Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2007 Describing Students' Pragmatic Reasoning When Using "Natural Mathematics Computer Interfaces (NMI)" Erich Nold Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF EDUCATION DESCRIBING STUDENTS’ PRAGMATIC REASONING WHEN USING “NATURAL MATHEMATICS COMPUTER INTERFACES (NMI)” By Erich Nold A Dissertation submitted to the Department of Middle and Secondary Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy Semester Awarded: Summer Semester, 2007 Copyright © 2007 Erich Nold All Rights Reserved The members of the Committee approved the dissertation of Erich Nold, defended on March 2, 2007. _______________________ Janice Flake Professor Directing Dissertation _______________________ Russell Dancy Outside Committee Member _______________________ Elizabeth Jakubowski Committee Member _______________________ Leslie Aspinwall Committee Member Approved: _________________________________________________________ Pamela Carroll, Chairperson, Middle and Secondary Education The Office of Graduate Studies has verified and approved the above named committee members. ii This dissertation is a gift both to and from the ones I love. iii ACKNOWLEDGMENTS I first want to acknowledge the blessing of my loving parents. Numerous colleagues have assisted me in the completion of this work. My Directing Professor Janice Flake contributed numerous key ideas and critiques through conversations over several years. This dissertation would not have been completed without her help and support. I also recognize my initial Directing Professor, Norma Presmeg, who introduced me to C. S. Peirce. Professor Emeritus Benjamin Fusaro patiently lent his ear and wisdom on numerous occasions. I want to extend thanks to Professor Elizabeth Jakubowski who initially attracted me to Florida State University and has supported me ever since. Dr. Aspinwall has energized my work both at school and on social occasions. I appreciated the critical guidance of Professor Wheatley, especially with respect to our study on Computer Algebra System use and mathematical reasoning—planting seeds for the present work. Finally, I thank Professor Russell Dancy who so graciously and amiably sat on my committee, in spite of my bull-headedness at times. I also wish to thank my brother Richard Ellsworth, and Professors Patricia Baldarez, Mary Sutherland, John Cannon, Larry Andrews, the late Emma Fenceroy, Roselyn Williams, Bruno Guerrieri, Wendell Motter, Osiefield Anderson, Darel Hardy, and Larry Dennis. I need also thank my subjects “Ann,” and “Ed,” Dr. Robert Watkins, Dr. Michael Capps, Dr. Michael Naylor, Michelle Hughes, Cindy, Leslie, & Mary Pugh, Sonya Dudley, Roy Dutton, Teresa Summeral, Mark Horner, Jannina Herring, Steve Christian, Cindy Hewitt, and the always helpful staff at MacKichan Software. Special thanks to the patient Reverend Dr. Michael Petty and to the beautiful St. Peter’s Anglican choir. I want to acknowledge the late Elmo Thomas and Barbara, for being like second parents to me. Finally, I have to thank Elizabeth Burkett for her family’s consistent moral support, gentle pushes, and always encouraging sense of humor. iv TABLE OF CONTENTS LIST OF TABLES…………………………………………………………………………………..xv LIST OF FIGURES……………………………………………………………………………...….xvi ABSTRACT…...…………………………………………………………………………………...xviii I. INTRODUCTION…………………………………………….…………………………... 1 Rationale……………………………………………………….….………………….……… 2 Learning and Reasoning using an Ideal Super-calculator……….……….…………...……… 3 Aspects of a Natural Mathematics Interface…………………….…...……………………… 4 Initiation of Study………………………………………………...………………………….. 5 Research Overview……………………………………………….…..…………………….... 5 Research Question……………………………………………….….……………………….. 6 General Introductory Remarks………………………………….….………………………… 7 Natural Language Mathematics Natural Mathematics, Learning, and Technology Mathematical Representations and Reasoning Reasoning and Pragmatism Natural Mathematics, and a Natural Mathematics Computer Interface Reasoning and Learning using NMI Tools Researcher’s Bias…………………………………………………………………………….. 12 II. LITERATURE FOUNDATIONS……...……………………...…….….……………….... 14 Organization of the Chapter………………………………………………………….….……. 14 Overview of Peirce’s Pragmatism……………………………………………………..