Idea Transcript
PROCEEDINGS OF THE 14TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE MICHAEL OEHRTMAN PORTLAND, OREGON FEBRUARY 24 – FEBRUARY 27, 2011
PRESENTED BY THE SPECIAL INTEREST GROUP OF THE MATHEMATICS ASSOCIATION OF AMERICA (SIGMAA) FOR RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
Copyright @ 2011 left to authors All rights reserved
CITATION: In (Eds.) S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman, Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education, Vol. #, pg #-#. Portland, Oregon.
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FOREWORD The research reports and proceedings papers in these volumes were presented at the 14 Annual Conference on Research in Undergraduate Mathematics Education, which took place in Portland, Oregon from February 24 to February 27, 2011. th
Volumes 1 and 2, the RUME Conference Proceedings, include conference papers that underwent a rigorous review by two or more reviewers. These papers represent current important work in the field of undergraduate mathematics education and are elaborations of the RUME conference reports. Volume 1 begins with the winner of the best paper award, an honor bestowed upon papers that make a substantial contribution to the field in terms of raising new questions or gaining insights into existing research programs. Volume 3, the RUME Conference Reports, includes the Contributed Research Reports that were presented at the conference and that underwent a rigorous review by at least three reviewers prior to the conference. Contributed Research Reports discuss completed research studies on undergraduate mathematics education and address findings from these studies, contemporary theoretical perspectives, and research paradigms. Volume 4, the RUME Conference Reports, includes the Preliminary Research Reports that were presented at the conference and that underwent a rigorous review by at least three reviewers prior to the conference. Preliminary Research Reports discuss ongoing and exploratory research studies of undergraduate mathematics education. To foster growth in our community, during the conference significant discussion time followed each presentation to allow for feedback and suggestions for future directions for the research. We wish to acknowledge the conference program committee and reviewers, for their substantial contributions and our institutions, for their support. Sincerely, Stacy Brown, RUME Organizational Director & Conference Chairperson Sean Larsen, RUME Program Chair Karen Marrongelle RUME Co-coordinator & Conference Local Organizer Michael Oehrtman RUME Coordinator Elect
VOLUME 3
CONTRIBUTED RESEARCH REPORTS
VOLUME 3 TABLE OF CONTENTS MAKING THE FAMILIAR STRANGE: AN ANALYSIS OF LANGUAGE IN POSTSECONDARY CALCULUS TEXTBOOKS THEN AND NOW .............................9 Veda Abu-Bakare THE EFFECTIVENESS OF BLENDED INSTRUCTION IN GENERAL EDUCATION MATHEMATICS COURSES ...........................................................................................13 Anna E. Bargagliotti, Fernanda Botelho, Jim Gleason, John Haddock, and Alistair Windsor OBSTACLES TO TEACHER EDUCATION FOR FUTURE TEACHERS OF POSTSECONDARY MATHEMATICS.....................................................................................19 Mary Beisiegel DESIGNING AND IMPLEMENTING A LIMIT DIAGNOSTIC TOOL........................24 Timothy Boester ASSESSING ACTIVE LEARNING STRATEGIES IN TEACHING EQUIVALENCE RELATIONS .....................................................................................................................29 Jim Brandt SURVEYING MATHEMATICS DEPARTMENTS TO IDENTIFY CHARACTERISTICS OF SUCCESSFUL PROGRAMS IN COLLEGE CALCULUS .....................................................................................................................33 Marilyn Carlson, Chris Rasmussen, David Bressoud, Michael Pearson, Sally Jacobs, Jessica Ellis, and Eric Weber TRANSLATING DEFINITIONS BETWEEN REGISTERS AS A CLASSROOM MATHEMATICAL PRACTICE.......................................................................................39 Paul Dawkins THE ROLE OF CONJECTURING IN DEVELOPING SKEPTICISM: REINVENTING THE DIRICHLET FUNCTION ........................................................................................44 Brian Fisher TOULMIN ANALYSIS: A TOOL FOR ANALYZING TEACHING AND PREDICTING STUDENT PERFORMANCE IN PROOF-BASED CLASSES ..............47 Timothy Fukawa-Connelly
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A MULTI-STRAND MODEL FOR STUDENT COMPREHENSION OF THE LIMIT CONCEPT .........................................................................................................................52 Gillian Galle AUTHORITY IN THE NEGOTIATION OF SOCIOMATHEMATICAL NORMS .......57 Hope Gerson and Elizabeth Bateman STUDENT UNDERSTANDING OF EIGENVECTORS IN A DGE: ANALYSING SHIFTS OF ATTENTION AND INSTRUMENTAL GENESIS .....................................61 Shiva Gol Tabaghi UNIVERSITY STUDENTS’ UNDERSTANDING OF FUNCTION IS STILL A PROBLEM! ........................................................................................................................65 Zahra Gooya and Mehdi Javadi
