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2007 Describing Students' Pragmatic Reasoning When Using "Natural Mathematics Computer Interfaces (NMI)" Erich Nold
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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 Interest in NMI use
Reflective Question’s Consideration
NMI Implementation Considerations……………………………………………………………… 120
Ed’s Use of MUI Relative to NMI
Overly Abstract and General NMI Representations
Ann, Ed and MUI Properties
Consideration of Ann and Ed’s NMI
Reflection on the Relevance of MUI Procedural Organization
Transition to Idealized NMI Discussion…………………………………………………………... 125
Ann’s work with the Geometer’s SketchPad (GSP) NMI
Perceptual judgments, Mathematical Embodiments, and Student reasoning
using NMI
VII. PERCEPTUAL JUDGMENTS, REASONING, AND NMI USE…………………………… 128
Introduction………………………………………………………………………………………... 128
Four Types of Mathematical Embodiments concerning NMI use…………………………...... 129
Generic Mathematical Embodiment (GE)
Ed and GE
Ann and two GE’s
Interface-procedural Embodiment (IPE)
Specific Examples of Ann’s use of IPE’s
Conclusions: Ann and IPE
Natural Mathematics Computational Embodiment (NMCE)
xi Ann, Ed, and NMCE
NMCE and Reasoning through Distinct Mathematical Approaches: Ed
Ann, Ed, and NMCE Conclusions
Application’s Embodiment (AE)
AE and Mathematics as a Tool
Students’ Weak Understanding of AE Analogs
Ed’s and other Students’ Common AE Experiences
Uncommon Extension of AE by Ed
Ann and Ed’s Interest in AE’s
Ann and an AE of Teaching
Ann, AE, Teaching, and Proof
Ann, AE, and Motivation
My AE Experiences
Reflection: Student’s AE use
VIII. ED AND ANN’S REASONING USING NMI COMPUTING: REFLECTIONS…...…….. 149
Introduction…………………………………………………………………………….…………. 149
Ed’s Use of NMI to work at Higher Mathematical Levels………………………….……………. 150
Ed’s Mathematical Explicitness and NMI use…………………………………….……………... 152
Ann’s Use of NMI to work at Higher Mathematical Levels……………………….……………... 153
Ann’s Mathematical Explicitness and NMI use…………………………………………………... 154
Ann’s Explicit Reasoning concerning the NMI Construction of Geometric Figures……...... ……155
Deduction and NMI Use…………………………………………………………………...... …...156
Certain Constructions and Proofs Studied by Ann
Ann’s Difficulty with Understanding the Proof of Theorem SB
IPE, NMCE, AE, and GE: Final Examples…………………………………………………...... 157
xii IX. SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH……...…..………………….... 160
Summary…………………………………………………………………………………………... 160
Conclusions and Future Research Questions…………………………………………………….... 163
Abstractions, Embodiments, and Reasoning using NMI
Empirically Tagged Theoretical Description: Ed
Empirically Tagged Theoretical Description: Ann
Exploitation of Mind-Independent Representations……………………………………………… 172
Kinds of Abstractions: Ann and Ed
Concrete to Abstract Perceptual Judgment
Future Research Considerations…………………………………………………………..…….… 176
Explicitness, Four Embodiments, and Mathematical Manipulatives………………………...... 178
Final Conclusions and Research Questions…………………………………………………...…... 180
Disclaimer……………………………………………………………………………….………………………….. 182
APPENDIX A SCREEN SHOTS: SCIENTIFIC NOTEBOOK NMI………………….…...... 184
APPENDIX B ED’S EXPOSITORY WRITING AND RELEVANT PROBLEMS……..……... 186
APPENDIX C MATHEMATICS EXPLORATION USING THE NMI: SCIENTIFIC NOTEBOOK…………………………………………………...……...…………. 212
APPENDIX D SCREEN SHOTS: GEOMETER’S SKETCHPAD NMI………..……...……….. 229
APPENDIX E ANN’S INITIAL LEARNING OF INTERFACE PROCEDURES……………… 233
APPENDIX F ANN’S FIRST PROOF………………………………………………….……….. 235
APPENDIX G POINTS EQUIDISTANT FROM TWO POINTS: CONSTRUCTIONS AND PROOF……………………………………...... ……... 238
APPENDIX H THEOREM SB: THE SIDE BISECTORS OF A TRIANGLE ARE CONCURRENT………………………………………………..……..……. 244
APPENDIX I THEOREM ABC: ANGLE BISECTOR CONCURRENCY……………...... 246
APPENDIX J PYTHAGOREAN CONSTRUCTION AND PROOF……………..…………….. 249
xiii APPENDIX K HUMAN SUBJECTS COMMITTEE FORMS……………….………………….. 256
REFERENCES AND BIBLIOGRAPHY……………………...……………….…………………. 259
BIOGRAPHICAL SKETCH………………….………………………………………………….. 279
xiv
LIST OF TABLES
1. Operational Terminology……………………………………………………………...... 90
2. Alphabetized cross-reference to terms as numbered in Table 1…………………...……... 106
3. Tall’s Natural Mathematics Theory………………………………………………………. 111
4. Reasoning Acts Leading from Concrete to Abstracted Embodiments……………...... 175
xv
LIST OF FIGURES
1. Calculations using the NMI Scientific Notebook………………………….…….. 184
2. Plotting using the NMI Scientific Notebook…………………………………..… 185
3. Markov Chains…………………………………………………………………… 187
4. Markov Chains: Applied and theoretical study……………………………….…. 195
5. An NMI facilitated exploration: Conjecture on finding Steady State………….… 213
6. An NMI facilitated exploration: “Serial Production”…………………………….. 220
7. “Serial Production”: State Space Flow Graph……………………………………. 228
8. Constructions using the NMI GSP…………………………………………….…. 229
9. Transformations using the NMI GSP……………………………………………... 230
10. Rotation using the NMI GSP………………………………………………...….. 231
11. Analytic Geometry using the NMI GSP…………………………………………. 232
12. Triangles within triangles……………………………………………………..….. 233
13. Equilateral triangle by radii………………………………………………………. 234
14. Pedagogical motivation: Angle sums………………………………………….…. 235
15. Construction: Sum of the inner angles of a triangle…………………………..….. 236
16. Formal proof concerning the interior angles of a triangle……………………...… 237
17. Construction for case 1 of 2pt……………………………………………………. 238
18. 2pt construction using vector translation…………..…………………………...… 239
19. Construction of a perpendicular bisector…………………………………………. 240
20. Triangle constructions facilitating proof……………………………………...….. 241
21. Forward case proof sketch………………………………………………………… 242
22. Case two construction and formal proof………………………………………..… 243 xvi 23. Side bisector concurrency construction………………………………………..…. 244
24. Triangle angle bisector concurrency diagram …………………………….....…… 246
25. Angle bisector equidistance: Lemma AB proven by two cases…………….…… 247
26. Angle bisector equidistance: Lemma AB proven by two cases…………….…… 249
27. Inner square construction: Reflections and rotations………………………….…. 250
28. Square within square: All reflections………………………………………….…. 253
29. Construction used (and animated) for proof of Pythagorean’s Theorem……….... 254
30. Ann’s (pre-scaffolded) proof of Pythagorean’s Theorem…………………….…... 255
xvii
ABSTRACT
The researcher characterized the pragmatic reasoning of students’ mathematics learning using certain technology. A “Natural Mathematics computer Interface” designation, NMI, was introduced and predicated on its virtual use of things like compass-rule, or pencil-paper traditional mathematical inscriptions. The NMI provided capacities for manipulative geometric constructions and transformations, or symbolic interfacing to a Computer Algebra System. Two separate case studies facilitated empirically-based characterization and reflection concerning students’ explorations, experimentations, and deductions in this NMI use setting. Over the course of a semester, one student studied Geometry proof (an elementary education major), and one Markov Chains (a lower division mathematics major). Four distinctive types of perceived mathematical embodiments were observed to be used by the students. These abstract embodiments, and related reasoning acts were described in the context of C. S. Peirce’s Pragmatic Reasoning theory. NMI interactivity, and the means of a mathematical semantics level organization (via interface lay-out), were seen to be important contributors to the students’ pragmatic reasoning. The abstract types of mathematical embodiments revealed were named: i) Interface-procedural, ii) Natural Mathematics Computational, iii) Applications, and iv), Generic. These mean, respectively, (i) interpreted merely as memorized interface procedures, (ii) resultant from interactive computation, interpreted as mathematical in a (sometimes) surface sense, as the student may not understand the underlying mathematics directing the computation, (iii) resultant from a student’s interest in a real-world application used to analogously consider a mathematical model and its interpretation, and (iv) clearly abstracted and generalized, internal or mentalesque mathematical explanations or systematizations.
xviii CHAPTER I INTRODUCTION
Mathematics is, in a sense, a language for, by, and of reasoning. Extending this analogy to language, mathematics can be taken to have proclivities to natural, or innate cognitive capacities, much like some linguists see natural language. Technology is being developed where most all the nuances of certain parts of this mathematical language (e.g., diagrams, symbols, and numbers) can be used to interactively communicate with computers. Compare the case of natural language implementation. Implementing natural language interaction or discussion with a computer would involve unheard of artificial intelligence capabilities (e.g., having a computer emulate conversational capabilities in English). Mathematical syntax, semantics, logographics, and diagrammatics are particularly suited for online implementation due to its formal deductive structuring. As a consequence of this, computers are now able to hold a written dialogue with users in a purely human based language. The term Natural Mathematics Computer Interface (NMI) was devised by the researcher to provide a distinct name for this kind of “written dialogue with users in a purely human based language.” Logographics describes a pictorial quality of written language. It contrasts with alphabetic, or phonetic based written languages. Logographical languages can involve a highly abstract pictorial quality. Chinese is considered logographic. There are many abstract pictorial, or diagrammatic aspects of written mathematics. Examples of mathematical diagrammatics include plots, and geometric figures, as well as (more standard taken to be) diagrams such as used to represent bijection or surjection relations. The mathematical written language forms and their underlying logical relationships help interweave deep cognitive learning experiences as well (Lakoff & Nunez, 1997). Such experiences range from subtle implicit subjective interpretation to scientifically explicit objective representation.
1 Mathematics has evolved so as to admit a natural cognitive sort of recursiveness.1 Aspects of mathematics explicitly understood in one context are implicitly understood in others, allowing for the great compactness or enfolding of mathematical elements in the both written and cognitive forms of this so-called language. The mathematical diagrams, symbols, procedures, numbers etc., are not unlike a natural language (e.g., English, Italian) in so-far-as they involve both implicit and explicit understanding—yet mathematics having a far superior degree of precision in both its logicality and meaning.2 The following had three goals. These goals were (a) to describe and discuss certain mathematics-based interactive technologies, (b) to describe a theoretical means for discussing the practical aspects of a student’s reasoning activity when using this technology, and finally (c) to use this method for researching such activity.
Rationale Research in mathematics education technology use is challenging. There are many different kinds of software, sometimes involving students in many divergent kinds of thinking processes. I qualified a software category particularly attuned to this kind of challenge. This general software category, although the programs within it change, are held together by an ideal goal of facilitating natural mathematics representation and interaction as previously discussed. Is there something essential in mathematical thinking that may help focus research on using new technology tools? It makes sense in this research context to choose to use a theory of pragmatic reasoning to organize one’s descriptions. Mathematics learning quintessentially involves reasoning, while technological tool use is a pragmatic exercise. Pragmatic reasoning is also inclusive enough to include formal deductive as well as plausible inference. This is a rationale for (a) defining a particular ideal for a class of software constantly under development, (b) focusing otherwise divergent discussion of students’
1 Recursiveness in cognitive natural language processing at a syntactic level has been proposed by Noam Chomsky. Recursiveness in mathematical thinking at a semantic level has been conceptually proposed by Tom Kieren and his colleagues. 2 Zoltan Dienes (interestingly here the grandson of Zoltan Dienes, the inventor of base ten blocks (Dienes, 1961)) has written extensively on tacit or implicit knowledge from a psychological perspective. Dienes generally describes implicit knowledge to be unconscious, in contrast with explicit knowledge which he describes as conscious (Dienes & Perner, 1999).
2 thinking through consideration of how they are reasoning, and (c) choosing to use a theory of reasoning framework which is flexible enough to capture new kinds of student reasoning relevant to the new technologies being developed.
Learning and Reasoning using an Ideal Super-calculator Much critical research on graphing calculator use has critically shown positive outcomes concerning the facilitation of exploratory and experimental frameworks for student’s learning. Following along these lines, assume a mathematics student has access to a calculus, geometry, linear algebra, differential equations super-calculator described in the introduction. Assume the students’ communicatively interactive interface to the super-calculator were as transparent, flexible, and admitting of detail as their using manual mathematical tools such as a pencil and paper, or compass and straight edge. How might students use such a device? They may not want to use it because of the precise mathematical formality required to address such a super-calculator, unlike the pre-packaged formulaic button formats of typical calculators. Rather than be precise and explicit in their mathematical denotations, some students might rather think through a problem using their own intuitive or qualitative intellectual resources. On the other hand, students may have neither the mathematical insight, nor the mathematical language facility to have an understanding of the context, orientation, or frame for the mathematics at hand—interactive super-calculator or not.3 Students might also hedge their reasoning with the super-calculator (e.g., trial and error use). This kind of guessing and checking may or may not prove fruitful for students—depending on their mathematical acuity as well as their general practical reasoning capabilities. How might a student intellectually frame, reason through, or reason about mathematical situations using such powerful computing tools? How might students react
3 Trying to describe this kind of issue parallels pragmatic and semantic considerations in linguistics. Linguistics has classically struggled with capturing language use pragmatics, and semantics in a general theory. Ernst von Glasersfeld early on explained that his constuctivist sense of the need for students to frame their own mathematical ideas was inspired from linguistics work he did trying to solve machine translation problems between natural languages (e.g., English to Italian). Machines have not the ability to frame their translational contexts, leading to high error rates in the translations.
3 to this potentially overwhelming power? On the other hand, how might they find ways of using the machine to genuinely benefit their mathematical understanding? Certain computer programs provide ideal approximations to the kind of ideal super-calculator interfaces described. Their general qualities described below can be seen to set them apart as a unique classification of mathematics educational software. This research teased out different aspects of two students’ reasoning, plausible and deductive, while using this software.
Aspects of a Natural Mathematics Interface Many user interfacing softwares are fashioned so that the user can interact with the computer within some prescribed framework, designed for one purpose or another (e.g., running a simulation of a biological cell model, mathematical computing, or the playing of a video game). This is typically accomplished using a computer command language or a point and click graphics user interface. The software user interface studied here specifically prescribes a natural mathematics language and representational framework. The typical computer command language is replaced by a kind of natural language mathematics—where mathematical diagrams are considered a natural part of this language. The graphical user interface is semantically organized around conventional mathematical areas, topics, subtopics, et cetera. Software designed around these properties is designated here by the term ‘Natural Mathematics Interface’ or ‘NMI.’ NMI is meant to designate an evolving software design ideal, where aspects of its functional properties are likely to be continually refined and reformulated as technology moves forward. Like the arithmetic, to scientific, to graphing calculator technology development, NMI just extends the migration of finer grained conventional mathematical representations of ever higher levels of mathematics onto automated computational platforms. The NMI ideally captures the kind of direct and naturally evolved interface one has with their paper, pencil, compass, straightedge, graphing paper, et cetera. From a practical computer program classification standpoint, NMI can ideally designate a specific class of actual stand-alone computer programs (e.g., consider
4 programs such as Geometer’s SketchPad, discussed below). Certain NMI programs will provide back-end interface options to varied computing implementations as well.4
Initiation of Study My explicit consideration of qualifying a distinct NMI format came out of working with Computer Algebra Systems (CAS).5 In 1988, I began using a CAS called MatLab in an engineering setting. Later, in 1997, I needed a different CAS, Mathematica, to complete some graduate work in applied mathematics. Mathematica though was too complicted to use in as educational setting, so I began learning Maple. Frustrated with having to learn yet another CAS language, I tried to find something more transparent to use, more fit for a general mathematics educational setting. This led me to the computer programs Scientific Notebook (SNB) 6 (Mackichan, 1996), DPGrapher (Parker, 1999) and Geometer’s SketchPad (GSP) (Jackiw, 1991; Scher, 2000). This group of programs sparked a classification of NMI as a distinct entity.
Research Overview The research began as a case study and action research with two undergraduate students whom I had the opportunity to lead through an affirmative action summer program funded for the purposes of encouraging African-American students to attend graduate school in the sciences. The students had funding to work on their own individual undergraduate mathematics research projects. One student worked on geometry proof, and the other worked on learning about Markov chains and their basic applications. The student’s research involved the use of two different NMI’s: Geometer’s SketchPad and Scientific Notebook.
4 For example, the Scientific Notebook NMI (MacKichan, 1998) has interfaced the Maple Computer Algebra System interchangeably with the Mu Pad Computer Algebra System, noticeably effecting the NMI performance in certain areas. Computer Algebra systems are immediately discussed in the following section, and footnote number 5. 5 Computer Algebra systems are high level computer languages which emulates a natural mathematical format. 6 SNB has been adopted by several universities (e.g., Texas A&M, Kennesaw State, C.S.U., and the U.S. Naval Academy) for teaching undergraduate calculus and engineering courses (Allen, Stecher, &Yaskin, 2002; Allen, Rahe, Stecher & Yasskin, 2000). Recently, Hardy, Richman, Walker, & Wisner (2005) have written a calculus text available on the internet using SNB (in publication).
5 The research was successful for both students. They were able to present their work at several conferences and as well as gain funding for ongoing research. One student received the first place prize in mathematics upon presenting his work at the national affirmative action competition, while the second student presented her work at both a national and international conference. The research involves two parts. The first part involves documentation of the specific case study with the students, and analyzing the data gathered. The second aspect of the study involves an action research reflection. Here, I would reflect upon my own general views concerning NMI use. Several simple grade school level computer programs were used in the discussion to give concreteness to the researcher’s reflections on pointed aspects of student reasoning and NMI related use.
Research Question The general guiding question for the research is: How might students reason mathematically when using NMI oriented tools? More specifically, how might students reason having immediate, interactive access to computers equipped with transparently direct natural mathematical capabilities? How might using detailed and compliant natural language mathematical syntax, semantics, and pragmatics effect a student’s reasoning as it concerns their learning of mathematics? The following review of literature cites much research already contributed in this general area of technology use. The research concerns the related foci. 1) A category of mathematics educational software, NMI, which may be useful for future research consideration. 2) A richer sense of the term ‘mathematical reasoning,’ as it applies to a student’s learning of mathematics.7 The work of C. S. Peirce is developed as a research framework in this regard.
7 Mathematical reasoning has recently been looked at from the point of view of making conjectures, or forming hypotheses (e.g., see the compilation in English, 1997). Polya (1954) elaborated on this topic as well. C. S. Peirce balances these kinds of plausible reasonings with deductive reasoning—therefore presenting a useful framework for the research.
6 General Introductory Remarks Following are several discussions which motivate the present study. Mathematical representations are discussed in the context of it comprising a language, as well as a context involving the processes and products of reasoning. The teaching, learning, and applications significance of conventional mathematical representations as opposed to general computational algorithms is considered. The section concludes with several remarks concerning the relevance of a pragmatic reasoning perspective to the present study.
Natural Language Mathematics One way to view mathematics is as a natural language. Natural language is a term generally used in cognitive science, and computer science to distinguish languages such as English or Italian from computer languages. Computers are unable to interpret or produce many aspects of natural languages. Typically in linguistics, one refers to the syntax, semantics, and pragmatics of a natural language. An analogy to what one might call ‘natural language mathematics,’ can be made in simple terms as follows. Syntax or grammatical structuring seems apparent enough (i.e., the syntactical form of well formed mathematical formulae). Semantics typically refers to the meaning of things in natural language, and so can be analogized to mathematics (Lakoff & Nunez, 1997). The pragmatics of natural language mathematics is seen to be analogous to the pragmatics of a natural language, noting Jaszczolt’s description of the latter as
…a view that advocates underdetermined semantics according to which a pragmatic aspect of meaning contributes to what is said. The distinction is made here between the semantics of natural language and the semantics of the conceptual representation system, with pragmatic factors bridging the gap. Pragmatic factors are understood as contextual information plus… including an account of [the] speaker's intentions. (Jaszczolt, 1999, p. 199)
7 Student’s interaction with NMI is analogous to consideration of a speaker’s intentions when communicating with a machine using natural language mathematics. A student’s intentions relevant to this research involve figuring out the conceptual representation system of the mathematics at hand. The NMI implementation itself is a pragmatic factor as well, providing certain contextual information. The relationship to pragmatic reasoning is implicit in this linguistic analogy in so far as the intention of figuring out certain mathematics involves the pragmatic factors pointed out.
Natural Mathematics, Learning, and Technology This research raises the question just what the term ‘natural mathematics’ means in a context of mathematics learning and technology. This is vital as the consideration of the ideal NMI has led to questions about what the term ‘natural’ means in this context.8 David Tall’s (2000a, 2000b, 2002, 2004, 2005a, 2005b ) works are applied in context at this point in order to elaborate of what is meant by natural mathematics in a technology use setting. Tall breaks mathematics into three classes: embodied, (what he calls) symbolic-proceptual, and formal.
The embodied world underpins all our activities. It begins with our perceptions and actions on the actual world and through the use of language for internal and inter-personal communication; it builds from perceptual representations to platonic representations. The proceptual world is the world of calculation in arithmetic and symbol-manipulation in algebra and symbolic calculus. It uses very special symbols (called ‘procepts’) which can function either as a process (such as addition) or as a concept (such as sum). (Tall, 2002, p. 1).
Tall goes on to describe embodied mathematics as the underlying natural framework of mathematics itself, within the context of learning. He then uses symbolic- proceptual mathematics in a way I find particularly relevant to this study. Ann and Ed made many process oriented interactive moves with NMI, and seemed to learn concepts
8 Peirce’s notions of pragmatic reasoning considers both the natural and the idealic as certainly distinct, and both considerably involved in discussions of logic—an essential part of any consideration of what mathematics is in some general sense (EP2, 250).
8 from them. Tall also considered a third conception of mathematics following on from the embodiments, and the proceptual: the formal-axiomatic. This was seen to play a role in Ann and Ed’s work as well, but to a lesser extent as concerned the immediate use of the NMI. Learning and reasoning are seen to shade into one another. They are also considered distinct at times in this dissertation in order to emphasize the cogent, fruitful, simple, and single theory of Peirce’s pragmatic reasoning, in contrast to a plethora of psychological theories concerning the learning mathematics.
Mathematical Representations and Reasoning There are other ways of looking at mathematical representations besides the linguistic orientation touched on above. Many (e.g., Wheatley (1997), Tall (2002)) consider mathematics education to be well served by considering the psychological or mental aspects of a particular person’s learning, sense-making, or reasoning in mathematical activity as being a defining characteristic of mathematics itself. Many mathematicians are Platonists, seeing mathematics as a kind of pre-existing form which is discovered through a method of conjecture, postulation, and deductive proof. Mathematics education is particularly interested in viewing mathematics through a lens which lends itself to descriptions of how it might be learned. Therefore, in addition to the (NMI relevant) linguistic qualifications in the previous section concerning a kind of natural language mathematics, considerations of mathematical reasoning qua mathematics learning ought to play a role in the present research.
Reasoning and Pragmatism Peirce’s work on pragmatism sketched general inter-relationships of signs and reasoning.9 The concern for this research involves how students use the representations of mathematics to learn it—the signs of mathematics in order to reason through it. What’s more, present technology provides explicitly dynamic signs (e.g., moving pictures, dynamic inquiry into calculational chains). The term ‘natural mathematics’ intends to
9 Signs are a term Peirce used to describe things which symbolized, or stood for other things. The area he developed here is now generally called Semiotics. This dissertation focuses on Peirce’s descriptions of reasoning, avoiding the complexity, and sometimes vague descriptions put forward in semiotics.
9 point to the naturalization of reasoning Peirce can be interpreted as putting forward concerning the primary role that perception plays in reasoning.10 However, Peirce equally emphasizes a method of logical regulation on this naturalized perspective. Peirce wrote, “…there are propositions about triangles in general, which… are either true or false…whether we have an idea of a triangle in some psychological sense or not…” (EP2, p. 227).11 However, Peirce qualifies a logic of pragmatism based on perceptual judgments, or perceptually affirmed statements. This motivates the researcher’s drawing of a relationship between NMI use and pragmatic reasoning, or Peirce’s so-called “logic of pragmatism” (EP2, p. 226).
Natural Mathematics, and a Natural Mathematics Computer Interface It was argued in the introduction that what might be usefully considered to be natural mathematics can be represented by an analogy to natural language (i.e., natural language mathematics). Also though, we touched on the strong connection between reasoning itself, its processes and products.12 and how those might be central to a moniker of ‘natural’ mathematics qua NMI. The researcher is then specifying ‘natural mathematics’ to convey a sense of both a kind of natural language mathematics, and a well reasoned logical system, impossible to understand without it being reasoned through. Natural language allows one to express reasoned arguments in a language which has little to do with the argument being presented, while natural mathematical language (or “natural mathematics”) is already a result of reasoning itself, therefore facilitating much more concrete and explicit expressions of well reasoned arguments. In broad terms, NMI allows for the doing of mathematics on a computer in ways that approach how one would use manual-intellectual tools such as paper, pencil,
10 Peirce saw what he called perceptual judgments as fading into what he saw as pragmatic reasoning. This is taken up in detail in the following chapter. 11 A citation in the form of “EP1, and “EP2” is adopted here, following common convention with respect to other scholars referencing the highly atypical, sprawling publications work of Peirce. EP1 and EP2 refer to the Essential Peirce: Selected Philosophical Writings, Indiana University Press. The numerals 1 and 2 used here, refer to the volume number cited. Six volumes of a projected thirty have been published to date under the name “The Peirce Edition Project.” 12 In the present context, one might think of “process” as cognitive, while the “products” are written. This interpretation of “product” is particularly apt to the NMI use context. NMI is defined through its capacity to provide a mathematics-logographic written language format for interacting with a computer.
10 compass, and straight edge. This made apparent in Appendix A, Figures 3-4, and Appendix D, Figures 8-11 computer display screen shots of the NMI’s used in this study. NMI enables detailed explication of purely mathematical representations which extend one’s memory and intellectual organization of concepts via an external media. The invention of writing extended a person’s ability to express and carefully study what previously was limited to verbalizations. Analogously, NMI extends natural mathematical facility. Natural refers to a property of being cognitively compliant or malleable—a property emergent from the human evolution of mathematical systems. NMI compliments many cognitively imaginable portrayals, visualizations, diagrammatizations, or images of typical mathematics. Automated computing makes the mathematics interactively live. NMI also extends certain external mathematical representations (e.g., continuously transformable graphics or geometric figures). And finally, its function as an interface ergonomically organizes mathematical topics in certain natural mathematical groupings (e.g., having a menu pull-down for the geometric transformations of translation, rotation, dilation, and reflection).
Reasoning and Learning using NMI Tools This research considers certain current research on mathematical thinking and learning under a theoretical rubric based on Peirce’s Pragmatism as it concerns reasoning. Pragmatic reasoning theory provides a platform for considering the interaction between various kinds of plausible reasoning with deductive reasoning. Pragmatic reasoning can be seen in the following chapter to be a basis for the scientific method as well as including the kinds of reasoning appropriate for describing mathematics learning. This outlook on reasoning is reviewed extensively in the next chapter concerning how it can guide a study of student reasoning while using NMI tools.
11 Researcher’s Bias Qualitative research methods encourage researchers to share their own biases. I have biases concerning (a) the importance of aspiring to objective knowledge, (b) an attempt at sidestepping (as much as feasible) the difficulties in describing subtleties in student’s thinking, and their interpretations of meaning, contrasted with describing the reasoning which emerges this thinking and interpretation, and finally, (c) the importance of emphasizing the difference between deductive and plausible inference. I believe that constructivist based theories of learning and objective knowledge complement one another. It is my view that mathematics is the best tool for aspiring to objective scientific knowledge. I also have a strong belief in the usefulness of constructivist-oriented pedagogy. Therefore, I have been determined to explicate a research framework that can explicitly accommodate both a constuctivist and an objectivist mathematics learning theoretic position. Secondly, I found during my pilot study and early research experiences that it was extremely difficult to describe a student’s thinking, understanding, and interpretations of meaning. I experienced the frustration of noticing that there were many more shades of distinctions in describing these things than I felt I would ever be able to capture. Also, I was never quite sure if I am describing their sense of things, or if I am projecting my own thinking onto theirs. Along these lines, I am not sure if my intermittent member-checking with my pilot student and case study students during the course of their long term (semester) NMI use might have been leading his subsequent articulations of his thinking. These kinds of issues have been discussed in many places concerning qualitative research in general. This influenced my particular interest in discussing pragmatic methods of reasoning which bear intimate relationship to practical thinking and meaning but have an organizational component that can be explicated somewhat independently from the subtle particulars of student’s psychological thinking processes. My final bias concerns the importance of making a distinction between deductive and plausible inference. This is a distinction that can become blurred as the use of computers in teaching mathematics goes forward. These concerns biased me towards the pragmatic reasoning stances of Peirce.
12 Peirce was able to relate the kinds of plausible reasoning employed by scientists and engineers to the deductive reasoning of mathematical proof. Peirce showed how distinctively different kinds of reasoning could none-the-less complement one another. Secondly, Peirce initiated the area of Semiotics which allowed the consideration of meaning and interpretation alongside considerations of methods of reasoning. Peirce’s perspectives facilitated views of learning which admitted both a constructivist and objectivist perspective concerning the how’s, why’s and what-for’s associated with mathematics and the mathematical sciences.
13 CHAPTER II LITERATURE FOUNDATIONS
This chapter covers three general topics: 1) mathematics education research literature involving technology use, 2) specific connections between mathematics educational literature and C. S. Peirce’s writings on plausible and deductive reasoning, and 3) a review of literature concerning mathematics educational constructivism and its connections to the pragmatic aspects of reasoning put forward by Peirce. Each of these topics is contextually motivated relative to considerations of how a student might use NMI related computer programs for reasoning mathematically. The researcher has defined NMI related programs as sharing the property of natural mathematical representation as regards notational artifacts, diagrams and procedures. Natural is meant in the sense of how a traditional mathematics student or instructor might notate or draw mathematical diagrams (e.g., using compass, rule, pencil, and paper), as well as be seen to display typical sorts of mathematical procedures. Natural mathematics is meant to be analogous to the term ‘natural language’ also used in other kinds of computational contexts (e.g., English or Italian).
Organization of the Chapter This chapter is divided into five sections. They are: I. a brief overview of Peirce’s pragmatism in so far as it is relevant to mathematics education and NMI use. II. a detailed literature review of NMI related mathematics educational technology use. III. a review of literature defining abduction, induction, and deduction from Peirce’s perspective in the general context of the research. IV. a Peircean guided review of mathematics educational research literature describing aspects of reasoning pertaining to psychological origins, as well as
14 the facilitation of drawing generalizations (e.g., visualization, use of metaphor, abstracting). V. a rationale for taking Peirce’s pragmatism as a research stance for the research.
Overview of Peirce’s Pragmatism Vile (1997) a mathematics educator, notes that Peirce “[attempts] a development of logic aimed at clarifying reasoning and formulating easily understandable general principles of reasoning.” Charles Sanders Peirce (1839-1914), an American, worked as a geological engineer, chemist and logician, while also publishing in a broad range of the mathematical sciences, mathematics, psychology, philosophy, and many other areas. He is renowned for work on the philosophical perspective called Pragmatism.13 His view of pragmatism drove his prolific writing concerning sentential, syllogistic, and predicate logic,14 reasoning in general, realism, objectivity, and scientific truth (always in consideration of practical cognition, experience and subjectivity) as well as pioneering the field of semiotics (Burch, 2001). Peirce was a prolific writer15. It is of general interest to mention that Peirce wrote of pure mathematics as an endeavor based in aesthetic spheres. Neither logic nor mathematics was seen by Peirce to emerge from merely technical motives.16 Peirce’s perspectives on reasoning provide important consideration of subjectivity, intersubjectivity, and objectivity without having to resort to the bottomless realms of psychology or sociology (see Rescher, 2005, p. 23-28). Peirce saw psychology and
13 William James and John Dewey are other well known members of the pragmatist school. 14 Predicate logic is the logic of mathematics. It allows for quantification (e.g., There exists an ‘e’ in the set X, such that for all elements x in X, we can write: x * e = x, where e signifies a right identity element for the operation *.) as well as includes all of sentential logic (e.g., If a and b, then c) and syllogism (i.e., all two premise deductions). Peirce is said to have had quantification worked out first, before others working near that time (e.g., Giuseppe Peano). Peirce developed diagrammatic means for simplifying logical long expressions called “existential graphs.” These are presently gaining interest in mathematics education (Vile, 1997). 15 It has been estimated that his publications would fill 500 volumes. 16 Peirce describes the urge for the pursuit of logicality in argument, as well as the endeavors of a logician to be of a normative nature. The editors of Peirce (EP2, p. 196) write, “The logically good, Peirce said, is a species of the morally good, and the morally good is itself a species of the aesthetically good… Now the aesthetically good involves choice of aims, or purposes… [and] pragmatism involves the conception of actions relative to aims.” Logic might be thought of in a moral light in the sense of the common phrase “equal judgment before the law.”
15 reasoning as separate, developing a broad sense of what reasoning entailed (EP2, pp. 226- 241). Pragmatism can be interpreted to turn on the premise that the seeking of knowledge involves the consideration of the process and product (or action and result) of self-controlled conduct. This conduct employs a well-defined method of experimentally- centered reasoning purposefully focused by anticipated practical consequences. The principle of pragmatism is that “the meaning of any conception in the mind is the practical effect it will have in action” (Radi (2002), citing Peirce (1992/1878a)). Peirce states that the “central result” of pragmatism is the explicit definition of the methods of scientific reasoning, which he disassembles to exemplify all matter of reasoning in general (EP2; p. 528, endnote 12).17 From an educational perspective, the use of the framework of pragmatic reasoning facilitates a cogent, fruitful discussion of otherwise complex psychological processes (see Byrnes, 2001; Bransford, Brown, & Cocking, 1999; Norman, 1981). Semiotics is Peirce’s means for describing how people conceptualize what they reason about (EP2, p. 4-9).18 Reasoning is central to the learning, doing, or using of mathematics, and so consideration of reasoning from the general Peircean perspective might complement research in mathematics education, especially as it regards the practical use of technology (e.g., Olive, Iszak, & Blanton, 2000; Presmeg, 1997a; Wheatley, 1997; Gross, Levitt, Norman & Lewis, 1996; Magnolia, 1992; and the entire compilation in English, 1997).
Mathematics as Necessary and Diagrammatic Reasoning: NMI Graphics Peirce studied deductive or logical reasoning extensively. He called this kind of reasoning “necessary,” and claimed that all deductive reasoning was mathematical (Peirce, 1998/1903d). Peirce’s father Benjamin (1809-1880), a renowned mathematician, is often quoted: “Mathematics is the science of drawing necessary conclusions” (Peirce, B.,
17 The editors point out that Peirce himself wrote that his three distinctions in types of reasoning were more evidenced than the three categories he defines in semiotics (i.e., often generally referred to as firstness, secondness, and thirdness, or icon, index, and symbol). 18 Semiotics is Peirce’s theory of signs. A sign is something which stands for something else. The most typical signs are icons, indices, and symbols. Mathematical examples of these three types of signs are mathematical diagrams, models, and symbols, respectively.
16 1870). This is to say that when mathematical arguments are valid—true premises force true conclusions, for example: all A is in B, all B is in C, therefore all A is necessarily in C, meaning there are sufficient conditions for A to be in C. Peirce (1992/1878a) often used syllogistic schemes in his discussions concerning deductive as well as plausible inference. Peirce used syllogism to discuss how deductive inference can be seen to be diagrammatic, or have a pictorial basis.19 By diagrammatic, Peirce meant a kind of visualization internally or in the mind. Deduction, according to Peirce, is always something mentally observable in an abstract form. Peirce (EP2, p. 212) wrote, “All necessary reasoning [(deduction)] without exception is diagrammatic. That is, we construct an icon [(pictorial representation)] of our hypothetical state of things and proceed to observe it.” He explained that this leads us to suspect the truth of certain things, and so devise a plan to prove or disprove our suspicions. Often times certain suitable abstractions are introduced so that by “transformation of our diagrams… [properties] of one diagram may appear in another as things” (EP2, p. 212-213). Discussions of the objectification and mental visualization of the concept of a function are seen in Thompson & Sfard (1994). Some aspects of internal or mental mathematical visualizations can be represented externally. This is most easily seen to be apparent in geometry. In calculus one visualizes the limit process, which again, in certain aspects, is externally representable. Critical discussions concerning mathematical representations, and how multiple mathematical representations of identical mathematical situations can be pedagogically useful are taken up in Janvier (1987) and Someren, et al., (1998) (e.g., a function as: a set of ordered pairs, a graph passing the vertical-line test, or simply a mapping between a domain and a range). NMI studied in this research can present many mathematical representations externally and instantaneously viewable via computer graphics. Further, in the section below entitled “Representations,” there is a discussion of external and internal mathematical representations or diagrams.
19 Vile (1997) briefly reviews Peirce’s work on existential graphs which diagrammatically represent many areas of symbolic logic.
