________________

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

Appendix B: Sample Assessment

National Council of Teachers of Mathematics. (2001). Principles and standards for school mathematics. Reston, VA: Author. Parker, L. E., & Lepper, M. R. (1992). Effects of fantasy contexts on children’s learning and motivation: Making learning more fun. Journal of Personality and Social Psychology, 62, 625–633. Parsons, R. D., & Brown, K. S. (2002). Teacher as reflective practitioner and action researcher. Belmont, CA: Wadsworth. Renninger, K. A., Ewen, L., & Lasher, A. K. (2002). Individual interest as context in expository text and mathematical word problems. Learning and Instruction, 12, 467–491.

(odd numbered problems personalized) 1.

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

2.

Tom has a ball. He passes it to Wally, and Wally passes it to Anne. Anne passes it back to Tom. If they continue in this order, 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? It’s the grand opening of Futura Florists! Every day for 8 days they give away 50 roses. How many roses in all do they give away?

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

4.

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

5.

Josh read 67 pages of a book. Jon read 32 pages. How many more pages did Josh read than Jon?

6.

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

7.

A toy maker can put together 1 Gameboy™ every 6 minutes. How many Gameboys™ can he put together in 60 minutes? There are 7 seats in each of 6 vans. How many seats are there in all?

Ross, S. M., McCormick, D., & Krisak, N. (1985). Adapting the thematic context of mathematical problems to students’ interests: Individual versus group-based strategies, Journal of Educational 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. Educational Communication 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 for teaching mathematics in the middle grades (pp. 121–142). Albany: NY, State University of New York Press.

8. 9.

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

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

Wright, J. P., & Wright, C. D. (1986). Personalized verbal problems: An application of the language experience approach. Journal of Educational Research, 79(6), 358–362. 1

Pseudonym.

Bates & Wiest

25

Appendix C: Sample Assessment (even numbered problems personalized) 1.

2.

Four students are collecting empty soda cans. Meg has more than Jo but fewer than Sid. Bart has the same number as Meg. Who has the greatest number of cans so far? Josh has a ball. Josh passes it to Jon, and Jon passes it to Robby. Robby passes it back to Josh. If they continue in this order, who will catch the ball on the 10th throw?

3.

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

4.

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

5.

Wendy read 67 pages of a book. Ellie read 32 pages. How many more pages did Wendy read than Ellie? Josh, Jon, Robby, and Warren are playing Battleship™. They take turns playing Battleship™ in pairs. If each kid plays one game of Battleship™ with each of the other kids, how many games do they play in all?

6.

7. 8.

A toy maker can put together 1 toy robot every 6 minutes. How many toy robots can he put together in 60 minutes? There are 7 seats in each of 6 Toyotas™. How many seats are there in all?

9.

Jen is older than Arnie. Paul is older than Jen. Who is the oldest of the three? 10. Josh made dented cars to sell at the fair. Josh made 1 on Monday, 2 on Tuesday, 3 on Wednesday, and so on, until Saturday. How many did Josh make on Saturday?

26

Personalization of Mathematical Word Problems

The Mathematics Educator 2004, Vol. 14, No. 2, 27–33

Mathematics Placement Test: Helping Students Succeed Norma G. Rueda & Carole Sokolowski A study was conducted at Merrimack College in Massachusetts to compare the grades of students who took the recommended course as determined by their mathematics placement exam score and those who did not follow this recommendation. The goal was to decide whether the mathematics placement exam used at Merrimack College was effective in placing students in the appropriate mathematics class. During five years, first-year students who took a mathematics course in the fall semester were categorized into four groups: those who took the recommended course, those who took an easier course than recommended, those who took a course more difficult than recommended, and those who did not take the placement test. Chi-square tests showed a statistically 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 those who take a higher-level course or do not take the placement exam. With achievement in coursework as the measure of success, we concluded that the placement test is an effective tool for making recommendations to students about which courses they should take.

There is a widespread recognition of the need for appropriate placement in the mathematics courses for undergraduate freshmen. Many colleges and universities around the nation have used the Mathematical Association of America (MAA) Placement Test; others have designed their own exams or used a combination of placement exams and other measurements, such as ACT or SAT mathematics scores and high school GPAs. Since the MAA has discontinued its placement test program, the responsibility has been put on individual institutions to develop their own placement exam. The purpose of our study was to determine the effectiveness of Merrimack College’s placement test by examining the connection between students enrolling in recommended courses and their success in those courses. Literature Review In this section, we investigate some of the specific methods reported in the literature for placing undergraduates in their first mathematics course. We point to some of the assumptions in these reports and suggest some of the drawbacks to the method of placement. Cederberg (1999) reported on the three placement tests administered at St. Olaf College in Minnesota. Norma G. Rueda is a professor in the Department of Mathematics at Merrimack College, North Andover, MA. Her research interests include mathematical programming and applied statistics. Carole Sokolowski is an Assistant Professor in the Department of Mathematics at Merrimack College, North Andover, MA. Her research area is undergraduate mathematics education.