THE LIMIT NOTATION: WHAT IS IT A REPRESENTATION OF?...........................68 Beste Güçler STUDENT OUTCOMES FROM INQUIRY-BASED COLLEGE MATHEMATICS COURSES: BENEFITS OF IBL FOR STUDENTS FROM UNDER-SERVED GROUPS............................................................................................................................73 Marja-Liisa Hassi, Marina Kogan, Sandra Laursen ON EXEMPLIFICATION OF PROBABILITY ZERO EVENTS ...................................78 Simin Chavoshi Jolfaee DIFFERENCES IN BELIEFS AND TEACHING PRACTICES BETWEEN INTERNATIONAL AND U.S. DOMESTIC MATHEMATICS TEACHING ASSISTANTS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!82 Minsu Kim IMPROVING THE QUALITY OF PROOFS FOR PEDAGOGICAL PURPOSES: A QUANTITATIVE STUDY ...............................................................................................88
Yvonne Lai, Juan-Pablo Mejia Ramos and Keith Weber PUTTING RESERCH TO WORK: WEB-BASED INSTRUCTOR SUPPORT MATERIALS FOR AN INQUIRY ORIENTED ABSTRACT ALGEBRA CURRICULUM...............................................................................................................92 Sean Larsen, Estrella Johnson and Travis Scholl STUDENTS’ MODELING OF LINEAR SYSTEMS: THE RENTAL CAR PROBLEM.........................................................................................................................96 Christine Larson and Michelle Zandieh
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NAVIGATING THE STRAITS: CRITICAL INSTRUCTIONAL DECISIONS IN INQUIRY-BASED COLLEGE MATHEMATICS CLASSES ......................................101 Sandra Laursen, Marja-Liisa Hassi, and Anne-Barrie Hunter STUDENTS PERCEPTIONS OF AN EXPLICIT CRITERION REFERENCED ASSESSMENT ACTIVITY IN A DIFFERENTIAL EQUATIONS CLASS ................105 Dann Mallet and Jennifer Flegg REACHING OUT TO THE HORIZON: TEACHERS’ USE OF ADVANCED MATHEMATICAL KNOWLEDGE...............................................................................109
Ami Mamolo and Rina Zazkis STUDENTS’ REINVENTION OF FORMAL DEFINITIONS OF SERIES AND POINTWISE CONVERGENCE .....................................................................................114 Jason Martin, Michael Oehrtman, Kyeong Hah Roh, Craig Swinyard, and Catherine Hart-Weber AN ANALYSIS OF EXAMPLES IN COLLEGE ALGEBRA TEXTBOOKS FOR COMMUNITY COLLEGES: OPPORTUNITIES FOR STUDENT LEARNING .....................................................................................................................119
Vilma Mesa, Heejoo Suh, Tyler Blake, and Tim Whittemore PROMOTING SUCCESS IN COLLEGE ALGEBRA BY USING WORKED EXAMPLES IN WEEKLY ACTIVE GROUP SESSIONS............................................125 David Miller and Matthew Schraeder RELATIONSHIPS BETWEEN QUANTITATIVE REASONING AND STUDENTS’ PROBLEM SOLVING BEHAVIORS ............................................................................129 Kevin C. Moore THE PHYSICALITY OF SYMBOL USE: PROJECTING HORIZONS AND TRAVERSING IMPROVISATIONAL PATHS ACROSS INSCRIPTIONS AND NOTATIONS...................................................................................................................134 Ricardo Nemirovsky and Michael Smith FROM INTUITION TO RIGOR: CALCULUS STUDENTS’ REIVENTION OF THE DEFINITION OF SEQUENCE CONVERGENCE ........................................................137 Michael Oehrtman, Craig Swinyard, Jason Martin, Catherine Hart-Weber, and Kyeong Hah Roh HOW INTUITION AND LANGUAGE USE RELATE TO STUDENTS’ UNDERSTANDING OF SPAN AND LINEAR INDEPENDENCE .............................142 Frieda Parker
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THE IMPACT OF TECHNOLOGY ON A GRADUATE MATHEMATICS EDUCATION COURSE .................................................................................................147 Robert A. Powers, David M. Glassmeyer, and Heng-Yu Ku STUDENT TEACHER AND COOPERATING TEACHER TENSIONS IN A HIGH SCHOOL MATHEMATICS TEACHER INTERNSHIP: THE CASE OF LUIS AND SHERI ..............................................................................................................................152 Kathryn Rhoads, Aron Samkoff, and Keith Weber PROMOTING STUDENTS’ REFLECTIVE THINKING OF MULTIPLE QUANTIFICATIONS VIA THE MAYAN ACTIVITY ................................................156 Kyeong Hah Roh and Yong Hah Lee HOW MATHEMATICIANS USE DIAGRAMS TO CONSTRUCT PROOFS ............161 Aron Samkoff, Yvonne Lai, and Keith Weber EXPLORING THE VAN HIELE LEVELS OF PROSPECTIVE MATHEMATICS TEACHERS.....................................................................................................................165 Carole Simard and Todd A. Grundmeier CLASSROOM ACTIVITY WITH VECTORS AND VECTOR EQUATIONS: INTEGRATING INFORMAL AND FORMAL WAYS OF SYMBOLIZING Rn ........170 George Sweeney CHANGING MATHEMATICAL SOPHISTICATION IN INTRODUCTORY COLLEGE MATHEMATICS COURSES ......................................................................175 Jennifer E. Szydlik, Eric Kuennen, John Beam, Jason Belnap, and Amy Parrott INDIVIDUAL AND COLLECTIVE ANALYSIS OF THE GENESIS OF STUDENT REASONING REGARDING THE INVERTIBLE MATRIX THEOREM IN LINEAR ALGEBRA..................................................................................................179 Megan Wawro USING THE EMERGENT MODEL HEURISTIC TO DESCRIBE THE EVOLUTION OF STUDENT REASONING REGARDING SPAN AND LINEAR INDEPENDENCE ...........................................................................................185 Megan Wawro, Michelle Zandieh, George Sweeney, Christine Larson, and Chris Rasmussen