17 The Breadth of Educationally Relevant Peircean Theory Peirce saw reasoning and learning as virtually the same intellectual activity. Peirce’s descriptions of the intellectual processes of reasoning and mathematical symbolization are closely related to contemporary views on reasoning and learning such as constructivism, cognitive science and semantics, or reification (English (ed.),1997; Lakoff & Johnson, 1999; Lakoff & Nunez, 1997; Thompson & Sfard, 1994; and Presmeg, 1997a). Peirce noted observational experiences as central to conceptualization, and is noted early in his career for the position that “nothing is in the mind which was not first in the senses.” He is often quoted, “The elements of every concept enter into logical thought at the gate of perception and make their exit at the gate of purposive action; and whatever cannot show its passports at both these two gates is to be arrested as unauthorized by reason” (CP 5.212). Peirce (EP 2, p. 209-225) discusses how elements of concepts come about through perception (central to his theory of semiotics)20—while reasoning involves the use of such perceptions and concepts21. A specific example of using perceptions and concepts in mathematics educational reasoning theory is seen in Sfard (2000a, p. 344-345). Sfard discusses how a metaphor is a transplanting across contexts (e.g., from every day contexts to mathematical ones) via new conceptualizations—thus initiating a new concept. Likewise, she then discusses analogizing as being a distinctive reasoning about such concepts. Later, Sfard (2000a) directly refers to Peirce’s semiotic theory in the same general context of conceptualization’s role in mathematics learning and reasoning—particularly as it occurs in students’ dialogues amongst one another and the instructor.
20 Peirce, in his theory of semiotics, discusses an icon as something directly perceived in a pictorial fashion. In so far as perception is involved in reasoning, Peirce sees what he calls perceptual judgments (i.e., deciding what one is perceiving) as fading into the abductive reasoning process. See several topic headings below for discussion of abductive reasoning. 21 Dunlop & Fetzer (1993) define concepts as “abstractions that classify individual objects or events together on the basis of some feature or set of features.” Perception is generally defined to be the acquiring of information about the external world by means of the senses. A broader notion of perception is described by Peirce in his development of pragmatic theory to include, for example, what might at this point be roughly called learned intuitive visualization.
18 Technology as a Pedagogical Tool There are many uses for technology in mathematics education. Describing certain pedagogical software as a cognitive tool, Thorson (2006, p. 15) wrote, “Computer tools can provide motivation, incentive, and even feedback… [while] students …gather, analyze, and present information.” Concerning mathematics education, Flake, McClintock, & Turner (1990) provide general conceptualizations which touch on how computers can be pedagogically put to use, e.g., (a) exploration, (b) development of heuristic and deductive reasoning, (c) visualization and imagery development, (d) virtual manipulatives implementation , and (e) modeling and simulation. Oldknow & Smith (1980), and Oldknow (1997) also discuss a similar set of pedagogical uses of computers specific to mathematics education. Research and discussion on technology use in mathematics teaching and learning has evolved in many directions in just the last two decades (Roschelle & Pea, 2002; Laborde, 1998; Schoenfeld, 1994; Lin & Hsieh, (1994); & Kaput, 1992). Thomas & Thomas (2003) write, In order to use technology effectively in mathematics education, its applications must enlighten and empower the learner relative to challenging mathematical questions. That is, the mathematics itself, rather than the enabling technology, must provide the motivation for and focus of the lesson. For this to occur, the mathematical context must be rich enough that its consideration will spawn an engaging, sustained dialogue among all participants. (p. 1) Wilson (2004) wrote, “Our interest is in empowering teachers through the use of technology in mathematics exploration, open-ended problem solving, interpreting mathematics, developing understanding, and communicating about mathematics” (p. 4). Garofalo, Drier, Harper, Timmerman, & Shockey (2000) offer the following five guidelines for using technology in mathematics teaching: 1. Introduce technology in context; 2. Address worthwhile mathematics with appropriate pedagogy; 3. Take advantage of technology; 4. Connect mathematics topics; and 5. Incorporate multiple representations.
19 The study group members of HEID (2003) echoed these kinds of guidelines, while Shannon (2003) described similar guidelines in the context of a need for future engineers and scientists. An important aspect of certain mathematics learning software involves mathematical conjecturing, exploration and interpretation (Heid, 1997; Berger, 1998). Cappuccio (1996) and Heid (1998) discuss how computer use can complement the actual organization of course curricula. Cappuccio (1996), Heid (1997), and Thomas, Tyrrell & Bullock (1996) discuss how the use of technology in the classroom can take the focus off of standard lecturing by providing a medium that allows the students to be in more control of their learning. Along these lines, Allen, Rahe, Stecher, and Yasskin (2000) have devised an online calculus course using Scientific Notebook (MacKichan, 1996), a NMI program. Allen, et al. (2000, p.2) point out, “students are a lot more proactive in learning their material” using the online facility. Students become “actively engaged in educating themselves.” Finally these authors have indicated that they spend more time “[walking] around answering questions as they arise” than they do lecturing (p. 2). Problem posing is also facilitated by technology use. Bransford, Brown, and Cocking (1999) point out how exploratory problem posing facilitates learning. Such problem negotiation and posing are facilitated by rich mathematical contexts provided through computer environments (Schoenfeld, 1989; Silver, 1987). There are specifically cognitive uses for mathematics learning technology, including NMI. Technology can raise the level of student’s thinking. This can be described as facilitating a meta-cognitive stance (Schoenfeld, 1987, 1992), or developing the capacity for explicit second-order thinking (Dienes & Perner, 1999). For example, students are able to look down on a mathematical situation (in real-time) from a vantage point unavailable to them when engrossed in working fine grained calculations by hand.
20 Wide Ranging Classes of Pedagogical Applications Software There are many wide ranging classes of pedadogical applications software being developed for mathematics educational use, and are reviewed below. This research qualified a class of programs as NMI. This facilitates research across the related aspects of these computer programs, rather than computer program specific research. Microworld Steffe & Olive (1995) provide research on an entire class of programs called Microworlds.22 Microworlds consist of mathematical situations providing a rich context to explore and work within in order to consider and solve certain mathematical problems. Students make sense of microworld problem situations by exploration and experiment using an interactive interface in order to negotiate their own paths to a solution (Steffe & Olive, 1990; Kynigos, et al., 1997). Logo is an example of a microworld program which facilitates a user friendly computer programming interface language that enable students to conceptualize vector-like movements of an object in various directions in two- or three-dimensional space (Papert, 1980; King, 1992, respectively). Microworlds have typically been geared to junior high school grade mathematics and lower. Other kinds of higher grade level microworld-like technology applications are discussed in Spicer (2002). Interactive Mathematical Situation The researcher terms an “Interactive Mathematical Situation” to represent a real- world, and real-time, 23 computed mathematical situation which can be manipulated through various parametric choices explicitly chosen through a windows-like (e.g., menu’s, toolbars, icons, or other screen-spatial layout) screen display. Lin & Heish (1994) provide examples such as a model or simulation of a real-world physics situation. At times when given the mathematical modeling equations with interactively variable parameters, students can change variable values within the equations and observe certain dynamic properties of the model in a pictorial format. Because software
22 Some might consider the Interactive Mathematical Situation class of programs to be synonymous with the microworld class. One distinction can be described as that being between school mathematics situational (microworlds) and higher level real-world mathematical model simulations. Seymour Papert initially coined the term “microworlds ” (Papert, 1980). 23 “Real-time” is a term describing a computation time which is brief enough to be able to be carried out seemingly in sync with the user input (e.g., joy stick tilt). This creates a virtual environment where the user’s sensory input timing-expectations are met by the computational input to output time requirement.
21 with attributes of this kind are moving into higher grade level applications, and employ greater use of mathematical symbolism and graphics, the researcher uses the term “Interactive Mathematical Situation” to describe the aspects of these programs. Such a program uses a screen display which is typically user-friendly (having a steep learning curve), and is typically interactive in real-time. It can be typically thought of as a computer interface program. Non-real-time Interactive Mathematical Situation A related way of using the computer to represent a mathematical situation, such as a mathematical model interpretation is typical of a computer command-language based simulations program. These programs stretch the limits of what might be termed pedagogical, however, the example is given because lower-division undergraduates who may have the opportunity to work as co-op students (cooperative opportunity education in industry) may use these programs.24 These computer programs also provide a sharp distinction between what might be termed a “computer interface,” versus what is typically termed a “computer command-language” based program—elucidating the meaning of “interface” in NMI usage. Such a computer command-language based program requires a relatively long time to learn, given a highly specific syntax requirement, differing from the spatial windows-like layout interface-type program described above. However, such simulation programs are far more flexible and powerful. A typical example here is a simulations language typically used in engineering-industrial applications such as a six-degree of freedom simulation of an aircraft’s stresses and strains as it maneuvers through its degrees of freedom at various velocities and accelerations. For example, this can include mathematically modeled wind drag, and gravitational forces, with their linear and nonlinear aircraft-component responses, e.g., wing rudders, or torque on the aircraft frame, respectively. The interactive nature of the program is typically limited to entering lists of mathematical model parameters before the program is run, i.e., the program is not real-time interactive.
24 For example, the researcher had an opportunity for two semesters to work with the Naval Research Labs, and one semester with Lockheed-Martin Aerospace Co. during his sophomore and junior undergraduate years.
22 Mathematical Object Constructor Another example of a class of mathematics pedagogical software is student built representations of a “mathematical objects.” For example, Dubinsky (1995) developed a program whereby students can directly construct mathematical operators. An example of such a mathematical object might be an algebraic matrix multiplication operator which is programmed by the student to do the appropriate row and column multiplications. Computer algebra systems can be considered to overlap the class of Mathematical Object Constructor programs (Wolfram, 1996). These programs offer many data manipulation commands which for example would allow the construction of a numerical trapezoidal integration routine using sampled data points. Spreadsheet programs can also be considered to exhibit characteristics of a Mathematical Object Constructor program. Spreadsheet programs are readily available to students, and have many applications. Spreadsheets allow students to use the spatial layout of the displayed columns and rows of data, in devising their own mathematical operations to work on this data. Natural Mathematics Interface Programs (NMI) The review below considers several NMI and related computer programs.25 A Natural Mathematics Interface program (NMI) refers to a computer interface which represents mathematics in ways that are common to manual mathematics representations such as those used on paper or blackboards, but also have the many computationally live interactive advantages detailed in the first chapter. Classes of programs with partial NMI characteristics. The following five computer program classes reviewed below each have certain NMI characteristics. If one were able to single out NMI characteristics in research studies, the specific NMI qualities of each of the software categories reviewed below could be distinctly considered in future research. 1) Computer Algebra Systems (CAS). A principle technology which emerged in the 1980’s is a high-level mathematical semantics programming language now referred to
25 The term “Graphics User Interface” (GUI) is widely used. This led my consideration of the term “Natural Mathematics Interface” (NMI )as well as the term “Mathematics User Interface” (MUI), where MUI is already a term the researcher found to desribe application specific interfaces that admitted a mathematical equation as input, e.g., a biological cell model (both NMI and NMI applications specific MUI interfaces are described in detail at the start of Table 1, Chapter IV).
23 as Computer Algebra Systems (CAS). Some research on CAS use in mathematics education concern teacher training (Lobanova & Glazov, 2002) as well as secondary and college student use (e.g., Herget et al., 2000; Ball & Stacey, 2001; Flynn, Berenson, & Stacey 2002; Tynan & Asp, 1998). Various types of research on CAS include: algebra teaching, student explorational inquiry, and general technical, calculational, and interfacing issues (Tynan & Asp, 1998; Pierce & Kendal, 1999; and Gathen & Gerhard, 2003, respectively). Gathan and Gerhard (2003) write, “Computer Algebra is a more recent area of computer science, where mathematical tools and computer software are developed for the exact solution of equations” (p. 2). The CAS is typically able to present numerical, graphical, or symbolic results (Wolfram, 1996). It should be noted that CAS are used as computational engines facilitating the computing component of certain NMI software (e.g., MacKichan, 1994; MacKichan 1996). CAS interfaces share some properties with NMI, providing some natural mathematics interface layouts as well (e.g., Wolfram, 2005). Some CAS are rapidly developing other NMI properties, such as presentations of step-by-step calculus calculations (Maplet, 2005). The mathematics education research on CAS use is extensive. Unfortunately, as NMI qualities are typically not identified in the research literature per se, it is not possible to distinguish which aspects of these studies discuss NMI use in particular. Some critical considerations in using CAS are very similar to NMI. Herget et al. (2000) point out that certain intermediate skill levels of hand calculation are necessary (also see Flynn, Berenson, & Stacey, 2002). Flynn, Berenson & Stacey (2002) report, “Twenty-four… educators with some CAS experience responded to a … questionnaire” concerning the estimation of time it would take students to do “particular items by-CAS or by-hand…” (p. 7). This research concluded in part, Syntactical and technical difficulties encountered with entering expressions, student execution time to complete a procedure with CAS, automatic simplification by CAS of expressions and the interplay between by-hand and CAS techniques are issues that teachers must assume familiarity with and develop strategies for… (p. 8)
24 concerning effective CAS use. Several articles included in Carroll (2002) also point out that learning specific computer languages (including CAS) can be difficult because users need to learn idiosyncratic syntax and program specific command or functional notation meanings. There are approximately forty general purpose CAS, and eighty special purpose CAS individual computer languages at the time of this writing (CAS list, 2005; Compalg, 2004; CAIN, 2004), though it must be said that only a handful are typically used in education through the undergraduate level. Ball & Stacey (2001) also point out that some “hand-held calculators (really hand-held computers)” (p. 55) have CAS capabilities. However, these capabilities are more difficult to use due to a compressed syntax, along with the physically small screen display, causing there to be many layers of hidden menus. Many positive considerations of CAS use provide a direct overview of how NMI can be used as well. For example, Ball & Stacey (2001) state, Pedagogically, including graphical and numerical approaches in the curriculum gives some students better understanding of what a solution of an equation means. They can lose sight of this under the weight of learning the algebraic manipulations. What are the new elements of mathematical literacy for solving equations in an era when CAS is available (p. 59) Their research specifies the following answers to this question: a) understanding basic “equation solving” operations b) being sufficiently familiar with algebraic manipulation to be able to modify an equation before input or to recognize calculator output that is in non-standard form c) being able to recognize the basic form of an equation (or set of equations) [e.g., implicit or explicit or matrix form] d) knowing the nature of the solutions of equations of various forms [e.g., asymptotic, exponential] e) recognizing that numerical, algebraic, or graphical solution methods are available f) dealing with accuracy… (p. 56-57).
25 Ball & Stacy’s list applies to NMI as well. Other considerations might involve students being able to interpret whether their solutions make any sense. This may entail checking dimensional units in applied problems, or simply making rough estimates of what might be a sensible solution range. 2) Mathematics User Interfaces. The term ‘Mathematics User Interface’ (MUI) is a sparsely used term, denoting devices which are related to NMI in spirit. Two examples of MUI are given. First, the term MUI has been used to refer to a component of a much larger simulations system. The MUI component controls the simulation’s behavior and can allow for the input of natural mathematical symbols (Virtual Cell, 2004). In the Virtual Cell, the MUI interface allows certain models to be input mathematically, say to model the shape of the 3-dimensional cell (e.g., spherical, ellipsoidal), or the densities of certain nutrients or wastes over space and time. The Virtual Cell is an example of an Interactive Mathematical Situation computer program interface. Secondly, MUI is a term used to refer to mathematical graphing programs. For example, Avitzur (2004) describes his program named ‘Graphing Calculator’ as “a tool for quickly visualizing math. Just type an equation and it is drawn for you without complicated dialogs or commands26” (p. 1). This second kind of MUI software is strictly for plotting and visually analyzing mathematical graphs (e.g., rotating, taking cross sections, or zooming in or out of 3-dimensional graphs). Avitzur (1990, 1995) discusses how these MUI facilitate students to “directly manipulate” two- and three-dimensional graphs, exploring them by changing plotting variables as well as constants and variables. Avitzur, contributing to Kajler (1998) wrote, “The [MUI] interface is live, active, and interactive, and most of all, is minimalistic both visually and functionally. There are very few menu items and dialogs. Most interaction is through direct manipulation grasping with the mouse and provides immediate feedback” (p. 43). However, Nold’s (1999) experience with such a minimalistic MUI demonstrated that although MUI has advantages within its extremely limited range of mathematical
26 Ron Avitzur began this programming project while in school in 1985, and finished most of it while working for Apple computers—perhaps leading to the coinage of MUI, similar in a sense to Apple’s GUI (see Avitzur, 2004).
26 functionality, its procedural command structures even limit these advantages relative to an NMI. Parker’s (1999-2004) MUI is identical in function and general interface qualities to Avitzur’s (2004) program. Nold (1999) presented research concerning a student’s use of Parker’s (1999) MUI graphing program. Nold (1999) served as a pilot study for this research, and is described in the next chapter concerning research methodology. A full NMI would ideally have the same qualities of MUI built into it, without the present procedural, or non-natural mathematical functionality. It was evidenced by Nold (1999) that the student using the MUI often turned to his graphing calculator for certain analyses. Presumably then, a full NMI would move the MUI technology into more useful spheres concerning their use in learning to analyze plots of mathematical expressions. 3) Mathematics on the Web: OpenMath and MathML. OpenMath is a [computer] language for communicating mathematical data between computer programs. Its purpose is to enhance different programs that handle mathematical data, such as symbolic algebra packages [(CAS)], typesetters, Web browsers, and the like, by providing a basis for their smooth co-operation. (OpenMath, 2005, p. 1) When interfacing a CAS, these programs have the potential to facilitate a complete NMI system. “MathML can be used to encode both the presentation of mathematical notation for high-quality visual display, with the capacity to display this in standard web browsers (MathML, 2005, p. 1).” MathML is intended to be used in conjunction with OpenMath when semantic issues concerning computer interpretation of mathematical expressions is an issue. Both MathML and OpenMath are intended to provide perfectly conventional NMI mathematical expression displays. 4) Graphing calculators. Graphing calculator research is prodigious and much of it relates to the graphing, calculational, and natural mathematical interfacing aspects of NMI. Ellington (2003), after evaluating over eighty research studies on graphing calculators wrote, "The findings of 54 research studies [(the only ones with the kinds of data needed for the methodology of this study)] were integrated through [inferential- statistical] meta-analysis to determine the effects of calculators on student achievement and attitude levels” (p. 433). Ellington found that graphing calculator use was essentially either positive or neutral concerning the parameters of the study.
27 Ellington (2003) went on to say, In spite of the fact that the NCTM (1989, 2000) has been advocating changes to the mathematics curriculum with computer and calculator technology as an integral component, the search for studies for this meta-analysis yielded only six studies in which special curriculum materials were designed for calculator use. Because this number reflects only eleven percent (11%) of the studies analyzed, this is an area in which more research needs to be conducted. (p. 460) Shannon (2003) drew similar conclusions concerning the need for integration of computational resources into mathematics courses at the lower division collegiate level in reference to science and engineering related mathematical applications (e.g., modeling interpretations or raw data analysis). The research provided insight concerning where NMI might fit into such a computational resources integration. Penglase & Arnold (1996) conclude that graphing calculators allow students to work at higher levels since graphs and calculations are provided on command, bypassing the need for performing them manually. On the other hand, they report that the sometimes cryptic notations of mathematical functions and hidden layers of menu options provided by these calculators can present difficulties in learning how to use the technology itself. This echoes the results of Ball & Stacey (2001) as previously discussed in the CAS review section above. Heid (2003) contains numerous articles on graphing calculator technology use. Some results concern graphing calculator use for function graphical exploration. By providing concrete hands on manipulation of functions and their graphs, the calculators were noted, in many cases, to help students grasp abstract concepts. Watkins (2003) discusses the utility of graphing calculators in conjunction with Calculator Based Laboratory (CBL) equipment (e.g., acoustic transducers can send and receive signals much like radar). This equipment allows students to run, walk, etc., towards or away from the device while the calculator graphs (a) student distance from the transducer, (b) student rate or velocity of movement, (c) student acceleration, along a direction normal to the transducer. His results indicate that students were more likely to visualize and imaginatively manipulate mathematical ideas when presented in what Watkins calls “kinesthetic experiences.”
28 Other aspects of graphing calculator research focuses exclusively on the graphical display aspect. Shoaf-Grubbs (1992) discusses how visual thinking is enhanced by the graphs presented on graphing calculator displays. Ruthven (1990) pointed out that these graphs enhance student’s understanding of the associated symbolic forms. This graphical display research describes how calculator graphics provide numerical displays of coordinates at points which the mouse can designate, increasing the student’s ability to connect the graph to the symbolic representation. Research results reported above concerning graphing calculators reflect the kind of important contributions NMI can potentially make. This is particularly true as NMI typically works at higher mathematical levels (i.e., grade levels) than graphing calculators. Therefore they employ the use of more intricate mathematical objects, which in turn makes the automated computational utilities provided more valuable. 5) Textbook formatted NMI facilitated interactive programs. There are programs which provide direct textbook formatted presentations. These kinds of instructional specific programs are relevant to the work of educational researchers such as Robert Gagne, or Marcy Driscoll. The programs reviewed here can provide single step or multiple step interactions. a) Single-step interactions: Many computer programs are being distributed with associated textbooks in order to provide more immediate feedback to students working more or less “single step” kinds of problems typical of freshman mathematics courses (e.g., MyMathLab, 2003). NMI contributes to these programs by allowing an avenue for student entered responses, as opposed to multiple choices. Educo (1990) has one of the most sophisticated programs of this sort. Students are able to answer by free response through pre-calculus level problems. Often problems are graphical, and enable students to mouse-over certain parts of the display in their responses to graphical representation problems. Educo (1990) is intelligent in the sense that it supplies remedial exercises of a related but simpler nature when a student misses problems. b) Multiple-step interactions: Some programs use NMI facilities to output step- by-step problem solutions to students in calculus, or similar level courses (Maplet, 2004; Beeson, 2001). Students can interact in terms of obtaining hints, both graphical or
29 symbolic. These programs are generally considered to be tutorial in nature as the students control the problems to be worked on. These programs also provide certain pre-defined kinds of inquiries that the program can respond to at various steps in the solution process. The Maplet (2004) program includes what can be seen to be Interactive Mathematical Situation programs. This program allows multiple parameters to be changed in a continuous fashion yielding a mathematical model or diagram that responds to changes accordingly (e.g., numerical Riemann sums approximating integrals using variable rectangle widths, for variously chosen functions). These programs provide NMI representational aspects so that students can interact directly through mathematical symbols and graphs as they would with pencil and paper. Maplets (2004), part of Maple (Heck, 1996; Geddes & Gonnet, 1982) at the time of this writing, also provide several single screen kinds of multiple input parameter button dialog boxes. For example, there are three separate but similarly formatted (after similar mathematical operations) dialog boxes, each performing either integration, differentiation, or the taking of limits. These Maplets provide step-by-step calculus solutions with hints, if necessary, and have standard operation names on the formatted dialog boxes. Specific NMI programs used. Following are the two programs which were studied in this research. They are two of the most complete and well laid out NMI to date. 1) Scientific Notebook (SNB). This section reviews the program named Scientific Notebook (SNB) (MacKichan, 1996). There is a small body of SNB specific literature related to mathematics education (e.g., Majewski, 2004; Wang, 2001; Majewski, 1999; Majewski, 1998; Bostock & Chandler, 1994). SNB originated as a computer program to ease the word processing of mathematics publications which used natural mathematics symbols, automatic natural mathematical semantic parsing, and user friendly layout mouse-over and click-into-place layout (MacKichan, 1984; MacKichan, 1992). The program cleverly provided linkage with a CAS, facilitating the algebra-calculus-linear algebra computational facilities for inclusion in this full NMI tool (MacKichan, 1994; MacKichan, 1996). SNB might be colloquially thought of as a kind of super graphing calculator.
30 The SNB natural mathematics display capabilities are similar to those which are targeted by OpenMath (2005), and MathML (2005) (discussed above). In addition, SNB provides a dialog box, menu pull down, and toolbar orientation. SNB provides the same natural mathematics notational display qualities (e.g., an automatic semantic parsing of natural mathematical expressions) that the program OpenMath (2005) facilitates for the display of mathematical expressions on the web. Mathematics textbooks centered on SNB exist, as well as online calculus and engineering courses (e.g., Hardy & Walker, 1997; Allen, Stecher, & Yaskin, 2002; Lewin, 1997; Szabo, 1996). For example, Hardy, Richman, Walker & Wisner (2005) is a self-contained calculus text (in publication) remarkably created in and unified with SNB, accessible either online or via CD removable media. Sterret (1990) discusses how having students explicitly compose mathematical explanations or essays can aid mathematical learning. Melita (1999) discusses the learning benefits accorded students who strive to clearly write their mathematical ideas. SNB provides a mathematics lexical medium where students can meticulously compose mathematical expositions—also allowing for reader initiated elaboration (interactive study) of the written text through graphical, numerical or symbolic representational/computational means. In other words, readers of the written text on a computer can use the text itself to command an instant display of numerics, plots, or expanded/simplified symbolics. In many cases, this is as simple as mousing over or placing the cursor on the mathematical expression of interest. This encourages students to think through and include such things as parameter range choices concerning particular models or theorems for the reader to explore. It is interesting to consider how future NMI scientific publications or mathematics text books might be routinely published in this “calculation/manipulation of text expressions,” or, “live mathematics text” mode. 2) Geometer’s SketchPad (GSP). Scher (2000) provides a history of the development of perhaps the most widely applied NMI to date, Geometer’s SketchPad (GSP). A significant body of research and educational literature exists concerning GSP (e.g., Hollebrands & Hollylynne, 2003; Laborde, 1998; Olive, 1998; Oldknow, 1997; and Jackiw, 1991). The GSP NMI emulates a typical high school geometry context. There are other NMI programs very similar to GSP—at least one of them implemented on a
31 graphing calculator platform. These NMI programs exhibits a manual mouse guided dynamic “redrawing of geometric diagrams.” The program holds the specified geometric characteristics constant as if another deformed figure or diagram were drawn by hand (e.g., using compass and straight edge techniques, including conventional symbolic labeling). In other words, by this automated “redrawing of geometric figures” capability, one can mouse over and drag figures which keep their fixed geometric relationships (as constructed), physically displaying how the construction generalizes through these deformations. Other NMI aspects include automation of the hand drawing of geometric transformations such as rotation, dilation, reflection, and translations of geometric figures. Also, numerical qualities of geometric measures such as length, angle, or coordinate axes are provided.
Reasoning and Mathematic Education Technology Use in a Peircean Framework What does Peirce refer to as ‘reasoning,’ and how might his descriptions apply to mathematics educational technology use? Peirce (1992/1878) separates reasoning into three non-overlapping, independent categories: abduction, deduction and induction. These three categories are claimed by Peirce to describe, in general terms, all modes of plausible and deductive reasoning. Peirce (EP2, p. 97, 235-238) describes induction as confirming one’s hypothesis, or testing of one’s hypothesis—a method of reasoning discussed below which can enhance student’s learning of mathematics by using certain technologies in experimental ways. Abduction is generally the coming to a hypothesis, or best explanation. Abduction is at the center of the learning process in mathematics (English, 1997), considering the need for students to make sense of their work beyond a blind following of procedures or rules (Blumenfeld, 1991; von Glasersfeld, 1983). This research employed the three Peircean categories of inference as they provide a theoretical description of reasoning useful for mathematics education and technology use research (e.g., Ali & Kee, 2003; Heid, 1997; Kaput, 1992). By alleviating lengthy manual calculations, technology use facilitates a student to take a wider view of their mathematics. Some of these wider views, as they concern
32 reasoning, are exemplified by a) intellectual use of metaphor, metonymy, and analogy (English, 1997), b) strengthening mathematical connections by practicing varied kinds of perceptual judgments as facilitated by mathematical graphics, c) forming abductive hypotheses, or conjectures, and testing them inductively, d) employing inductive generalization, e) using strict mathematically derivable relationships motivated by these abductive/inductive strategies, or f) virtually (computationally) observing mathematical model and concrete real-world relationships which serve as embodiments of mathematical ideas (e.g., a see-saw and the multiplicative nature of a lever/fulcrum force, testing of a linear system model by inputting various functions and observing the output). Mathematical embodiments are related to Peirce’s notions of semiotic icons, indices, and symbols (e.g., perception of a pictorial situation, and its link to a symbolic meaning). These kinds of reasoning oriented activities are related discussions in Allen et al. (2000), Presmeg (1997a), Lakoff & Nunez (1997), English (1997), Presmeg (1992), & Wagon (1991). If one accepts Peirce’s analysis concerning the splitting of plausible reasoning into two distinct categories, certain theoretical generalizations might be made use of in mathematics education literature. Further though, as technology use greatly increases the opportunity for the employment of plausible reasoning in the learning and doing of mathematics, this distinction becomes more relevant (Borwein & Bailey, 2004; Heid, 1997).
Peirce’s Three Distinct Types of Reasoning This section describes the distinctions and interrelationships of deduction, abduction, and induction. Peirce’s only hesitation in distinguishing the three types of reasoning involves analogy, which is explained in the section detailing abduction below. Peirce (EP1, p.189) classifies in taxonomic form “all inference as follows:…Deductive or Analytic, [and the] Synthetic [kinds of inference as] Induction and Hypothesis [or Abduction].” Peirce (EP2, p. 206) stated his assured opinion that “abductive and inductive reasoning are utterly irreducible, either to the other or to deduction, or deduction to either of them, yet the only rationale of these methods is essentially deductive or necessary.” He gives a sense of what he means elsewhere by
33 relating deductive necessity to perceptual judgments (i.e., the initiation hypotheses, or abduction—discussed later), and induction. Peirce said, “perceptual judgments [cum abductions] contain general elements so that universal propositions are deducible from them in the manner to which [induction] shows that particular propositions usually, not to say invariably, allow universal propositions to be necessarily inferred from them” (EP2, p. 227). Peirce discusses both reasoning and semiotics in terms of three concepts: potentiality, actuality, and regulation--relating to abduction, induction, and deduction, respectively (Peirce, 1998/1903, p.289-299). An abductive inference concerns reaching a hypothesis which is potentially ‘the case,’ or a fact based explanation of the matter. Peirce sees induction as inferring a verification of a hypothesis to actually be the fact of the matter in the highest of plausible terms Deduction is completely defined by rules, or laws which can be thought of as motivated by a principle of truth preservation in argument from premises to conclusion. This is why deduction is seen as regulative, or meaning to control or direct according to rule, principle, or law.
Abductive Reasoning Abduction is a term, first introduced by Peirce, which he explicitly calls “the logic of pragmatism” (EP2, p. 226). Peirce (EP2, p. 205) cites Aristotle’s Prior Analytics27, “…Aristotle… [describes] that mode of inference which I call by the otherwise useless name of Abduction,--a word which is only employed in logic to translate Alpha pi alpha gamma omega gamma, etc.” Abduction is the kind of inference best known today as the ‘coming to a scientific hypothesis’. In briefest terms, Peirce (EP1, p. 194) describes abduction as reasoning “from effect to cause.” Peirce equates the coming to explanations, theories, or conjectures as abductions. This general category of inference, although not by name, has since been studied in great detail by mathematics educational researchers (English, 1997). Shank & Cunningham (1996) have outlined detailed considerations of different general classes of abduction to be discussed below. Abduction is the least procedurally regulated class of inference, and
27 Peirce cites the applicable section of Prior Analytics as Book 2, Chapter 25, 69a30-36. There a hypothesis is explained which concerns the possibility of squaring of a circle.
34 thus requires much conceptual insight in its application. Although abduction amounts in essence to a mere educated guess, the practice of using many associated cognitive mechanisms, developed habits of and confidence in capacities for reflection, and the rational force of clearly understood and focused intellectual purposes allow for general methods and organizational principles to be described for this form of reasoning. Josephson (1994), from an artificial intelligence perspective, explicitly calls abduction “inference to the best explanation.” Josephson, following Peirce (EP1, p. 188) provides the example: D is a collection of data (facts, observations, givens), H explains D (would, if true, explain D), No other hypothesis explains D as well as H does. ------Therefore, H is probably correct.” This kind of inference amounts to “adopting a hypothesis [or an explanation] as being suggested by the facts” (EP2, p. 95). Peirce (EP1, p. 188) distinguishes abduction from induction and deduction using the following concrete example. “Rule: All the beans from this bag are white. Case: These beans are from this bag. Result: These beans are white.” (EP1, p. 188) This ordering of the premised rule, and case, to the concluding result represents a deductive argument. An induction proceeds from Case to Result concluding in a Rule. “Case: These beans are from this bag. Result: These beans are white. Rule: All the beans from this bag are white.” (EP1, p. 188) One can think of the Result of an induction as being the data gathered in a statistical sampling of members of some population. Notice that the above deduction began from a general rule and proceeded to a particular conclusion. The inductive argument, on the other hand, begins from particulars and concludes with a general rule. The analogy to a statistical sampling problem is not arbitrary, and motivates much of Peirce’s discussion of distinguishing three kinds of reasoning.
35 The Case, “These beans are from this bag,” designates a certain set membership. In the case of induction, the members of the set are from whom the resulting samples are drawn. The Result, “These beans are white,” is the conclusion resulting from the sampling. The Rule, “All the beans from this bag are white,” represents an inductively inferred rule concerning a population where each member plausibly has the common property of whiteness. The abductive inference proceeds from Rule to Result, concluding in the Case— precisely the reverse ordering of the inductive inference above. “Rule: All the beans from this bag are white. Result: These beans are white. Case: These beans are from this bag.” (EP1, p. 188) As far as deduction is concerned, this is the fallacy of the converse. However, an abductive interpretation casts a different light: i.e. If B then W. W.______Therefore, one might hypothesize B. This example casts a statistical sampling interpretation distinguishing Peirce’s three kinds of reasoning. This interpretation definitively distinguishes abduction from induction, while offering an explanation for one proceeding with this form of inference. Peirce (EP1, p. 189) remarks in reference to this example, “Hypothesis is where we find some very curious circumstance, which would be explained by the supposition that it was a case of a certain general rule, and thereupon adopt that supposition.” Peirce’s over-simplified description of abduction above is meant only to distinguish the three kinds of reasoning in a discussion that discretely points out general differences in the three types of reasoning. Finer grained discussion is presented below.
36 Detailed Senses of Abductive Reasoning Educational theorists Shank & Cunningham (1996) discuss six modes of abduction they have derived from Peirce’s semiotic theory of signs (Peirce, 1998/1903, p.289-299). These six modes of abduction are: 1) hunch, 2) symptom, 3) metaphor/analogy, 4) clue, 5) diagnosis/scenario, and 6) explanation. All senses of abductive inference are merely ampliative, meaning they go beyond the data, or beyond what the premises guarantee to what is only a plausible conclusion (Rescher, 2005, p. 95). Shank & Cunningham (1996) use the terms rule, case, and result below in the same sense of Peirce’s example above. Consideration of Shank’s (i.e., Shank & Cunningham (1996)) six modes of abduction is taken up below. Hunch Shank & Cunningham consider a hunch or omen as “[dealing] with the possibility of possible resemblances…a merely subjective act” (p. 3). An archeologist might infer that examining the banks of an old stream bed might lead to finding artifacts. This particular example of hunch offered by Shank may not be the best of one as the archeologist may have previously had many successful experiences with old stream beds. If we were to assume a successful experience with perhaps only one stream bed previously (or even a river bank), we can discern how the resemblance in this case leads to merely a hunch, instead of an inductive inference. Symptom Shank explained that a symptom “deals with possible resemblances… [as] reasoning in order to determine whether our observations serve as symptoms for the presence of some more general phenomenon” (p. 3). If the archeologist finds a smooth stone whose smoothness is not clearly natural or man-made, she can make an inference based on this symptom. Metaphor/Analogy A metaphor/analogy “ deals with the manipulation of resemblance to create or discover a possible rule” (p. 3). Shank goes on to “describe [this mode of reasoning as] inference that uses analogy and metaphor to create new potential rules of order” (p. 3). As the rules are merely potential rules, they are not thought to be verified, and so the inference can be seen as abductive.
37 Presmeg (1997a) describes metaphors to be “implicit analogies.” For example, “An equation is a balance,” provides an analog to weighing things, where a person’s knowledge of the concept of weight may not be something they might be able to explicitly discuss—however—the analogy works for them just as well. Peirce finds analogy to be the only clear trouble spot in his distinction between induction and abduction. Burch (2001) wrote, “Peirce modified his views on the three types of arguments, sometimes changing his views but mostly just extending them. He seemed to have some hesitation, for example, about whether arguments from analogy were inductions (on properties of things) or abductions” (p. 4). For example, “Branch is to tree, as arm is to body” can be an induction on the properties of: (i) the typically vertical tree and body, (ii) the typically horizontal branch and arm relative to the tree and body, (iii) the branches and arms being thinner than trees and bodies, (iv) the branches and arms subtend more frequent to and fro displacements than do entire trees or bodies, etc. Holland et al. (1986) considers analogy to simply be inductive, but not with regards to individual properties. Holland considers repeated experience with many cases of the analog as facilitating analogical inference. However, English (1997) provides many examples specific to mathematics learning where significant distinctions between analogies and metaphors are drawn along several lines. Clue Shank stated, This type of inference deals with possible evidence. A more concrete way to characterize this type of reasoning is to describe it as reasoning in order to determine whether or not our observations are clues of some more general phenomenon. (Shank & Cunningham, 1996, p. 4). Our archeologist finds pottery shards next to the smooth stone. She must decide “Is there a connection between the two, or is it just a coincidence?” (p. 4).