Rueda & Sokolowski

She explained that the placement recommendations were based on a large number of regression equations that required considerable expertise in development and periodic redefinition. The placement test also required the coordination of numerous categories of student data used in the equations. Approximately 85% of students who enrolled in a calculus course based on the 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 was less technically sophisticated than St. Olaf’s, but still required considerable background data about students. They recommended multiple criteria methods, which included a placement test customized to an institution’s curriculum. They started with sixty variables, and found the eight best predictors: high school graduation status, number of hours employed, units planned, age, high school grade point average, mathematics placement test score, reading placement test score, and English placement test score. Cohen et al.’s study was based on (a) thousands of surveys completed by students and faculty members at eight California community colleges, (b) a comparison between student scores on assessment tests and grades in different courses, and (c) the relationship between student characteristics and their grades. Krawczyk and Toubassi (1999) described a simpler placement procedure used by the University of Arizona. The University of Arizona used two placement tests adapted from the 1993 California Mathematics Diagnostic Testing Project (see http://mdtp.ucsd.edu/). Students chose which test they felt was most appropriate for their ability and choice of 27

major. One test covered intermediate algebra skills and placed students in one of three levels of algebra or a liberal arts mathematics course. The second test covered college algebra and trigonometry skills and placed students in courses from finite mathematics through Calculus I. In the fall of 1996, their data indicated that approximately 17% of freshmen placed in College Algebra through Calculus I failed or withdrew from their respective courses, compared with a 50% attrition rate in the early 1980’s before the mandatory testing and placement. Apart from a test, they also considered other factors, such as high school GPA. A number of studies have investigated the use of standardized tests, such as ACT and SAT. Bridgeman and Wendler (1989) found that the mathematics SAT score was a relatively poor predictor of grades compared to placement exams. Their results were based on grades from freshman mathematics courses at ten colleges. Odell and Schumacher (1995) showed that a placement test used in conjunction with mathematics SAT scores could be a better predictor than SAT scores alone. Their conclusion was based on data from a private business college in Rhode Island. Callahan (1993) studied the criteria followed at Cottey College in Missouri to place students in the appropriate level course, as well as their placement success rates. As with the studies mentioned above, Cottey College used several variables to achieve their results – the MAA Placement Test, ACT and SAT mathematics scores, and years of high school mathematics taken. Each of these studies assumed that the success rates were based on students following the placement advice. Mercer (1995), on the other hand, conducted a study to compare pass rates in a college-level mathematics class for mathematically unprepared students who enrolled in developmental courses and those who did not. The results of this study showed a statistically significant relationship between passing the course and following placement advice. Background Merrimack College, located in North Andover, Massachusetts, is a small four-year Catholic college offering programs in the liberal arts, business, the sciences, and engineering. Among the college’s distribution requirements, students must complete three mathematics or science courses, with no more than two courses from the same department. Most of the students take at least one mathematics course. Most liberal arts majors usually choose Basic Statistics, Finite Mathematics, or Discrete Mathematics to satisfy 28

the mathematics/science requirement. During data collection for this study, business administration majors were required to take Applied College Algebra, Calculus for Business, and one other mathematics course. Students majoring in science or engineering generally were required to take more mathematics; for example, engineering students were required to take three calculus courses and one course in differential equations. They also took Precalculus if they did not place out of this course on their placement exam. Since some incoming freshmen are not prepared to take a college-level mathematics course, a non-credit developmental mathematics course, Math I, has been offered at Merrimack College since the fall of 1994. Before 1994 we administered a mathematics diagnostic exam to incoming students, but were unable to accommodate students who were not ready to take a college-level mathematics course. Instead, they enrolled in a mathematics course at a higher level than their exam score indicated they should. There was a high failure rate among these students. Because students at Merrimack College often have difficulty in other courses if they have not completed Math I, proper mathematics placement has become important to all of our departments. For example, the Chemistry Department now requires the students who place into Math I to complete the course before they take several of their chemistry courses. Krawczyk and Toubassi (1999) have found similar results at the University of Arizona in which all chemistry students and 90% of biology students—whose placement test scores indicated they should be placed in intermediate algebra or lower—received grades below C– in their chemistry or biology courses. This interest in correct mathematics placement extends beyond the chemistry department as evidenced by the many questions from the business and liberal arts faculty, as well as science and engineering faculty at the meetings in preparation for orientation advising. A major reason for our concern about student success is that successful students are more likely to remain in their studies. A high level of student retention is not only academically and socially desirable at a school, but it makes sense economically. We do not place students according to their prior high school GPA or whether they have taken calculus. We do not assume that these factors indicate whether or not they need algebra. In fact, many students placed into Math I have had four years of high school mathematics, including precalculus (and occasionally calculus), but according to our placement test they do not appear to understand the basic concepts of algebra. Mathematics Placement Test