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Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education
Making the familiar strange: An Analysis of Language in Postsecondary Calculus Textbooks then and now Veda Abu-Bakare Simon Fraser University Three calculus textbooks covering a span of about 40 years were examined to determine whether and how the language used has changed given the reform movement and the impetus to make mathematics accessible to all. Placed in a discourse analytic framework using Halliday!s (1978) theory of functional components –ideational, interpersonal and textual, and using the exposition of the concept of a function as a unit of comparison, the study showed that language is an integral indicator of the author!s view of mathematics and an important factor for textbook adoption in the pursuit of student success. Keywords: discourse analysis, calculus textbooks, language of mathematical discourse INTRODUCTION In the late 1980s, the Calculus Consortium at Harvard (CCH) was funded by the National Science Foundation to redesign the Calculus curriculum with a view to making Calculus more applied, relevant, and accessible. The intent was to re/think and re/present the content so as to focus on real-world applications, to emphasize concepts and graphical representations, and to take advantage of the increasingly sophisticated technology. Calculus is now presented in a manner radically different from the traditional approach of abstraction, formal notation and symbolism, and algebraic conventions. The goal of this research is to see whether and how calculus textbooks designed for the postsecondary level in „regular! Calculus courses have changed over the years with respect to the language used in the exposition and by inference, the view of mathematics manifested. One concept, that of a function and in particular its definition, is chosen and used to trace the dimensions of the language over the years and the consequent shifts in the view and presentation of mathematics in calculus textbooks. The research questions are: Has the language of calculus textbooks changed over time and if so, in what ways? Has the language changed from one that is exclusive (mathematics as an elite subject with an elite community) to one that is inclusive and accessible to all? From the language, how are the authors! views of mathematics characterized and how have they changed over time? The three textbooks I have chosen are Calculus by Spivak (1967), The Calculus of a Single Variable with Analytic Geometry, 5th edition by Leithold (1986), and Single Variable Calculus: Early Transcendentals, 5th edition by Stewart (2003). Textbooks may be studied subjectively to describe the interaction between the student and the written material or to describe teachers! use of textbooks and the subsequent effect on the teacher (Remillard et al, 2009). However, following Herbel-Eisenmann (2007), I seek to examine the „voice! of calculus textbooks over the years as objectively given structure (emphasis in the original, p.396). This examination will be placed in a discourse analytic framework which attends to the aspects of text relating to language, voice, agency and identity. ANALYTIC FRAMEWORK Language has been increasingly seen as an important issue relating to mathematics teaching and learning. Rowland (2000) emphasizes two principles in studying language: the linguistic principle (language as means of accessing thought) and the deictic principle
Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education
(language as a means of communication and a „code to express and point to concepts, meanings and attitudes!) (p. 2). In his Language as a Social Semiotic, Halliday (1978) identifies three functional components or functions of language– the ideational, the interpersonal, and the textual –from which meaning is apprehended. The ideational functional component of the text answers the questions: What is the view of mathematics as presented in the text? How is the subject of mathematics envisioned in the mind of the author of the text and in what style is it rendered? The interpersonal functional component describes the social and personal roles and relationships among the authors and readers. Evidence of this function is discerned by considering the use of personal pronouns (first, I/we/us/our, and second person, you), imperatives, and modality. The textual functional component describes the content matter or the mathematics presented in the text, the theme and modes of reasoning, the arguments and their forms, and the narratives of mathematical activity. Each of the textbooks will be examined as to the “voice” that emerges, the extent of agency, and the construction of the identity of the reader by the text. METHOD The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
Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education
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