38 Diagnosis/Scenario Shank describes a diagnosis/scenario to involve “the formation of a possible rule based on available evidence” (p. 4). The archeologist might find the broken pottery to be placed in a shallow pit—with smooth stones around the pit’s edge—allowing here to be constructed a possible scenario. Explanation Finally, the sixth kind of abductive inference for Shank is explanation. Explanation is described as dealing “with a possible formal rule [which one infers or back tracks from]… in order to form a general plausible explanation”28 (p. 4). The archeologist’s sum total of these discoveries above and additional relevant finds may help her abduce an explanation in terms of some sort of religious ceremony. General details concerning various kinds of abduction provided above are helpful for qualitative research. Most all of the work in English (1997) refers to abductive reasoning processes at great levels of empirical, practical, and theoretical detail—though not making specific reference to it as being abductive (e.g., Presmeg, 1997a; Wheatley, 1997; Lakoff & Nunez, 1997).
Inductive Reasoning According to Peirce, (EP1, p. 32) inductive reasoning “proceeds as though all the objects which have certain [properties] were known.” He comparatively defines abduction as “the inference which proceeds as if all properties requisite to the determination of a certain object or class were known” (EP1, p. 32). In other words, abduction is retrospective, in that one imposes an explanation in unconfirmed hindsight— hoping that one knows the characteristics of what is being considered. 29 Induction, like science, is generally empirical in nature. Characteristics or properties of something are discovered to hold by clever, incessant, and repeatable
28 One is reminded of a student’s creative back-tracking speculation from a single solutions manual entry, or result taken from a calculator output. 29 Peirce used the term ‘retroduction’ interchangeably with the term abduction. Consider the term ‘retroactive.’ One common definition of retroactive is: the extending in scope or effect to a prior time or to conditions that existed or originated in the past. This notion of retroactive can be imagined to parallel the ‘fallacy of the converse’ discussion: abducing that “these beans are from this bag.”
39 experiments. One can inductively learn that a property exists, independent of there being some general rule that might explain it. There may be one or more plausible explanations or abductions, but nothing has been verified. It is inductive reasoning that accomplishes such verification of the general scientific rule. In the case of mathematical sciences, a hypothesis might be encouraged by deduction within a theoretical system (e.g., mathematical physics). However, it is the inductive inference from experiment that confirms or denies the deduced theoretical possibility. The deduced theoretical result is seen as physically plausible, and so doubles as an abduction. It is a mathematical deduction serving as a scientific abduction. For Peirce, along with others (e.g., Holland, et al.,1986), inductive inference means that by observing only some number of things of a certain class, we can infer what would be observed to hold for each member of the whole class—all things being equal. As was mentioned, it is common to think of induction as reasoning from particular cases to a plausible generalization. Peirce (EP1, p. 79) wrote, “Induction holds because if we accept the presumption that certain things can be identified as being in some possible general class with others of similar nature, and then as we check the properties of a few, we can—all things being equal—make general claims about the presumed group of things.” Peirce pointed out this reasoning hinges on the presumption that “any one [thing] would be [chosen] as often as any other.” Statistics is an inductive mathematics application. Hypothesis testing is the testing of abductions.
How magical it is that that by examining a part of a class we can know what is true of the whole of the class, and by study of the past can know the future; in short, that we can know what we have not yet experienced! … Is not this an intellectual intuition! Is it not that besides ordinary experience which is dependent on there being a certain physical connection…[through sense organs] and the thing experienced, there is a second avenue of truth dependent only on there being
40 a certain intellectual connection between our previous knowledge and what we learn in this way?30 (EP1, p. 75)
This description of induction offers some explanation for what the NCTM (2000) calls “making connections.” Although Peirce applies these explanations to the regularity of the observables of nature, NMI makes mathematics itself an observable with its own kinds of regularities. Peirce (EP2, pp. 44-47) denotes that “[the] self- correcting property of Reason, which…belongs to every sort of science, although it appears as essential, intrinsic, and inevitable [exists] only in the highest type of reasoning, which is induction.” Peirce then also considers induction the most important of the three modes of reasoning as it is the only one of the three types that is self-correcting. “The only way of attaining any satisfactory general knowledge of experiential truth is by the inductive testing of theories [or hypotheses]” (Peirce, 1998/1907, p. 432). Induction plausibly confirms observed patterns whether they be mathematical or scientific. Per statistics, hypothesis (abduction) testing is inductive inference, strictly regulated by the mathematical postulations and deductions of probability theory.
Induction as the Logic of Relatives Our final consideration of induction is Peirce’s referring to this type inference as the logic of relatives. By this, Peirce generally meant in a statistical kind of sense. For example, an inductive inference allows one to sample over and over again, thereby obtaining a relative accuracy on this account (EP1, p. 33; EP2, p. 97, pp. 235-238). Peirce often discusses the logic of relatives in the context of continuity and generality (EP2, pp. 206-207). Imagine several externally drawn, geometrically constructed right triangles. One can organize these images internally (as a mental visualization) so that they can be imagined to continuously deform, one into another into another, etc., while constantly maintaining the properties of a right triangle. Here is a way of understanding how induction moves from particular cases to general rules. By
30 Some critique Peirce to be a Voluntarist (the world is willed, actuated by volition). I read him to be an Intellectualist (the world can be rationally/cognitively known about).
41 accepting continuity, and imagining contiguous figure snapshots which can be seen to move into one another continuously, the general class of all right triangles is plausibly imagined (visually induced) to possess the identical properties which the several externally drawn right triangles were geometrically constructed to have. NMI provides user commanded presentations of mathematical artifacts which can be taken as empirical data—such as the externally drawn right triangles. These mathematical presentations can suggest internal visualizations to students. When a student is able to internally visualize, they can then test their hypotheses inductively— employing what Peirce has called the logic of relatives.
Abduction and Induction Compared and Contrasted Peirce worked to draw comparisons and contrasts between what he took to be two distinct types of plausible reasoning: induction and hypothesis (or abduction). By induction, we conclude that facts, similar to observed facts, are true in cases not examined. By hypothesis, we [plausibly] conclude the existence of a fact quite different from anything observed, from which, according to known laws, something observed would necessarily result” (EP1, p. 194). Induction demonstrates, from particular facts, a similar general fact. Abduction infers a fact in no way similar to those under consideration, or an alien fact to the situation. An abduction is literally a “new idea” concerning the existence of a potential fact and so at first may seem alien to the situation at hand. Yet from this alien fact, certain consequences would follow.31 Peirce explained that the method of abduction and induction to be “the very reverse” of one another. Abduction makes its start from the facts [e.g., observations], without, at the outset, having any particular theory [or hypothesis] in view, though it is motivated by the feeling that a theory is needed to explain the surprising facts. Induction makes its start from a hypothesis [supposed theory] which seems to recommend
31 Hookway (2004) points out that Peirce’s early work, as just quoted (EP1, p. 194) was minimalistic in the sense of presupposing starkly direct idealistic frameworks (e.g., the context of the mathematical sciences) in order to make his ideas clear. Peirce seems to be supposing such a minimalist framework in saying that “according to known laws, something observed would necessarily result (italics added).”
42 itself [e.g., from observation], without at the outset having any particular facts in view, though it feels the need of facts to support the theory. Abduction seeks a theory. Induction seeks for facts. In abduction, the consideration of the facts suggests the hypothesis. In induction the study of the hypothesis suggests the experiments which bring to light the very facts to which the hypothesis had pointed (italics added). (EP2, p. 106). Notice that abduction emerges from a motivational bias based in one’s feelings, while induction is seen to be unbiased, in that “at the outset [it has no] particular facts in view.” Peirce went on to further clarify a difference in modes of suggestion to the intellect: abduction involving a suggestion of a hypothesis from the facts, and in induction, hypothesis has suggested these self same facts under consideration. Abduction makes its suggestion by resemblance of the facts to the consequences of the hypothesis.32 Some set of facts seem to suggest a general picturing of the situation in hypothetical terms resembling something already familiar. This indicates how analogy and metaphor (in the visualization sense of researchers such as Presmeg (1996), Presmeg (1997b) and Wheatley (1997)) might play a role in abductive inference. Induction makes its suggestion to thought by what Peirce calls contiguity— “familiar knowledge that the conditions of the hypothesis can be realized in certain experimental ways (EP2, p. 106).” One can picture a theory as establishing (or an abduction as suggesting) a sweeping or overall general rule. Peirce’s sense of contiguity implies the opposite of an overall generalization—instead—there is careful consideration of each condition of the hypothesis and how it can be experimentally controlled and realized in successive particular cases. The experimental procedure or situation involves knowledge which is already familiar—it provides contiguity (adjacency) between the theoretical or hypothetical situation presumed or guessed to hold, and the familiar, common knowledge from which the experimental situation is constructed.
32 Here, the iconic aspect of semiotics is implicit—Peirce made this claim about inductive reasoning in other places.
43 Deductive Reasoning Peirce (1998/1903; p. 205) wrote, “Deduction is the only necessary reasoning. It is the reasoning of mathematics. It starts from a hypothesis the truth or falsity of which has nothing to do with the reasoning; and of course, its conclusions are equally ideal.” What’s more, Peirce declared that, “all necessary reasoning…is mathematical reasoning” (EP2, p. 206). Deduction proceeds from facts (hypothetical postulations, or general rules), to particular conclusions by certain definite rules of logic.33 One of the greatest value of these rules is that they are truth preserving. In other words, what is taken to be ideally true at the outset remains true for all successive deductive conclusions. Likewise, all ideal conclusions are just as truth preserving as the premises, given that the deduction is logically valid (discussed immediately below).34 Some Formalized Systems which Guide Deductive Inference Deduction is a form of reasoning which has lent itself to many kinds of formalizations. The central formalization of deductive reasoning is the proposition. A proposition is a statement which is either true or false.35 Two important concepts of formal deductive systems is the notion of validity and soundness. A deductive argument whose premises, if they were all true, would provide conclusive grounds for the truth of its conclusions, is said to be valid …Validity is the logical certification of a legitimate reasoning process whereas soundness is the empirical certification of facts claimed in a legitimate reasoning process...[and] a logic is sound with respect to its semantics if only true sentences are derivable
33 It is interesting to note this ongoing work of the discovery or construction of formal (detached or abstract) deductive rules (by abductive and inductive reasoning in the present context) occurred in two spurts: during the tail end of Greece’s golden age, then to lay completely dormant until the generation just preceding Peirce’s. Under the pragmatist interpretation reasoning, one can imagine how long mathematical reasoning was primarily abductive/inductive, while only perhaps beginning with the ancient Greeks has it become more and more deductively codified (Ernst Cassirer wrote about how primitive peoples before the ancients reason out their mathematically related ideas). 34 Deduction can be sharply contrasted with dialectical thought or reasoning. Here, a thesis is presented, an anti-thesis (or denial of the premised thesis) moves to a synthesis (or conclusion). This conclusion then can become a new thesis, etc. 35 Lofti Zadeh, and Bart Kosko have done work on extensions of logic which have been coined “fuzzy logic.” They assume shades of truth of a statement, which in some instances can be defined with continuous distributions. It is the researcher’s recollection that it has been shown that this type of continuous distributional system can always be implemented in nothing more than a three-valued discrete logic system.
44 under the inference rules from premises which are themselves all true (italics added). (Copi & Cohen, 1994). For example, three common logic formalizations are: (a) propositional or sentential logic, (b) syllogism, and (c) predicate logic. See footnote 32. In predicate logic, certain properties can be said to hold of the objects of a domain of discourse. The primary distinction between logic and mathematics is that the domain of logic can be any objects whatsoever, while mathematics holds only over the domain of mathematical, and mathematizable objects. First-order predicate logic is known to be powerful enough to formalize set theory and therefore all of mathematics (Wolfram, 2002; see footnote 4). Deduction as Regulative Peirce has this to say about deductive reasoning:
Logic is a science little removed from pure mathematics. It cannot be said to make any positive phenomena [e.g., the quality of redness] known, although it takes account of and rests upon phenomena of daily and hourly experience, which it so analyzes as to bring out recondite truths about them. One might think that a pure mathematician might assume these things as an initial hypothesis and deduce logic from these; but this turns out, upon trial, not to be the case. The logician has to be recurring to reexamination of the phenomena all along the course of his investigations. But logic is all but as far remote from psychology as is pure mathematics. (EP2, p. 311).
Psychology differs from logic in that logic is a theoretical system following fixed laws. It serves a regulative function. Deductive reasoning, which follows the rules of logic, can add no new data beyond what is in the premises; it serves the function of truth- preservation. True premises force true conclusions in a valid deductive argument. Although no new data is added by deduction, as it merely mechanically follows, new information can be made sense of, or learned from not only the deduced result, but the experience of leading one’s own deductive path to this result. This informative aspect of deductive reasoning fully depends upon the imaginative thinking of the agent.
45 Deductive Reasoning versus Plausible Reasoning Plausible inference assumes more than the premises provide—it is said to go beyond the data. Rescher (2005; p. 94-101) describes plausible reasoning as the drawing of inferences which are likely or viable when taken in the context of a typical situation. Deductive results on the other hand follow from fixed rules wholly independent of the context of the premises. Polya (1954) and Hadamard (1945) corroborate the key role plausible reasoning plays in a mathematicians’ problem solving. Conjecture is an example. Mathematical experimentation with the aide of a computer also involves explicit use of plausible reasoning methods (Borwein & Bailey, 2004). For example, one can use a computer to aide one’s imagination by making sense of the information gleaned from one’s computed results. Computers can be used to explore various contexts which are conjectured to be relevant (e.g., graphically or numerically) to some formally cast problem. The informal computed results can help one to draw plausible inferences guiding one to a deductive chain of logical rule applications demonstrating a solution to the formal problem. Specific Examples of Deductive vs. Plausible Inference Rescher (2005, p. 94-104) provides some examples which explicitly compare deductive and plausible inference. Consider the compound statements X and Y, where X is “If p, then q” and Y is “If (p and r), then q.” Deductively then, “If X, then Y” must also hold.36 However, for a plausible inference, if ‘r’ relates to the situation X in an unanticipated way, the inference can be false. Rescher gave an example. Compare: If you greet him, he will answer politely. with: If you greet him and kick him in the knee, he will answer politely. Rescher pointed out that an inference like the initial one above implicitly means: If you greet him in the usual and ordinary way, he will answer politely. There is a presumption of typicality, or normality in plausible reasoning. New information can disrupt the typicality of the context upon which the inference depends. Plausible
36 This can be seen by constructing the eight different combinations of p, q, and r in a truth table for the statement “If X, then Y” or “If (If p, then q) then (If (p and r), then q).” This statement is seen to be true in all cases.
46 reasoning is therefore seen to be context-dependent; meaning, nongeneralizable over atypical contexts. An indication of this contextual dependence in plausible reasoning is conveyed by use of the phrase: all things being equal (Rescher, 2005, p. 99). Consider the plausibly true statement, “If there’s smoke, there’s fire.” But, now let the antecedent statement “There is smoke” be conjoined with the statement “There is a dust storm”. We see again that the veracity of a plausible inference often depends upon statements to have implicit inter-relationships of typicality. On the other hand, deductively valid arguments are built up from independent logical atoms (simple statements) which have individual truth values independent from one another. In this sense, the arguments themselves can be said to be context independent—they do not depend on any special implicit “typicality of situation.” Analogies or metaphor can be drawn from presumed to be typical mathematical contexts (English, 1997). However, a student can be caught up in an over-generalization of such an analogy or metaphor, and draw plausible, but deductively false conclusions. Elsewhere, a student may notice a pattern repeatedly among certain particular cases and plausibly conclude the pattern to hold for all cases. However, it may be the student was under-generalizing, leaving out the testing of cases which did not hold. Student’s can also gain from their own errors when they are recognized, by the insight they provide. This is further discussed under the heading ‘Constructivist Learning Theory Research Tools,’ and ‘Perturbation’ in the following chapter entitled ‘Methodology.’.
Summary of Peirce’s Views on Deduction A brief summary of the positions of Peirce cited above concerning deductive reasoning are presented. Peirce used the terms ‘necessary reasoning’ and ‘deduction’ synonymously. He claimed that all deductive reasoning is mathematical, and that all mathematical reasoning is diagrammatic. Therefore, all deductive reasoning is diagrammatic. It also follows then that internal visualization of diagrams may be completely deductive activities, or plausible reasoning, or both. This reasoning through use of imagery is particularly integral to current mathematics education research in a broad sense (English, 1997), as well as regards computer generated external
47 representations (e.g., NMI) in aiding the development of such internal imagery as related to mathematics learning (Stylianou, 2002; Someren, et al. 1998; Goldin & Kaput, 1996; Lin & Hsieh, 1994; Thompson & Sfard, 1994; Janvier, 1987).
The Three Types of Reasoning Taken Together The editors of EP1 (p. 33) emphasize Peirce’s emphatic position that “All valid reasoning is either deductive, inductive, or hypothetic; or else it combines two or more of these…” . Peirce also claimed that all reasoning is organized around these three kinds (EP1, p. 245). Peirce (EP1, pp. 245-246; EP2, p. 233) claims that each form of reasoning is organized around a triad, where abduction may stand alone as a hypothesized expectation, and induction infers a likelihood of this expectation. He then describes deduction as follows. “All [deduction] without exception is diagrammatic. That is, we construct an icon [(pictorial representation)] of our hypothetical state of things and proceed to observe it.” (EP2, p. 212). For example, we imagine an analogy, and proceed to observe each of the analog’s properties, and checking them over and over from different perspectives in order to determine if the analogical diagram is a valid theory for what is actually true—from any given perspective. Say one has conjectured some hypothetical relationship and by virtue of the fact that (a) all the properties repeatedly tested which are predicted by the hypothetical case turn out to be true, and (b) the hypothetical case began as a completely abstracted analog. If then one applies the abstracted or theoretical rule to the particular case which has been sufficiently tested, it must necessarily fit or have the expected properties because it was those properties which were used to form the theory at the start (the abstracted analogical properties). So then hypothetically speaking, what has been analogically diagrammed must necessarily follow whether or not it actually follows or not. It is an entirely hypothetical result necessitated by construction, removed from whatever the facts may be. The premises are now the general rule speculated to hold theoretically—and the case which determined this rule (EP2, p. 212).
48 Affirmed Perceptions and the Origination of Reasoning How students initialize their mathematical reasoning actions is particularly salient for this research. The following lists a general sampling of the body of literature concerning intuitive, heuristic and perceptual aspects of reasoning. Considerations in this literature also include how students communicate and reflect upon (or self-communicate) aspects of their mathematical ideas. The sampled list includes Askew (2000), Byrnes (2001), Carroll (2002), Cobb and Bauersfeld (1995), Cornoldi, Logie, Brandimonte, Kaufmann, and Reisberg (1996), Deines (1961), English (1997), Golledge (1999), Janvier (1987), Lave and Wenger (1991), Littleton and Light (1998), Middleton and Goepfert (1996), Noddings and Shore (1984), Sperber (2000), Steffe, Mesher, Cobb, Goldin, and Greer (1996), Sutherland and Mason (1995), and Zimmermann and Cunningham (1991). Peirce generally viewed reasoning, particularly mathematical reasoning, as having its inception in perceptual judgments (EP 2, p. 227). Peirce (EP2, p. 227, 231) claimed to support Aristotle’s view that all inference is grounded in perception: “every element of general thought is grounded in perceptive judgment.” He specifically said one’s asserted perceptions fade into their abductive reasoning. Thought processes discussed below are more or less taken by Peirce to originate the general objects one considers in their reasoning (e.g., mathematical objects). Notions below concerning Peirce’s mathematical reasoning initiatory senses of (i) abstraction, (ii) intuition, (iii) understanding, and (iv) colligation, are reviewed. Mathematics education research literature is reviewed concerning (a) representation, and (b) psychology and reasoning. These are reviewed in so far as they might facilitate the action research employed.
Psychology and Reasoning It is useful here to consider a distinction between implicit and explicit awareness. Implicit awareness or tacit knowledge can be generally thought of as subconscious (Dienes & Perner, 1999 (this Dienes is the grandson of the Dienes (1961)); Sternberg & Horvath, 1999; Stadler & Frensch, 1998; Underwood, 1996). Following on from this in mathematics learning, one might refer to that which has already been consciously
49 learned, compressed and packed in to the background unconscious as tacit knowledge for later indexing by the foreground conscious (Schoenfeld (1992) might be interpreted along these lines). Generally mathematics educational constructivist interpretations often discuss what might constitute a sort of mental construct—or an object of one’s reasoning. For example, Presmeg (1997a) and Sfard (2000b)37 consider notions such as metonymy, metaphor, simile, or analogy in the context of mathematical reasoning and conceptualizations concurrent with students learning mathematics. Other examples concern reification or the concept formation of mathematical objects (Sfard, 2000a; Presmeg, 1986, Presmeg, 1997b, pp. 275-277; Sfard, 1994), processes of concrete (concerning situational contexts) or abstract (concerning mathematical embodiments) mathematical visualization (Presmeg, 2000b; Wheatley, 1997).38 Presmeg, in particular, elaborates on use of visualization or mental imagery, semiotic chaining, and metaphors, metonymies, and prototypes (Presmeg, 1986, 1992, 1997a, 1997b, and Wheatley, 1997). Metonymies are how one generally views mathematical symbolization (e.g., “Let ABC be any triangle,” or, “Let ‘y’ be a sinusoidal forcing function.”). A student, working with a damped linear second order ordinary differential equation might metaphorically hypothesize, “This damping is dropping a pebble in a pond, but this one is dropping it on Jell-O!” These ways of thinking create percepts cum concepts when within contexts which then facilitate the growing objects of reasoning.39 Rapid computer processing allows for more real-time self-guided inductive inference, where only abductions/conjectures or deductions were feasible (in real-time) concerning complex mathematical models before the advent of NMI-like programs.
37 Sfard (English, 1997, p. 345) discusses metaphor as the creation of a new conceptual system, while analogy comes later as a kind of inference between similar concepts. 38 Consider the situational context of see-sawing in a playground. Visualization here can be seen to perhaps involve other senses of perception such as kinesthetic. This situational context may serve as a mathematical embodiment of the general mathematical abstraction of ration and proportion (e.g., picture a see-saw where one person sits three times as far from the fulcrum point as another). 39 Peirce takes a percept to be something accepted preceding any need of reasoning. A perceptual judgment then can be seen to be an accepted perception (not inferred, but implicitly accepted). Certain concepts then can be taken to be inferentially accepted or affirmed arrangements or schemata concerning perceptual judgments. Peirce said though that perceptual judgments fade into abductive inferences. Peirce (EP1, p. 198-199) discusses psychological relationships to reasoning: abduction and emotional feelings; induction and the formation of rules of habit; and deduction and volition.
50 Abstraction Peirce (1998/1903, p. 270) made three distinctions concerning the common meaning of the term abstraction: dissociation, discrimination, and abstraction (or “precission”) (EP2, p. 352).40 They each generally refer to separation of certain things. Peirce (1992/1867, pp. 2-3) explained that abstraction is a stronger separation than discrimination, but dissociation is even stronger. One discriminates when one takes things in a different sense, or to have differing meanings. Dissociation is separating things normally associated, as in a sand dune from a beach or steam from the air. Abstraction “arises from attention to one element and neglect of the other.” Peirce went on, “[Abstraction] is not a reciprocal process. [Frequently] while A cannot be [abstracted] from B, B can be [abstracted] from A.” For example, one can abstract circular motion from a spinning gear, where conversely we get a concrete embodiment. See Haaparanta (1999) for further discussion of Peirce’s three distinctions of abstraction.
Intuition and Abstraction Charles Parsons (1993) wrote that “There is… a notion of intuition of objects… particularly in Kant. Kant insists on the fundamental role of such intuition in mathematical knowledge…” Page (1993) quotes Parsons concerning that what is essential to mathematical objects "is the relations constituting the structures to which they belong." White and Mitchelmore (2003), uses Sfard’s work on reification (e.g., Thompson & Sfard, 1994) to advance a teaching method which keys on students gaining familiarity with several abstract similarities (relations) of mathematical objects (e.g., aspects of corners and rectangular and square shapes related to circles, and central angles; and the subtended “angle” diagram are related still again to notions of north related to south south west et cetera). Then the student’s multiple perspectives of a similar relation can be reified into an encompassing, stand alone mathematical object in its own right, e.g.,‘angle’.
40 It appears that Peirce spelled this third sense of abstraction “prescision” in 1867 (EP1, p. 2), and “precission” in 1905 (EP2, p. 352). Peirce (EP2, pp. 351-352) explains why he thinks English should use “prescind, presciss, prescission, and prescissive on the one hand to refer to dissection in hypothesis [as discussed above], while precide, precise, precision, and precisive [should be used in the familiar sense] to refer to a…determination which is either full or made [contextually] free for the interpreter” (p. 352).
51 Peirce (EP1, p. 262) wrote, “Intuition is the regarding of the abstract in a concrete form, by the realistic hypostatisation of relations; that is the one sole method of valuable thought.” This is key in diagrammatic reasoning,41 where Colapietro (1993) defined Peirce’s general use of the term ‘hypostatic abstraction’ as “the process by which a predicated quality or formal operation is converted into—as it were, an entity in its own right.”42 Peirce (1992/1887-88) went on to say that hypostatic abstraction “furnishes us the means for turning predicates [e.g., qualities of things--] from being signs that we think through, into subjects thought of.” Colapierto explained that these kinds of abstractions “forge connections between things that would otherwise be separate or unrelated.” This is a more explicit way of thinking about what the NCTM (2000) simply refers to as the importance of “making mathematical connections.” Peirce (1992/1887-88) implies that hypostatisation is a form of cognition “eminently characteristic…of a consciousness of a process, and this in the form of a sense of learning (italics added).”43 Here there is a pronounced second order thinking on one’s thinking—a recursion if you will (Dienes & Perner, 1999; Eddington, 1998). Peirce brought his thoughts on conscious abstraction, and tacitly (unconsciously) indexed intuition—hypostatic-abstraction (or one might say, abstraction fixed by intuition):
…The work of the poet or novelist is not so utterly different than the scientific man. The artist introduces a fiction; but it is not an arbitrary one; it exhibits affinities to which the mind accords a certain approval in pronouncing them beautiful, which if it is not exactly the same as saying that the synthesis is true, is something of the same general kind. The geometer draws a diagram, which is not exactly a fiction, is at least a creation, and by means of observation of that diagram he is able to synthesize and show relations between elements which before seemed to have no necessary connection. The realities compel us to put
41 See the discussion on diagrammatic reasoning and computer graphics at the start of this chapter. 42 For example, “The dress is white.” By hypostatic abstraction, one thing that can be said is, “The whiteness of the dress is brilliant (Colapietro, 1993).” 43 Recall previous footnote on “water and jell-O” example regarding conscious concepts and subconscious or sensory percepts.at the start of this section. Hypostatic abstraction can be interpreted to “fix” conceptions into percepts, or into “post” conscious tacit knowledge.
52 some things into very close relation and others less so, in a highly complicated, and in the sense itself unintelligible manner; but it is the genius of the mind that takes up all these hints of sense, adds immensely to them, makes them precise, and shows them in intelligible form in the intuitions… Intuition is the regarding of the abstract in a concrete form, by the realistic hypostatisation of relations; that is the one sole method of valuable thought. (EP1, pp. 261-262)
NMI allows a student to physically see a concrete diagram (e.g., geometric figure, or a plotted graph) in coordination with other mathematical representations (e.g., compass measures, or symbolic equation, y = f(x)) which might then facilitate explicit second- order thinking whereby there is a hypostatic abstraction of the mathematical relationships. Others might describe this hypostatic abstraction as a meta-cognitively looking down upon abstracted objects of thought (Dienes & Perner (1999); similar to an executive monitoring process, Schoenfeld (1987). Thompson & Sfard (1994) discuss this in terms of reification. Piaget has called this kind of thinking a reflective abstraction process. Essentially, one can describe a concrete cognitive creation, or mental construction, constitutive of a relational object. This is a good example of how Peirce has found useful general ways of describing things more contemporary researchers have found to be important—but that revolve around the centrality of pragmatic reasoning— providing a useful structure for these kinds of discussions of how students may think or learn in terms of their methods of reasoning. Hypostatic abstraction occurs when one takes a predicated quality or a formal mathematical operation and considers it to be a noun substantive, or a “thing” in itself (Thompson & Sfard, 1994; Sfard, 2000a; Goldin & Kaput, 1996). Peirce considered such an abstraction to be the result of reasoning, while Sfard explicitly (English, 1997, p. 345) considered the same kind of abstraction to be the result of discourse. Sfard (2000a) quoted Vygotsky (1962, p. 125), “Thought is not merely translated in words; it comes into existence through them.” Both Sfard and Peirce would agree that there are noun substantive representations which are imagined to be mathematical objects. We say, “Addition is commutative,” and “Differentiation is a linear operator” as if processes related to combining things, or
53 finding the slant of a tangent line to a curve at a point are actual things in and of themselves. Sfard (Thompson & Sfard, 1994) would say they are reified into objects and that mathematical discourse and mathematical objects create one another (or are co- emergent) (Sfard, 2000). Peirce, however, argues for both the reality (in the sense of hypothetical facts) of mathematical objects,44 as well as for their subjectivity in regards their idiosyncratic infusion of meaning by individuals (EP2, pp. 342-343). Peirce described this—in far more meaningful detail-- as a distinction between objective and subjective generalization (EP2, p. 342).
Understanding Pragmatism leads to interesting descriptions of how one can reason through, and distinguish, certain kinds of understandings. These are touched on below. Quasi-Empiricism and Differing Kinds of Understanding Velliers (2004) made direct connection to mathematics education and the NMI GSP discussing non-deductive methods of mathematical experiment, or quasi- empiricism.45 He pointed out that “the objects of mathematics, though largely abstract and imaginary, can be subjected to empirical testing much as scientific theories are” (Velliers, 2004, p. 2). Peirce parallels this point as regards the methods of scientific reasoning in the context of mentally doing mathematics. For example, Peirce’s discussion of the “logic of relatives” (induction) concerning how one visualizes a continuously changeable mathematical diagram, stepping through case after case, plausibly verifying one’s intuition (EP2, p. 206-207). 46 Polya (1954) quotes Leonhard Euler, As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet in fact… the properties of the numbers known today have been mostly discovered by observation, and discovered long before their
44 Peirce semiotically views interpretant symbols as laws. An interpretatant is a noun substantive representing a particular, though general interpretation. Elsewhere, Peirce describes a symbol to be “an embryonic reality endowed with power of growth into the very truth, the very entelechy [actuality] of reality (EP2, p. 324).” Entelechy “[in the philosophy of Aristotle, is the condition of a thing whose essence is fully realized]”(Martin, 1991). 45 Velliers attributes this term to Lakotos (1983). 46 Maddy (1980) links mathematical intuition to perception via a realism discussed by Rescher (2005).
54 truth has been verified by rigid demonstrations.”(Polya 1954, p. 3, quoted from Villers, 2004, p. 4) Notice that one can make connections in certain intuitive aspects of understanding, but have no explanation for why a mental inductive visualization (or NMI representation) fits together as it apparently does. One might though be led to a separate form of reasoning, abduction, to guess or provide a conjectural explanation (one that can later lead to proof). Finally one must turn to deduction for moving past this only plausible and possibly vague connection, to not only verify as mathematical fact, but to gain deeper understanding of why such a connection exists.
Besides not providing sufficient certainty, quasi-empirical evidence also seldom provides satisfactory explanations; that is, insight into why something is true in mathematics. In other words, a drawback of quasi-empirical investigation is that it does not tell us how a result relates to the other results or how it fits into the general mathematical landscape. (Velliers, 2004, p. 17)
Ordering Different Types of Reasoning and Personal Understanding The above discussion of quasi-empirical inductive results and its interrelation with abduction and deduction lead to speculation of how a person might have their own preferred way of ordering their uses of deduction, abduction, or induction. One can easily imagine this in the general cases of the theoretician, and the engineer. The theoretician may prefer to use visualization and NMI exploration to learn something and then discard its use for what they personally may find more efficient means of reasoning, such as deduction. An engineer working on one particular sprawling interrelated system might find visualization, NMI exploration and experimentation more efficient, and only turn to deductive reasoning for final verification of his plausible reasoning. And although abduction alone cannot verify a fact, it is the only means for a genuinely new idea (e.g., an inductive visualization leading to a speculative explanation which may end in a deductive verification—or to an inductive/visual plausibility check, etc.).47 In any case,
47 As regards three distinct categories of reasoning, one might note the individually idiosyncratic randomness of combinations of these categories might be modeled as the orderly (yet unpredictable)
55 these orderings of compositions of types of reasoning may prove to be more a matter of personal preference than a generalizable pattern.
Colligation Colligation is being reviewed as it relates to what the NCTM (2000) called “making connections.” Colligation is a term that Peirce uses to describe a particular thought process which precedes the making of any inference. Presumably, in so far as mathematics learning is concerned, if one were to try to isolate a certain kind of mental action for practice and exercise, colligation is a prime candidate. A colligation is an association of ideas under one umbrella—that umbrella being one’s intention to make an inference. For example, in order to make a deductive inference, one first studies the premises to be assumed in the argument. Registration of these premises as some mentally associated pattern organized by the intended inferential orientation or direction constitutes a colligation. Colligation is commonly defined to be a tying together of some collection of things. In logic, it is defined as “That process by which a number of isolated facts are brought under one conception…” (Webster’s, 1998). “Colligation is not always induction, but induction is always colligation,” J. S. Mill quoted by Webster’s (1998). Describing colligation, Peirce (EP2; p. 45) gave the example that we “begin a deduction by writing down all the premises. Those different premises are brought into one field of assertion, that is, are colligated…” Peirce (EP2; p. 293) provides a recursive justification of colligation in the sense of kind of a regression from the premise, P, to a less determined premise, or judgment O, etc. Then P becomes an association in one’s thinking of O and P. Therefore, Peirce concluded that instead of discussing an “Argument of Colligation,” one considers the collection and interrelation of judgments which underpin the premises for that inference. 48
trajectories of a nonlinear chaotic dynamical system. Chaos cannot arise without at least three dimensions (e.g., rabbits, foxes and cabbage; or the three body problem). Skarda & Freeman (1987) attempt to mathematically model the physiology that underlies thinking, learning, and consciousness. They find there to be an explicit dependence in their models on a necessarily chaotic dynamic for pattern recognition to occur in their mathematical models of physical brains. 48 Rescher (2005; p. 94-99) discusses plausible reasoning as depending upon the presumption of normal, commonplace, typical contexts in which the plausible (inductive or abductive) inference is to take place. This presumption is like a premise of the plausible inference. The colligation is then the content of the
56 A mathematics education researcher such as Sfard (2000a) might see colligation as a particular kind of conceptualization. Sfard (2000a, p. 344-345) discusses a case of analogical (plausible) inference intended to connect two concepts as being facilitated by the initial creation of the involved concepts (i.e., the base and target of the analogy (English, 1997; Holland, et al., 1986) as being the result of metaphorical thought). This “initial creation of the involved concepts” would seem to be what Peirce generally described as the act of colligating.
Representation It has long been apparent that students have trouble interpreting mathematical symbols (Kaput & Sims-Knight, 1983). More recently, research concerning mathematical representations has ramified considerably (Ainsworth, Bibby, & Wood, 2002; Goldin, G. & Kaput, 1996). The multiple representations of NMI (i.e., numerical, symbolic, graphical, and diagrammatic) when taken together, have known potential for aiding students learning (Janvier, 1987). These results hold as well as in the Microworld or Interactive Mathematical Situation class of computer program environments discussed above (Watkins, 2003; Lin & Hsieh, 1994; Steffe & Olive, 1990). Girard (1997) noted that the multiple- representational-view-of-concepts model of instruction, coupled with integral use of the graphing calculator, allowed students to demonstrate knowledge of a multiple representational approach to problem solving in calculus. Although there was no overall significance between graphing calculator use and correctness, there was a significant association of correctness when the calculator was used for exploratory purposes on unfamiliar calculus limit problems. Thompson & Sfard’s (1994) discussion focuses primarily on internal versus external representations. As researchers must pay attention to student’s internal constructions or conceptualizations, it is also worthwhile to consider the external construction especially as interactive computer use grows (Goldin & Kaput, 1996; Kaput 1992; Kieran, 1992; Kaput & Simms-Knight, 1983). NMI research involves external
perceived judgments implicit to the typical context of the intended inference. See the section entitled “Deduction and Plausible Reasoning” above.
57 representations, while this research pays careful attention to what students themselves may be mentally visualizing, when feasible. Research on internal visualization is well developed (Stylianou, 2002; Shoaf-Grubbs, 1992; Zimmermann & Cunningham, 1991; Presmeg, 1986).
Pragmatism as a Research Stance The stance taken by C.S. Peirce’s pragmatism, with regard to reasoning, is reviewed as a research framework for this study. Peirce’s description of methods of reasoning are primarily considered here along with certain of their ties to other mathematics educational research frameworks (e.g., Ernest, 1997; English, 1997; Presmeg, 2001; Sfard, 2000a).