Mathematics Placement Test All incoming freshmen at Merrimack College are expected to take a mathematics placement test developed by members of the Mathematics Department. There are two versions of the exam, one for students who will major in Business Administration or Liberal Arts, and one for students in Science and Engineering. The version for Business Administration and Liberal Arts consists of two parts, elementary and intermediate algebra. The version for Science and Engineering students contains a third part that tests students’ understanding of functions and graphs. Students are instructed not to use calculators. From 1994 until 1999 the placement exam was taken at Merrimack College in June during orientation or at the beginning of the academic year. Since 2000 the exam has been mailed to students at home. A Scantron form with the students’ answers is mailed back to Merrimack College and graded. Students are informed that using outside resources or calculators may result in their being placed in a course for which they are unprepared and may result in their failing or withdrawing from the course. There is not a difference between these mail-in results and the previous monitored exam results with respect to the percentages of students who place into the various mathematics courses. Thus, we believe that most students heed our warnings. The recommendations are available to students and advisors during June orientation. See Appendix A for some problems similar to those given in the mathematics placement test. Part I of the placement test consists of seventeen questions on elementary algebra. If a student does not answer at least fifty percent of these questions correctly, then that student must take Math I. For students who score above fifty percent on Part I, liberal arts majors may take any mathematics elective course; business administration majors are placed into a college algebra or a business calculus course, depending on their overall score; and science and engineering students may be placed into a college algebra, Precalculus or Calculus I course, depending on their total score. The specific recommendations resulting from the mathematics placement exam are as follows: Science and Engineering: Score below 9 in Part I ⇒ Math I For 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 Rueda & Sokolowski

Business: Score below 9 in Part I ⇒ Math I For 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 Business

Liberal Arts: Score below 9 in Part I ⇒ Math I Any other score ⇒ Enroll in a mathematics elective course

Data Analysis We have performed statistical analyses since 1997 to study whether there was a relationship between the score on the placement test and how well first year students did on the first mathematics course taken at Merrimack College. In order to determine this relationship, we first examined the correlation between total score on the placement test and students’ grades. A preliminary study with n = 372 showed that the correlation between the grade earned in a mathematics course and the total score on the placement test was 0.466. For the same study, the correlation between grades earned and SAT scores was 0.334. A multiple regression model to estimate the final grade based on the SAT score and the placement exam score gave the equation: Final Grade = 0.80 + 0.00122(SAT) + 0.0641(Placement Exam). In addition, a t test for each of the variables in the model indicated that the SAT had a t value of 1.01 (p = 0.314) and the Placement Exam had a t-value of 6.58 (p < 0.0005). Based on the t test values and the p-values, we removed the SAT variable from the model and concluded that the placement exam was a better predictor of student success. Although we first used t-tests to compare the total score on the placement test with SAT scores as predictors of students’ final grades, we ultimately decided against them as a means to further examine the effectiveness of our placement test as a placement tool for two reasons. First, there was not a true linear relationship between the total score on the placement test and placement. It was more like a step-function, with a range of scores in each part of the test being considered for placement. Second, there was a relatively weak relationship between total score on the placement test and final grades because students are placed into so many different levels of courses. For example, a student may have a very low placement test score, be properly placed into an elementary algebra class, and earn a high grade in that course. Thus, a simple correlation between total score on the placement test and grades earned was considered inappropriate to determine the effectiveness of the 29

placement test, and therefore we decided to categorize the data. Each first-year student was categorized according to the level of mathematics course taken: (1) the course was easier than that recommended by the placement test score; (2) the course was the recommended course; (3) the course was more difficult than that recommended (higher-level); or (4) the placement test was not taken. Although the test was required, some students—usually transfer students—were allowed to take a mathematics course based on their mathematics grade(s) at a previous institution. This policy has not worked well and is being changed to require all incoming students to complete the placement test. Within these categories, students were counted according to whether they (a) did well (received a grade of C– or better) or (b) did poorly (received a grade below C–). Chi-square tests were performed to determine whether there was a relationship between the level of the course taken and the grade received in the class. Given the eight possible categories previously described in this study, the chi-square test indicated whether the percentage of students, say, who did well in each category of course level taken was significantly different from that in any of the other categories. The use of the chi-square test assumes that a random sample, representative of the population, was taken. In this study, we used the entire population of freshmen students who took mathematics in their entering fall semester at our college for the years from 1997 to 2001 (n = 1710). The null hypothesis stated that there was no association between the two variables. The alternative hypothesis stated that the grade depended on whether

or not the student followed the placement test result recommendation. Results We wanted to know whether there was an association between the level of the course taken and the grade earned. Table 1 shows the number and percentage of students who did well in the class (C– or higher) and the number and percentage of students who did poorly (D+ or lower, or withdrew from the class) from 1997 through 2001. It was generally accepted that grades of D and F were unsatisfactory, as evidenced by the fact that almost all comparable studies used cut-off grades of C or C–. Those that used the C cut-off often had a minimum grade requirement of C for a student to move on to a subsequent course. Our department does not have such a requirement, and thus the choice of C– for this study was somewhat arbitrary. We felt that our professors might be less likely to slightly inflate the grade of C– to C than would those at schools with the minimum requirement. As described above, students were classified according to the level of the course taken: easier than the one recommended, the one recommended, or a higher-level course than the one recommended. A fourth category was used in order to include 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 relationship between the two variables for first year students who took a mathematics course in the fall of 1997 [χ2 (3, n = 369) = 24.66, p < 0.0005]. The same conclusion held for the data corresponding to the fall of 1998