……… 15 Mathematics as Necessary and Diagrammatic Reasoning: NMI Graphics The Breadth of Educationally Relevant Peircean Theory v Technology as a Pedagogical Tool………………………………………………………………… 19 Wide Ranging Classes of Pedagogical Applications Software Microworld Interactive Mathematical Situation Non-real-time Interactive Mathematical Situation Mathematical Object Constructors Natural Mathematics Interface Programs (NMI) Classes of programs with partial NMI characteristics Computer Algebra Systems (CAS) Mathematics User Interfaces (MUI) Mathematics on the Web: OpenMath and MathML Graphing Calculators Textbook formatted NMI facilitated interactive programs Specific NMI programs used Scientific Notebook (SNB) Geometer’s SketchPad (GSP) Reasoning and Mathematics Educational Technology Use in a Peircean Framework Peirce’s Three Distinct Types of Reasoning………………………………………………………. 33 Abductive Reasoning Detailed Senses of Abductive Reasoning Hunch Symptom Metaphor/Analogy Clue Diagnosis/Scenario vi Explanation Inductive Reasoning Induction and the Logic of Relatives Abduction and Induction Compared and Contrasted Deductive Reasoning Some Formalized Systems which Guide Deductive Inference Deduction as Regulative Deductive Reasoning versus Plausible Reasoning Specific Examples of Deductive vs. Plausible Inference Summary of Peirce’s Views on Deduction The Three Types of Reasoning Taken Together Affirmed Perceptions and the Origination of Reasoning………………....……………………….. 49 Psychology and Reasoning Abstraction Intuition and Abstraction Understanding Quasi-Empiricism and Differing Kinds of Understanding Ordering Different Types of Reasoning and Personal Understanding Colligation Representation Pragmatism as a Research Stance…………………………………………………………………. 58 Clarification and Caveats Relevance Student Motivation Peirce and Constructivism vii Objectivity, Reality, and Mathematical Conceptualization Socially Constructed Objectivity Constructions of Mind Independent Outcomes Learned Construction of Mathematical Facts through Pragmatic Reasoning Closing Reflection III. METHODOLOGY……………...………………………………………….............................. 68 Qualitative Research Approach………………………………………………………………..….. 68 A Research Framework based on Pragmatic Reasoning…………………………………….…… 68 Specific Research Methods………………………………….………………………………….… 69 Case Study Action Research Recursive Qualitative Approach………………………………………………………………….... 70 Triangulation…………………………………………………………………………………......... 71 Determining a Sense of a Student’s Mathematical Reasoning using an NMI Constructivist Learning Theory Research Tools Perturbation Zone of Proximal Development Scaffolding Pilot Study…………………………………………………………………………………………. 73 Setting and Research Subject Description NMI Displayed Images Conic Sections, Mathematical Embodiments and NMI use Trigonometric Functions and Scientific Analogs Lessons Learned--Methodological Contrasts with Present Research Data Gathering Techniques viii Use of Homework Use of Scaffolding The Study…………………………………………………………………………………………... 76 Tabulation Present Case Study Initiation Setting Selection of Participants and their Backgrounds Selection of NMI Founding Literature Review concerning Theoretical Terminology Selection Data Gathering Procedures Initial and Ongoing Interview Direct Student Observation Thinking aloud Ongoing assessment and evaluation Three Kinds of Scaffolding Instructor scaffolding and participatory observation Student self-scaffolding using NMI Instructor and NMI scaffolding Teaching and Learning Research Activities Student Activity Procedures Sessions with individual students Learning the interfaces Homework Interview Student perturbations ix Working independently of NMI use Procedures for the general scaffolding of activities Encourage explicit mathematical dialogue Assign related problems to be worked by hand Assign simpler related NMI problems Encourage applications oriented analogical creativity Reasoning prompts Data Recording Procedures Field Notes Audio Taping Computer Screen Shots Validation of Study IV. EMERGENT RESULTS TABULATED FOR SUBSEQUENT REFERENCE………………. 89 Introduction………………………………………………………………………………………... 89 Discussion of Case Study Relevant Terminology…..…………….……………………………….. 89 Research Results: an Emergent Tabulation.. ……..……….………………………………………. 90 V. THEORETICAL CONSIDERATION OF THE EMERGENT RESULTS…............................ 108 Introduction……………………………………………………….….………………….…........... 108 Mathematical Embodiments………………………………………………………………………. 108 Theoretical Considerations of Natural Mathematics and NMI…………………………………… 109 Theoretical Connection of Conceptual-Embodiment and Perceptual Judgment…………….......... 112 Mathematics Educational Centrality of Reasoning………………………………………………... 113 Summation of Key Assumptions………………………………………………………………….. 114 VI. INITIAL CASE STUDY REPORTING AND THEIR INTERPRETATIONS……………….. 116 Introduction……………………………………………………….….………………………......... 116 x Ann and Ed’s Initial Distractions due to NMI use………………………………………………… 117 Reflective Question………………………………………………………………………………... 118 Ann’s Initial Interest in NMI use Ed’s Initial