Clarifications and Caveats One clarification and one caveat is in order concerning an adoption of Peirce’s perspectives. The clarification involves senses of the terms mathematics and mathematical reasoning, while the caveat concerns (a) mathematical fact, (b) the pragmatist sense of realism, and (c) the external mathematical representations or artifacts produced by NMI. This needed pointing out as regards the researcher’s senses of the terms ‘mathematical reasoning,’ and ‘realism’ when taken from a strict Peircean perspective. The first caveat is easier to make than the second. Peirce viewed mathematics as a purely deductive science—but, as a theoretical (or mathematical) science, it involves abductive, inductive, and deductive reasoning. Although Peirce also tends to view mathematical knowledge and mathematical reasoning as one, this is not surprising from the standpoint of pragmatism, where knowledge and use are so closely intertwined and only concerns; one regarding what is taken to be mathematical reasoning in the context of mathematics education—and Realism, or mathematical fact, by clearly distinguishing three aspects of mathematical reasoning.
58 Relevance Literature which demonstrates the relevance of pragmatism to NMI use is reviewed. Mathematical reasoning and NMI tool use makes pragmatic considerations highly relevant. Peirce (CP 5.8, c. 1905) provides the description "... pragmatism does not undertake to say in what the meanings of all [semiotic] signs consist, but merely to lay down a method of determining the meanings of intellectual concepts, that is, of those upon which reasonings may turn." Pragmatism concerns gaining the meaning of things and situations—the principle concern of educational constructivism as well. The editors of EP2 (1998/1893-1913, preface) write that Peirce “is not an epistemological pessimist: indeed, quite the opposite. He tends to hold that every genuine question (that is every question whose possible answers have empirical content) can be answered in principle, or at least should not be assumed to be unanswerable.” A significant paper in mathematics education pointed out that students learning through a programmed learning approach could sometimes learn the program itself, without learning the intended mathematics purported to be taught by that program (Erlwanger, 1973). In other words, students would learn an unintended (by the program writers) routine in the programmed learning format. However, when using NMI, any figuring out of the computer program itself is a step in the direction of learning mathematics. What of other kinds of pragmatic uses? One need look no further that Shannon (2003) to see the relevance of NMI in its “super-calculator” form (e.g., SNB) to college science and engineering preparation. These students can be considered more directly with the use of NMI—especially under a Peircean interpretation of mathematical reasoning.
59 Student Motivation Following is a discussion of how pragmatism might serve as a useful theory in describing how NMI use might motivationally enhance a student’s mathematics instruction. Objectivist-oriented considerations coincident with the sciences are first discussed. Lomas (2001) pointed out that certain relativist positions in science [or mathematics] education can be an issue which “conflicts with the convictions of these students with regard to the propositions they have themselves verified, sometimes with “their own eyes” in classroom experiments.” Understanding a Pythagorean Theorem proof as a deductively necessary result, sensed to be an objective, mind- independent/hypothesis dependent actuality can be extrinsically motivational (see Skemp, 1987). Understanding mathematical results as austerely dependent only upon postulations and logic, provide students with intellectual experiences beyond mere academics, computed results, or an instructor’s subjective opinion (Rowlands & Graham, 2001; Lomas 2001). The enormous weight of useful scientific discoveries brought into intellectual grasp by deducing from completely hypothetical mathematical postulations bears consideration. For example, Wigner (1960) explicitly discusses the unreasonable, or surprising effectiveness of mathematics in physics. Students are likely motivated by considering an objective real world beyond their immediate experiences—much of which cannot be grasped without mathematical explanation. NMI use can aide both implicit (intuitive) and explicit mathematical understanding in the process of its application to modeling things of genuine interest in the real world. Complementing this motivationally objective perspective in view of pragmatism, see Holma (2004) for a presentation of Israel Scheffler’s (a former high school mathematics teacher) synthesis of the pluralist educational philosophy of Nelson Goodman with the realism of Peirce’s pragmatism (Rescher, 2005). Now consider subjectivist-oriented pedagogy. Sanson & Morgan (1992) point out that students who choose their own paths in problem-solving experience higher senses of personal achievement leading to an intrinsic motivation concerning their learning
60 (Skemp, 1987). Personal interaction with instructors is motivating to students as well (Middleton, 1995). Gelman & Greeno (1989) pointed out that having opportunity to craft a project around one’s familiar self-interests is motivating. Schiefele & Csikszentmihalyi (1995) specifically pointed out that small group projects integrated with computer use facilitate learning and levels of interest. Blumenfeld (1991) pointed out that engaging in long-term projects on topics students find authentically interesting can boost motivation. Finally, Nolen (1988) pointed out that students become motivated when there are clear reasons for what they are studying, meaning that they can discern clear goals for why certain mathematics is being taught to them.
Peirce and Constructivism Many educational researchers and theorists often write from an essentially constructivist viewpoint when referring to Peirce’s pragmatism in one respect or another (e.g., English, Ernst, Holma (on Israel Scheffler and Nelson Goodman), Lomas, Presmeg, Sfard, Shank & Cunningham, and Vile; to name those cited here). Some would say that the central tenant of mathematics educational constructivism is that learning solely involves the mental action of making sense for oneself. “Construction” of one’s own mental intuitions, concepts, and models intersubjectively negotiated with others is another apt description of educational constructivist learning theory (Greer, 1996; Cobb & Bauresfeld, 1995; von Glasersfeld; 1983). The mathematics educational orientation provided by constructivism has led to significant strides in the field’s research concerning learning theory and pedagogical practice (e.g., Sfard, 2000a; Thompson & Sfard, 1994; Wheatley, 1997; Phye, 1997; Steffe & Olive, 1995; Steffe et al., 1996). Cobb & Bauersfeld (1995) tie social constructivist oriented views (negotiation of intersubjective meaning) into complementary views as concern an individual’s learning. Presmeg (1986, 1997b) and Wheatley (1997) were led by a constructivist outlook in describing mathematics learning as aided by visualization. Peirce (1992/1885, pp. 227-228) agreed in describing mathematical visualizations of “objects of reasoning, [in order to experiment] upon [these images] in the imagination, and observing the result so
61 as to discover unnoticed and hidden relations among the parts.” It is likely that the increased feasibility for inductive and abductive experimentation and exploration of mathematics facilitated by NMI might aide in what the NCTM (2000) called “making connections.” Although a central philosophical tenant of radical constructivism dismisses the possibility of objective, as well as mathematical knowledge (von Glasersfeld (1983)), non-radical constructivists do not necessarily hold to these tenants. Further related discussion follows. Objectivity, Reality, and Mathematical Conceptualization The following literature review specifically addresses a concern of current trends in mathematics educational theory. These trends undermine claims about obtaining objective knowledge of a mind-independent reality through certain reasoning processes, as well as the truths of validly deduced results from self-evidently assumed mathematical postulations (e.g., Rowlands & Graham, 2001; Lagemann, 2000; and Gross, Levitt, Norman, & Lewis, 1996). 49 The following review limits itself to aspects of constructivist learning theory which embrace this trend (e.g., von Glasersfeld, 2001; Ernst, 1999; Cobb & Bauersfeld, 1995, von Glasersfeld, 1983). Careful attention is given to Peirce’s views relevant to the aforementioned trends. This has specific relevance to how one frames an approach to the research question. Mathematics is the primary tool of reasoning for discerning and predicting objective scientific fact (e.g., Wigner, 1960). In the context of pragmatism, this can be explained as a purposeful, or practical triangulation between strongly independent ways of reaching conceptualizations (e.g., inductively vs. deductively). Rescher (2003, 2005), assuming a pragmatic epistemological perspective wrote that “the miracle of mathematics explaining nature,” is that “[m]athematics must characterize reality on conceptual issues alone” (Rescher, 2003, p. 281-282). Peirce commented: "I understand pragmatism to be a method of ascertaining the meanings, not of all ideas, but only of what I call 'intellectual concepts,' that is to say, of those upon
49 Gross, Levitt, Norman, & Lewis (1996) are particularly concerned with a postmodern influence on views of reasoning, and the sciences. Peircean pragmatism rejects the wholly relativistic, anti-realist positions of postmodern writers (e.g., Richard Rorty, Paul Feyerabend, Jaques Derrida, or Jean-Francois Lyotard) which have been explicitly embraced by certain well respected mathematics educational theorists (e.g., see the collection of work in Walshaw, 2004).
62 the structure of which, arguments concerning objective fact may hinge.” (CP 5.467, 1907) Peirce’s pragmatism is then undoubtedly seen to be central in ascertaining the meanings of concepts facilitating objective discoveries about the world. The triangulation of empirical fact, mathematical conceptualization, and methodical reasoning make certain knowledge of a mind-independent objective reality possible. Socially Constructed Objectivity Rescher (2003, 2005) clarified these aspects of pragmatism in a perspective supportive of socially constructive, yet aspiring to objective communicative activity. 50 He wrote: [A] commitment to objectivity… affords us an effective practical instrumentality that facilitates communication and cognitive collaboration. Specifically we require [such a commitment]: 1. As a presupposition to make communication possible by way of agreement and disagreement. To principle a commonality of focus. 2. As a contrast conception that enables us to acknowledge our own potential fallibility. 3. As a regulative ideal whose pursuit stops us from resting content with too little. 4. As an entryway into a communicative community in which we acknowledge that our own views are nowise decisive.” (Rescher, 2003, p. 184; also see Rescher, 2005; p. 23-28, 43-44). Students similarly gain the confidence and competence required to create, test, refine, and validate reasons as cogent. From a social standpoint, Rescher concluded that the process of …rational selection… [impels] reason-concerned communities in the direction of a commitment to normatively cogent reasons. The social and communicative practices of an interactive group of intelligent agents are bound to manifest an impetus to objectivity… [A] good or cogent reason
50 Vygotsy, a father of social constructivism, also supports a view to objectivity interwoven with his positions as is similarly done in pragmatist theory. On a related note, Sfard (2000a) locates Peirce’s semiotic perspectives in a social dialogue learning context.
63 [for Rescher is one where] the beliefs substantiated by it are generally true. (p. 300). Habermas (1984) & Habermas (1987) offer similar considerations. He describes a so-called lifeworld which facilitates a social construction of rational, objectively oriented perspectives demanding critical, interpersonal communication. Views of Habermas have been referred to extensively in the mathematics educational constructivist oriented literature concerning the shaping of mathematics curriculum (Grundy, 1987).51 Constructions of Mind Independent Outcomes Peirce’s pragmatism viewed reality as more or less, that which is mind- independent. “That is real which has such and such characters [(properties)], whether anybody thinks it to have those characters or not” (EP2, p. 342). And again, “That which any true proposition asserts is real, in the sense of being as it is regardless of what you or I may think about it” (EP2, p. 343). Rescher (2005, p. 59) explained “The components of a pragmatic validation of a truth criterion… takes the route of an appeal to experience.” The core of this criterion hinges on the “methodological appropriateness” of considerations—which in turn hinge on the methods of pragmatic reasoning. Rescher did not divorce reasoning from the practical aspects of human actions, but pointed out that reasoning predicts outcomes, and it is these outcomes which are wholly independent of (although may parallel) what is cognitively dependent. Rescher gave the anecdotal example of a man who walks headlong into a large plane of glass. The outcome of his cognitive predictions can turn out wholly independent of what was, quite rationally thought to be. Burch (2001) aptly wrote: Peirce's insistence on fallibilism, [as] he is not an epistemological pessimist; indeed, quite the opposite: he tends to hold that every genuine question (that is, every question whose possible answers have empirical content) can be answered in principle, or at least should not be assumed to be unanswerable. (p. 4)
51 Grundy discusses what Habermas (1984) calls three spheres of knowledge constitutive interests (and so then curricular of interest): the normative, aesthetic, and technical. These interests are considered by Peirce’s pragmatist discussions concerning an ethical, moral basis of logic, an aesthetic basis of pure mathematics, and the facilitation of science by specific methods of reasoning, respectively (EP2, p. 51, pp. 196-207, pp. 242-257; Hookway, 2004)
64 For this reason, one of Peirce’s most important dicta was, “Do not block the path of inquiry!” Rescher (2003, p. 175) drew the distinction between someone claiming, “The cat is on the mat, [and] I think/believe that the cat is on the mat… [No] sort of statement specifically about you and your beliefs can ever be equivalent with a claim regarding the you-independent arrangement of the world.” Rescher concluded: The long and the short of it is that the circumstance that our ideas and the words in which we formulate them are human artifacts does not prevent them from applying [to] the real world. Neither our physical nor our mental artifacts are severed from reality in virtue of their man-made condition. (Rescher, 2003, p. 175) A shovel, though man-made, can certainly be used to shovel (non-man made) earth or sand. But how can we understand the transition from subjective thoughts of “inner” personal experience to that of “outer” impersonal objective fact? Rescher (2003) continued, contending, People who care about giving and having reasons for their beliefs—will in the course of time tend (under the pressure of “rational selection”) to increasingly offer and require good reasons for beliefs (i.e., reasons of a kind that will, at least by and large, be available when, and only when, beliefs that are based upon them… [provide] true [outcomes]). (p. 300) In conclusion, Peirce’s views clarify an independent reality where, “So long as these ‘reals’ are ‘external’ and are not created by our beliefs that they obtain, it is no obstacle to their reality that they are relative to [or paralleling] our perspectives and their peculiarities” (Hookway, 2004, p. 3). The researcher interprets these cited authors to admit a constructivist learning perspective side by side with an embracing of an external, objectively knowable reality.
65 Learned Construction of Mathematical Facts through Pragmatic Reasoning Rescher (2003) and Peirce above have pointed out that mathematics is not a natural science but a theory of hypothetical possibilities.52 However, it can be argued that the triangulation of pragmatic forms of reasoning allow students to approach their learning of mathematics much like they approach their learning to understand how we know various scientific facts. A familiarity with the plausibility of certain mathematical facts can help a student in later making deductive inferences which concern, or support those facts (or mathematical propositions). Pragmatic reasoning then supports a means for understanding mathematics both plausibly, as well as deductively. Abductive hypothesis, inductive verification, and deductive necessity triangulate to make a student’s mathematical knowledge well rounded, robust, and well connected (NCTM, 2000). From a pragmatic perspective, students can get insight into mathematical facts without yet knowing how they were deduced. Then in retrospect from these insights, or experiences using NMI, students are better equipped to put together chains of deductive mathematical inferences. Then students are able to deduce and understand the factual truth of certain mathematics, so long as they understand and accept the premises of their inferences (Angeli, 1998).53 The analogy to science learning holds in that students can first gain a plausibly reasoned knowledge of certain mathematical facts in isolation through experimentation. At other times, students might use this knowledge to draw deductive inferential connections between these facts—analogous to a theoretical understanding in the sciences.
Closing Reflection In conclusion, pragmatism offers a theoretical perspective which places central importance on clearly distinguishing, and elevating both practical (“what is to be done?”) and theoretical (“what are the facts of the matter?”) reasoning. NMI make mathematical results and renderings directly accessible and instantaneously presentable. Computers
52 Mathematics is a hypothetically based consistent system of facts deduced from its most carefully chosen hypotheses (EP2, p. 51; Rowlands & Graham, 2001). 53 Rescher (2005, p. 35) develops a parallel argument concerning the ascertainment of external objective
facts.
66 more generally provide a means for experimentally interacting with, and plausibly reasoning about these mathematical results and renderings (Borwein & Bailey, 2004; Wolfram, 2002). Computation facilitates reasoning about certain mathematical situations as if they were akin to the empirical sciences.54 Peirce himself reflected, Mathematics appears to me to be a science, as much as any science, although it may not contain all the ingredients of the complete idea of a science. But it is a science, as far as it goes; the spirit and purpose of the mathematician are acknowledged by other scientific men to be substantially the same as their own. (EP2, p. 86) Mathematics learning has also been seen to coincide with what Peirce (EP1, pp. 1-10) has described to be the three distinct reasoning methods of science: abduction, induction, and deduction. Peirce (EP1, p. 120) states, “Everybody uses the scientific method about a great many things, and only ceases to use it when he does not know how to apply it.” It would seem to make sense then not only to describe student’s reasoning in these terms, but also to teach the application of these methods of reasoning in the course of using NMI. Finally, Peirce’s semiotics, an integral part of Peirce’s theory of pragmatism, has also been pointed out as playing a significant role in mathematics education research more generally. The role of cognitive representational conceptualization, or semiotics, is central to mathematical reasoning. NMI diagrammatics can play a significant role here. Therefore it can be seen that the pragmatic stances of Peirce provide a sensible research framework. The kinds of reasoning that NMI oriented programs have been discussed as facilitating are the same kinds of reasoning that the growing area of the computational mathematical sciences use: the methods of scientific reasoning applied in support of the more exacting deductive demonstrations necessary in certain mathematical reasoning.
54 It is relevant to note that the editors, Houser & Kloesel (EP1, p. xxxiv) point out that Peirce’s sense that “experience is our only teacher,” a fundamental empiricist tenet. The editors went on, “Yet although an empiricist, Peirce rejected the notion of tabula rasa, claiming that there “is not one drop of principle in the whole vast reservoir of established scientific theory than has sprung from [the mind separate from experience].”
67 CHAPTER III METHODOLOGY
The research question studied was “How might students reason mathematically when using NMI oriented tools?” This question was posed to consider a student’s approaches to, and aspects surrounding their reasoning when using such a tool in a mathematics educational setting. Peirce’s pragmatism is an area of theoretical language that has been selected here to provide a fabric for holding together a practical look at NMI use without pre-deciding if one is going to take an empirical, rational, or pragmatic view of mathematics learning more generally speaking. That is to say, in selecting the use of Peirce’s language, my intention is to keep as neutral a position as possible in describing student’s reasoning. Pragmatism was shown to give credence to widely varying aspects of reasoning which have their roots spanning rationalism to empiricism. This is in keeping with the intention of this methodology to be broadly considerate of these other forms of reasoning—not to simply cut off consideration of empirical or rational approaches or dispositions simply because one is focusing on use of a practical tool, an NMI, for learning mathematics.
Qualitative Research Approach Qualitative research was used in this study. Qualitative research requires a theoretical framework, or a set of perspectives which provide a lens for focusing the research. This theoretical lens is employed within the generic methodological forms of case study and action research.
A Research Framework based on Pragmatic Reasoning Mathematical reasoning in light of computer use is a vital area of research in mathematics education. Current research concerning student’s mathematical thinking has resulted in many subtle and multi-faceted theoretical frameworks. Some of these theories involve concepts such as horizontal and vertical thinking, reification, constructive thought (making meaning, or sense making), enactive co-emergence of thought, constructionist thinking involving computer use, reflective abstraction, metaphorical
68 thinking, visual thinking, procedural thinking and memorization, making mathematical connections, and analogical thinking concerning global interpretations, one-to-one mapping constraints, target and model parallels, etc. (e.g., Ratterman, 1997, p. 250-251). This list illustrates many interwoven aspects of mathematical thinking, some in psychological, philosophical or metaphorical terms. The framework chosen for this research concerns how Peirce’s forms of pragmatic reasoning might be used to describe how students learn when using certain aspects of natural mathematics interfaces (NMI). Peirce’s theories were careful to delineate descriptions of thinking from methods of reasoning. This provides a framework for discussing student’s thought processes in terms of the more general descriptions laid out by Peirce concerning aspects of pragmatic reasoning. Finally, the application of a pragmatic reasoning framework to student’s using NMI is particularly relevant in so far as the NMI allows for experimental interaction with external representations of conventional mathematical artifacts and procedures. Peirce saw the methods of pragmatic reasoning to be the practical foundation for pursuit of both experimental and theoretical inquiry.
Specific Research Methods This research constitutes a qualitative study further described below. Centrally, a hybrid of the two approaches, case study, and action research, were used.
Case Study Case study is a fairly generic qualitative research method applied in the social sciences. Interpretive and critical methods of case study facilitate descriptions of research in natural pedagogical settings. Qualitative studies can emerge from independent pedagogical settings, with there only being a roughly defined research focus and design at the start (Stake, 1995; Bogdan & Biklen, 1998; Guba & Lincoln, 1989). “The general design of a case study is best represented by a funnel. Good questions that organize case studies are not too specific” (Bogdan & Biklen, 1998, p. 62). This research began with the not so general research question, “How might students mathematically reason when using NMI oriented tools?” However, attempts to
69 capture a sense of depth and breadth for this research question acted as a funneling process none the less. Following are some of the considerations made thus far in the narrowing down of the research goals.
Action Research Action research allows for the researcher to be “actively involved in the cause for which the research is conducted (Bogdan & Biklen, 1998, p. 223).” It is considered to be applied research. In particular here, the research can be carried out considering improvement of teaching methods with computers, or consideration of the curricular effectiveness of certain computer use. This research encouraged plausible and deductive reasoning for students using NMI. The researcher actively participated in the research as an instructor. Action research not only provides for purposeful researcher participation, but it depends upon the participant to carefully reflect upon their results in a specific applied context. Action research is simply a form of self-reflective inquiry undertaken by participants in social situations in order to improve the [reasonableness and justification] of their own practices, their understanding of these practices, and the situations in which the practices are carried out. (Carr & Kemmis, 1986, p. 162) McNiff (1993) notes that action research requires: (a) stating the problem, (b) devising, implementing, and evaluating certain pedagogical strategies, and (c) offering reflections for possible solution to the problem. This study combines elements of case study and action research methodologies. The term ‘case study’ is generically used below to describe either.
Recursive Qualitative Approach I used what I would call a recursive approach during the final evolution of my research reporting methodology. My method involved changes in reflective perspective, and methods of presenting case study data and interpretation. The analogy to recursion is iteration on the same algorithm—the algorithm here being beginning to end edits. This seemed inevitable during my editing process, this iterative process from first draft to final edit.
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Triangulation Applying the three methods above, and real-time member checking with students, provided a validation of the research, its reporting, and interpretation. Validation in the sense here means that the research resulted in as unbiased a case study reporting as possible, so that my own interpretations did not color the factual parts of the study—the parts that did not involve my own interpretive reflections..
Determining a Sense of a Student’s Mathematical Reasoning using an NMI The qualitative methods described admit of an approach to capturing the subtle nuances of a student’s reasoning. A student’s mathematical reasoning, although similar to a mathematician’s, had differing shades of emphasis. There was a plain difference in the kinds of reasoned products each is most often responsible for in their reasoning activities—the student’s learning process versus the final product of the mathematician.
Constructivist Learning Theory Research Tools Certain constructivist instrumentalities were used in the research analysis. Some of the instrumentalities are part of a social constructivist framework. Part of the method for the action research is to encourage students to have a kind of mathematical dialogue with the machine. The primary constructivist ideas used are described below. Perturbation Perturbation, in the constructivist learning context, describes how student’s errors and misunderstandings can be used as opportunities for students to learn. Rather than a teacher telling students what they did wrong, student’s errors can perturb them to reflect on their own misunderstandings. This approach to learning is especially applicable to NMI use. Firstly, students can explore and experiment with the NMI concerning what they do and do not understand. This also works in reverse in that by experimenting with the NMI, students have ample opportunities to run across situations that perturb them, leading to further experimentation.
71 Zone of Proximal Development The zone of proximal development (ZPD) concerns the potential of a student when helped by others.55 This zone bridges what a student can do with help to what he or she can do when left to work autonomously. This requires two attributes of the helper, (a) the helper needs to know more than the student, and (b) the helper needs to be aware of what the students are able to do (i.e., is able to diagnose where the student is in their understandings). One can consider a ZPD relative to NMI use. Here the student self-diagnoses through NMI interaction, and learns to work with less and less help from the NMI. Scaffolding Scaffolding is a method of instruction which focuses on the intersubjective understanding shared between the teacher and student, and is typically thought of in terms of social interaction. Scaffolding is a means for keeping students within their ZPD’s. NMI allows students to communicate with machines in the shared natural language of mathematics. NMI then facilitates a kind of intersubjective communication between the student and the NMI—the NMI scaffolds the student, while the students control their own ZPD. The ideal NMI would allow for fine grained interaction, rather than, for example, the facility to merely enter equations and obtain completed solutions, without any means for inquiry into intermediate mathematical steps, representations, or procedures. From a diagrammatic perspective, the NMI would ideally provide fine grained interaction as well. This interaction might self-guide students’ development of their own means to mathematically visualize. A possible disadvantage in using the NMI as a scaffold involves the student making automated calculations at a higher mathematical level than they understand (i.e., exceeding their ZPD). Students may do a lot of guessing and checking (abducing and inducing), but never understand underlying derivations, calculations, or proofs.
55 The term was coined by Lev Vygotsky, a father of what is now referred to as social constructivism in mathematics educational theory (Aidman, 1995). Social constructivism and individual constructivism as used in the mathematics education literature are combined under the general term ‘constructivism’ here.
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Pilot Study Summarized below is a pilot study which led to the present methodology. This study helped focus a perspective for so-called natural mathematical interfaces, and how they might be used in both broadening and deepening a student’s mathematical understanding.
Setting and Research Subject Description A male junior civil engineering major, minoring in graphics design agreed to do a pilot study with me in exchange for tutoring in a differential equations course. I will call him Alan. Alan and I worked in an office cubicle with the following mathematical tools: one computer running the program DPGrapher, a TI-83 graphing calculator, white board, and pencil and paper. DPGrapher (Parker, 1999) has the NMI property that it allows mathematical expressions to be entered in a syntax very much like naturally written mathematics, and graphs these expressions in up to 3 dimensions. DPGrapher rotates, zooms, and allows user friendly controls over graphing ranges, axis tick marks, etc. A video camera set on a tripod was used to record the research sessions, with only intermittent field notes taken, and no organized means for saving of computer files created by Alan. The meetings lasted for 8 weeks, at approximately 60 to 90 minute sessions. There were two meetings per week. About 25% of our meeting time involved interview, informal discussion about the project, and member-checking activities (see the section below entitled Member-checking). Only intermittent field notes were taken, as a full transcription of the audio was planned.
NMI Displayed Images DPGrapher facilitated Alan’s artistic creativity in how he navigated the many graphical controls. It seemed every session there was a new tool to explore. This was an exploration not of the mathematics directly, but indirectly via Alan’s interest in the software capabilities, and the displayed graphics. For example, he was able to create and fine tune some truly breathtaking kinds of graphical journeys through relative variations
73 of the function or relation being graphed (e.g., fix various mathematical parameters, and vary others). This was clearly a plausible reasoning approach to contriving and manipulating mathematical diagrams. One of the most analytical set of explorations accomplished concerned conic sections. Alan was able to take various cross sections of the 3-dimensional displays and come to understand mathematical relationships between the algebraic expressions and the images. Conic Sections, Mathematical Embodiments and NMI Use The concept of a mathematical embodiment involves a kind of analogy between mathematical and nonmathematical objects (e.g., 2 apples and 3 apples make how many?). NMI use was observed to provide a kind of recursion in one’s reasoning with embodiments—where one level of mathematics itself served as an embodiment for the learning of a higher level. Consider the following sequenced phrases as examples of qualitative mathematics, mathematical embodiments, and reasoning recursing upon less sophisticated mathematics, mathematical embodiments and reasoning. Two apples, and two apples make…, 5 groups of 5 is 5, 5 times, which can be concretely arranged in the shape of a square shape with 5 columns… square planes of area can be physically seen to grow parabolically with their side length…, squared variables are seen concretely… and how does a vertical and a corresponding height horizontal slice of a simple cone relate to this abstracted parabola? to concrete visualizations of circles, radial distance and area? to linear and quadratic proportionality? And how can we find the volume of our cone? Consider the next reasonable complement of the height and cross-sectioned width of our cone. We add a consideration of breadth by making a horizontal cross-sectional slice. We can then concretely note a stacking of ever shorter, concentric-sized cylindrical volumes to approximate the cone volume… this final embodiment leading us to a sense of an integral. Alan was scaffolded through the later part of the particular conic section oriented tasks described. Alan’s excitement at making mathematical connections through (a) the NMI images, and his mathematical control of them, (b) his own graphing calculator use and (c) a pencil and paper nailing down of ideas, all served to demonstrate a genuine
74 striking of his analytical ZPD. Although much of the analyses was qualitative, he was able to verbally explain relationships between the displayed conic slices and his own algebraic and calculus understanding. Trigonometric Functions and Scientific Analogs Plotting trigonometric functions was fascinating for Alan in terms of an applied mathematics orientation. We discussed modeling of physical phenomena like vibration. Alan guessed and checked concerning his holding certain things constant and changing others.56 During an interview, Alan indicated that his engineering background provided his interest and capability here. Both engineering and science analogs were seen to act as mathematical embodiments as well for Alan’s qualitative consideration of the trigonometric displays which he created through use of the NMI.57
Lessons Learned--Methodological Contrasts with Present Research The pilot study indicated that students needed guidance, and better defined goals for their investigations. This is especially true when using software that is aesthetically fascinating, and liable to entice exploration of intellectually unmanageable mathematical situations. Mathematical investigation using NMI requires guidance as well as homework/analysis time away from the computer. The methods used in the pilot study were similar to those discussed below with the following noted exceptions. Data Gathering Techniques Video tapes were made of the student’s work, and approximately 6 hours of the tapes were transcribed word for word. I found that it was more effective to carefully watch the video’s and transcribe only what I found to be relevant to the research. I decided field notes indexed to saved computer files was a better data gathering methodology.
56 Guessing and checking is a term used by Polya (1954) and is a cliché for aspects of what Peirce has described in greater detail as abductive and inductive reasoning respectively. 57 Typically, an analog is an analogical interpretation of something real, but not wholly representable in terms of a mathematics model. The analog is as close the real situation as feasible, and is often brought into relation with certain mathematical modeling tools. An embodiment is also an analogy but for different purposes. A student projects their familiarity with an embodiment onto yet unfamiliar mathematics, while a scientist projects their familiarity with mathematics onto a physical analog. Students make use of what are typically referred to as an analog in the way they make use of an embodiment—they can serve identical purposes to the student.
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Use of Homework Alan did not work on the NMI or the mathematics considered outside of the meetings we held together. He did not have the time or incentive to do this. The pilot study prompted me to look for a research situation where I could depend upon students to do homework with the NMI devices. Use of Scaffolding The research was minimally guided. The goal was to understanding how Alan might go about exploring and analyzing the 2- and 3-dimensional functions and relations he was able to easily plot from expressions he contrived himself. I decided that in the future I needed to scaffold the student more, and steer him towards computer renderings which were simple enough to lend themselves to certain mathematical analyses. Aesthetics was seen to be more of an underlying motivation for the research subject than mathematical analysis. Alan was interested in exploring colorful 3- dimensional graphical representations. The kinds of analysis he did was in thoughtfully stepping through differing variations of parameters and observing the results each time— considering how the mathematics determined new variation in the graphics. Much was learned from the pilot study concerning the present design described below. The subsections above have tried to capture a brief description of the lessons learned in the pilot study in-so-far as they applied to the research below.
The Study This study aims at providing a theory based description of how students might reason using NMI. Relevant aspects of Peirce’s theory of pragmatic reasoning were tabulated as their relevance emerged to the researcher. The specific research question is then answered through action research reflection in consideration of the tabulated points of theoretic distinction. The reflection responded to the research question employing the tabulated terminology where it might clarify both (a) the specific research situation being described, and (b) how this theoretical language of Peirce may apply to specific mathematics educational discussion of NMI use.
76 Tabulation The tabulation is an instrumental or operational device (i.e., definitions or descriptions of terms) which allows for interpretation of specific pedagogical eventualities in terms of pragmatic reasoning theory. The pedagogical eventualities considered are how students might reason using NMI. The instrumental descriptions developed are useful for engaging substantive discussion of how a student might pragmatically reason when using an NMI to study mathematics. Certain terms used in the description of student’s reasoning which were not put forward by Peirce are noted separately in Chapter 4, Table 1. These separate theoretical designations are necessary in order to keep with (a) the theoretical development of tabulated encodings or descriptions, and (b) the specifics of this particular research context and question.
Present Case Study Initiation The detailed case study procedures are now considered. The study’s initiation and context are discussed, along with the research setting, research subject selection, and NMI selection. Setting I was given the opportunity by a prominent affirmative action state university to act as a research guide for two undergraduate students in a funded research project.58 We initiated work on a mathematics digital library to be used for study by other students. Student participants in the project were responsible for splitting their research time between our face to face meetings, and work done on their own. Our meetings were in office space provided on campus, just down the hall from a computer laboratory which the students spent some of their own individual time working, often concurrent with my own office hours. I will call the two research subjects Ann and Ed. Ann had her own laptop which she brought to the office. She worked in a GSP (Geometer’s Sketchpad) environment. Ed did most of his work using the computer facilities at school. Ed also worked at times with
58 The university was in the southeastern United States, and the research funding was in part provided by the National Science Foundation. There was no funding for the educational research or myself as prefector—all funding went to the students in support of their summer mathematics research projects.
77 laptops provided by the mathematics department for various presentations. His work with SNB (Scientific Notebook) also ran on a Microsoft Windows operating system.59 The formally paid research spanned one full semester. Other activities emergent from the research, such as presentations made by the students of their work at various mathematics conferences, are discussed later as they present themselves as relevant in the case study research analysis (e.g., student motivation, demonstration of student’s facility at explaining and answering questions on their research, student’s use of the software to explain their reasoning). Selection of Participants and their Backgrounds I attended a meeting where university faculty were informed of the funded research monies available to engineering, science and mathematics students. I selected two undergraduate students (after interviewing several). They were selected based on our mutual interests (e.g., convenience sampling, Gay, 1996). I had Ann as a student in two mathematics courses I had previously taught, while Ed was an undergraduate tutor for the mathematics department’s tutoring and computer laboratory. The students are called Ann and Ed respectively. The students were selected for their fit with the NMI programs I had interest in using, as well as the mathematical topics they had interest in studying. Ann’s further qualifications for selection was her interest in becoming a teacher, as her children were young, and she was interested in school oriented mathematics activities which might engage their interest. Through our classes together, we discussed mathematics as an area where one “knows that they know,” independent of what a teacher may tell them, or not. Ann is planning to attend graduate school in education. She is a national merit scholar based on her outstanding SAT undergraduate entrance exams. She had finished the first semester of calculus but said she did not get very much out of the course. She said the proof orientation of the instructor made it difficult. This led her to a natural
59 Three technical notes on the NMI discussed are of interest. SNB is not a “What-you-see-is-what-you- get,” but a logical indexing program. The internal program structure is more like a relational database than a program whose only function is to create computer screen renderings. This facilitated its insertion as an interface to MapleV. DPGrapher was written by a free lance programmer who did work for Mathematica and Maple. DPGrapher is an elegantly written 500 Kilobyte program (e.g., Microsoft Word is on the order of 100 times the size). Finally, because Geometry is so explicitly defined, GSP is an intelligent program in terms of what is pale gray on the menu’s and what is an applicable choice in the present working context.
78 interest in geometry proof—wanting to understand mathematical proof, and resolve some of her anxiety about her negative calculus experience. Ed’s background made him a good candidate for using SNB. Ed is preparing to be an actuarial statistician, with interests in operational research. Ed was educated under the European school system in Jamaica, and is extremely skilled at calculus. I observed him in his tutoring of the subject, and his skills were by far the best of the ten or twelve tutors observed. Ed also expressed an interest to me in learning about real world application of his mathematical skills. When Ed was not tutoring, he was working on his own course work. He preferred to work independently on his mathematics, asking as few questions as possible. Ed seemed humble about his mathematical skills. I once complimented him on his talent, and he replied, looking down as we were walking together, “Not really, I just work at it.” Ed had been selected for another funded mathematics program offered through the university, and he was sharply discerning over what was good mathematics, and what was not. Ann told me she had to overcome certain challenges growing up. For example, she told me she grew up in the projects in Detroit. She was sometimes given a difficult time by her classmates growing up, as she was extremely studious. Her family placed great value in education (her sister is attending Harvard). She once told me that whenever she missed a spelling word at school, her grandfather would have her write it one hundred times. Ann has many talents. She works sometimes as a model, and has appeared on MTV music videos. She writes poetry, and has an impassioned drive to encourage the people around her to achieve their highest goals (e.g., she often speaks at an organization on campus concerning young women’s academic, social, and family involvements). Part of the reason for selecting Ann and Ed was their work ethic, the confidence I had in their completing the research, and the tenacity they had exhibited in their mathematical studies. They both have goals to attend post-graduate school—Ed in the mathematical sciences, and Ann in education, particularly as it pertains to the teaching of mathematics to young children. Ed and Ann are African-American, in their early
79 twenties, and although they both have earned certain university support, are working supplementary jobs to maintain their financial viability for attending college. Selection of NMI The NMI SNB and GSP were chosen to be used. Ann expressed interest in the GSP NMI itself. Ed on the other hand Ed was interested in Markov chains, and SNB happened to be a good match for his mathematical interests. SNB is an Algebra and Analysis program, while GSP is primarily a geometric diagram representation program with numerical measure capabilities. These programs provided a wide coverage of their areas of mathematics in a single, uniformly designed, and mathematically cogent interface. Founding Literature Review concerning Theoretical Terminology Selection A foundational literature review included description of theoretical terminology selected for use in the case study discussion. This was completed both during and subsequent to considerable reflection upon previous observation of the student’s activity with the NMI. Applicable terminology descriptive of the case study’s subjects reasoning using NMI included Peirce’s (CP1, CP2) descriptions of pragmatic reasoning, and Tall’s (2002, 2004, 2005a, 2005b) description of natural mathematics.