Table 1 Number and Percentage of Students Each Year Disaggregated by Grade and Course Category 1997 Course Easier

Recom.

HigherLevel

No Exam n

30

1998

1999

2000

2001

< D+

> C–

< D+

> C–

< D+

> C–

< D+

> C–

< D+

> C–

6

23

2

17

1

12

5

21

2

8

21%

79%

11%

89%

8%

92%

19%

81%

20%

80%

47

168

57

168

55

220

57

205

50

200

22%

78%

25%

75%

20%

80%

22%

78%

20%

80%

30

30

33

34

6

16

4

12

7

15

50%

50%

49%

51%

27%

73%

25%

75%

32%

68%

28

37

22

28

4

18

12

16

18

16

43%

57%

44%

56%

18%

82%

43%

57%

53%

47%

111

258

114

247

66

266

78

254

77

239

Mathematics Placement Test

Table 2 Contingency Table and Chi-Square Test (Expected counts are printed below observed counts; shaded cells indicate expected counts less than 5.) 1997 Course Easier Recom. HigherLevel No Exam n

< D+

1998 > C–

< D+

1999 > C–

< D+

2000 > C–

< D+

2001 > C–

< D+

> C–

6

23

2

17

1

12

5

21

2

8

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

200

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

15

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

16

19.55

45.45

15.79

34.21

4.37

17.63

6.58

21.42

8.28

25.72

111

258

114

247

66

266

78

254

77

239

Chi-Square test

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 not perform a chi-square test for the data corresponding to 1999 because there were 3 cells with expected counts less than 5 (see shaded cells in Table 2) and Moore (2001) does not recommend the use of the chi-square test when more than 20% of the expected counts are less than 5. The relationship was not significant for the fall 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 statistically significant. The relationship was again significant for the data corresponding to the fall of 2001 [χ2 (3, n = 316) = 18.42, p < 0.0005]. In sum, the percentage of students who did well in their first undergraduate mathematics course was higher for those students who followed the advice or took an easier course than the one recommended based on their placement test score. The chi-square test only showed evidence of some association between the variables. We then looked at the tables to determine the nature of the relationship or association (Moore 2001). Table 3 shows the number of students who took the recommended course, an easier one, a higher-level course, as well as the number of students who did not take the placement exam, and the mean and median grades on a 4.0 scale for each group from 1997 to 2001. The following correspondence between letter-grades and numbergrades was used at Merrimack: A

4.0

B

3.0

C

2.0

D

1.0

A– 3.7

B– 2.7

C–

1.7

D– 0.7

B+ 3.3

C+ 2.3

D+

1.3

F

Rueda & Sokolowski

0.0

In addition, students who withdrew from a course were counted and were assigned a number grade of 0 for this study. Discussion and Conclusion From Table 1 we saw that students who took the recommended course or an easier one did much better than those who took a higher-level course or did not take the placement exam. The same conclusion can be drawn from Table 3, with the exception of the year 2000. In 2000 there was no significant difference among the average grades received by those students who took the recommended course and those who took a higher-level course. An explanation for this may be that the proportion of students who took a higher-level course dropped from 19% in 1998 to 7% in 1999, and to 5% in 2000. The few students who took a higherlevel course seemed to know that they would be able to succeed. In addition, the percentage of students placed into our developmental course, Math I, has been decreasing. Twenty five percent of the 521 who took the mathematics placement exam in 1998 were recommended to take Math I. Twenty percent of the 532 who took the mathematics placement exam in 1999 were in that category. Those figures went down to 14% out of 525 students in 2000 and 16% out of 517 in 2001. One of the reasons for this decrease was that students were allowed to retake the placement exam and to place out of our developmental mathematics course. Even though that possibility was available to students before, more effort has been made in the last 31

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

3.1

3.3

26

2.4

2.5

(%) 10

Median

13

n

Mean

2.7

(%)

Median

2.7

n

2001

Mean

19

(%)

Median

3.0

n

2000

Mean

2.8

(%)

Median

29

n

1999

Mean

(%)

Median

n Course

1998

Mean

1997

2.4

2.7

2.6

3.0

1.9

1.7

1.6

1.2

Easier 8% 215 Recom.

HigherLevel

n

2.4

2.7

58% 60

225

4% 2.4

3.0

62% 1.7

1.5

16% 65

No Exam

5%

67

2.0

50

1.6

1.7

2.7

22

1.7

2.0

22

262

3% 2.5

2.7

79% 2.2

2.0

7%

16

79% 2.4

2.5

5% 2.4

2.2

28

250

22 7%

1.8

2.2

34

18%

14%

7%

8%

11%

369

361

332

332

316

few years to avoid improperly placing students in a non-credit course (Math I). With the exception of 1999, there is no significant difference, using z tests, between the proportions of students who did well or poorly among those students who did not take the mathematics placement exam. It is not surprising that students who enrolled in the recommended course or an easier course performed better than did students who enrolled in a higher-level course than the one recommended. What is important here is that approximately 80% of these students who took the recommended or easier course succeeded with a grade of C– or higher. We have found the mathematics placement exam to be a useful tool to place students in the appropriate mathematics course, and we have been successful in convincing most of our students to follow our advice with respect to which courses to take. A challenge for us has been persuading students to take Math I, the non-credit class, when they are not ready for a college level course. While there is no perfect placement method, we have found that our test is better than SAT scores in placing students into the appropriate course. In addition, our multiple-choice test is easier to administer than methods used at other schools mentioned in this paper. From our experience, a well-designed in-house placement test geared towards our curriculum is a simple and powerful tool for placing incoming students in an appropriate mathematics course. Keeping 32

2.5

83%

19% 1.8

275

8%

adequate records and analyzing them with regard to the placement test’s effectiveness, as we have done in this study, is a key component in maintaining the validity and 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, the entire Merrimack community appears to view the test score with increased respect because of the results presented in this paper. These ongoing statistical validations of the connections between proper placement and successful achievement have served to legitimize the placement test as part of a larger effort to increase retention on our campus. REFERENCES Bridgeman, B., & Wendler, C. (1989). Prediction of grades in college mathematics courses as a component of the placement validity of SAT-mathematics scores. (College Board Report No. 89-9). New York, NY: College Entrance Examination Board. Callahan, S. (1993, November). Mathematics placement at Cottey College. Paper presented at the Annual Conference of the American Mathematical Association of Two-Year Colleges, Boston. (ERIC Document Reproduction Service No. ED 373813) Cederberg, J. N. (1999). Administering a placement test: St. Olaf College. In B. Gold, S. Keith, & W. Marion (Eds.), Assessment practices in undergraduate mathematics (pp. 178−180). Washington, DC: Mathematics Association of America.

Mathematics Placement

Cohen, E., Friedlander, J., Kelemen-Lohnas, E., & Elmore, R. (1989). Approaches to predicting student success: Findings and recommendations from a study of California Community Colleges. Santa Barbara, CA: Chancellor’s Office of the California Community Colleges. (ERIC Document Reproduction Service No. ED 310808) Krawczyk, D., & Toubassi, E. (1999). A mathematics placement and advising program. In B. Gold, S. Keith, & W. Marion (Eds.), Assessment practices in undergraduate mathematics (pp. 181−183). Washington, DC: Mathematics Association of America.

Mercer, B. (1995). A comparison of students who followed mathematics advisement recommendations and students who did not at Rochester Community College. Practicum prepared for Nova Southeastern University, Ft. Lauderdale, FL. (ERIC Document 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 college mathematics courses using scores on SAT-math and a test of algebra skills. PRIMUS, 5, 61−72.

Appendix A: Sample Problems The 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|

2.

b.

3

−8

a. 12 x + 3x 3x

b.

− 2x3 (−2 x) 3

e. 5x + 18 − 4(x + 7)

€

c. 3.42

d. 8 − 5

.02

9 12

Simplify the following: € 2

3.

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

c. (25 x 4 y 8 ) −1 / 2 3 2

(3 x )

d.

a 4 + a+2 a−3

f. log( x 2 − 1) − log( x + 2) + log x

6.

Let f (x) =

7.

Find the zeros of the function f ( x) = 2 x − 3 .

€

1 . Find the domain of f. x −1

x +1

−1

( x), if f ( x) = 3 x + 2.

8.

Find the inverse function, f

9.

Which of the following points is not on the graph of y = e x

2

−1

?

Solve the following for x: a. ax − b = 5

b. | 2 x + 1 | = 5

c. x = 2 x + 15

d. 1 = 64 4x

e. x + 2 y = 5 x + 4y = 7

f. x 2 + x < 6

e −1 ), (1, 0), (2, e 3 ), (3, e 8 )

10. Convert 135° to radians.

4.

Solve, then simplify the radical: x 2 − 2 = −4 x

5.