Data Gathering Procedures Following are procedural descriptions of how pedagogy was blended with research observation and data gathering. The procedures involve interview, direct observation of students using NMI, and the use of scaffolding as a means to observe student progress using NMI. Initial and Ongoing Interview Goldin & Gerald (1998) and Seidman (1991) discuss how interview can be integral to qualitative research—allowing for detailed information to be gathered concerning student’s mathematical thinking, and other relevant information. Ann and Ed were interviewed about their backgrounds—both general and academic. This allows a richer contextual interpretation of the case study results. Interviewing has also been used intermittently during the course of the research. This allows real-time discussion about how students are thinking about the mathematics
80 and their NMI use (Goldin & Gerald, 1998). The researcher’s questions about their thinking are intended to guide the student to consider things that were noticed during work with the NMI. More general topics concerning student’s overall interests in mathematics were also discussed. Direct Student Observation Student observation includes (a) watching how they interact with the NMI, (b) listening to them thinking aloud, (c) considering how they respond to questions posed by the researcher, and (d) evaluating their mathematical progress as they go along. The second major observable related to student observation is studying the actual computer work in progress (their files) that they produce as they create their guided investigations using NMI. Following are two methods of student observation employed. Thinking aloud. ‘Thinking aloud’ is a technique often used in educational qualitative research (Gay, 1996; Goldin & Gerald, 1998). I ask the student to tell me what he or she is thinking about when puzzling over a problem. As Gertrude Hendrix pointed out though, this technique does not by any means give a complete picture of a student’s thoughts (see Usiskin, 1999 on ‘nonverbal awareness’). Verbalization at times may even interfere with a student’s thought processes. Care was taken in this regard. Ongoing assessment and evaluation. Student’s reasoning techniques and mathematics learning are assessed and evaluated as the research progresses. This provides the opportunity for the researcher to fine tune the kinds of activities to be presented to the student for ongoing consideration (e.g., to better meet their ZPD, and to encourage broad ranges of reasoning studied in the research). This evaluation of what is observed influences emergent teaching and learning activities described below. Three Kinds of Scaffolding Three kinds of scaffolding are described below. They involve scaffolding by the instructor, by the student’s independent use of the NMI, and finally the combination of the instructor aided by the NMI. Instructor scaffolding and participatory observation. Students were asked guiding questions when they reached an impasse in their perturbations. This kind of method is similar to an erotetic dialogue. Erotetic dialogue concerns the logic of questions and answers (MacMillan & Garrison, 1988). Erotetic logic introduces
81 inferences which have questions as premises and conclusions.60 The teacher, in an erotetic dialogue often answers a student’s question with another question (Burbules & Bertram, 1998). Care was taken not to give too much away in the instructor’s dialogical sequence of questions.61 The goal of this kind of scaffolding is to lead students to questions related to their own genuine questions, but falling within their ZPD. Then the student is not only able to answer the final question autonomously, but may have a sense of how to proceed given the sequence of questions that preceded. Student self-scaffolding using NMI. The most interesting part of this research was observing when the student’s autonomously used the NMI to answer their own questions. Often these were fine grained NMI inquiries that would not normally be asked as questions to an instructor. Sometimes the inquiries to the NMI involved contextual senses of the mathematics at hand, apparent only to the student.62 Of equal interest is when the student uses the NMI in order to ask questions that they may otherwise have trouble explicitly articulating. These are discussed immediately below. Instructor and NMI scaffolding. The NMI can be an integral part of an interactive dialogue with the instructor (researcher). Students can use the NMI to implicitly refer to, without yet being able to explicitly articulate their questions (i.e., “point and grunt”). This is also a kind of implicit erotetic dialogue among the student, instructor, and NMI—one driven by the logic of progressive questions, rather than final answers. The student (and instructor as well) have more freedom to ask more wide ranging questions by using the NMI as an intermediary point of reference. Some NMI can even make inquiries of the user (e.g., an interrogative statement of the NMI perhaps requiring a parameter to be entered in response). NMI use is a flexible
60 I took a course with the late CJB Macmillan who introduced Erotetic Logic in the context of teacher, student dialogue. He attributed the method in part to Hintikka’s work on the logic of interrogative inquiry, applying it to pedagogical dialogue (Macmillan & Garrison (1988), Hintikka (1982)). Erotetic logic is basically the study of inference through the posing of questions. 61 An apt analogy and metaphor is to a variation of Zeno’s paradox—moving half the distance towards a point each time, yet never reaching the point. The remaining distance to “the point of the dialogue” is left for the student to bridge. 62 This is similar to a student autonomously working out a problem that they know they are capable of finding the answer to with a little work. Ann in particular was observed to use the NMI as a kind of extension of her own intellect (see Papert & Harel, 1991).
82 means of scaffolding students’ own erotetic approach to their own questions—one question leading to another—however implicit or inarticulable they may be.
Teaching and Learning Research Activities The student’s selection of mathematical topics to be studied gave overall focus to the research activities. Ann and I discussed her interests studying certain geometry proofs, while Ed independently chose to study Markov Chains and their applications. As my own primary background was in engineering, I knew these topics would be interesting and challenging for both myself and the students. Progressive mathematics content in their respective areas of interest was negotiated with the students in an ongoing fashion—tailored to their (a) emerging interests, (b) background and ability, and (c) growing facility with the software. The software was learned on an as needed basis as the research progressed. Further, the problem selections negotiated were based on the anticipated relevance to the NMI being used. There was an emergent design of the methods described in detail below (Stake, 1995; Bogdan & Biklen, 1998; Guba & Lincoln, 1989). Student Activity Procedures Following are the methods which emerged regarding how the student’s learning activities were facilitated and studied. Greater detail is provided as regards procedural protocol than has been touched upon above. Sessions with individual students. Students worked with me individually for approximately 8 - 10 hours per week. This varied in that Ed would spend much more time working alone, coming intermittently to ask questions—where we would interact at 15 to 30 minute periods. Ann would often put in a full 10 hours a week with me, one on one. She and I spent a lot time on background geometry, and various ways to approach the proofs we chose to study (e.g., working explicit examples of biconditionality). I would often leave the office to run errands, and in part to give her time and space to work more deeply on her own. Learning the interfaces. Generally the interfaces were learned on an as needed basis. Ann and I, at the very beginning, sat down for about two hours together at the very outset. She was rapidly learning, and began predicting what I was going to show her by
83 completing the task sometimes faster than I was able to do it myself! SNB and GSP genuinely are “natural” interfaces. The students picked them up very well, while learning certain purely mathematical ideas in the process of figuring out how to use the interface for their personally intended purposes. Homework. Students were required to work half-time on their projects per their research funding. Their total hours of work put into the project fulfilled this requirement as there were conference presentations which were prepared for in the following semester. Both students, as far as I could tell, were working at the minimum an average of 5 to 7 hours a week outside of the 10 hour blocks of time we spent together. The funding required a midterm and final report. This required the students to put in far more outside work at those times. Interview. Students were interviewed at least three times (average 30 minute discussions) near the start of the research concerning their backgrounds and interests. We discussed what mathematical content they were interested in as the project moved forward. I guided in their negotiation of topics with regards to my interests in their use of the NMI. There was ongoing assessment of the student’s progress in order to tailor their tasks appropriately (Goldin, 1998). Explanations of assigned paper and pencil problems were part of their assessment. The interviews were informal to put the student at ease. Sometimes I would just sit and listen to Ann reflect on her motivations, general concerns about her future career, or immediate concerns only vaguely related to mathematics and NMI use. Ed on the other hand was strictly business, and a very quiet individual in the research setting. It was a challenge at times to get him to talk. Student perturbations. I paid careful attention to what the students were perturbed over concerning their understanding of the mathematical content. The students were scaffolded as to how the NMI might help them reason through difficulties. Methods for gaining insight into their perturbations included (a) listening to their thinking aloud as we work together, (b) engaging in erotetic dialogue, and (c) intermittently interviewing them.
84 Working independently of NMI use. I purposefully negotiated particular ongoing mathematical tasks to be worked on their own with or without the NMI, but that lent themselves to NMI use. Observing their NMI use and mathematical progress as described above facilitates this procedure. The assignments were negotiated according to our mutual interests. This proved to be the perhaps three quarters of the time we spent working on mathematics together. Procedures for the general scaffolding of activities. I refrained from telling the student’s procedures or methods of problem solution. Instead, I scaffolded them with leading questions, leading them into their ZPD. The discussions above on erotetic scaffolding dialogue explain this procedure in detail. I scaffolded students in their work in such a way that they might find ways to make use of the NMI in order to answer their own questions. I encouraged explicit use of the NMI as a reference in order to ask and answer their own questions, including the one’s they had trouble articulating. Finally, I found that students could use the NMI as an intermediary in implicitly approaching their questions—directed to me or to the NMI— ultimately stimulating their own puzzling out the mathematics involved. These things I tried to employ in a disciplined, procedural fashion wherever applicable throughout the research. Encourage explicit mathematical dialogue. We discussed the student’s mathematical progress. They explained much of their work, and perturbations. I encouraged their responses to be mathematically explicit. I often used the phrase, “I am not quite sure what you precisely mean about this or that. Can you explain it a little further?” I encouraged the student to use verbal, or pencil and paper presentations/derivations of their understanding and work. Assign related problems to be worked by hand. I assessed the student’s understanding by giving them related problems to solve and then explain. We discussed the kinds of reasoning they used in their manual problem solving, and explored its possible relation to their NMI use. The great majority of the direct mathematical work done by the students was by hand. This was by their choice, and it seemed to be the best media for their deepest reasoning and learning. The computer more or less served as a supplementary tool to be used interspersed with a lot of manual work. However, the times
85 the NMI was used, it made a huge difference in how we were able to communicate, and how they could find flaws in their own thinking. As I found manual work to be a natural means for the students to reason through things at a deeper level, I encouraged this as part of the research methodology. Assign simpler related NMI problems. The students were mostly working on larger project oriented investigations. However, along the way I assigned them simpler related tasks which necessitated either some kind of NMI use that helped them see things in a simpler light, or facilitated them to break their larger project into manageable tasks. Encourage applications oriented analogical creativity. The student was sometimes asked to contrive imagined or real world analogs, or applications for aspects of the mathematics they were studying. If the student could not imagine a real-world analogy, then a partially or purely made-up one was fine. This provided a way to (a) prompt students to abduce and attempt abstract visualization, as well as (b) motivate the students, encouraging them to think how the mathematics they were learning might be used. This became part of my procedural toolbox as it became apparent that both Ed and Ann sometimes were wondering what the mathematics they were doing might have to do with real-world situations. Although they both expressed—just a few times—great fulfillment concerning their mathematical understanding (Ed more implicitly, Ann more explicitly), they very often had questions about mathematics application concerning their project. The NMI added an even greater sort of mystique or opaqueness about what the mathematics might be applied to—primarily it seemed because the machine was so abstractly powerful compared to general human reasoning facility. When the students grasped analogs of their projects, even if they were only contrived, they were better able to make sense out the complex NMI contraption before them (analogs such as scientific, business, artistic, fanciful, or in combination). Reasoning prompts. Throughout the procedures described above concerning the student’s mathematical work, I subtly suggested and probed ways of encouraging various kinds of reasoning techniques in association with their NMI use. I used a technique of asking them questions to lead them to their own understanding. Perturbations were prime opportunities for this activity.
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Data Recording Procedures Following are particular data gathering techniques. They are independent of the student’s mathematical project work. Field Notes Gay (1996) discusses how the immediate taking of detailed notes on research episodes facilitates the capturing of salient features while they are fresh in one’s memory. The field notes were used to richly describe relevant contextual information. They also included dates of the research episode, and indexing to other data gathering media such as audio tapes or student generated computer files. Audio Taping Audio taping is used in order to capture details that might be lost in field notes (Gay, 1996; Patton, 1987). As they are indexed to field notes, the particular parts which may be necessary to clarify certain conversations, or to use as quotes in the research analysis are readily available. Only short, specially selected transcriptions of the tapes were used. Computer Screen Shots As the case study progressed, students were to build upon their work using the NMI. Screen shots of various stages of their work were to be used in analyzing their reasoning.
Validation of Study Qualitative research does not make general or universal claims of validity. This research has internal validity based on the triangulation of several viewpoints and contexts. Triangulation here specifically meant that I considered the research data gathering, case study reporting and reflections, each from several somewhat independent perspectives in order to see if there is an internal consistency of findings among these various contexts. Furthermore, there was a disciplining of the case study reporting and reflection upon the research findings from two independent theoretical perspectives, (i) Peirce’s theory of pragmatic reasoning, and (ii) Tall’s theory of mathematics from the perspective
87 of it being learned in a natural way. This facilitated a validation in the sense that the research was constructed in a fashion such that it reflected other person’s critically considered theoretical perspectives beyond my own means of ad hoc description. The final validation criterion involved the research subject’s reflections on their own learning and reasoning as regarded their NMI use. My interpretations of their activities were member-checked through regular discussions with the research subjects.
88 CHAPTER IV EMERGENT RESULTS TABULATED FOR SUBSEQUENT REFERENCE
Introduction The purpose of this chapter is to present terminology found useful in describing my observations of student reasoning acts using NMI. The table contains the researcher’s interpreted results of the empirical case study. Separated out is contextual terminology interpreted through case study relevant reflection of the literature review. These reflective interpretations were made to discipline, or ground the study in relevant theoretical terms. The chapter is centered around Table 1 (Table 2 is its index). Motivating the sense of the table, as regards the research question, is Peirce’s pragmatist descriptions of reasoning. Constructivism taken as a learning theory is also a central orientation. Finally, three conventional epistemologically oriented terms ( numbers 38, 39, and 40) are described in cogent, fruitful, student centered contexts. These ideas are re-iterated below, emphasizing the centrality of the organization and information in this chapter.
Discussion of Case Study Relevant Terminology This section brings together descriptions of operational terminology for the discussion of case study subjects’ pragmatic reasoning using NMI as a tool for learning mathematics. Many of the terms below come from Peirce’s work on pragmatic reasoning and have been previously elaborated in Chapter 3. Some have been created by the researcher and are marked with asterisks. The researcher calls the terms (beyond number 2) to be ‘reasoning acts.’ This delineates their intended relationship to practical matters, or actions (e.g., to pragmatically reason, “guess and check,” or estimate, or aspire to an objective point of view) rather than theoretical (e.g., the theory of knowledge or epistemology (Blackburn, 1996)). A separate table of alphabetical cross-referencing to Table 1 is provided in Table 2.
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Research Results: an Emergent Tabulation Table 1 presents emergent operational terminology, wholly interpreted by the researcher. These came to be used in the manner described during the case study, and subsequent reflection on the research question. They consist of interpretations of what I call reasoning acts of the students. Terms wholly created by the researcher are marked with a single asterisk. These emerged through consideration of empirical observation of the students in the context of the literature reviewed. Table 1 is resultant from the empirical and theoretically study, and considered to be central results of the dissertation. Terms marked with a double asterisk were interpreted from Peirce’s theory, and those with no asterisk were interpreted from general consideration covered in the review of the literature. All descriptions in Table 1 emerged from ongoing researcher interpretation. As mentioned, they appear here in the fourth chapter so that they may be referred to during the reading of the subsequent work. Table 1 of the emergent results was intended to be placed at the start. It facilitates a referencing back to it as the subsequent work progresses in the following chapters V through IX. It also allows the reader to get an immediate scope of the subsequent work.
TABLE 1. Emergent Operational Terminology Term Operational Description
Kinds of Graphics-User-Interfaces
1) NMI* Natural Mathematics Interface > a person-computer interface organized around, and facilitating user interaction with natural mathematics representations (see Table 2). > interfaces differ only in their chosen natural mathematics content orientations, i.e., they are implementationally standardized >facilitates use of conventional and explicit mathematical terminology, symbols and diagrams typical of using a paper and pencil—however—on user command, can be theoretically compacted, simplified, or expanded in representation, e.g., expanded graphing capabilities or compacting of notation. > typically outruns a students’ cognition at time of use in its subject matter > descriptions listed above are ideal, where the MUI described just below list practical limitations
90 TABLE 1. Continued 2) MUI* Mathematics User Interface. > a person-computer interface which facilitates certain state of the art representations of natural mathematics, but is necessarily limited by the computer implementation details at the time. > interface procedures may be nonstandard from interface to interface (e.g., use of interface dependent command language syntax). See Interface- procedural Embodiment (IPE). >attempts to facilitate use of conventional mathematical terminology, symbols and diagrams, but may involve some artificiality interfering with the typical compressions and expansions indicative of a natural mathematical form (i.e., typical paper and pencil form). May suffer from implicit assumptions about user’s mathematical competence. > may not outrun students’ cognition—may be designed mathematically narrowly for certain particular applications (e.g., computer simulations, or a pedagogically circumscribed lessons). > Some MUI properties will always exist to some extent in NMI due to implementation details (NMI is an ideal).
Peirce’s divisions of pragmatic63 reasoning
63 Scruton (1994, p. 104-105) said he went out on a limb to make the succinct description: “Pragmatism is the view that ‘true’ means useful. A useful belief is one that gives me the best handle on the world: the belief which, when acted upon, holds out the greatest prospects of success… “ Rorty says here, “For pragmatists, the desire for objectivity is not the desire to escape the limitations of one’s community, but simply the desire for as much intersubjectivity as possible…[italics added]” Scruton goes on discussing seminal pragmatists, “Simple definitions of truth in terms of utility seem transparently absurd. More complex positions tend to become indistinguishable from the coherence theory (Quine) or the correspondence theory (Peirce)…” Scruton (1994, p. 105-106) says pragmatists say, “…’we’ are the test of truth, and final court of appeal in all our scientific judgments [italics added].” Scruton then goes on to ask, “On whom does the Onus lie. Does the pragmatist have to show how we can dispense with the idea of a [mind-independent] reality? Or does his opponent [e.g., Rorty] have to show that this idea is—well, true?” Scruton then explains Kant has argued in this matter, Scrutton believing Kant to have “the best word that has ever been uttered on this [topic of objectivity]… We cannot step outside our concepts so as to know the world ‘as it is in itself’, from no point of view. Nevertheless, our concepts are shaped by the belief that judgments are representations of reality: our concepts are concepts of objectivity, and apply to the realm of ‘objects’. Without that…we could not begin to think.” “We cannot attain God’s perspectiveless view…but the thought of it inhabits our procedures as a ‘regulative idea’, exhorting us always along the path of discovery [italics added].” (Scruton, 1994, 107-108).
91 TABLE 1. Continued 3) Perceptual An assertion, or circumstantiation of a perception. The initiation of any Judgment** new line of reasoning, as well as the origin of conceptions. The description here is directly derived from a reading of Peirce’s theory of pragmatic reasoning. One may think of it, in the most general sense, as an all of a sudden intellectual awakening of an idea. Mathematical embodiments are special cases of perceptual judgments. As this study is focused on mathematical reasoning, perceptual judgments and (mathematical) embodiments are, taken to be equivalent. An uncontrolled association (not an inference) to a mentalesque representation—an apprehension. Uncontrolled, as in an immediate reflection on a sensate experience. Likely dependent on present or past contexts, experiences, or knowledge. Thus the notion of a percept follows; it presents itself to one’s attentional focus analogous to how we see, or hear. Pierce specifically describes it as a point of uncontrolled intellectual action which cannot, at the time, be considered a bad judgment. Something inconceivable to deny (e.g., similar to sense-datum, first-hand experience, a feeling). He said such a self-declared percept is often difficult to distinguish from hypothesis formation (abduction). As an example, consider the case of physically seeing black, but suddenly, it happens to noisily fly away in all directions. I infer that it was a covey of black crows huddled directly in front of my face. This inference is “something whose truth can be questioned or even denied” (CP2, p. 230); to ask if I saw blackness is inconceivable in so far as its already been asserted. Likewise, I may infer a just cause to re-assess my apprehended perceptual judgment. Such perceptual judgments are immediately fed back upon through deliberation of arising inferences—they “fade” into one another. It follows then that it may be difficult to assert, or judge a percept, and not associate an immediately suggested inferential explanation (abduction), e.g., I see black, therefore: my eyes are closed, its dark, I’m in a box, I’m surrounded by something black, etc. Consider the learning of addition through suddenly perceiving and judging there to be two groups of apples (the student counting four and seven). The student then counts eleven in all. Suddenly, in an indistinguishable moment (a “fading”), 64 the student infers the hypothesis of a general concept—all groupings of numbered objects “add up.” This generalizing action also serves as an example for “precissive abstraction,” number 15 below. Operationally, perceptual judgments, and the special case of mathematical embodiments, are interpreted to involve abstraction.
64 There are highly specific explanations on this process of learning addition by mathematics educational theorists such as Leslie Steffe. Such detailed analyses are left out here as Peirce’s pragmatism describes a theory of reasoning in general. This is further discussed in the “Future Research” section of Chapter XI.
92 TABLE 1. Continued 4) Abductive Forming a hypothesis, conjecturing, or construing an explanation which Inference** parsimoniously covers the facts at hand. An inference that “fades” into perceptual judgment; its purest limit. Occurs during mathematical exploration. Examples are metaphor, and holistic analogy (rather than an inductive analogy over several successive, related properties).
5) Inductive Verifying a generalization, law, or rule concerning certain facts or Inference** properties. Inference led by a hypothesized pattern, relationship, or theory, so as to make an inference to a general rule or principle. Moving from specifics to a generalization. May involve consideration of accruing information in deciding if a hypothesis is plausible. Examples are experiment, checking one’s guess, and analogizing over a number of common properties.
6) Deductive Purely logical reasoning. Inferring conclusions following from fixed Inference** premises justifiable by rules of a certain logic. Nothing is in a deduced conclusion that was not first in the premises. Compare to induction, which generates, rather than follows rules. Deduction progresses from a rule and presented results (premises) to a particular inferred case. Comparatively, induction moves from a result, to several related cases, finally inferring a plausible rule. Finally, abduction moves from a known rule, and observed case, which leads to the inference (conjecture) of a plausible result. Note then that abduction can lead to an initiating result for an induction, which may provide a general rule for initiating a deduction. Mathematical in several senses: a) often yielding to theoretical representation such as symbolization, b) truth-bearing (e.g., consistent), c) having formalizations such as, i) propositional logic, ii) first-order predicate logic (exhibiting quantification), iii) modal logic of necessity and possibility (McCarthy, 1996), iv) multi-valued logic (e.g., truth, falsity, and neutrality).
7) Colligation/ Colligation means the bringing together of one’s thoughts in new Copula** relation to one another, or in new combination, with the intention of drawing together an attentional focus in a deliberate, or controlled manner. Colligation is a deliberation having the purpose of gathering the premises of one’s intended inference to be under consideration at once. Peirce thought this to be an activity that may employ significant creative genius. This gathering of ideas forms the noun substantive Peirce calls a “copula.” It is a conjunctive proposition. The copula is the interconnected ideas, assumptions, previous inferential results, or perceptual judgments which are brought into awareness when making an inferential step. Colligating can be seen as intellectually making connections.
Higher level descriptions of pragmatic reasoning
93 TABLE 1. Continued 8) Pragmatic Scruton (1994, p. 104-105) wrote, “Pragmatism is the view that ‘true’ reasoning** means useful. A useful belief is one that gives me the best handle on the world: the belief which, when acted upon, holds out the greatest prospects of success…Practical, and purposeful reasoning.” Inferring to, depending upon, relevant to; situational or intentional contexts. Action oriented as concerning: real-world implications, conceivable sensible effects, or intentional goals. Pragmatic reasoning includes both plausible and deductive inference. Plausible reasoning includes both inductive and abductive inference. Discussion in the context of plausible reasoning is useful in mathematics learning as it provides a middle ground between procedurally oriented deductive inference, and the slipperiness of the plethora of psychological explanations.
9) Multiple-chain Developing multiple step arguments from a fixed, minimal pool of reasoning*65 elements (a copula) by logical or plausible inference. These developments of arguments, when fully deductive, often concern intuitively obvious conclusions deduced by in-obvious arguments. Often one employs middle- out reasoning (see below) to connect links of an inference chain. The process of multiple-chaining typically involves holding in mind several completed links, while examining a potential link.66 Good examples are the reasoning acts of doing proofs, or, mathematical derivations. Multiple-chain reasoning is what is typically thought of as the reasoning necessary for doing mathematical proof. It involves making strictly logical connections (deductions) without a dependence on any plausible inference (e.g., analogy).
65 This term, and the next, “middle-out reasoning,” was derived from private conversation with Janice Flake. 66 It is interesting to note that any plausible link in the chain may involve many justificatory remarks for each plausible step emerging from the simple question, “But why is that?” Deductive multiple-chain reasoning does not suffer this hazy regress as all statements are well defined from the start, or taken to be true or false, period.
94 TABLE 1. Continued 10) Middle-out Working backwards from things one knows to what one is trying to reasoning* find. For example, proving a geometry theorem by writing down most everything one can justify, and then a) going back and ordering a chain of reasoning from what has been justified to deductively draw the necessary conclusion, and b) discard all the extra information that was not needed. Another example is solving a problem or proof in a plausible way by making assumptions that one cannot yet justify, but that one has confidence. The term intuition, in pragmatic reasoning terms means a) an abductive inference, or b) a perceptual judgment (i.e., a) an educated guess, or b) an immediate insight into a special case, or an immediate overly general insight concerning the problem at hand). A justification for middle-out reasoning in this case is that if a person were to pause to justify each step of their intuitions, they may lose their own individual or personal train of thought. Working outwards from what a particular student finds familiar, or feels certain of. This is opposed to having a student draw mathematical conclusions or theorem proofs derived from minimal first principles (e.g., typical of a geometry course which starts from Euclid’s axioms). Working in a sense backwards from more well known mathematical object properties to more sparsely propertied, or, “closer to the postulates” first principles. Working to solve a problem (e.g., write a proof) by building on what one is already familiar—rather than deducing particulars from given premises which are closer to minimal assumptions. One can describe middle-out reasoning in plausible reasoning terms. Middle-out reasoning is comparable to non-demonstrable reasoning. Non-demonstrable reasoning means “a [mesh] of ‘intuitive’ comparisons of ideas, whereby a principle or maxim can be established by reason alone (Blackburn, 1996, p. 98).” A distinction of middle-out from multiple-chain reasoning is one of personally idiosyncratic association of potential inferences. Examples are devising a related but simpler problem (Polya, 1954), or triangulating from contexts. An example of middle-out reasoning from a different perspective is working one’s inferences backwards or outwards from what one is given (but has not yet understood) to what one feels comfortable to be taken as a first principle (e.g., reverse-engineering).67 Here, one picks up their reasoning from a mathematical fact one is told, or given, but, which is unfamiliar. From here, students find “nearby” mathematical knowns in order to (eventually) make logical (multiple-chain) connection to a given.
67 The terms ‘multiple-chain reasoning’ and ‘middle-out reasoning’ were thoughts conveyed to me in private conversation by Janice Flake. These terms, in the context of writing computer programs for teaching and learning, are discussed in Flake, McClintock, & Turner (1990).
95 TABLE 1. Continued 11) Triangulation Reasoning from varied contexts, or perspectives relevant to a situation. NMI facilitates this as it helps students rapidly attempt differing solution approaches to be compared; as well as a single solution approach triangulated with something completely different (e.g., an interactive graphic display of a computer simulation). 12) Framing Regarding what can be taken to be implicit given certain contexts. Orienting oneself to a fresh mathematical situation by drawing on previous related knowledge not necessarily in one’s attentional focus (i.e., tacit or implicit knowledge). NMI facilitates the provision of many fresh mathematical contexts. 13) “Guess and Same as abduction and induction, respectively. Making a plausible check” inference (e.g., the compromise of an estimation) and verifying that inference (e.g., by triangulating). This phrase was coined by Polya (1957).
Peirce’s descriptions of abstraction
14) Dissociative Separating things known only at a pictorial, or more generally, a Abstraction** heavily memory dependent level (e.g., algorithmic knowledge). E.g., “Find all the triangles among these shapes (where all the shapes are well- defined, and discretely different from one another, i.e., circles, squares, etc.).” Separation of things typically seen to be associated, e.g., a sand dune from a beach or the plastic “control” (Cntrl) key ‘dissociated’ (disassociated) from a computer keypad. As regards NMI use, there is a needed dissociation of keyboard commands from the pictures it might make on the display (typical of a geometry program). Also useful kind of abstraction for describing the separation of one algorithmic sequence of buttons pushed on a computer with another—procedural activity--done at a purely memorization (rather than reasoning) level. It is the weakest or simplest form of abstraction. Typically involves abstracting (separating) one thing or object from another—meaning physical or real objects as recognized in pictures (e.g., icons), or, things such as memorized sequences or procedures.
96 TABLE 1. Continued 15) Precissive Separating away a “picture-likeness” (memory intensive clutter) Abstraction** towards a precise abstraction (having simple clarity and definiteness, allowing generalization due to its minimality). A mentalesque representation which was related before to a “picture-or-memory” (context cluttered) situation. Moving from real things to representations related to those things (e.g., forming mathematical embodiments—4 apples and three, and then suddenly a mentalesque representation of addition). Creating, or to a lesser degree, recognizing, a representation (more technically speaking, a construction, see 19 below), such as separating object from ground (i.e., separating out some representation of a thing from what is taken to be its background). E.g., “I claim these several somewhat straight curves made in the sand to comprise four triangles,” or, “I see these curves in the sand to represent a mountain chalet,” or, “I mathematically interpret this sine wave to be a projection of this revolving propeller tip (see 23 below).” One might analogize here, in the context of NMI use, the following of a rule and the finding of a rule— both are making precissive abstractions—one more creative, while the other more recognition and selection (see 30 and 31 below). Peirce describes precissive abstraction as paying attention to one property or action while intentionally neglecting others. This kind of abstraction is highly typical of mathematics. For example, abstracting ‘counting’ from an abductive explanation involving putting one apple after another in a line (see 4). Here, one perceptually judges what it is to count (see 33). How one might conceptualize a situation to be mathematical (i.e., separation of a mathematical representation (construction) from this or that situation). See Chapter IX, Conclusion for discussion on constructs vs. constructions vs. representations. The final example considered in describing perceptual judgment in the process of learning addition, of number 3 above, provides an example of precissive abstraction. Apples here are merely icons or concrete “pictures.” They are not representative of the construct (a “representation” in Peirce’s pre-mathematics education detailed theory, e.g., Janvier (1987)) of “number” until this is perceptually judged, hypothesized (abduced), and perhaps inductively verified. This move from dissociatively abstracted icons (the abstraction of grouping here—a separation of “pictures” of individual apples, to “pictures” of groups of apples) to mathematical constructs (or representations to one’s self) is a precissive abstraction. The concept of number is constructed by the student (this is related to what is described in footnote 62 concerning Steffe’s notions of learning addition).
97 TABLE 1. Continued 16) Discriminatory Discerning, or separating representations via other representations. Abstraction** Working with meta-representations (i.e., representations (constructs) of representations (constructions). Typically thought of in the context of linguistic semantic distinctions, however, mathematics involves this type of abstraction to a very large degree. Astute, judicious, or perceptive noticing of the shades of meaning in mathematical language. Separating one representation from another. E.g., discerning numbers from algebraic variables. The strongest form of abstraction, in that it can do anything the other two can. See Chapter IX, Conclusions concerning constructs, constructions, and representations, also 19, 20, and 25 below). An example here is found at the end of numbers 15 and 3 above— concerning precissive abstraction and perceptual judgment). When the student, upon learning addition with the apples, makes the abstraction from numbers to addition, a construct of a construct (or internal representation of an internal representation) is formed—resulting in the knowledge of the concept of addition.
General pedagogy
17) Social A learning situation where students are guided by a social situation of construct instruction that helps them develop their own constructs. This instruction scaffolding is typically done by teachers, or through social interaction, however, NMI interaction provides another avenue for students to seek a kind of dialogical interaction while learning. 18) Quasi- Thought experiment involving plausible reasoning. Doing empiricism mathematics using a kind of scientific method, where one hypothesizes and tests one’s ideas. In the case of NMI, this is further facilitated by the externalizing of previously mind-dependent objects as in the facilitation of an interaction with the NMI concerning one’s plausible reasoning. 19) Constructions External manipulatives (e.g., interactive computer diagrams) capturing or embodying certain mathematical ideas. Interactivity is key, in that it presents a perspective of an external manipulative to be, in a sense, mind-dependent. The constructions may or may not be mathematical embodiments for different students, or the same student at different times. This term is difficult to describe as it attempts to be a bridge that spans the internal and the external—constructs and representations (see numbers 20 and 25, also, Papert (1980)). The term ‘embodiment’ also captures this notion of such a bridge, but ‘construction’ further focuses the notion of an embodiment to be something the student can interactively create using virtual manipulatives.
98 TABLE 1. Continued 20) Constructs Mentalesque, or mind-dependent mathematical conceptualizations and meanings. Emerge from sense-making activities concerning mathematics problems, activities, or embodiments; especially contrasted with memorization of procedures and rules. Mathematical objects and mathematical constructs are used interchangeably here. One may say that ‘constructs’ are in the eye of the beholder, but are--intersubjectively speaking--a taken-as-shared meaning. 21) Constructivism An educational learning theoretic stance that knowledge is a taken-as- shared set of meanings among groups of people. Distinguished from a notion that there exists some fixed, mathematical knowledge existing outside a person’s mind. Mathematics learning is seen as a negotiation of meaning—a sense making activity between people--rather than as if someone merely transferred ‘The knowledge’ to a learner (e.g., colloquially; pouring ‘It’ from one head into another). Also seen as a fully developed philosophical stance rather than merely a learning theoretic stance. The latter position is taken in this dissertation.
Applied Mathematics
22) Modeling Abstracting a mathematical representation, pattern, or relationship holding of a situation to some level of generality. 23) Interpretation Considering a mathematical model in some particular case. Concretizing a model.68 24) Estimation A rough idea of what a calculational result should be. Choosing apt simplificational limits on one’s mathematical considerations. Making compromises in the accuracy of a calculation to draw plausible inferences, e.g., determining a bound on a solution.
NMI relevant distinctions
25) Representation Mathematical, mind-independent displayed graphics (e.g., diagrams), numerics (e.g., data) or symbols (e.g., logographics) typical of NMI use. Not a mentalesque construct or abstraction. See pp. 21, 22, 60, 65, 67, 72, and 77 concerning Thompson & Sfard (1994) for elaboration. Peirce’s description of representation concerning reasoning conflates construct, construction, and representation. However, it is typically clear from the context in which Peirce uses the term ‘representation’ whether he is referring to mentalesque situations, or mind-independent. Janvier (1987), and Stenning (2002) tend to write about the external representation, not that internal constructs are any less irrelevant, but that research is more easily done in reference, or anchored, to mind-independent situations.
68 This distinction between modeling and interpreting was pointed out to me in private conversation with Benjamin Fusaro.
99 TABLE 1. Continued 26) NMI Actions involved in forming an abductive hypothesis while reasoning exploration through an exploration of a mind-independent NMI construction, guided by mind-dependent interaction. May involve colligating perceptual judgments perceived through NMI interaction, which, may, then “fade into” abductive reasoning. 27) NMI Typically an inductive, verificational action involving a quasi-empirical experimentation interaction with the fully empirical data provided by an NMI. Testing hypotheses, or verifying certain plausible inferences via interaction with NMI. 28) Implicit A media which facilitates a knowledgeable tacit assumption representation concerning a mathematical model, interpretation, or context. For example, such a representation can “bring to mind” an implicit assertion of a mathematical embodiment via interaction with NMI (see “Implicit cognition” just below). 29) Implicit Involving thinking or reasoning which depends on the use of cognition knowledge that is not in the conscious awareness of the agent at the time of its use. Related to perceptual judgments which depend in large part on tacit knowledge—their intellectual actions are not consciously controlled, but yet are knowledgably informed. NMI allows implicit communication between students and instructors, i.e., the NMI serves as a mathematical mediator. One might colloquially say that teacher and student may “point and grunt,” as is quite commonly done in using blackboards. 30) Finding a Conjecturing (abducing) the usefulness of an algorithm. The Procedure algorithm might be created, or simply selected. 31) Following a Carrying out an algorithm or rule-following. Procedure 32) Diagram** Generally, a drawing, construction, or graphic with mathematical significance, e.g., geometric figures, a function drawn as a box with inputs as the domain and outputs as the range, a graph. Often interpretable as part of conventional mathematical canon.
Mathematics Learning and NMI Relevant Embodiments
100 TABLE 1. Continued 33) Mathematical A mind-independent situation having mind-dependent mathematical Embodiment, meaning. Something “out there” which indicatively helps one make sense of mathematical ideas. For example, manipulating groups of apples, and or, seeing them in light of number and arithmetic. Resulting from any kind of interactive activity (mentalesque or physical manipulation) which Perceptual facilitates one to make sense of mathematical ideas. Judgment** A perceptual judgment (author’s interpretation of Peirce), the initiation of (mathematical) reasoning. Although a perceptual judgment happens all at once as an apprehension, the activity that builds up to it can be a long process. This time lag is especially true of NMI use, as it can depend upon years of mathematics study beforehand. As the interaction with the NMI is with the mathematical symbols and diagrams themselves, depending on the student to come up with their own more abstract perceptual judgments, because they are not immediately presented to them in some concrete form. Typically involves looking down on a familiar situation, and taking it to be a mathematical heuristic—however, when using an NMI, much background mathematical knowledge is necessary for a student to do such “looking down.” 34) Interface- A procedural embodiment that requires much memorization, but procedural whose procedures may at least be logical. Its being an embodiment Embodiment depends on perceiving it as an overall pattern which may have some (IPE)* purpose to the perceiver. A procedure determined by the idiosyncrasies caused by the implementation details of NMI, but that has the potential for bearing some usefully perceived relationship to natural mathematics. For example, in learning the non-mathematical NMI procedures, there may come a perceptual judgment, where the student notes a particular pattern, or relationship. Here, they may begin to abduce these patterns or relationships to have mathematical import. The procedures of NMI implementation details (MUI procedural details) might later proceed to serve as an operational or instrumental embodiment for certain mathematics. IPE is the most indirect and artificial of the embodiments as concern mathematical content. Examples of mind-independent IPE characteristics are a) the menu layout of an interface in how it groups together mathematical operations, objects, or concepts, b) using syntactic commands for the naming of independent variables, or, number of rows in a matrix. IPE is analogous to memorizing a formula, and then later figuring out what it might mean. Interpreted here to be at a Peircean icon or pictorial level of abstraction. IPE’s are concrete, rather than being abstracted representations of something else. This is literal in the sense that the spatially laid out visually oriented tool, menu bar, and cursor pointers of windows environments were designed to be seen, from a practical standpoint, as objects in themselves (e.g., a printer picture on a tool bar).