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

Rueda & Sokolowski

(0,

11. Simplify in terms of sin θ : 1 − cos 2 θ = ? 12. Which of the following is greatest? sin 30° , sin 45° , sin 90° , sin 180°

33

The Mathematics Educator 2004, Vol. 14, No. 2, 34–37

In Focus… Can the Ideal of the Development of Democratic Competence Be Realized Within Realistic Mathematics Education? The Case of South Africa Cyril Julie As is the case in any country there is a constant search to improve the mathematics offerings presented to school-goers in South Africa. The activity surrounding this search was intensified after the attainment of democracy. The primary aim of this search was to establish a mathematics curriculum that would result in productive learning and the mastery of the goals set for the curriculum. These goals are predetermined and are embedded within the country’s ideological intent of its school educational endeavors. Explicitly it is stated that school education should be a mechanism to contribute towards the development of “a participating citizen in a developing democracy [who has] a critical stance with regard to mathematical arguments presented in the media and other platforms” (South African Department of Education, 2003, p. 9). This goal was already proffered during the struggle for liberation against apartheid and encapsulated in the alternative school mathematics program during the latter periods of that struggle. The alternative mathematics curricular movement found expression in People’s Mathematics (PM). People’s Mathematics was an independent development in South Africa during a particular historical moment but shared commonalities with the varieties of Critical Mathematics Education (Skovsmose, 1994; Frankenstein, 1989). It differed from other varieties in that it adopted the stance of critique but also emphasized action against those practices which inhibit human possibility. The broad umbrella goals of People’s Mathematics were “political, intellectual and mathematical Cyril Julie obtained his doctorate in Mathematics Education and Computer-Based Education from the University of Illinois at Urbana-Champaign. He teaches courses at undergraduate and graduate level in Mathematics Education at the University of the Western Cape. He is currently involved in a major project related to the development of research capacity building in Mathematics Education in Sub-Saharan Africa. His major research centers around the application and modeling of school mathematics and the relevance of school mathematics for students in grades 8 to 10.

34

empowerment” (Julie, 1993, p. 31). It is with these goals in mind that Realistic Mathematics Education (RME) was found a viable approach to school mathematics with which the People’s Mathematics movement could “live.” The particular characteristics of RME that PM found useful were: (a) It has a Lakatosian research program nature (Gravemeijer, 1988). Being such a research program there was some certainty of sustainability due to Lakatos’s notion of strong research programs fulfilling their predictions. (b) The retention of the integrity of mathematics through RME’s vertical and horizontal mathematization (Streefland, 1990). (c) The centrality of applications and modeling in RME (De Lange, 1987). (d) The seamless integration of the history of mathematics and educational contexts from extramathematical domains (De Lange, 1987). (e) Mathematics curriculum development that is continuous and not of a once-off tinkering nature. (f) A curriculum development research methodology that is classroom-based and action-oriented (Gravemeijer, 1994) with an accompanying reporting strategy of research findings that is understandable to practitioners. In terms the umbrella goals enunciated above in RME fulfilled the mathematical and intellectual ideals ((b) and (d)) of PM but not the political. This does not imply that contexts of an overt political nature are not included in the RME program. For instance, in the following activity (De Lange & Verhage, 1992) around national budgets and military expenditure, the overt political dimensions are clearly discernable. In a certain country, the national defense budget is $30 million for 1980. The total budget for that year was $500 million. The following year the defense budget is $35 million, while the total budget is $605 million. Inflation during the period covered by the two budgets amounted to 10 per cent. Democratic Competence

A. You are invited to give a lecture for a pacifist society. You intend to explain that the defense budget decreased over this period. Explain how you would do this. B. You are invited to lecture to a military academy. You intend to explain that the defense budget increased over this period. Explain how you would do this.

What is not clear from these and other similar activities is how these activities are to be used in classrooms or how and whether there are follow-up activities that take the intent of the activities beyond the purely mathematical. The political empowerment ideal within Peopleâ€™s Mathematics implied a movement beyond this purely mathematical treatment of issues of political import. This begs the question of where does the political reside in mathematical activity? The political find expression in at least three areas of mathematical activity. They are all within the arena of the applications and modeling of mathematics. The first of these is akin to the activity of De Lange and Verhage above. What is added to this is opportunity for overt reflection on those issues that relate to inequality and discrimination on the basis of race, sex, social class and economic developmental level of countries. This is aptly illustrated by the following activity (Frankenstein, 1989, p. 140): Review the comparisons made in the following three tables and write briefly about the connections among the data in each table and any conclusions and any questions you have about the given information. (Note: Only one table is given here.) Median incomes of full-time workers by occupation (persons fourteen and above) Major occupation group Professional and technical workers Non-farm managers & administrators Sales workers Clerical workers Operatives (including transport) Service workers (except private household)

1976 income ($) Women Men 11,081 10,177 6,350 8,138 6,696 5,969

16,296 17,249 14,432 12,716 11,747 10,117

A second area where the political is overtly present is during the model construction process. When a mathematical model is constructed, interpretations and translations take place. The given reality situation from outside of mathematics is stripped down to make it amenable for mathematical treatment, and the resulting mathematical model is more a mathematical representation of a stripped-down version of the Cyril Julie

situation. In essence, there are three domains involved in mathematical model making. These are the extramathematical reality, the consensus-generated reality domain, and the intra-mathematical domains. The characteristics of these domains are summarized in Figure 1. ExtraMathematical Reality Domain Issues of a technical, physical, financial, social, political, environmental, and so forth nature are at stake.

ConsensusGenerated Reality Domain

IntraMathematical Domain

Issues are stripped of some of the influencing factors.

Mathematical procedures and ideas are developed and used.

Consensus is reached based on purposes and interests.

Mathematical conclusions are reached.

Issues are complex and under a variety of influencing factors. Figure 1. The translation of reality issues through different domains.

It is during this process of translation that issues of interests, ideological preferences and power are at stake and contestations manifest themselves. These contestations occur prior to the subjucation of the issue for which a mathematical representation is to be constructed to mathematical treatment. They occur between the domain of the real and that of mathematics. The resolution of conflicts, interpretational variances, and interests render a different reality. This reality is realized through consensus and hence the postulation of a consensusgenerated reality domain as outcome of deliberations on differences, interests, and intentions. It is within the consensus-generated reality domain that ideological intentions are explicitly revealed. A resulting mathematical model is always a product of its consensus-generated domain and thus different and non-equivalent models might result for the same phenomenon. Lastly, the political rears its head during the final phase of the modeling process when the adequacy-offit of the model to the reality situation is considered. Here the issue is whether the derived mathematical conclusions should be accepted or not. Consider, for example, the mathematics of voting. In this instance a 35

dictatorship is defined as “[a member] in [a voting set] A is called a dictator in A if and only if {x} is a minimal coalition” with a minimal coalition “a subset of K…if and only if K is a winning coalition and no proper subset of K a winning coalition” (Steiner, 1968, p. 189, 184). From this definition, if the results of a general election in a country is such that a political party has a majority such that they need to form no coalition with any other party to carry an issue, then that political party, and by implication, the government of the day, is a dictatorship, at least mathematically. For example, for the South African 2004 election, the African National Congress (ANC) commanded that 69.75% of the parliamentary seats and a two-thirds majority was needed to carry any decision. Thus, according to the mathematical definition of a dictatorship, South Africa is under the dictatorship of the ANC. However, the lived experiences in South Africa are such that this is not the case. Here then it is clear that mathematical conclusions and lived experiences might at times be in conflict and for all intents and purposes it is sometimes wiser to be guided by lived experiences rather than by the dictates of mathematical conclusions. One of the features that stand out in modern school mathematics curricula in terms of the goals that are offered is the development of democratic competence. This competence is an individual’s (and a collective’s) ability to make sound judgments about those issues which structure and steer the affairs and practices of humankind. The judgments are about the appropriateness or not of the development and implementation of the mechanisms that guide these affairs and practices. These mechanisms are profoundly driven by worldviews on issues such as race, gender, and class differentials and generally the kind of world that is envisioned. During the model construction process issues of this political nature come into play and hence the need for democratic competence as stated. Further, as enunciated above, part of this competence is a considered skepticism towards being convinced through mathematical argumentation. Democratic competence is normally captured in the definition of Mathematical Literacy as, for example, given below for the Programme for International Student Assessment (Organization for Economic Co-Operation and Development, 2003, p. 24): Mathematical literacy is the individual’s capacity to identify and understand the role that mathematics plays in the world, to make wellfounded judgments and to use and engage with 36

mathematics in ways that meet the needs of that individual’s life as a constructive, concerned, and reflective citizen.

A question that can be asked is whether the goal of the development of democratic competence can be realized within the Realistic Mathematics Education framework. What was indicated above is that RME falls short as a paradigm in this regard. It does allow for reflection on issues of a political nature but remains at an awareness and conscientization level. It does not allow for a consensus generation phase in modelconstruction. Neither does it explicitly allow for the questioning of mathematical conclusions in relation to lived and other experiences. It is suggested that RME needs to be broadened to incorporate at least these three issues in order to contribute more to the goal of development of democratic competence. In doing so there is a need to move beyond awareness and conscientization. This “moving beyond” is what Ellis, a leading South African and internationally recognized cosmologist, suggested about 20 years ago. He stated: …the only true basis of freedom is a realistic vision of the alternative possibilities for change before us. Mathematical studies can sometimes help us in understanding what these alternative possibilities are. But such an understanding is quite valueless unless it affects our actions. An understanding of the causes of any social wrong, which does not lead to some corrective action to right that wrong, is meaningless. (Ellis, 1974, p. 17)