101 TABLE 1. Continued 35) Generic A purely mathematical embodiment that mostly arises from previous Embodiment mathematics reasoning and learning. Same as a mathematical embodiment (GE)* (above), but, because of its association here with NMI use, it is interpreted more particularly from the Pragmatic Realist perspective (Rescher, 2005). Rescher (2005, p. 116) refers to a minimalistic or “generic” mathematical form; a “lowest species,” as being a GE. NMI elicits GE through quasi- empirical interaction with the NMI. GE elicits an implicit, tacit, or inarticulate knowledge of some natural mathematical situation. Seen as a lowest common mathematical form—a form which embodies the most cogent, fruitful pragmatic inferential connections within an overall mathematical scheme. 36) Natural A mathematical embodiment that is a result of intensive Mathematics computation. A natural mathematics calculational, diagramming, or Computational derivational use of NMI which can be looked down upon as a perceptual Embodiment judgment. Analogous to how calculator interactivity can be used in (NMCE)* learning (e.g., graphing, zooming, checking zero’s, etc). Also somewhat analogous to how hand calculations, done procedurally without meaning, can all of a sudden make mathematical sense. Facilitated by the immediacy of the feedback of the computational results—this computational feedback eliminating the tedium of calculations or the plotting of graphs. May arise through inductive preparation of working many problem examples—facilitated by NMI interaction--leading to a perceptual judgment of a perceived pattern in these types of problems. Subsequently, the student has opportunity to inductively reason, and abductively hypothesize (building on previous perceptual judgments), leading to the NMCE perceptual judgment. Newly gathered NMCE’s require substantial prior mathematical knowledge to guide one in their conjecturally, and inductively driven computations. NMI facilitates immediate interactive feedback associated with quasi-empirical exploration and experimentation enabling students to achieve perceptual judgments concerning higher levels of theoretical mathematics because of the computational support. A perceiving of calculational, diagrammatic, or derivational themes “end to end.”
102 TABLE 1. Continued 37) Applications A mathematical embodiment arising from considerations of a real- Embodiment world situation, or some application of mathematics as a tool. For (AE)* example, watching a weight bounce up and down on a spring, and perceptually judge the analogy to a trigonometric sinusoid. AE is a perceptually judged, end to end, colligation (a copula) for a reasoning backwards from some practical purpose (e.g., boot-strapping), via a mathematical model’s interpretation—thus facilitating opportunity for considering the actual mathematics of the model. AE, unlike the other embodiments is looked up to. It involves considering a mind-independent, real-world situation with which the student is familiar enough to be curious about, but which does not understand the underlying mechanics (ergo, it is looked up to). The Applications Embodiment is not seen to be reasonable, in the sense of it being no more than a perceptual judgment. AE is a concrete perceptual judgment concerning an attempted apprehension of a special case, or interpretation, of a mathematical model. AE involves previous reverse-engineering kinds of reasoning. AE involves apprehending certain mathematics only in so much as it can be analogized from a physical, scientific, engineering, etc., interpretation of a mathematical model. AE’s are typically derived from NMCE’s when working with an NMI. Using NMI to represent the model, one has a less tedious cognitive load to make the AE analogy. Lesh, Post, and Behr (1987, p. 675) state that “by reducing the conceptual energies devoted to "first-order thinking," higher-order "thinking about thinking" becomes possible. Otherwise, students frequently become so embroiled in "doing" a problem that they are unable to think about what they are doing and why.” Students are typically given applications problems to help them learn mathematics unfamiliar to them. The NMI is almost indispensable for anything but either very simple (e.g., toy), or highly abstracted (e.g., classical mechanics) applications problems. NMI facilitates longer term learning projects facilitating a deeper immersion in the mathematical theory behind the application at hand. Mathematical model interpretations, or particular real-world situations, have many handles, or footholds from which one may tether their mathematical considerations. The real, mind-independent anchors (the concrete AE) hold the tether fast. NMI’s themselves can provide an anchored tethering from the other side of the analogy; the mathematics to be learned. There is an implicit motivational factor as well with many real-world applications. Often, the mathematics learned from AE’s is an afterthought of a drive to intellectually penetrate how a real-world situation works. ‘Analogical’ here means abductive reasoning—but a reasoning attempting to reduce one’s abductive explanation of how something works into a (mathematically) compressed, but encompassing perceptual judgment. This encompassing is most commonly pertaining to a particula
103 TABLE 1. Continued model’s interpretation. Below is discussed an encompassment concerning, more generally, the mathematical model itself. There is a more advanced AE, one that if a student grasps a capacity to look down upon their mathematical model, they are able to broaden their non-general, one-to-one corresponding limitation of the analogy described (i.e., one interpretation of a many possible interpretation model, e.g., the differential equation having one interpretation being the oscillation of a weight on a spring can be further generalized). This kind of AE would seem to be more distinctive of NMI use, differing from the more common AE described in detail above. I believe students must move through the common AE described above before they can hope to grasp this second, far more general AE (a perceptual judgment concerning a general model, rather than a concrete interpretation of that model). The AE arises from concrete analogs (either on the applications, or the NMI side) rich in perceived contexts, and rich concepts, allowing many cognate’s to grasp and analogize to the mathematics (e.g., grasping a closed form model by considering a numerical representations of real-world data). AE is the most difficult use of an embodiment as it looks up to (instead of how all the other embodiments are looked down upon) to analogize two widely different situations, one a concrete or real analog, likely only plausibly understood, the other a logical mathematical situation not yet learned to the necessary theoretical depth. AE is a common way engineers and scientists might be seen to gain insight into mathematics they have applied but may have eluded their full theoretical grasp. Applications Embodiments are at the root of the reasoning involved in science and engineering. They are not reasoning per se, as they are a perceptual judgment—an assertion of a mentalesque perception. This applications embodiment may come about (take an implicit cognitive formation) via any of the three types of pragmatic reasoning: abductions, induction, or deduction.
104
TABLE 1. Continued 38) Objective A perspective taken as mind-independent, factual, or a best approximation to the facts. Typically thought of as viewing a situation in a detached, dispassionate, or disinterested way. “Of, relating to, or being an object, phenomenon, or condition in the realm of sensible experience independent of [an individual’s] thought…(Webster, 1998).” Rescher (2005) looks at the aspiring to an objective view as a “regulative idea” that is commonly shared. This idea of objectivity regulates how one may approach a certain view of the world. 39) Reality “That is real which has such and such characters [(properties)], whether anybody thinks it to have those characters or not” (EP2, p. 342). Something that is ‘real’ is mind-independent; not dependent upon an individual’s thought. The quality of being actual or true. ‘Reality’ outruns cognition, i.e., there are always aspects of real things that can never be known. 40) Truth Some common descriptions of truth are (a) an objective, mind- independent reality constituting what can be said to be ‘true’, (b) a “correspondence between the things that are true, and the things that make it true, [i.e.,] ‘to say of what is, that it is, and of what is not, that it is not, is to speak truly’[Aristotle] (Scruton, 1996, p. 99),” and (c) the coherence, harmony, or ‘hanging together’ of things; “each of its components supporting and supported by every other (Scruton, 1996, p. 102).” The pragmatic scientists’ truth is reasoned methodologically by: a) coming to a hypothesis often based in perceptual judgments, b) verifying the hypothesis by repeated experimental trials, and sometimes, c) modeling this plausible truth by deducing it from much simpler premises which are already accepted as true. Scruton (1996, p. 104) on pragmatism and truth most generally, wrote “…true means useful. A useful belief is one that gives me the best handle on the world: the belief which, when acted upon, holds out the greatest prospects of success.”
*Originally created terminology by the author to better explain student reasoning in this study.
**Terms interpreted from Peirce’s pragmatist reasoning theory.
105 Each of the forty terms enumerated in the table played some role in the student’s NMI use reasoning, and are discussed in various places in the case study, or reflection upon that study. Please note the asterisk explanations below which distinguish reasoning act observations original to this research. The lack of an asterisk denotes common terminology which though, required a narrowed re-interpretation to serve the specific case of the description of the case study students’ reasoning while specifically using the NMI.
TABLE 2. Alphabetized cross-reference to terms as numbered in Table 1. Term Number Abduction 4 Applications Embodiment (AE) 37 Colligation/Copula 7 Natural Mathematics Computational Embodiment (NMCE) 36 Constructions 19 Constructivism 21 Constructs 20 Deduction 6 Diagram 32 Discriminatory abstraction 16 Dissociative abstraction 14 Estimation 24 Finding a procedure 30 Following a procedure 31 Framing 12 “Guess and check” 13 Generic embodiment (GE) 35 Implicit cognition 29 Implicit representation 28 Induction 5
106 TABLE 2. Continued Interface-procedural Embodiment (IPE) 34 Interpretation 23 Mathematical embodiment 33 Middle-out reasoning 10 Modeling 22 MUI (Mathematics User Interface) 2 Multiple-chain reasoning 9 NMI (Natural Mathematics Interface) 1 NMI experimentation 27 NMI exploration 26 Objectivity 38 Perceptual judgment 33 Pragmatic reasoning 8 Precissive abstraction 15 Quasi-empiricism 18 Reality 39 Representation 25 Social constructivism 17 Triangulation 11 Truth 40
107 CHAPTER V THEORETICAL CONSIDERATION OF THE EMERGENT RESULTS
Introduction Explanations concerning Table 1 in Chapter IV are presented here. Some new terminology presented in Table 3 concerns a clear description of what is meant by “natural mathematics.” The chapter focuses on supporting several key theoretical ideas which motivate the tabulated theoretical terms used in the presentation of the case study and its reflective interpretations. The ‘Introduction’ section to Chapter VII might be of interest for putting this specific chapter into a broader perspective.
Mathematical Embodiments Although the term ‘mathematical embodiment’ has taken on many meanings (e.g., psychological, emotional) I would like to present a description made by Lesh, Post and Behr (1987) prior to these other interpretations; directly of Dienes (1961) notion of a mathematical embodiment.
“Part of what Dienes meant when he claimed that mathematical ideas must be constructed is that when arithmetic blocks, for example, are used to teach the structure of our numeration system, the relevant systems must first be "read into" the set of materials in the form of concrete activities. Only after this "reading in" has taken place can the structure be "read out" as abstract relational/ operational networks [e.g., concept maps].” (Lesh, Post, and Behr, p. 650)
Lesh, Post and Behr (1987) emphasize a movement from perceived actions to asserted abstractions as capturing the notion of mathematical embodiment. They go on,
…the mathematical structure is pictured not simply as a network of relations and operations; according to Dienes, it is the pattern itself (and properties of the pattern) that must be abstracted. So, in activities with blocks, it is not so much the blocks that the child must organize as it is his or her own activities on the blocks;
108 and it is the pattern of the activities that forms the basis for the abstraction. When individual activities cease to be treated as isolated actions and start to be treated as part of a systematic pattern of activities, the student begins to shift from playing with blocks to playing with mathematical structures. (Lesh, Post, and Behr, p. 649-650)
I interpret “the mathematical structure…pictured not simply as a network of relations and operations…is the pattern itself (and properties of the pattern) that must be abstracted” to be what is entailed in the judgment of a perception. Therefore, I purport that the identifying of a perceptual judgment with a mathematical embodiment is part and parcel to Diene’s (1961) original use of the term.
Finally one might argue that embodiments (purported by Deines above), being a basis of abstraction, cannot be the result of abstraction. This is a fine point. My observations in the case study below found that certain mathematical embodiments could in fact be the result of abstractions. This was due to the fact that NMI presents already abstracted representations to students, i.e., computer presented natural mathematical representations. It was observed that because the students had such readily access to interaction with these external abstract representations, they were able to take them as concrete embodiments, relative to the further abstraction and generalization concomitant to the reasoning through of mathematics, or, the making sense of mathematics, using the terms of constructivist theorists (Table 1, 21), e.g., Steffe & Olive (1995).
Theoretical Considerations of Natural Mathematics and NMI Tall (2002, 2004, 2005a, 2005b) has described a way of looking at mathematics which is grounded in the psychology of the student. He calls his theory “The Three Worlds of Mathematics (Tall 2005a, 2005b).” Tall saw such things as mathematical abstraction, symbolizing, and formalizing as natural occurrences of disciplined thinking and learning. Tall’s work was developed for mathematics education use, and is used as an anchor to provide a concrete meaning for the term ‘natural’ in my description of a Natural Mathematics Interface (NMI).
109 Tall (2005b) describes a fundamental, all encompassing, and underlying, first world of mathematics he calls a “conceptual-embodiment.” Tall (2004, p. 3) discusses an “object-based conceptual-embodied first world reflecting on the senses to observe, describe, define and later, deduce properties developing from thought experiment to Euclidean proof and beyond. Later, in Tall (2005b, p. 2), he writes this first world “refers to conceptual embodiment in which we reflect on our sensory perceptions and imagine relationships through thought experiment.” This reflects how I understand the general term “mathematical embodiment” described in table 1. Tall (2004, p. 3) describes a second world as an “action-based proceptual- symbolic world that compresses action-schemas into thinkable concepts operating dually as process and concept (procept).” Tall (2005b, p. 2) further elaborates,
Symbolism here [in the second world] refers not to symbols in general, but to those symbols used in mathematical calculation and manipulation in arithmetic, algebra, and subsequent developments. These arise through actions on objects (such as counting) symbolized and manipulated as concepts (such as number). A symbol used dually to represent process (such as addition) and concept (such as sum) is called a procept… (Tall, 2005b, p. 2)
Tall (2004, p. 3) then discusses a third world of mathematics as, “a property-based formal-axiomatic world of concepts definitions and set-theoretic proof.” Tall (2005b, p. 2) writes again about this world, “Formalism refers to the formal theory defining mathematical concepts as axiomatic structures whose properties are deduced by formal proof.” Tall (2005b) goes on to make clear that his theory, which concerns itself with education and learning, centers on a natural agency of reasoning, rather than viewing mathematics as strictly a formal systematization.
110 TABLE 3. Tall’s Natural Mathematics Theory. Operationally descriptive summaries of Tall’s (2002, 2004, 2005a, 2005b) three components of natural mathematics, and a resultant interpretation the researcher takes to be “natural mathematics.” Aspects and Interpretations of Tall’s description of natural mathematics Conceptual- Object-based. Underpins all mathematics. A reflection on our embodiment sensory perceptions in order to imagine abstracted mathematical patterns and relationships through thought experiment. An example activity is visualization. NMI is particularly distinguished by offering sophisticated embodiments that require higher levels of implicit mathematical knowledge. However, the interactivity of the NMI balances this fact. Proceptual Action-based. ‘Procept’ is a concatenation of the terms process -symbolic and concept. Concepts are taken as mentalesque mathematical objects, these objects serving as an indexical69 compression of certain mathematical operations or actions. These objects are typically represented by mathematical symbols and diagrams. An example activity is the process of calculation concerning mathematical concepts, or the plotting (action of substitution) of a function (object). NMI is particularly distinguished by offering higher levels of interactive proceptual mathematics. Formal- Property-based. A “Formalis[ation] refer[ring] to the formal theory axiomatic defining mathematical concepts as axiomatic structures whose properties are deduced by formal proof (Tall, 2005b, p. 2).” Intellectually working with hypothetical postulations serving as premises for a strict logical (or formal) deduction of relevant mathematical objects, actions, and their properties. An example is a formal proof. The interactivity of the NMI helps with conjectural and inductive aspects of finding plausible paths to multiple-chain deductive demonstration (i.e., middle-out reasoning). NMI can be instructive concerning learning needed implicit and background knowledge to do a proof, as well as motivating in that one can interactively “guess and check” relevant cases of a general theorem (i.e., middle-out reasoning). Because deductive inference is the least ‘natural’ of mathematical reasoning, it is difficult to formulate computer programs that would take what could be judged to be the simplest, or directly comprehensible inferential path that would be ‘naturally’ communicated person to person. There is a cogent, fruitful choice of inferential paths in the third world. Fruits such as aesthetic beauty or cognitive handles are difficult to merely computationally manufacture. In summation, that which is pragmatic to actual human cognition is difficult to program into what would be an NMI as regards its capacity to provide demonstrations of what would be seen as ‘natural’ deductive proofs. Distinct from Artificial Intelligence (AI) computing, NMI explicitly purports to require as integral, interactive control by users.
69 Indexing here is the researcher’s common sense interpretation of Tall and relates to Peirce’s semiotics.
111 TABLE 3. Continued Natural Psychologically under-pinned by mathematical embodiments,70 and Mathematics characterized by three components: embodiment-based, procept-based, * and formal property-based. The researcher directly interprets Tall’s three worlds to be descriptions of “natural mathematics.” “Natural” means in most part, to be of a psychological basis, while “pragmatic reasoning” stands against this as something supervenient upon a “natural” description. Tall means to describe his third world to pivotally concern an agent’s deductive reasoning, as opposed to nothing more than computationally arbitrary formal systematizations. 71 Therefore, Tall’s Three Worlds of Mathematics has been considered an apt description of “natural mathematics,” as intended in the term NMI.
*Researcher’s terminology.
Theoretical Connection of Conceptual-Embodiment and Perceptual Judgment Tall’s theory is firmly intended to have roots in both a psychological and educational context. I have interpreted Peirce’s notion of a perceptual judgment to be a generalization of what Tall calls a mathematical conceptual-embodiment, within the framework of pragmatic reasoning. This is justifiable in at least two ways. Firstly, Deines (1961) justifies consideration of the term ‘mathematical embodiment’, under my interpretation, as purporting a process which initiates mathematical reasoning, albeit supervenient upon a psychologized use of the term. Backburn (1996) gives the example of biological properties purportedly supervening on chemical ones. Blackburn (1996, p. 368) goes on concerning the term, that “by its means we can understand the relation of such different layers of description without attempting a reduction of one area to the other.” Secondly, Tall has explicitly termed a ‘conceptual-embodiment’--an expression, under my interpretation, explicating the reasoning through of a concept. I see Peirce’s theory to firmly parallel Tall’s, when ‘conceptual-embodiment’ is taken as a special case of ‘perceptual judgment’—the initiation of reasoning (see Table 1, 3 and 33). Behind every good quality mathematical explanation lays a clear concept, i.e., good quality mathematical abductions require clear conceptualizations. And in so far as
70 Rescher’s (2005) description of cognition outrunning reality can be seen as analogous to how an embodiment, or perceptual judgment indicates hypothesized directions for inquiry with high densities of new information for the inquirer, likely to lead to more, and so on. 71 Agency is meant here in the sense of “natural agency.”
112 the clarity of one’s mathematical conceptualizations admit a compressed mentalesque representation (e.g., visualization), Peirce’s description of abductive reasoning being continuous (confluent, non-discretely connected) with perceptual judgment makes sense. Therefore, the term ‘embodiment’ is noted to do exactly what Deines (1961) intended—it bridges ‘things perceived’ into mathematical concepts. Peirce’s insightful discussions of perceptual judgment then can be seen to eruditely elaborate contemporary educational discussion of mathematical embodiment (Deines, 1961), and Tall’s taken to be synonymous use of the term ‘conceptual-embodiment.’72
Mathematics Educational Centrality of Reasoning Why use the terms ‘perceptual judgment’ and ‘abduction’ when the term ‘embodiment’ already has a firm foothold in mathematics educational discussion? The terminology serves as a means of separating a more direct or “ordinary language” means for consideration of mathematics learning, from more complex, and sometimes slippery psychological descriptions. Here, there is a clear epistemological –psychological separation concerning terms such as ‘knowledge,’ or ‘knowing’. Why might having a substitute term for a ‘mathematical embodiment’ be important in mathematics education discussion? The concept of a perceptual judgment seen as a mathematical embodiment facilitates mathematics education discussion purely in terms of reasoning proper. It allows clear discussion of the roles of perception (e.g., visualization) and abduction (e.g., analogy or metaphor) as distinctly concerned with reasoning, aside from discussion in mixed psychological (or cognitive) and reasoning terminology. Reasoning is clearly admitted as both a sufficient means, and ends, of discussion concerning mathematics learning. I believe distinct notions of reasoning are fundamentally simple and fruitful for certain mathematics educational discussion. Concepts of reasoning can simplify and focus mathematics educational discussion in an inclusive manner as regards learning theory. For example, a student’s mathematical construct can be taken to be a construct of
72 Tall’s Three Worlds of Mathematics dove-tail with Peirce’s three levels of semeiotics (CP2, pp. 274-275, 313-314, 316-317, 326-327, 413-414, 443). Tall’s object-based, action-based, and properties-based worlds can be viewed in the light of Peirce’s icon-direct-image-based, indexing-action-based, and symbolic-laws- and-properties-based semeiotic system. However, semeiotic theory is unending in its unfolding (by Peirce’s own description) and so falls far outside my scope here.
113 something. A construction of knowledge, by both means, and educational ends: reasoning. The concept of ‘perceptual judgment’ allows straight forward educational discussion as a study in how students learn to reason, rather than training students in a mere exercise theoretically founded in this or that psychological description. I believe if teachers are provided a clear understanding about how mathematics learning need not slide down this slippery slope, they may have confidence in sharing the most simple, fruitful essence of their student’s studies of mathematics in general. There is a provision for a clear insight into answering one of a mathematics student’s most common and fundamental questions, “Why are we forced to study mathematics in such great detail if computers can do these things for us?”
Summation of Key Assumptions Following is a summation of key assumptions made in the dissertation. 1) An ideal for a segment of softwares which particularly concern mathematics education practice and research are here called Natural Mathematics Interfaces (NMI), and their development is assumed to continue into the indefinite future.73 2) Tall’s “Three World’s of Mathematics” is taken to provide a theoretical description of what is meant by ‘natural’ in the name Natural Mathematics Interface (NMI). 3) Reasoning is preeminent in learning and doing mathematics. 4) Psychologically based descriptions of reasoning can be complex, and occur in unbounded multiplicity. This may inhibit focused, forward moving theoretical agreement among certain mathematics education researchers, and may detract from its practical translation to school teachers and their students.74
73 Kaput and Thomas (1994), in a scathing critique of mathematics education research in the technology area over the previous 25 years, stresses the need for “research that anticipates rather than trails the technological curve.” 74 Psychological description of thought is a science of real phenomenon, and by virtue of this fact shall always outrun cognition. It is not necessary to understand a psychological explanation to be able to understand an explanation of inference patterns. This is not to dismiss the huge empirical contribution of psychologically based description of so many widely varying aspects of mathematics learning. It is only to say that a cogent, fruitful area of research which has greater potential for theoretical agreement among researchers due to its comparative simplicity is the study of student inference patterns.
114 5) Peirce’s Pragmatism provides a simple, fruitful theory which cogently describes reasoning as it pertains to learning and doing mathematics using NMI.
115 CHAPTER VI INITIAL CASE STUDY REPORTING AND THEIR INTERPRETATIONS
Introduction The purpose of this chapter is to provide descriptions taken from the case study which discuss my two students’ reasoning with NMI. Also presented is my interpretation of these case study descriptions. My interpretations are in large part disciplined by the theoretical constraints of Peirce’s Pragmatism as concerns students’ reasoning and Tall’s Three World’s of Mathematics in describing what is specifically meant by the term ‘natural mathematics.’ The presentation here takes the form of brief descriptions and reflections organized around my case study observations of Ann and Ed. Theoretical terminology is used to (a) allow more explicit, exacting descriptions, (b) give empirically based consideration to its relevance in describing students’ NMI use, and (c) to provide a means for uniformity and generality of description which might be useful in bridging to future educational NMI research, while taking advantage of a wealth of past theoretical discussion (e.g., CP1, CP2). Although I am biased by my applied mathematics background, I have great interest in deductive reasoning, and how one learns this distinctive spirit of reasoning, i.e., purely logical ratiocination. This research stems from (a) my interest in how computational devices such as NMI affect such learning, e.g., how fixed, stable mathematical rules, consistently implemented computationally, might help students, and (b) an interest in re-exploring mathematical deduction in a constructivist light, i.e., using contemporary mathematics learning theory to study how one learns to use the rules of logic75 in a sense-making way.
75 There has been a purposeful lack of emphasis on logical deduction in mathematics education research in the constructivist vein. This is merely my personal sense, but I see it as in part being due to a kind of clash with certain constructivist principles as regards “rule following,” or “following a memorized algorithmic procedure,” seemingly skipping over sense making, or an asserted reasoning through of the material. I have selected Peirce’s work on inference as it pointedly addresses and deeply analyzes aspects of this very issue. I observed this to be of paramount importance considering the proliferation of computational devices used in mathematics learning.
116
Ann and Ed’s Initial Distractions due to NMI use Perceptual judgment and abduction is more difficult using NMI than software which provides problem contexts, or interpretations of purely mathematical models. When Ed and I first began with the project, I wanted to immediately try to figure out what it looked like to take a regular Markov matrix to higher powers, as this related to the steady state theorem (see Appendix C, Figure 5). This was thoughtful play on my part; seeking for some kind of pattern or relationship. I treated the exploration with as much reasoning as possible, but since I initially had very little idea of an outcome, my freedom in reasoning unfortunately could only be led by the merest hypotheses. I realize it would have been better to introduce an exploration to Ed for which I already knew the conclusion—but his mathematical topic was something I was learning alongside him. I became excited in noticing a well-defined pattern in the calculations, and resultantly conjectured a theorem (see Appendix C, Figure 5). I wanted to reason my way through to its plausible truth using the NMI. The pattern noticed turned out to be a distraction to Ed at that time as we could not make immediate sense as to why this pattern should always hold. Ed, I believe, was frustrated that I was not providing an immediate explanation (let alone a proof) of what was perceptually judged, or, he may have generally felt exploration was simply not the right way of approaching (let alone learning) a new mathematical area. Whatever the case may be, he was unduly polite, and never explicitly shared his thoughts on the matter—even upon my questioning. This became somewhat of a pattern. He spoke very minimally most of the study. I believe (besides his quiet, serious demeanor) this was because a) he was somewhat uncomfortable with the plausible reasoning methods encouraged in my research, and b) he was used to having a learning guide who knew the mathematical and computer-user knowledge inside and out in advance—falling outside my interest in having students make their own explorations in both these areas. Neither (a) nor (b) are criticisms in any way, just observations of what likely was outside the norm for Ed’s previous mathematical and computer work. I believe now that Ed thought I was going to push him through far more material concerning
117 peripheral aspects of the NMI’s use, and consideration of Markov chains. Therefore, he may have felt frustrated in the sense of being behind in his research. There are other forms of distraction an NMI might cause in its initial use. For example, Ann did not always make connections to the mathematical nature of problems. She seemed to be enjoying the aesthetic aspects of computationally drawing geometric shapes which had interesting looking patterns. This activity I saw as sometimes distracting her from direct consideration of general mathematical constructs. Ann’s initial use of the NMI was similar to my pilot study with Bob.
Reflective Question How does one keep attention focused on curricular activity—or even define curricular activity—when computational devices such as NMI advance? I will discuss in greater detail below that NMI use can be a significant advance over graphing calculator use because of (a) vivid, technically advance interactive graphics, (b) the ergonomic ease of use of full-up computer NMI, but most importantly, (c) higher levels of mathematics admitting exponential growth in pathways to what can be learned using NMI. A machine that does linear algebra (as opposed to matrix manipulation) and differential equations (e.g., plotting vector flow fields, or automatically “assumes a solution of the form…”) can be looked at as either (a) to give students opportunities to build more sophisticated mental constructs through their reasoning, or (b) to be snowed and want to give up—as NMI computational power adds levels of sophistication to direct natural mathematics representations that have never before existed. There is nothing new about students “being snowed,” but for pedagogist’s considering NMI use, it is pivotal to provide clearly defined curricular activities that admit of intellectual reasonableness, boundedness, and clear academic expectation as curriculum drives instruction.76
Ann’s Initial Interest in NMI use Ann’s initial perceptual judgments grew from seeing the Geometer’s SketchPad (GSP) NMI as providing some thought provoking manipulative shapes within a sort of
76 The idea that “curriculum drives instruction” was a consideration Elizabeth Jakubowski relayed to me in private conversation.
118 computer drawing program. The NMI exhibited symmetries and facilitation of iterative possibilities through simple interactive click, drag, double click, and pull down menu commands, repeat, etc. (see Appendix E, Figure12-13). Very soon, Ann began to see things as mathematical patterns, rather than aesthetically created objects using the NMI procedural shell of commands. This occurred immediately with Ann’s iterative construction of interior triangles. Ann’s perceptual judgment, to my view, led her to hypothesize what to do next--anticipating what the next pattern in the iteration ought to be. In this case, the outcome was an infinite regress of triangles within triangles. This observation was made when Ann told me about how the inner triangles would become “infinitely smaller if the computer could draw them so we could see them.” Ann and I then discussed the pixel limitations of the drawing part of the NMI program display (Appendix E, Figure12).
Ed’s Initial Interest in NMI use Ed had a completely different perception of the NMI from Ann’s. This was not surprising as his NMI was more symbolic than diagrammatic. From the very beginning of Ed’s use of the NMI, he had completely different kinds of perceptions and hypotheses— more of a mathematical-linguistic nature. Scientific Notebook (SNB) NMI afforded Ed free expression of his own detailed and explicit mathematical reasoning in conventional, formal symbolism, all provided in a word processor-like computational implementation. This seemed to motivate him to apply his proceptual mathematics capabilities (Tall, 2005b).77
Reflective Question’s Consideration I reflected above, “How does one keep attention focused on curricular activity— or even define curricular activity—when computational devices such as NMI advance?” One way to keep student’s attention focused on the mathematics seems to be to allow them problems which at first just encourage play, but facilitate a pre-planned guidance
77 Proceptual here means Ed’s learning through use of symbols representing conceptual objects—the actions these concepts present through their linguistic-like (logographic) symbolization.
119 which leads them into more technical considerations. This was evidenced as effective in Ann’s case. Ed needed a sense of reason as to why to initiate play from the start. One can provide questions which guide students towards the technical objectives sought after. This is not an easy endeavor—probably a reason why there exists such a strict, almost universal mathematics canon of coursework (i.e., algebra (basic, intermediate, college), geometry (figure-based proof), trigonometry and analytical geometry--including matrix manipulation, pre-calculus, calculus I, II, and III, differential equations or linear algebra or vector calculus, etc.). I believe that long term careful efforts of many educators, one piece at a time, will be required in refining each course (extending the kind of things already accomplished with graphing calculator capabilities) to incorporate NMI use, so that it relevantly stimulates the type of reasoning necessary to move through each higher level of mathematics to follow.
NMI Implementation Considerations An NMI is an ideal interface, having computer implementation limitations which keep it from being an ideal natural mathematics device. I believe these limitations, differing over computer programs, will always exist in some idiosyncratic form or other. This is guaranteed by new natural mathematical computational features sure to follow, traded off against implementation challenges due to software-hardware limitations (e.g., graphing calculator improvements that unfortunately added the complexity of having to embed menus with often obscurely abbreviated names). Therefore, I have chosen to use the term ‘Mathematics User Interface (MUI)’ to describe implementation complications of the otherwise taken to be ideal NMI.78 Ed and Ann were observed (to what Peirce would call) dissociatively abstract among these non-reason oriented complications from the intended NMI. Some computer procedures have no other reason behind them other than they connect this part of a keyboard or icon click procedure to that kind of pictured screen display, or concrete action. Work with MUI was a distinction I observed which needed distinction. Peirce’s dissociative abstraction perfectly fits the description. In the
78 An NMI and a MUI are the same device having the different properties described above. The parsing of these properties will always have some shades of gray which is to be expected when considering idealizations of actual implementations.
120 case study it was observed in the students separating this procedure from that one (e.g., where the menu tab was for “toolbars,” or “find”), these procedures having no other cognitive demand than memorization, as they had no more sense behind them than that, and the ability to read the nonmathematical meanings.
Ed’s use of MUI Relative to NMI Ed had many different kinds of questions about how to use NMI. As an introduction, I present a generically representative example related to Ed’s difficulty with the NMI Scientific Notebook (SNB). Say a student wants principle valued solutions for a trigonometric equation. How might he use the NMI, given that he understands “principle value’s” mathematical employment, just not what they are conventionally named. It is somewhat conventionally known how to specify a domain of interest (i.e., 0 < x < 2 Pi). There is a nonmathematical Mathematics User Interface (MUI) procedure however that must be learned using SNB. Further as regards the help menu, a student may know the natural phrase “specify a trigonometric domain,” but find that a MUI help menu word-phrase search engine may not be sufficiently developed to use this student’s indexing phrase.
Overly Abstract and General NMI Representations Ed and I found the NMI at times to provide solutions which were overly general. For example, a MUI might provide all solutions to a trigonometric equation, rather than those over the principle domain, in a form indecipherable as regards a necessary discriminate abstraction at the appropriate particular curricular level (e.g., a set over the complex numbers with several conjunctions and disjunctions of number theoretic descriptions). These overly challenging NMI representations exemplify the point of constructivist learning theory. When a student has no construct (sense of meaning or reasonableness) for what a computer is representing. This is especially true for a raw exploratory NMI usage, lacking the scaffolds and orienting guidance of an instructor capable of approximating a classes’ zone of proximal development.
121 Ann, Ed and MUI properties A theme which requires critical consideration regards mathematical representations, and nonmathematical procedures learned by NMI users. The idea of NMI is to reduce the latter, so that there becomes a transparency here, allowing one to work directly with natural mathematical representations. In the case of SNB, Ed and I found that the parts of the interface program that we needed were learned and carried out with ease, and so were not distracting from the NMI idealization considered. However, it took me some time to learn and set-up the program (e.g., tool bars, worksheet with task pains) for our work. SNB has a few idiosyncratic command structures needing set-up in a windows format which pertain to the kinds of mathematics one engages. However, upon set-up, and our beginning our mathematical work, SNB approached the mathematical transparency ideally sought in an NMI. Firstly, SNB allows graphs to be overlaid, rescaled, zoomed, cursored over for the domain range values, and more. Secondly, Ed and I found the mathematics textbook oriented information in the help section to be incredibly well written and to the point. Not only would the help section explain which idiosyncratic key sequences needed to be employed, but it covered the actual mathematics involved in a concise manner with many examples. Many of these help section explanations included MUI key-strokes required for using SNB to perform mathematically related processes or to use certain mathematical objects or concepts, related to Tall’s proceptual mathematics (Tall, 2000a, 2000b). These “textbook/key-stroke/menu-option layout” features not only helped Ed or myself possibly better integrate our own mathematical constructs, but by learning these procedures, there was another layer of structural learning provided by the NMI. Ed seemed to have found these textbook oriented features, with their direct linkage to their computer interface procedural requirement’s use very useful, as Ed enjoyed figuring things out on his own. The fact that the interface procedures (MUI necessities) and natural mathematics (at times) were intimately related, Ed was able to learn mathematics in tandem with necessary MUI procedures. The procedural features of the MUI provided a common interface use pattern or set of rules, and, a common window organization and usage that gave a helpful cross-
122 sectional face to the mathematics studied. In other words, the interface procedures had a pragmatically useful consistency about them all on their own, facilitating Ed’s reuse of similar MUI procedural patterns. This was even more apparent in Ann’s work with GSP as this NMI is extremely well organized in its MUI procedural implementation.
Consideration of Ann and Ed’s NMI There is a comparison between these two NMI worth mentioning. Ann’s NMI took far longer to learn how to use, but once learned, was extremely consistent, and compact concerning the mathematics involved, and so over time became almost transparent to her. Ed’s NMI (SNB) had many differing patterns to learn relative to Ann’s GSP. Ed’s NMI had a complex overall organization scheme, but our time using the program did not approach the level of sophistication in using SNB’s MUI related procedures as did Ann’s use of GSP. However, SNB covered many differing kinds of higher level mathematics, where GSP had less mathematical ground to cover. Therefore, it makes sense that the learning curve for each NMI’s MUI aspects was completely different. Appendix B, Figure 3, and Appendix E, Figures 12-13, show the work of Ed and Ann in their very first instances of using their respective NMI’s, showing what can be learned in only a very short time. Unfortunately, because of time, my lack of expertise, and my wanting to remain focused on the most transparent natural mathematical qualities of SNB, Ed had to settle for use of only the fraction of the computer program’s capabilities that I thought sufficient to do our Markov Chain work. NMI features will always outrun, and implementationally differ from previous pencil and paper facility (e.g., rapid automated calculation). NMI outruns students simply because of the nature of them being a mathematics student of the NMI contingent mathematics they are studying.