Democratic competence is thus about an individual’s (or collective’s) capacity to interact with mechanisms which affect their lives and those of society and to act where such mechanisms are to the detriment of humankind. Where these mechanisms have a mathematical base, or where they can be explained and understood through mathematical means, necessitates that schooling in mathematics be called upon to provide the spaces for such interactions and actions. REFERENCES de 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 third R – Radical maths. London: Free Association Books. Gravemeijer, K. (1988). Een Realistisch research programma. In K. Gravemeijer & K. Koster (Eds.), Onderzoek, ontwikkeling Democratic Competence

en ontwikkelingsonderzoek (pp. 106–117). Utrecht University, Utrecht, Netherlands: OW & OC. Gravemeijer, K. (1994). Educational development and development research in mathematics education, Journal for Research in Mathematics Education. 25(5), 443–471. Julie, C. (1993). People’s mathematics and the applications of mathematics. In J. de Lange, C. Keitel, I. Huntley, & M. Niss (Eds.), Innovation in maths education by modelling and applications (pp. 31–40). Chichester: Ellis Horwood. Organisation for Economic Co-operation and Development (OECD). (2003). The PISA 2003 assessment framework—Mathematics, reading, science and problem solving knowledge and skills. Retrieved November 23, 2004 from http://www.pisa.oecd.org/

Cyril Julie

Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht, Netherlands: Kluwer Academic Publishers. Steiner, H. G. (1968). Examples of exercises in mathematization in secondary school level. Educational Studies in Mathematics, 1, 181–201 South African Department of Education. (2003). The national curriculum statement: Mathematical literacy. Pretoria, South Africa: 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 Joint Meeting of the Mathematical Association of America and the American Mathematical Society http://www.ams.org

Atlanta, GA

January 5 – 8

AAMT 2005 Australian Association of Mathematics Teachers http://www.aamt.edu.au/mmv

Sydney, Australia

January 17 – 20

AMTE Association of Mathematics Teacher Educators http://amte.sdsu.edu/conf_info_2005.shtml

Dallas, TX

January 27 – 29

RCML Research Council on Mathematics Learning http://www.unlv.edu/RCML

Little Rock, AK

February 24 – 26

Mα The Mathematical Association http://m-a.org.uk/resources/conferences/

Coventry, UK

March 30 – April 2

NCTM National Council of Teachers of Mathematics http://www.nctm.org/meetings/index.htm#annual

Anaheim, CA

April 6 – 9

AERA American Educational Research Association http://www.aera.net/meeting/

Montreal, Canada

April 11 – 15

AMESA Eleventh Annual National Congress http://academic.sun.ac.za/mathed/AMESA/Index.htm

Kimberley, South Africa

June 27 – 30

PME-29 International Group for the Psychology of Mathematics Education http://staff.edfac.unimelb.edu.au/~chick/PME29/

Melbourne, Australia

July 10 – 15

JSM of the ASA Joint Statistical Meetings of the American Statistical Association http://www.amstat.org/meetings/jsm/2005/

Minneapolis, MN

August 7 – 11

GCTM GCTM Annual Conference http://www.gctm.org/georgia_mathematics_conference.htm

Rock Eagle, GA

TBA

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The Mathematics Educator (ISSN 1062-9017) is a student-produced journal published semiannually by the Mathematics Education Student Association (MESA) at The University of Georgia. The journal promotes the interchange of ideas among the mathematics education community locally, nationally, and internationally, and presents a variety of viewpoints on a broad spectrum of issues related to mathematics education. TME also provides a venue for the encouragement and development of leaders and editors in mathematics education. The Mathematics Educator is abstracted in Zentralblatt für Didaktik der Mathematik (International Reviews on Mathematical Education). The Mathematics Educator encourages the submission of a variety of types of manuscripts from students and other professionals in mathematics education including: • • • • • • • •

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

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The Mathematics Education Student Association is an official affiliate of the National Council of Teachers of Mathematics. MESA is an integral part of The University of Georgiaâ€™s mathematics education community and is dedicated to serving all students. Membership is open to all UGA students, as well as other members of the mathematics education community.

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TME Subscriptions TME is published both online and in print form. The current issue as well as back issues are available online at http://www.coe.uga.edu/tme. A paid subscription is required to receive the printed version of The Mathematics Educator. Subscribe now for Volume 15 Issues 1 & 2, to be published in the spring and fall of 2005. If you would like to be notified by email when a new issue is available online, please send a request to tme@coe.uga.edu To subscribe, send a copy of this form, along with the requested information and the subscription fee to The Mathematics Educator 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

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In this Issue, Guest Editorial… Researching Classroom Learning and Learning Classroom Research DAVID CLARKE The Consequences of a Problem-Based Mathematics Curriculum DAVID CLARKE, MARGARITA BREED, & SHERRY FRASER Impact of Personalization of Mathematical Word Problems on Student Performance ERIC T. BATES & LYNDA R. WIEST Mathematics Placement Test: Helping Students Succeed NORMA G. RUEDA & CAROLE SOKOLOWSKI In Focus… Can the Ideal of the Development of Democratic Competence Be Realized Within Realistic Mathematics Education? The Case of South Africa CYRIL JULIE