123 Reflection on the Relevance of MUI Procedural Organization The procedures specific to an NMI (its MUI realization) can be useful in and of themselves.79 For example, in basic algebra, students are typically taught mathematical processes and objects grouped together in ways that help students make sense of how certain operations are interrelated (e.g., multiplication’s relation to division). This is seen for example in SNB’s Combine submenu command groupings: exponentials, logs, powers, or trigonometric functions; while the Rewrite submenu similarly contained the listing: rational float (decimal number), exponential, logarithm, sine and cosine, polar, rectangular, equations as a matrix and matrix as equations. The Solve menu integrated the solution representations of exact, integer, numeric, and recursive. Upon reflection on my own College Algebra teaching experience, these grouping structures would have aided my students in their abductive explanations to themselves, as well as their making sense in natural mathematics proceptual terms. Algebraic deduction (e.g., the necessary reasoning outcomes of most calculations), in my student’s cases, were often obscured, or brute force over-ridden by memorized procedures. There is a kind of forest for the trees aspect in these NMI sub groupings; a larger picture of related ideas perceived in a spirit of facilitating and encouraging orientational questions. Perhaps these orientational aides can help students find plausible connections to a bigger picture which can help organize their approaches to mathematical reasoning. Afterall, NMI is a computer interface, inviting a distinctly logical structuring and orientation. This research describes how such an interface might affect students’ pragmatic reasoning—where deduction is a core goal to the pragmatics of mathematical reasoning (e.g., being able to integrate, or find connections between different presentations of mathematics).
79 This discussed in great detail in following sections concerning what has been called “Interface- procedural Embodiments” (IPE).
124 Transition to Idealized NMI Discussion This section makes a cross-over from MUI implementation discussion, to speaking in the ideal sense of NMI. NMI is nothing more than a pat categorization for representing the aim or goal presumed by the researcher to be existing in the software development community as computational platforms grow in ergonomic accessibility and technical power (e.g., screen display sizes, and amount of memory, respectively).
Ann’s work with the Geometer’s SketchPad (GSP) NMI Critically speaking, the Geometer’s SketchPad (GSP) NMI seemed at first to obscure Ann’s sense of necessity to mathematically prove what was apparent through the interface constructions, seen through dynamic screen displays facilitated by visually interactive mouse dragging motions. As Ann became comfortable with building her own constructions using the interface, she gained a solid sense of confidence in the truth of the theorems stated (not their validity, as in terms of a demonstrative proof). The constructions seemed to prompt her own mental constructs (interpreted here as perceptual judgments), gaining a sense of self-explanations or understanding, as well as gaining a capacity to verbally articulate explanations of how the theorem could be visually seen to hold (a fading into an explicit abductive reasoning move). Then, by moving the diagrams relative to themselves, she saw case after case that held together, gaining, through inductive reasoning, a confidence in the theorem’s veracity.80 The strong visual plausibility of these truths, unfortunately, seemed to overshadow an ambit for Ann to approach the more difficult rigor of a multiple-chain deduction from first principles. On the other hand, these diagrams eventually proved a useful implicit representation for social construct scaffolding. On the one hand, the visuals evoked strongly implicit intellectual actions (i.e., “seeing is believing”).The visuals admitted opportunity for scaffolding in a kind of “point and grunt” communication between
80 The case to case visual feedback gave her an inductive means to see that the theorem ought to likely hold in general.
125 instructor and student by utilizing the dynamic screen displays and their concomitant mathematical designations.81 There were times when Ann, by herself, would figure out relationships using GSP alone, i.e., a student’s self-scaffolding. There were means for her to form constructs from constructions directly apparent from the GSP implementation itself. One might describe this as a kind of direct social dialogue, a natural mathematics one, with the NMI. The interface helped her figure out the meaning of the pointed, explicit mathematical language in the NMI menu bars, used subsequently in her deductive reasoning. More importantly, Ann used the continuously seen motions of figures (interactively caused by mouse drags) to admit an inductively plausible generalization of an otherwise logically deduced consequence of the construction. These facilitations as mentioned, seemed to come at a cost in terms of ambit for a deductive proof of the otherwise plausibly reasoned situation. Ann sometimes had trouble motivating the reasoning of what the particulars of a subsequent proof might require. I must clarify though that it is not really possible to observe in this study a teasing out of whether or not this trouble was amplified by the NMI use, or was simply a natural pre-existing difficulty of writing proofs independent of NMI use.
Perceptual Judgments, Mathematical Embodiments, and Student Reasoning using NMI Four sections immediately following express distinctions of how I observed and interpreted Ann and Ed’s use of mathematical embodiments when working with NMI. I describe them as concomitant to Peirce’s sense of perceptual judgment. Peirce emphasized that all reasoning begins with perceptual judgment. He believed that perceptual judgment “fades” into abduction, or the coming to a hypothesis.
81 The colloquial terminology of “pointing and grunting” was communicated in private conversation with Janice L. Flake. It is a powerful metaphor for how by simply referring to some merely iconic representation, students have opportunity to take steps towards building actual constructs. Pointing and grunting means that attentional focus is (possibly) not dragged down by excessive verbalization. Gertrude Hendrix wrote on these kinds of ideas in learning mathematics. Also in private conversation, Herbert G. Wills III talked about how if a centipede had to think what number leg he was using while crawling, he might surely falter. This also relates to the common distinction in computer science between explicit and implicit use of knowledge in writing computer programs (i.e., specific data providing commands, versus a neural network algorithm that makes its own “implicit” associations).
126 These hypotheses need to parsimoniously cover any facts or judgments at hand that may concern subsequent reasoning about them. The gathering of these facts and perceptual judgments is done by colligating them into one’s attentional focus--resulting in what is (non-psychologically) called a ‘copula’. A copula, according to Peirce, is the interrelated or contextually interwoven operational premises which one employs when embarking on one’s reasoning. This could typically be interpreted to include one’s internal (mentalesque) mathematical connections, constructs or integrations. In deductive reasoning there exists an explicit set of premises—although this does not bar one from an internal colligating process or action of forming the copula, or informal collection of reason-initiating beliefs, connections, etc. This leads to a view of a mathematical embodiment, not in the physical sense of three apples and four, but here, in terms of the computationally manipulative environment of the NMI. The typical notion of a mathematical embodiment is elevated above a purely sensory or external perception. In terms of NMI, it was observed that Ed and Ann’s perceptions were not straight empirical observations of external objects, but instead required intellectual involvement in their perceptions, to a highly abstract degree.82 That is, the students had well developed intellectual constructs for (and preceding) the NMI representations, or constructions they worked with. Here, this kind of perception by Ann and Ed is what explicates the difference between empirical and quasi-empirical observation.
82 See CP1, pp. 2-5, for a related in depth discussion.
127 CHAPTER VII PERCEPTUAL JUDGMENTS, REASONING, AND NMI USE
Introduction It is important that I introduce the intent of this chapter, as it brings into focus the primary result of the dissertation. There were four NMI-distinctive types of mathematical embodiment observed to be used by the students. Recall that the “Natural Mathematics computer Interface” (NMI) used in the study, was characterized a priori, based on things like compass-rule, or pencil-paper traditional inscriptions of mathematics. The NMI provided capacities for manipulative geometric constructions and transformations, or symbolic interfacing to a Computer Algebra System. NMI interactivity, and the means of a mathematical semantics level organization (via interface lay-out), were seen to be important contributors to the students’ pragmatic reasoning. The four abstract types of mathematical embodiments revealed were named: i) Interface-procedural, ii) Natural Mathematics Computational, iii) Applications, and iv), Generic. These (roughly) mean, respectively, (i) interpreted merely as memorized interface procedures, (ii) resultant from interactive computation, interpreted as mathematical in a (sometimes) surface sense, as the student may not understand the underlying mathematics directing the computation, (iii) resultant from a student’s interest in a real-world application used to analogously consider a mathematical model and its interpretation, and (iv) clearly abstracted and generalized, internal or mentalesque mathematical explanations or systematizations. The previous three chapters lay the ‘pragmatic reasoning’, and ‘natural mathematics’ descriptive groundwork for the introduction of the four distinct NMI embodiment types presented just below. Table 1, Chapter IV provided a foreshadowing overview of the research results concerning the two students’ reasoning. Chapter V explained that Peirce described perceptual judgments as the initiation points of reasoning. My argument in Chapter V was then that Peirce’s notion of a perceptual judgment, under my interpretation, contained the notion of a mathematical embodiment. Paralleling this, Chapter V noted that Tall’s theory of the “Three World’s of Mathematics,” described
128 mathematical learning as initiatory in student’s grasping the lumped term “mathematical embodiments.” There are a plethora of mathematical content area distinguished embodiments. For example, base ten blocks are used for learning certain kinds of content knowledge; while algebra tiles, and geometric tangrams are as well. Mathematical embodiments are typically not distinguished by any general class of embodiments independent of their mathematical content (aside from perhaps commonplace distinctions such as “plastic vs. wooden,” or “computer virtual vs. manual,” etc.). Therefore, the distinguishing of the four embodiments mentioned have general significance (please refer to Table 1, 34-37). These atypically abstract embodiments cross mathematics-content bounds, yet are perceivable through identically identified media— the general computer interface characterization of a NMI. The students’ use of these four implicitly cognitive embodiments (please refer to Table 1, 28-29), as well as related explicit reasoning acts, are considerate of the pragmatic (tool) use of the NMI. Allied aspects of pragmatic reasoning are introduced through case study observation below.
Four Types of Mathematical Embodiments concerning NMI use Four distinct kinds of mathematical embodiments were observed when Ann and Ed were using the NMI. I have named and abbreviated them: Generic Mathematical Embodiment (GE), Interface-procedural Embodiment (IPE), Natural Mathematics Computational Embodiment (NMCE), and Applications Embodiment (AE). Each of these distinct mathematical embodiments relate to pragmatic reasoning through Peirce’s sense of a perceptual judgment. They are described below.
Generic Mathematical Embodiment (GE) The term “Generic Mathematical Embodiment” is motivated by Rescher’s (2005) description of mathematics as being a kind of lowest common relationship or pattern—a “generic” representation.83 A GE is parsimonious to a mathematical embodiment, and allows one to stand above, looking down upon it.
83 Rescher (2005) saw mathematics itself as both generic and “fictive.” However, NMI constructions and representations are quasi-empirical—they are actually perceivable through use of the computer interface.
129 GE’s in name are intended to be identical to the typical ‘mathematical embodiment,’ in that they are seen from a pragmatic reasoning perspective as a perceptual judgment. They are the root of mathematical reasoning, NMI use or not. Finally, being generic, generalizations can easily spring from them. All embodiments discussed here signify a perceptual assertion, a cognitively implicit display of knowledge (see Table 1, pp. 28-29), or intuitive capacity to not not immediately comprehend a definitive mathematical meaning. GE’s are quintessential mathematics-learning goals, which can give ambit to a student when off-handedly noticed, in an otherwise puzzling mathematical situation. GE’s are noted when little need be said about something that much can be said. Ed and GE Ed demonstrated having a clear GE perception of the transition matrices central to Markov chain theory. He explicated his understanding in proceptual form in Appendix B, Figure 4, Section 1.4. Initially, a notational inconsistency across multiple texts was noticed in off handed NMI calculations, and reinforced in its novelty by a dumb-founding notational explanation I received from a statistics professor. Ed was then seen to focus on this novelty through several revisions of NMI mathematical word processing. The NMI, additionally, seemed it may have had an appeal to Ed in so far as it presented an elegance in its presentational format (see Appendix A). I interpreted him to have a firm GE in a) the ease and detail of his explanation (needed for me at this point) concerning the notational obfuscation, and b) the interest and initiative he displayed in this task of creating his own notational representations using the NMI icon palettes. Ed reached a point of authentic perception—his notation was intuitively obvious to him. 84
Since this interface runs mind-independently, it has a strong sense of being real. This is quite a different type pf characterization as it relates to computational results, not the cognitive results often associated with Platonism. NMI representations have nothing whatsoever to do with Platonist senses of mathematics itself being real. 84 This is a good example of how Peirce saw perceptual judgment as “fading into” abduction, specifically; perceptually (in the cognitive sense) recognizing, and then (in the abductive sense) mathematically explaining what a Markov matrix signifies in terms of how it works.
130 Ann and two GE’s Two final examples of a GE came out of Ann’s proof that “The angle bisectors of the angles of a triangle are concurrent.”85 The first one was a mistaken GE at first, as it involved Ann only paying attention to a single modus ponens, or forward direction a biconditional statement Ann and I called ‘The Two-point Equidistant Theorem,’ “A line is the angle bisector of an angle if and only if the line contains all points which are equidistant from the sides of that angle.” Discussion Ann’s presentation of the Two Point Theorem and proof is included in Appendix G. Discussion of her mistaken GE is included in Appendix G-H, so as to allow the reader to refer to the diagrams. The second GE involved Ann correctly looking down on the transitive property of equality learned in grade school, in terms of an equality of distances between points. The use of these GE’s in Ann’s reasoning through a proof is discussed in the last section of Chapter VIII.
Interface-procedural Embodiment (IPE) Of the four kinds of mathematical embodiments discussed, the “Interface- procedural Embodiment” (IPE) is the most indirect, and artificial with respect to natural mathematics. IPE involves merely the procedures of the interface taken as the non-ideal MUI computer implementation. These procedures can hopefully be understood to be somehow contingent—yet in some cases may not even be relevant--to natural mathematics. Further description of IPE is presented in Chapter 4, Table 1. Consider what Ed’s use of the idiosyncratic command structure of the NMI SNB might be. For example, the designation of independent and dependent variables is necessary to solve certain equations, and the SNB procedures for doing this distinctly depend upon (or are functions of) mathematical relationships. More generally, the interface logic, however, took Ed astray from natural mathematics in having to search user help, learn abstruse interface specific procedures, etc. However, as the interface logic was more or less contingent on the mathematics at hand, Ed had opportunity to interpretively learn certain mathematical necessities through figuring out how to use the
85 Concurrency denotes when more than two lines intersect at a single point.
131 MUI interface itself. IPE’s then were interpreted to be mathematical embodiments which had not yet been abstracted away from the interface procedures. Consider Ann’s use of the NMI GSP. Her marking of lines and line segments, and the organization of these procedures is a MUI aspect of the NMI.86 However, many “menu select, click, and drag” interface procedures to construct geometric diagrams were implicitly figured out by Ann, given only the necessary learning of a small number of such procedures as mere operational activities (e.g., the means to construct a geometric diagram such as a triangle from learning only how to place points and draw segments). Ann progressed extremely rapidly in the learning of certain drawing level kinds of procedures—many of them having relation to general “paint program” point and click operations. Ann’s learning here involved implicit cognition—a perceptually judged generalization of past computer interface experiences irrespective of mathematical content. Over the long haul, the mathematics was being implicitly derived from the learned procedures—but only operationally, or non-generally (i.e., the procedures were not yet abstracted away from the NMI’s intended mathematical representations). This operational mathematics embodiment formation was observed to involve visual and tactile perceptions judged and re-judged by Ann via abductive inferencing (e.g., Ann’s guessing which procedures sufficiently connected up a draggable triangle on the GSP computer image display). It was later that I noticed that this seminal “playing” with the interface helped Ann enjoyably learn rule patterns that later served her in abstracting out generic, or general mathematical embodiments (GE’s). This was apparent in Ann’s work in building the many NMI constructions used in her proofs. She would recognize—or perceptually judge--an IPE procedural pattern that allowed her to boot-strap a mathematical understanding. The IPE allowed her to anticipate certain natural mathematical ideas to be learned later. It provided a small number of cubby holes that could be filled in, connected, and re- connected as she learned.87
86 A truly ideal NMI would be generic in so much as mathematics is—there would be no deviations in procedures from one NMI implementation to another. 87 This not unlike how some students go about learning mathematics procedurally first, logically understanding what they are doing, second.
132
Specific Examples of Ann’s use of IPE’s Ann figured out an extremely clever method using geometric transformations for constructing a rotating square within a square for a proof of the Pythagorean Theorem. Ann admitted to me that she had no clear path to how she intended to construct the inner rotating square from a mathematical standpoint. However, by running through several procedural applications of various transformations, she hit upon a pattern that was successful. She devised a second one as well that was a variation on a theme. It was extremely impressive in its complexity. What was interesting to me was that Ann learned the interface procedures to a high degree of facility before she learned the mathematics intended by those procedures. Ann explained her reasoning on how she devised the Pythagorean diagram of a square within a square fully in terms of interface procedures (even though she was by fiat applying geometric transformations such as reflection, rotation and translation). Please see Appendix J. Ann explained that in first creating an end to end procedure for creating the square within a square, there was much trial and error (guess and check, or abduction and induction). She said she was paying more attention to the pictorial changes (or actions) she saw caused by the procedures she was doing, not paying attention at first of any mathematical meaning associated. However she was doing more in her play—she was unintentionally operationalizing the underlying mathematics, which enabled differing combinations of operations to get the same result (this paralleling the nature of mathematics). This was evidenced by her using a variation on the same set of procedures (i.e., geometric transformations) to create the same construction of a square dynamically moving within a square. It was then, after she, in essence, successfully “programmed” the interface to make the automated moving square—did she go back and explicitly study (precissively and discriminatively abstract) the mathematical transformations and meanings of the interface-procedures she employed. Such “programming” served Ann as an implicit operationalization of the underlying geometric transformations. By operationalizing, Ann gained action oriented perceptual judgments (unified objects in her mind’s eye), or IPE’s.
133 This differs only in magnitude from memorizing mathematical procedures. Memorizing a small number of pre-configured idiosyncratic computer-procedural branchings that are explicitly laid before you as interactive actions is much easier than memorizing the enormous number of algebra branchings without the aide of external interactive devices. This is the nature of being able to signal, through these computer procedures, the doing of a vast number of mathematical procedural steps (e.g., using a calculator for taking a square root). Ann indicated that the procedures themselves, in connection to the dynamic graphics activity on the screen helped her plausibly reason their mathematical significance; therefore serving as an IPE. Ann and I had discussed these geometric transformations beforehand. She told me later however that she just “figured it all out” without using the mathematical ideas we discussed, but by using her own procedural choices, interactively viewed screen actions, and reading the mathematical terminology associated. Her reasoning amounted to abductions and inductions—conjecturing and checking what the mathematical transformations meant through her quasi-empirical use of the interface. Conclusions: Ann and IPE Finally, Ann’s IPE’s faded into becoming GE’s as her associations with the interface procedures fell away. This involved Ann dissociating the interface procedures from the mathematics intended. Ann also needed to precissively abstract from the visual display her own compacted, intellectual visualization. From these two abstractions, Ann finally had to discriminately abstract the meanings of the mathematical terms associated (e.g., reflection from a 180 degree rotation).88 The IPE’s served as boot-straps to GE’s. Once the IPE’s were implicitly seen as a perceptually judged object, all that was needed were the three types of abstractions discussed above to move to a GE. Further discussion of Ann’s use of IPE is presented at the conclusion of Appendix G.
88 Deines (1961) discusses a “reading in” to an embodiment, and a “reading out” from the embodiment. These intellectual activities have been detailed above in Ann’s reasoning through mere procedural interactions to make intended pictures (a “reading in” to the IPE), and then performing the abstractions to “read out” from the IPE.
134 Natural Mathematics Computational Embodiment (NMCE) NMI was observed to provide Ed, Ann, and many of my previous College Algebra students with a facility to form perceptual judgments called Natural Mathematics Computational Embodiments (NMCE) here. To give an immediate sense of what is meant by NMCE, consider student’s using graphing calculators. As a hypothetical example derived from previous classes, consider a student using the graphing facility to learn to plot higher order polynomials. Certainly, the deductive reasoning behind theorem’s such as De Carte’s rule (e.g., concerning even and odd numbers of zero’s and shapes of polynomial curves), and the application of the rule is useful to students. De Carte’s rule itself is quite complicated though to colligate and use as an “end to end” copula for reasoning. However, the facility to explore many instances of De Carte’s rule using the rapid feedback of the graphing calculator facilitates understanding via plausible reasoning qualitative connections to many graphical instances (an induction). So the graphing calculator NMI gives an immediate example of how students can colligate a NMCE copula for De Carte’s rule. Students can also take intentional control of their inquiries using NMI, facilitating their own colligational approaches, and their resultant component parts and interconnections of their copula. I observed my students choosing to work in their own zones of proximal development. The NMI was able to feedback to the student precisely what it is the student needed to answer concerning their own questions. Please see Table 1 for an explicit description of NMCE. NMCE are distinct from IPE in that they are perceptually judged using the semantics of the natural mathematics being computed, rather than the idiosyncratic level syntactic procedures used in IPE. NMCE however requires distinctly relevant previous mathematical knowledge in order to guide one’s quasi-empirical explorations and experiments moving forward.
135 Ann, Ed, and NMCE NMCE was observed to come about due to the plethora of student initiated natural mathematical problem examples so easily accessible via NMI.89 NMI facilitated Ann and Ed to be prolific in their subjectively intentional, inquiry based computations. For example, while Ann was dragging around a geometric figure in order to view its generalization of specific properties (e.g., see Appendix F, Figure 15), there were “behind the scenes” computations which facilitated each frame that she viewed. In the end, in using the NMI on separate subsequent occasions, she achieved immediate perceptual judgment concerning what properties would hold of certain constructions, or composite pieces of a construction (e.g., concerning a perpendicular bisector needed for a larger construction; Ann “seeing” the equidistance distance from end points of the segment to the bisector). Ann’s process of forming NMCE involved in this instance was observed as a coarse grained to fine grained graphics interaction. As Peirce explains, one’s abductions can be taken to such a refined level as to finally arise through pure perceptual judgments—or noticing of embodiments. Ann, through careful guessing and checking, was observed to arrive at such an ends. She explained in learning about perpendicular bisectors of segments, a recollection of a distance formula derivation we previously worked through together in her trigonometry class. She said, “the proofs by SAS triangle congruency [concerning perpendicular bisectors of segments] could be seen instantly, so, why didn’t you show us this then?” Ed developed NMCE concerning how recursive formulations of Markov chains numerically moved towards steady state (See Recursive example Appendix B, Figure 4, section 1.6.2). The NMI also facilitated insight into the explicit proceptual recognition of the process of the Markov property itself, as the random variable X at step i “hopped
89 SNB only provides input and output natural mathematics solutions, i.e., not including the intermediate steps. There is though the capability of manually calculating intermediate steps—in fact this can be done by mousing over an inner, or part of a larger mathematical expression. Beeson (2001) provides a program that works to provide intermediate steps; algebra through calculus. GSP works by construction, and so intermediate steps are forced. Proofs are not provided by GSP, but some mathematical programs, such as Wolfram’s (1988) Mathematica has the beginnings of such capabilities concerning larger general areas of mathematics (e.g., algebra), yet not as natural mathematics (to the researcher’s knowledge). Wolfram’s Mathematica is a Computer Algebra System (CAS) very similar to the computing engine behind the NMI interface of SNB. Through private conversation with Mackichan, developer of SNB, he conveyed that SNB would be joined by many other CAS engines developing their own NMI—it was just a matter of time.
136 around” according to the histogram realized in the deterministic state vectors— determined by the initial vector and the constant transition matrix (see the particular example in Appendix B, Figure 4, section 1.3). At a conference presentation, Prof. Henry Jones asked Ed to parse the difference between how the random variable instance were not deterministically backwards- computable (the Markov property), although the vectors were fully determinate forward or back. This raised a question leading to our combined efforts to make the process clear (Appendix B, Figure 4, section 1.3). This abductive explanation unfolded from refining explanations—write and delete, write and delete further until a clear, yet not terse explanation was relayed in mathematical and natural language sentences. After composing this explanation, it was potentially refined to the level of a pure perceptual judgment concerning the inner workings of the Markov process, and how one might clearly write consistent symbolic notation. Ed and I had differing methods of perceiving mathematics. Ed preferred concise mathematical-symbolic explanations (See Appendix B, Figure 4, section 1.4) while I preferred written explanations of particular instances which indexed general mathematical patterns.90 These patterns facilitated colligated, end to end thematic perceptions of the explained concepts—allowing for compressed perceptual insights (which included pre-inferred concepts). As mentioned Ed preferred more concise mathematical expressions as opposed to verbal explanations (a deductive versus an abductive orientation to understanding). Ed at times stood back from the proceptual formulations to larger theoretical perspectives which were facilitated in part by guesses and checks on the NMI. Results of this are evident in the explanation below concerning a comparison of steady state vector approximations by brute force computations, vs. a closed form eigen-equation approach. NMCE and Reasoning through Distinct Mathematical Approaches: Ed Described is an example of gaining comparisons of higher theoretical perspectives via two distinct NMCE’s emerging from a rare occasion of my scaffolding Ed. Our interaction concerning T u = u recursive formulations (Appendix B, Figure 4, section 1.3
90 Semiotically, Ed preferred explanations at the highest law-like, or symbolic level, where I preferred indicating, indexing, or pointing out particular instances which connected imagistic or perceptually judged reason with the law-like reason of Ed’s preference.
137 and 1.6.2) called attention to the eigen-equation with ‘Lamda’ = 1 (Appendix B, Figure 4, section 1.11, 4-3) (c)). A numerical approach of brute force vector matrix multiplications to reach only approximate solutions (Appendix B, Figure 4, section 1.10.2, 4-3) (b)) was now compared to a completely differing, more elegant theory of computing eigen-vectors and values. I tossed out an interpretation of Ax = Lamda x as reducing a matrix to a mere scalar value (Lambda). Although this was apparent from the eigen-equation on its face, Ed was intellectually unsatisfied with merely making this observation. He wanted to proceptually (see Table 3, Chapter V) move through the relationships that supported the peculiarly observed quality. This was where I believe the NMI gained Ed’s attention. He was observed to be critical of just what the NMI calculated eigen-vectors and values meant. This was an area he had not yet studied in his degree program, and found its derivation more satisfying as the theory presenting closed form solutions through sound deductive derivation. It was critically observed by the researcher that certain NMI use was observed to have little influence on Ed’s ambit for deductive validation of what the NMI provided “second hand,” so to speak. He reached for perceptual judgments through hand (or proceptual) calculations and reading of derivational text material. At times he would be challenged to want to validate by hand the NMI’s outputs, and other times he seemed a bit overwhelmed and critical of NMI results in terms of his own proceptual understanding. Ann, Ed, and NMCE Conclusions Finally then, NMCE’s were apparent with Ed and Ann when they were well within their zones of proximal development. The advantages provided by rapid computation controlled by Ed and Ann’s previous semantic knowledge of natural mathematics helped them move forward in their perceptual judgments within the natural language of mathematics, as opposed to some procedural substitute having its own idiosyncratic syntax (e.g., IPE). Their inquiry from rough to finer grained steps until solid insight was gained analogously paralleled a compacting of abductions to end point limits of instantaneous recognitions (perceptual judgments), facilitated by natural mathematics based calculations, and achieving end to end copulas of higher theoretical bite sizes. This
138 insight facilitated Ann or Ed to work at higher mathematical levels without having to do as many intermediate computations to reach what they could perceptually judge as the punch line of their inquiry. In the end, without these kinds of perceptual judgment steps forward, the students would have had less powerful use for the NMI. Ann stated that “watching what I was doing while I was doing it” gave her confidence through acquiring certain NMCE’s that anchored her later studies.
Applications Embodiment (AE) The last NMI distinctive mathematical embodiment I observed in Ann and Ed is named “Applications Embodiment,” (AE). These embodiments turn out to be motivating actions of curiosity, driving the quasi-empirical use of the NMI, and abstractions from analogies of real-world mathematical applications.91 Sometimes these analogies are incomplete or not fully comprehended, however, this was observed to increase Ann and Ed’s motivation, as discussed below. Therefore, I describe the AE to concern an (real- world) analog that cannot be looked down upon (fully comprehended). Instead, it is an analog that holds interest enough for the efforts involved in its being looked up to—these interests being a motivator to approach such a complicated (not fully, nor easily understood) concrete part of the mathematical embodiment (e.g., the apples, not the numbers—although these apples may correspond the thermodynamic workings of a jet engine). AE and Mathematics as a Tool AE typically involve learning mathematics as a tool in order to understand real- world phenomena. Ann and Ed were observed at times to emphasize their interest in the phenomena more than learning the necessary mathematics. This was seen to leave the learning of the mathematics as an afterthought. However the power here is to a great motivation to understand things about the world, considerations of scientific reality, as well as, finally, the mathematics, which ultimately merely facilitates the interpreted
91 From the mathematics teacher’s perspective, clearly, these analogies have the real-world application as their base, and the mathematics to be learnt as their target (Presmeg, 1992, 1997a). I have observed that from the student’s perspective, it seems this base-target ordering can swap back and forth as they learn— but definitely to start with, the application serves as the target of the analogy.
139 explanation. Example AE analogy targets (see Presmeg, 1992, 1997a)92 are a) environmental or ecological models, b) refraction, diffraction, or reflection of light or sound waves, c) biological systems such as concern cells, genetics, or plant or animal extinctions, or, d) means for developing alternate forms of clean, renewable energy sources. Students’ Weak Understanding of AE Analogs It was specifically noted with Ann, Ed, and certain other students I have taught, how AE analogies functioned as embodiments which were striven for, or looked up to, instead of what was typical of the other embodiments—which were well understood and looked down upon—meaning they were so second nature, as to provide a perspective “from above,” intellectually speaking. The possibility of connecting up (acting on an NMI quasi-empirically and then abstracting) an AE held out encouragement of learning something new, as AE were practically impossible to get a handle on without the mathematics still to be learned to its necessary depth. Ed’s and other Students’ Common AE Experiences My experience with other students and their graphing calculator use was comparable to how Ed came to his AE’s by NMI use. The graphing calculator power made it easier to devise real-world problems that a great many students were more apt to take an interest. Further, these problem situations had more natural footholds than the abstract mathematics associated—attractive to this very large proportion of students as well (such student population consideration is explained below). I observed in Ed and most other students an initial grasping of the mathematics associated with the real-world problems they did with NMI to be unnatural. The associated mathematics which modeled the real-world problems were, at first, taken for granted as the machine provided them with the answers for which they were looking— and so they perhaps felt they were done quite before they had a proceptual grasp of the mathematics involved. Afterall, the applied problem had interpretations which held the
92 Emphasizing the footnote above, it is typical that the base of an analogy used to learn mathematics is the concrete concept—where in the cases above, these are the targets, those things the students are aiming at learning via mathematics as the base tool for their learning about this target. Presmeg (1992, 1997a) explains how one typically analyzes the use of analogies in the learning of mathematics.
140 answer, even so that they could be checked by estimates, regardless of further underlying mathematical understanding concerning the actual model. Students though, had achieved an AE (a perceptual judgment) to which they could look up to via a perceived to be plausible analogy to the mathematics they hoped to learn. The lack of proceptual perspective was observed in that the mathematics involved fully depended on the NMI to do the calculations, facilitating only fragments of mathematical understanding of the model underpinning their merely interpreted, AE dependent answer (i.e., no GE was formed at this point). As described, students were looking up to this model’s interpretation to obtain their answer to the specifically stated problem. This I would have predicted given my definition of what I called a common AE (please note Table 1). Uncommon Extension of AE by Ed My own initial considerations of AE changed when Ed and I came up with an alternate approach to the steady state vector problem. The particular example involved the dual theories: a numerical and an eigen-vector and value solution approach (Appendix B, sections 1.10-1.11). Suddenly, there became two, completely distinct theoretical approaches to mathematically model the same problem. The problem was then eventually seen by Ed as model independent. As discussed above, student’s typical or common AE reasoning involved attempts to map an analogy both from and to the mathematics, as well as from and to the associated applications interpretation—sort of like a plausible iff analogy if you will. Both the interpretation and mathematics involved were vague, and there were only fragments of very particular, problem dependent, mathematics learnt. However, I interpreted my observations of Ed’s AE to grow into showing NMI use facilitating a reasoning far from a mere fragmented, model interpretation dependence. Ed’s Markov problem (Appendix B, sections 1.9-1.11: Faversham’s Hyena problem) admitted a more concise, theoretical, top-down view facilitated by the NMI. Theoretical here means say, linear algebraic, calculus based, or trigonometric. Ed’s (and my) use of the NMI eliminated calculational tedium which can be distracting from the overall structure of the mathematical theory applied. This “tedium” aspect was no different than common AE reasoning already discussed. Additionally though, the NMI
141 facilitated a complex enough problem so that one could appreciate the mathematical theory facilitating the modeling of the problem, and of course, it’s more concrete interpretation.93 This multi-level understanding though required a long term immersion in the problem’s interpretation, mathematical model and the supporting theory (or mathematical area) facilitating the model. Right at first, Ed took the NMI facilitated numerical approach to modeling—as it directly followed from his theoretical derivations (recursive representations) (Appendix B, section 1.3). These mathematical approaches were simply taken to be the means for solving this problem. In other words, the problem itself determined the theory to be applied. This particularizing of a problem not just to a theory, but for common AE’s, to a mere interpretation of a model within a theory is worth noting. There was a (typically loosely plausible to the student) analogy between interpretation to the mathematics of the model that held in the typical AE case, excluding further consideration (e.g., deductive aspects of a model’s theoretics). Through Ed’s and my discussion, I came to believe that the interpretative demands of the AE solution approach left little intellectual head room. I must mention here that Ed’s pure mathematical capabilities were more apparent to me than his applications bent, although this opinion may be biased by my engineering experience. I mention and qualify this for a significant reason. Ed, and other more logically, or pure mathematically minded students are rare, while applications oriented students make up a far greater number of those taking calculus in community colleges and state universities-- this at least being my experience. Therefore, I conclude that AE’s are a vital consideration for NMI use in these environments. So the NMI facilitated Ed having a broader, top down scope, in large part due to an elimination of the tedium of calculation that can rapidly diffuse one’s focus and creativity concerning larger theoretical perspectives on one’s applied model. Ed’s AE broadened to a point where he could look down upon the problem, and focus on the
93 Via a private email and conversation, Prof. Emeritus Benjamin Fusaro, pointed out the distinction between a model and an interpretation of that model. This interpretation is the concrete, real-world application. For example, the quadratic equation, or the wave equation, as models, can serve interpretations in physics or instead a biological modeling purpose (interpretation). The wave equation might be a model interpreted in the particular context of a wave-like electrical transmission down a biological nerve fiber.
142 differing modeling approaches applied. This is what I aptly call an advanced AE. Please see the concluding remarks on AE in Table 1 above. Here then, Ed had the headroom to step back and consider the mathematical model and aspects of its supporting mathematical area. This was evidenced by the interest he took in the linear algebraic approach of eigen equations, and how they could model a Markov steady state vector model—something new to him. Ed saw now the numerical modeling of his initial recursive theoretic approach as non-unique, opening him up to this advanced AE—a perceptual judgment at the highest level—compacting theory, model, interpretation and problem as one perception asserted. This list of compaction served as a mathematical embodiment for other similar situations—something I interpreted through brief conversations with Ed, that he had seldom if ever run across in his studies this level of depth and breadth of clarity that our long term project, his diligent studying, and the NMI facilitated. A further example of how the NMI allows a higher level of mathematics to be employed and understood in problem the solving of challenging problems is the “Serial Production” problem which was explored in Appendix B, Figure 6. The following Figure 7, although not composed on the NMI, pictorially diagrams Ed’s higher level of mathematical modeling considerations. Ann and Ed’s Interest in AE’s One of the more interesting aspects of AE was that it was interpreted to give Ed better focus in his estimates. Confidence in how typical steady state vectors converged (noted through NMI calculation), allowed Ed to estimate solution vectors without distractions that there might typically be a pathological behavior during convergence. Ann seemed amazed at how the analytical geometry and trigonometry fit with Euclidean geometry (e.g., notions of distance, circles, and trigonometry), as well as how symbolic logic was applicable to mathematics. 94 Ann and I discussed deductive proof, and its relation to mathematics and computer science, in addition to applications oriented mathematics problems (e.g., circular motion and vibration and wave modeling, the use of fine grained real-world data arrays) at length.
94 Ann took an analytical geometry and trigonometry course from me, as well as a mathematics for liberal arts majors course covering sets, symbolic logic, geometry, probability and statistics, and combinatorics. Ann seemed to display a similar kind of amazement to Wigner’s (1960) at how mathematics is peculiarly related to so many real situations.
143 Although Ed’s AE interests grew at times, it seemed always in a tempered way. I interpreted this to be due to his interest in the clarity of his thought—as AE seems to require a difficult kind of bimodal application of the intellect—a mixing of plausible and deductive reasoning—understanding one as tied to the other. Estimations often involve this kind of reasoning. While NMI technology develops, it seems vastly more applications oriented problems will become accessible to students. Because of this, greater mixtures of plausible and deductive reasoning approaches will be required for these problem’s solutions. Ann and an AE of Teaching Ann was interested in how to represent and re-represent geometric situations with the real-life application of teaching in mind—a qualification of AE quite peculiar in terms of the applications analog; but an undoubtedly lively and important one for school teachers. 95 She often would plausibly reason which way to present ideas to younger students. She had definite opinions about which ways were simpler, or more direct to grasp, dynamically understand as an action, and generalize to its pure mathematical idea. This is somewhat how Lesh, Post, and Behr (1987) describe Dienes original notions of mathematical embodiment—ideas we had explicitly discussed, following her questions on her own ideas of how to use manipulatives. Ann herself here was putting her own GE’s together. The NMI motivated her in this regard as she felt that students would learn much more readily with this type of hands on, dynamic interface. These kinds of pedagogical interests lessened as we moved into proof related reasoning, but even then she sometimes expressed her thoughts on how this or that related concept might be more effectively taught to lower grade students. A concrete example of how Ann often thought in terms of teaching others is displayed in Figures 14 and 16 in Appendix F.
95 This peculiar AE, that as regards motivation to teach mathematics in certain ways, can be thought through in terms of Scheffler’s (1994) work concerning the inseparability of desire and reasoning (consider what Hume famously indicated concerning reasoning to be a slave to the passions). Here, with Ann and her imagined students, there is a passionately driven recursion of reasoning—a recursion on the algorithm “thinking about another’s reasoning” is iteratively applied. Ann is reflectively thinking about how she would reason if she were a student. At the same time though, Ann’s reasoning (as teacher) about the students’ reasoning pushes the envelope of her own reasoning and understanding further. Ann, as a teacher would reason more carefully through the mathematics so that it might be taught to curious students, who then reason in the direction of her furthered understanding (which was, and is constantly motivated by the students’ perceived reasoning to begin with), etc.
144 Ann, AE, Teaching, and Proof As Ann eventually began putting together her multiple-chain deductions on her own, this kind of reasoning became her passion for the end of our summer research together. She explicitly told me after grasping a proof (and we had not had conversation in these terms beforehand) that “doing proofs was fulfilling.” Ann said this in a quiet, serene way, expressing to my mind a significant reflective moment for her; verbalizing a kind of meditation on that moment. I will now go back to the beginnings of our work together, and describe some of her reasoning that led to the feelings expressed above. During conversations early on in the research, Ann continually expressed interest in noticing how she could teach children in ways they may find interesting—in ways related to the hands on, perceived as purposeful approach we were using with NMI.96 Because of a bad experience Ann relayed to me about a previous calculus I course, she asked many questions on the order of, “What can we do with all this mathematical deduction? My Calculus course involved a lot of proof and I got hardly a thing out of the course. What is calculus used for?” Discussions about proof held little interest to her at first. Her questions instead revolved around real-world mathematical situations or applications, many of which she felt her future students might take an interest in as well. These student teaching oriented applications motivated her to form teaching applications AE’s, some of which led her to new concrete GE’s for herself. Although Ann’s eventual deductive reasoning, on the surface, seemed to have little directly to do with AE, it is my interpretation that the kinds of applications oriented perceptions of geometric ideas mentioned above provided a sense of purpose for her to approach formally oriented geometry. As mentioned, Ann’s AE related questions occurred long before the proofs were approached. This was at a time when we were comfortable with the NMI use--after she had gone through a phase of IPE and NMCE formations—a time when the NMI itself was evidenced through conversation with Ann to have some greater connection with real- world situations—beyond pedagogical applications (e.g., certain analytical geometry
96 I pointed Ann to the National Science Foundation’s National Library of Virtual Manipulatives website for her expressed interests here.
145 relationships concerning equidistances). I considered these questions to be seeds for a possible future AE—not yet an AE per se. “Seeing” a mathematical AE through simply conversation is not enough. AE formation as I am describing takes immersive intellectual involvement over time. One must make analogies where both sides of the analogy— mathematical and applications-- are in the process of formation, and typically involve the coordination of both deductive and plausible reasoning. Ann, AE, and Motivation These sorts of real-world considerations discussed above led to interesting discussions beyond what her students might engage, and likewise increased Ann’s motivation in our proof oriented mathematics project. She began to see how mathematics could be applied, and formed questions which I again believed to have pertinence to her possible formation of AE. Ann began looking upwards towards grasping mathematical interpretations that she had no sure capacity to mathematically reason through (i.e., understand the underlying mathematical model). This was in contrast to the GE’s she herself formed in thinking through what might be interesting to her future students. Ann though, wanted to know more as an individual. Therefore sometimes, we had off-handed discussions on centers of gravity, which concerned the different kind of “centers” capable of being found within triangles— a focus I was introducing to her concerning the proofs we planned to put together. The NMI proved enormously useful in this regard, as these different triangle centers (e.g., circumcenter) were able to be constructed using the NMI. The closing point here is that although mathematics and reality are quite different, they share similar constitutive interests. My AE Experiences I now can see more clearly, from my own experience as an engineer, some of the reasoning I initiated through my applications work using computers. AE is a good description of my own engineering reasoning when immersed in a problematic mathematical situation long enough to begin to “see” what it was that I used to have to take much time and effort explaining (abducing) to myself. As these explanations became crystalline in their organization and compactness—they became a mathematical embodiment, or AE. It was the confidence I had from previous engineering modeling
146 situations (e.g., how far could a certain specified infrared camera see through a certain specified rainy or a dry weather condition?), that if I kept pushing, such a clear mathematical applications understanding was actually there to be had. Often, it (eventually) became clear that either I had enough data to solve a specified applications problem, or not. In this regard, AE most certainly depends upon previous senses of what it is to know things in a mathematically specified way. Reflection: Students’ AE use GE and AE understandings are stretching further apart as one moves into higher engineering, science or business mathematical modeling requiring computation. This is due to the complexity of models mathematically accessible through computer use. Using NMI (and other computing means), the student or engineer has opportunity to perform their own quasi-empirical induction on math models for which they completely lack a GE. However, they may have an AE, and by abductive and inductive reasoning, in part through NMCE, they may gain an understanding of how to interpret given mathematical models in differing parametric situations—and how they may need to reframe such plausibly accurate models. In other words, the rigors facilitated by pure mathematical GE’s are not always available in the practical work which requires computers (e.g., numerical solutions). This is particularly important for the educator to realize. I believe they should explicitly make students aware of the students’ differences between a direct, closed form solution, and a “brute force,” necessarily computational result. Students ought to realize that their framing of a problem, the hypotheses which form the premises for their “number crunching” are typically plausible models. All the computing which comes afterwards is deductively set, but the results are always suspect with regards to one’s framing. Here is where the ability to estimate the outcome of a model’s calculated interpretation is a necessity—and the problem is insoluble to a needed accuracy without a solid AE (even if it takes model after refined model to be computed so one may form the AE to the level of detail needed). The plausibility of the premises are always subject to refinement, or down right discarding. This negotiation of a problem situation is part and parcel to a constructivist’s view of how to learn mathematics. Science or engineering problems have many paths to solutions. The reasoning by analogy allowing a well framed model’s creation and in situ interpretation, is part and
147 parcel to AE formation. A kind of independent success here (finding adequate models for mind-independent processes “out there”) may lead to a kind of intrinsic motivation—if not inspiration concerning a striving for an objective framework for knowledge concerning the greater world around us. It is my interpretation of AE contexts so prevalent in pragmatic reasoning, for me, prompt an interest in the comment made by Aleksandrov (1963).
But the rigor of mathematics is not absolute; it is in a process of continual development; the principles of mathematics have not congealed once and for all but have a life of their own and may even be the subject of scientific quarrels. In the final analysis the vitality of mathematics arises from the fact that its contents and results, for all their abstractness, originate … in the actual world, and find widely varied applications in the other sciences, in engineering, and in all the practical affairs of daily life; to realize this is the most important prerequisite for understanding mathematics. (Alecksandrov, 1963, p.3)
148 CHAPTER VIII ED AND ANN’S REASONING USING NMI COMPUTING: REFLECTIONS
Introduction This chapter considers the work Ann, Ed, and I did as evidenced by their NMI computer files (Appendices B, C, and E-J). Upon several iterations of reflection on the data, I see two additional themes (to the four NMI embodiment distinctions) which are explored below. The themes complement one another as a) NMI has the capacity to force students to work at high degrees of mathematical explicitness, while b) NMI free’s students to focus their attention on working with higher level mathematical theory because of what they may take as implicit—given their level of mathematical understanding. The latter facilitates a taking for granted of, or a release from the tedium of certain levels of calculations and graphical drawing work. The NMI then was seen to not only allow Ed and Ann to work at higher levels of mathematics, but forced them to be explicit at such levels. Analogously, a central property of the NMI software itself is a complementariness of providing explicit natural mathematics representations to work with, while also implicitly providing real-time (or “live”) computations behind the interface. Calculators, and graphing calculators many times constrain mathematical expressivity—limiting it severely by the machine implementation at hand. My observations of Ed and Ann in working with NMI’s, is that their mathematically creative productivity is more a function of their actual mathematical capabilities and limitations, and less a function of the machine implementation.97 This introduction then touched on explicitness and implicitness of both student use, and to a lesser degree, machine implementation, of NMI in an educational context.98 The NMI has explicit and implicit mathematical features, while students can explicitly
97 This is the crux of the idealization, ‘NMI’ from a mathematics education perspective—an aspiration towards an ideal of natural mathematics representation that is explicit at any chosen level, and does not burden student’s with learning computer program specific syntax oriented material--whether it be GUI or command line.
98 NMI implementation considerations were taken up in greater detail in Chapter VI.
149 show their work, or take lower level mathematical objects for granted—as they are implicit to their own reasoning—a part of their tacit understanding. It is important to keep in mind that mathematical representations can be implicitly assumed by the NMI that are in no way implicitly understood by the student. This can be a negative of NMI use--but was not of note in this particular case study.
Ed’s Use of NMI to Work at Higher Mathematical Levels Ed’s work was primarily concerned with algebraic manipulation of large matrices at levels that were not feasible to do by hand. The screen shots in Appendix A, Figure 1, shows the matrix manipulation choices available with the SNB NMI, while figure 2 shows certain graphing capabilities. These screen shots show some of the capability the NMI had to allow Ed to work at higher levels of mathematical theory—leaving the underlying or implicit calculations to be handled by the NMI. Appendix C demonstrates some of the kinds of exploration facilitated by this algebraic capability. As the data in Appendix C (Figures 5 and 6) can be readily interpreted, Ed and I were able to reason through, and concretely pursue mathematical ideas at higher levels than would have been feasible by hand. Figure 5 shows how when we could take for granted the calculation of extremely high powers of matrices, our attention could be focused on the higher level computed results, and conjectured properties of those results. Appendix C, Figures 6 and 7 show, after several weeks of consideration, a final pass at an operation’s analysis problem that would have been unapproachable without the NMI. We had no solution for this “Serial Production” problem, but put our best Markov chain model and its interpretation forward (see Appendix B, Figure 3, p. 5, and Figure 4, section 3.4).99 Ed and I strove to do the problem from a purely mathematical perspective—as it would be taught and learned in a mathematics course, not an operations analysis, or engineering applications course. The Markov chain transition matrix and circle graph, or state space flow graph, makes sense according to our interpretation of the problem. (Appendix C, Figures 6 and
99 The problem was written apparently to be solved with the use of an online MUI cast in the parlance of operations analysis.
150 7). We had our suspicions though as the solution had a low production throughput of 56%, while the disposed of product was only 2%. Although we are uncertain about the correctness of the model, we did feel confident that our abduction concerning the explanation of how the model was derived made a lot of sense. If the research time allowed, we felt confident we could abduce either (a) an explanation of the throughput versus discard percentages, or b) a reconfiguration of the transition matrix if necessary which would deductively lead to more sensible results. This working out of purely theoretical or mathematical models (i.e., Markovian theory) provided Ed and I with a perceptual judgment of an Applications Embodiment (AE) for a Markov process. Before a perceptual judgment was formed we needed to abduce a clear explanation—in our own words—of a) the serial production problem itself, and b) the relevant aspects of the theory of Markov chains. A Markov modeling theory explanation is seen to be abduced in the “Particular Example” of Appendix B, Figure 5, Section 1.3. We were then able to look at the “Serial Production” problem as an AE bridge between a real-world motivating curiosity of the problem, and the purely general mathematical modeling explanation of its implementation (Figure 5, Section 1.3). Ed was interested in the theoretical “circle graph” diagrams used in introducing Markovian properties. Ed’s work on these kinds of circle graphs (e.g., Appendix C, Figure 7) seemed to lead to stronger AE’s, as we discussed them providing a kind of spatio-geographical representation which laid out an explanation of the model in less compact, or symbolic terms. Although our NMI did not provide the capacity to produce circle graphs, this is definitively a natural mathematics diagram that future NMI’s might support. Ed and I shared a common sense of what was left implicit in the circle graphs— but this awareness inspired us to work that much harder in seeking an explicit solution to the problem. The AE which the graphs helped bring into focus were “looked up to” (please see AE in Table 1). They were intellectually perceived as a) interesting in the palpable reality of the particular problem situation, b) remarkable in their imaginable mathematical generality to similar complex real-world situations seemingly approachable (i.e., looked upwards towards) given only our fledgling knowledge of Markov chains,100
100 This is to say that the diagrammatic models held much implicit meaning—a kind of geographical knowledge component that symbols do not provide.
151 and c) an extremely logical schema—a kind of minimal visual representation for a theory that is complex to imagine without such diagrams (see the “Particular example,” Appendix B, Figure 4, section 1.3, p. 3-4). This AE perceptual judgment facilitated a new cognitive-organizational structure of tacit, or implicit knowledge (i.e., see “implicit cognition,” number 29, Table 1). Without the NMI, Ed and I would have had no means to find concrete, Matrix calculation dependent results for the circle graph model abduced in Appendix C, Figure 7 (see Appendix C, Figure 6, for the mathematical analysis).
Ed’s Mathematical Explicitness and NMI use In contrast to the higher levels of mathematical freedom rewarded Ed and I as discussed above, Ed was found to have painstakingly worked to make his Markov chain expository document as complete and clear as time allowed. This is demonstrated by seeing the progress Ed made from his first draft to last as displayed in Appendix B, Figure 3, to Figure 4. Ed wrote 11 drafts in all. The way I scaffolded Ed was by supplying him relevant texts, and acting as a kind of second author on his expository writing--asking his clarification on certain points, offering hyperlink footnote suggestions, etc. I found that the NMI encouraged Ed and I to clarify notation—as there is no standard from text to text on Markov chain theory (e.g., Appendix B, Figure 4, section 1.4). Further, it provided the capacity to introduce hyperlink footnotes—allowing middle-out reasoning backwards to more fundamental mathematical foundations concerning Markov chain theory, e.g., stochastic processes, probability theory, etc. (see Figure 4, “hyperlinked endnotes,” 1-12, pp. 14-17). Finally, the NMI forced us to re-think some of our work—helping us uncover mistakes that would not have been realized without the NMI providing us concrete numerical results (see Appendix C, Figure 6). This helped Ed be more deliberate and explicit in his write up of certain problem solutions (e.g., Appendix C, Figure 6). The NMI then was seen to facilitate Ed’s work at a very explicit level of detail in his mathematical exposition and problem solving. I noticed that the NMI would allow Ed to choose at which mathematical level he aimed to make explicit (e.g., compare the body of Appendix B, Figure 4, with the endnotes, or hyperlinked footnotes). The NMI then facilitated the computations which logically follow from that particular level to be computed (see Appendix C).
152
Ann’s Use of NMI to Work at Higher Mathematical Levels Appendix D display’s screen shots which make it somewhat apparent how the NMI worked to help Ann work at higher mathematical levels. Figure 8 shows 15 basic tools of Euclidean geometric construction accessible through the NMI (e.g., construct: parallel line, angle bisector, or segment’s midpoint). Figure’s 9 and 10 show how the NMI explicitly facilitates a student’s application of certain geometric transformations. Finally, Figure 11 shows certain analytic geometry capabilities of the NMI. Ann and I had several conversations reflecting on the representation of a circle analytically (Figure 11), and how that derived from the work she was doing with Pythagorean’s Theorem, and other purely diagrammatic work (see Appendix J, particularly Figure 30, and Appendix E, Figure 13, respectively). Ann remarked that the NMI “equations of a circle looked familiar from [our] trigonometry class” which led to an exploration using the NMI where we discussed equations for circles at various center points and radii (e.g., Appendix D, Figure 11). This exploration facilitated the kind of top down, or high level mathematical considerations which allow the relating of different kinds of mathematical ideas (e.g., here Ann related the Euclidean definition of a circle, and the distance formula learned in algebra). Appendix J, Figures 26 – 29 show how a simple drawing of a square inside another square can lead to creative mathematical work. Please refer to Appendix J. Ann was taught one way of producing the figure in Figure 26 by the method used in Figure 29. Ann demonstrated a solid NMCE for this square inside a square computational rendition. I interpreted her to see the diagram (Figure 26) in terms of what procepts (see Table 2) were brought to bear in her composing it (e.g., Figure 28 is constructed using only reflections about lines of symmetry). The visual interaction and feedback facilitated by the NMI, according to Ann, offered “interactions” unavailable to her using compass and straightedge. I interpret her description of these interactions as NMCE. Ann was able to bring more abstractly organized, or higher level mathematical ideas (geometric transformations) to bear on constructing the “computationally live” (or “drag-able”) diagram of the inner squares through the creative means described in Figures 27 and 28.
153 Ann discussed her creation in Figure 28, to my interpretation, to be a NMCE for the mirror reflection transformation. She said the visual interaction and feedback with the NMI gave her the ideas for making this simple kind of elegant construction. Ann also indicated it was intriguing for her to learn this way in that the NMI allowed her the freedom to create her own mathematical constructions and discover her own mistakes, and successes. The NMI was seen here to cause Ann many perturbations in trying to construct her mathematical content diagrams. Through NMCE interaction, Ann was observed to work through herself and learn from those perturbations.
Ann’s Mathematical Explicitness and NMI use When Ann first began to learn the NMI, she was faced with having to deal with very explicit mathematical activities, generally seen to be at the proceptual natural mathematics level (Table 2). Appendix E, Figure 12 shows a recursive diagram of triangles within triangles which Ann picked up on constructing within her very first hour using the NMI. She was observed to learn several layers of NMI articulation—each having proceptual content. For example, she was a) highlighting specific mathematical objects, while non-highlighting others (precissive abstraction), b) proceptually identifying and applying construction commands from the menu options (see Appendix D, Figure 8), and c) pointing, dragging, and noticing invariant figure properties—another proceptual conceptualizing activity. Later, Ann showed me several creative, interesting diagrams such as in Appendix E, Figure 13. This diagram I believe we had seen in a book, and she was able to duplicate as a geometric construction with the NMI. I asked her how she did it. She said she knew there must be a way to do it because the triangle in the picture was equilateral. So she constructed a segment, and by using the rotation transformation at 60 degrees, she could construct an equilateral triangle. From there, she used the “radius, center point” command at each pair of vertices of the triangle. The point here is that Ann was forced to think of making her drawing as a construction in explicit geometric terms.
154 Ann’s Explicit Reasoning concerning the NMI Construction of Geometric Figures GE’s are the most abstract mathematical embodiments relative to AE, IPE, or NMCE. Generalizing a GE from arithmetic to geometric distance was a fine display of explicit mathematical abstraction for Ann. Ann expressed excitement over how Theorem SB, “The side bisectors of a triangle are concurrent,” was proven using transitivity over equality (see Appendix H). This excitement heightened as Ann realized her GE concerning the transitivity of equality also was key in proving Theorem ABC, “The angle bisectors of a triangle are concurrent,” as seen in Appendix I. Finally, in discussing the circumcenter and incenter which occur as corollaries of these two theorems, respectively, Ann began to see the power, beauty, and generality of explicit logical proof (Appendices H-I). By way of her explicit constructions (just as with compass and straight edge) Ann was able to see what was logically necessary to complete the proofs. Appendix J shows results of Ann’s detailed reasoning concerning the construction of a square in a square—used in Appendix J, Figure 30, to prove Pythagorean’s Theorem. At first Ann was straightforwardly taught how to make the NMI construction used in Figure 30. After this, Ann creatively invented several other ways to construct this drag- able inner square which involved applications of geometric transformations significantly different from the one she was taught (see Figures 27-28). These multiple constructions she said helped her to see how to prove that the inner square was actually square (having 90 degree corners). She recognized that she had four identical right triangles, by Side Angle Side (SAS) congruency of triangles. I scaffolded Ann to look at the straight angle formed by a side of the larger square. She then told me that since the non-90 degree angles of the right triangles must add to 90 degrees, there must be 90 degrees left from the 180 degree straight angle to give the measure of the angle appearing as the corner of the inner quadrilateral—now shown to be a square—having 90 degrees at each of its corners. Ann told me that it was the painstaking creative constructions of Figures 27-28 that gave her the intuition to see this relationship—she dragged the inner square and saw the relationship between the three angles making the straight angle. In this case, I would interpret that the explicit transformations applied to make the four right triangles served
155 as an NMCE which gave her the insight to apply SAS—as she intuitively knew the triangles must be congruent by this relation of adjacency.
Deduction and NMI Use Ann’s work was centered on creating geometric constructions using the NMI, for which there were associated proofs. Ann was generally given a theorem to prove, whose proof was then negotiated through scaffolding over time. During this time, Ann would create relevant NMI constructions—some being quite intriguing (Appendices G, Figures 17-20, and J, Figures 27-28).
Certain Constructions and Proofs Studied by Ann Ann’s work on certain proofs helped me observe her approaches to deductive reasoning in light of her NMI use. Initially, Ann was able to explain, or abduce how it could be seen that the inner angles of an arbitrary triangle sum to 180 degrees (Appendix F, Figure 15). She expressed her own pedagogical interests, as an education major, concerning the motivation and teaching facilitated by the NMI’s diagrammatic relationship (Appendix F, Figure 14). She indicated that the movements of the diagram helped to understand the formal proof we studied and wrote in Appendix F, Figure 16. Next, Ann worked on a proof concerning equidistance from two points (Appendix G). She challenged herself to find a non-typical geometric construction which turned out using a higher level concept of a vector, and its translation (Appendix G, Figures 17-20). Appendix G details Ann’s work on the proof of this theorem (Figures 21-22), and how that work can be reflected upon concerning the four mathematical embodiments, or perceptual judgments discussed in the previous chapter. Work on the proofs of Ann’s work with respect to proving the Two Point Equidistance Theorem (2pt), Theorem SB (side-bisector concurrency), Theorem ABC (angle-bisector concurrency), and the Pythagorean Theorem are discussed below (Appendices G, H, I, and J respectively).
156 Ann’s Difficulty with Understanding the Proof of Theorem SB This section refers to Ann’s work on proving Theorem SB: The side bisectors of a triangle are concurrent; whose relevant construction is displayed in Appendix H, Figure 23, with a “Discussion and Proof” section following the figure.101 The discussion in Appendix H points out how Ann had trouble in seeing the need for both cases of the biconditional statement of 2pt (recall Appendix G). Upon reflection, I would interpret this difficulty to be caused by the NMI construction procedure. Ann was only required to apply the reverse direction of the biconditional in her quite creative, dynamically drag- able construction, and so apparently may have not realized the need for the forward direction to resolve the question asked of her in the Appendix H discussion. This was only my interpretation, as Ann had nothing specific to indicate about why she had trouble with the question about the proof discussed in Appendix H.
IPE, NMCE, AE, and GE: Final Examples The perceptual judgment called IPE is an initiation of reasoning point—even if it is nothing more than a rudimentary procedure which can be used pragmatically to initiate (bootstrap) an understanding of mathematical content quite apart from it (the other perceptual judgments NMCE, AE, and GE are likewise initial points of reasoning as discussed below). The IPE is a “proceduralizable,” a course of action that can be perceptually judged in a way that leads to (offers handles to grasp) an understanding of the situation it supports (i.e. certain mathematics content).102 As with GE, AE, and NMCE, IPE is a perceptual judgment relevant to what one wants to pragmatically accomplish through their subsequent reasoning (colloquially, one may call these four embodiments “intuitive hooks”). These embodiments are said to be perceived for two reasons, a) there is something mind-independent that is actually part of the perception, i.e., the NMI representations, and b) they involves tacit knowledge more than they do explicit reasoning per se, as mathematical embodiments are the initiation point of
101 Proofs and their specific discussion are generally found in the appendices containing their diagrammatic constructions. 102 Both mathematics and logic can also be seen as proceduralizables, however the semantics of these systematizations are on a different plane than the trivial MUI procedures which support the implementation of mathematical content. One might distinguish proceduralizable content (i.e., logical derivation from axiomatic hypotheses) from procedures which support the learning or implementation of proceduralizable content (i.e., formal mathematics or symbolic logic).
157 reasoning.103 Perceiving a mathematical embodiment is like seeing and judging there to be a bridge, which is quite obvious to the observer (requiring practically no reasoning), between a mind-independently perceived situation, and a mind dependent interpretation of the external situation.104 An example is Ann’s inner square work (Appendix J). Here she uses IPE to grasp a mechanistic sense of what can potentially be drawn independent of mathematical content (consider an etch-a-sketch, or a computer software “paint program”). Once Ann had this level of manipulative capability, she was free to begin consciously reasoning beyond what she had memorized, or knew would follow mechanistically, or logically, given her choice of “buttons to push.” Following this mechanistic familiarity, Ann was freed up to begin to develop NMCE’s through guesses and checks (abductive and inductive reasoning). This allowed her to form concepts about the successful results (e.g., noteworthy properties) of her IPE initiated reasoning. Finally, her learning of geometric transformations in this special case (Appendix J) provides a GE for future mathematical reasoning involving concepts of geometric transformations. Finally, one might consider how AE’s might come into play in this situation. Students who have built up insights concerning certain crystal structures—but be at a loss of how to explicitly express their understanding in an elegant way105--might have ambit to pursue mathematical explanations for what they know only implicitly. This would be an example of how geometric transforms might be learned, or provided a motive to learn, where the crystal structures (similar to groups of apples) can be seen as AE’s. It became apparent that Ann was using the NMI for something more than a procedural vehicle to represent diagrams having little mathematical meaning. This was apparent in many cases. For example, in Appendix F, Figure 15, Ann viewed—via the
103 If we were to look at mathematics as a consistent set of proceduralizables, then it would be natural to think of learning about them, with them, and through them to be a working with manipulables—a way of thinking of the NMI. 104 We can think of certain mind-independent situations as being proceduralizable (in so far as it is in fact actually proceduralizable, e.g., formalized mathematics), while its mind dependent counterpart is the observation, manipulation, meaning, visualization, reflection, reasoning, use etc., as concern such procedures.
105 It seems likely that when deeply imagining something as discretely symmetrical as a crystal structure, one might be inspired to find a proceduralization (an orderly explicit representation) to express one’s implicit knowledge.
158 computational capabilities of the NMI—a way to choosing to construct a diagram capable of the mathematical dynamics involved in dragging a vertex of the yellow triangle in ways that demonstrated its ability to represent all possible triangular shapes. This “way to choosing” was an NMCE perceptual judgment, initiating the plausible reasoning concerning the abstraction and generalization necessary to comprehend the representation of all possible triangle shapes. There are also symbolic mathematical dynamics when reasoning through cases involving successive matrix manipulations seen in Ed’s work in Appendix C. These are proceptual dynamics—observation of the changing of the symbolic state vectors (Figure 6), or transition matrices (Figure 5) approaching steady state. These activities serve as examples of NMCE use. The specific use above was discerned through discussions with Ann about what she was learning using the NMI. Ann said once, “It’s the dragging around that helps me see the general things [properties] about the geometry [geometric constructions].” Ann emphasized the “general.” She also said that “this program helps me think about what I’m trying to prove… it helps me understand things from different angles.” She said she was referring above to a) the numerics, b) the learning through the drag-ability of the diagrams, and c) the properties she knew she had had to explicitly build into the constructions to provide them those drag-able properties. Ann’s “see[ing] of general things” due to the computing capacities of the NMI is a concrete example of what lead me to describe a NMCE. IPE and NMCE are seen as closely tied to NMI use. I observed Ann both create new embodiments (e.g., her intuitive grasp of 2pt noted in Appendix G), as well as generalize old ones (e.g., her use of the transitive property in two proofs; Appendices H and I). I consider these Generic Embodiments (GE). The NMI role here was to force explicitness in the building of the geometric constructions—but to also allow higher level mathematical concepts to be computationally available in making ones mathematical reasoning explicit (e.g., the vector translation shown in Appendix G, Figure 18).
159 CHAPTER IX SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH
Summary The research considers the question, “How might students reason mathematically when using Natural Mathematics Computer Interface(NMI) oriented tools?” Notions of a Natural Mathematics Interface (NMI) are considered proleptically as a segment of mathematics user software under the assumption that NMI will continue to be available, and more so, refined, into the indefinite future. NMI was basically analogized to a computer interface consisting of conventional mathematical representations. An explicit description of what is meant by mathematical “naturalness” (i.e., psychological) was introduced in the theoretical context presented in Chapter V. This was summarized in Table 3; Tall’s “Three Worlds of Mathematics,” a psychologically-based theory. This summary was presented in Table 3, Chapter V. Consideration of both metalesque ‘constructs’, and external ‘representations’ (see Table 1, Chapter IV; 20, 25) helped to make sense out of the subtle discriminations necessary for understanding what a mathematical embodiment is, and how it relates to what Peirce, more generally calls a “perceptual judgment (see Table 1; 3, 33).”106 A mathematical embodiment is portrayed as a particular class of perceptual judgment. Chapter V discusses how Tall’s psychological consideration of mathematical embodiment parallel Peirce’s epistemologically-based description of a perceptual judgment, in that they both describe an initiation point of one’s reasoning. An example of Peirce’s epiPeirce’s three senses of abstraction, especially in how they related to using an NMI are described in Table 1, 14-16, and further discussed in the conclusions below. Chapter 2 reviews literature concerning different aspects of technology use in mathematics education. It then relates these uses to Peirce’s general descriptions of pragmatic reasoning (perceptual judgment, abduction, induction, and deduction). Please note the descriptions in Table 1 (see index in Table 2) and the citations of C. S. Peirce.
106 Seymour Papert (1980) combines the external representations with the internal constructs under a rubric he calls “constructions.” See Table 1, 19.
160 Finally, a review of literature which draws relationships between Peirce’s pragmatic reasoning and contemporary constructivism in mathematics education research was presented. A research methodology of case study, and action research reflection allowed the freedom, but required the discipline, for reflecting on student’s mathematical reasoning via the particularity afforded by Peirce and Tall’s theoretical lenses. The research reporting and reflections took on a recursively emergent style—as particular threads emerged from reflection on the case study, focusing and re-focusing the operational descriptions of theoretical terminology, and how that theoretical terminology was applied in addressing the research question. The foci of theoretical considerations are tabulated in Table 1 (Chapter IV) and Table 3 (Chapter V). The point of recursive emergence here was in the feeding backwards and forwards applications of continuously re-written operational terminology to facilitate specific description of case study results, as well as to use the case study results as exemplars for that theoretical terminology. Therefore, if one attempts to describe a student’s mathematical reasoning while using NMI, there has evolved here a means for such description, and further, there are provided examples of observations and reflections concerning the case study which exemplify just what these theoretical terms mean in this instantiated research context. Ann and Ed, my two undergraduate research subjects, were made accessible to my research through a program at their university for facilitating these students’ summer study of some mathematics topics of their choosing. We decided to put together material that might be useful to other students, and so embarked on creating what we called a mathematics digital library. I worked as their research mentor, and guided them each to a use of particular NMI software packages to facilitate interactive forays into their mathematical topics. Ann used the program Geometer’s SketchPad to study high school geometry proof. Ann was an education major considering the teaching profession, and graduate work in education. She wanted to gain a deeper understanding of what mathematical proof was, what it was for, and how she might better teach her anticipated students in preparing them for their high school geometry work.
161 Ed, a mathematics major anticipating graduate work in financial mathematics, chose to study Markov chains. It turned out the NMI software program we used (Scientific Notebook) was particularly applicable to this work. Ed was not only facilitated to do calculational work with large matrices, but he was facilitated in composing his own mathematical exposition using the symbolic and graphic representation utilities of the mathematics word processor built into the program interface. The essential reason why these two programs were selected to represent what are called NMI was their ability to not merely externally represent mathematical diagrams and symbols, but to interactively carry out student signified mathematical operations (computations, mathematics word processing, and graphics manipulation). The program interfaces designated and interpreted mathematics in a way that was considered natural. That is, its naturalness could be inferred according to a) Tall’s educational sense of how students learn mathematics, as well as b) close imitation of the conventions used with pencil and paper to represent the mathematics on which they were working. For example, Scientific Notebook facilitated students breaking down their higher level mathematics work (e.g., calculus, differential equations, and linear algebra) into the simplest terms of grade school beginning algebra and arithmetic. These are areas of mathematics that calculus level students could be assumed to have fairly well developed mathematical embodiments—a requirement of Tall’s theory. The resulting observations of these two students showed significant interactive use of the NMI. A summarizing analogy made to their learning and reasoning using the NMI is that to scientific experiment, exploration, and theorizing—which were described as inductive, abductive, and deductive inference respectively. The students were also interpretively observed to be using Peirce’s three types of abstraction: dissociative, precissive, and discriminatory (see Table 1, 14-16). Peirce’s inter-related theory proved invaluable in describing students’ pragmatically oriented mathematical reasoning. An original result of the study was describing how I interpreted Ann and Ed’s use of the NMI as a multi-faceted, interactive mathematical embodiment. I was able to distinguish between four types of such embodiments. These four embodiments were interpretively observed to elicit, or initiate aspects of mathematical reasoning, and so
162 were interpreted to be ‘perceptual judgments’ as Peirce described (see Table 1, 34-37; and Table 4 below).
Conclusions and Future Research Questions Several conclusions and research questions are presented. Each section below describes conclusions and prospective future research questions from differing perspectives. Some of the ideas are dense, so they are written in several contexts. In other words, some redundancy was included to provide enough contexts to make the ideas clear. Following then, the prospective research questions are also motivated by the varied contexts of the conclusions. Several references are made to pragmatic reasoning terminology in Table 1 (Chapter IV), which is alphabetically indexed in Table 2, and natural mathematics terminology in Table 3 (Chapter V).
Abstraction, Embodiments, and Reasoning using NMI There are several general conclusions the research support as to, “How might students reason mathematically when using NMI oriented tools?” Those primarily concerning I consider of more general significance and are discussed below, while more specific conclusions are drawn in the following section. It was striking how much spontaneous abductive and inductive reasoning paths were opened to students using NMI. I believe the mathematician Polya (1957), considering his discussions of “guessing and checking” would find NMI extremely useful and pertinent to mathematical reasoning. Deductive reasoning though, was not something seen to be of direct consequence in the use of NMI. However, it was observed that NMI use facilitated the learning of mathematical content necessary to move ahead from plausible reasoning to deductive reasoning.107 Following are concluding descriptions of how Ed and Ann were interpreted to move through widely varying reasoning acts using the NMI.
107 This is reminiscent of van Heile levels—the sequencing of what needs to be known before other types of reasoning can take place as regards the approach to high school oriented geometry proof.
163 Empirically Tagged Theoretical Description: Ed Peirce’s triadic distinctions between modes of abstraction (see Table 1, 14-16) were observed in both Ed and Ann’s reasoning. Ed was very private about his precissive abstraction activity—he did not talk much about it, but it was apparent to me that he was making progress here. This was apparent in his (as well as my own) concept formation, explicitly noted through how our NMI use in scaffolding compared to discussion of text book material. Recall that Peirce’s term “precissive” designates a separating out, or purifying of one’s representation to oneself (i.e., construct formation) from so much of the non-representative (of the purely abstracted concept) clutter involved in reaching such a point of understanding. Our discussion throughout Ed’s eleven drafts, beginning from his first to final (see Appendix B, Figures 3-4), evidenced much refinement of particular concepts in the sense of his moving from implicit glimpses of concepts to the explicit mathematical exposition, exploration, experimentation, casting problems in different theoretical frames, and attacking problems of greater complexity as the semester progressed. Each advance above was directly facilitated by his NMI use—as described in Chapters 6-8 above. The advances in exposition was evidenced in his setting up of clearly explained and presented premises for deductive arguments concerning proof of the Markov chain steady state theorem (a proof that was beyond our grasp). This was facilitated by the mathematical word processing made available through NMI. Abductively inferred exploration, inductively inferred experimentation, and a deductively derivable eigen-value approach were evidenced in his use of NMI.108 The Hyena problem
108 The eigen approach is a sophisticated hand computation oriented approach. It is my experience (which may further make certain terms clear) that hand calculations are dissociatively abstracted—separated in “piece parts” of memorized algorithms—a Peircean analogous “icon” level of discernment. The beauty is that there are precissively available abstractions available (concepts of the mathematics of beams buckling, or violins and their strings resonating), which further, facilitate discriminatory abstraction of the highest type. Separating logical inferences from mathematical objects and their properties—doing proofs. There is a distinction between deductive proof, and procedural calculation. One is so generally useful, and the logical deductive paths of it are used so often, that they are called calculation rather than deduction (they are thought of as algorithmic). Typical deduction in terms of proof is the reaching, by idiosyncratic logical chains, of what might seem like obvious conclusions (shown valid in nongeneric logical paths). Also, calculations typically involve ends in themselves, where deductions involve to coming to a possible means for more deduction. The point here is thought that both calculation and proof deductively follow, they have necessary results if they have well formed propositional premises (ones that are either true or they are false). I believe calculation fades into proof—only in proof one already has the (abduced) conclusion before starting—but such is the same in checking an estimation as regards calculation. But now the discriminatory abstraction at this high a level (proof) is so subtle that one can argue that either the logic (as
164 (Appendix B, section 1.9), and the Serial Production problem (Appendix C, Figure 6-7), along with its recasting from a numerically recursive problem to a closed form eigen problem evidenced his work on more complex problems and their theoretical reframing each greatly facilitated by the NMI.109 Ed’s reasoning using the NMI facilitated precissive abstraction in conceptualizations moving from pictorial-like explanations (Appendix B, figure 4, examples in 1.1 and 1.3), to refined mathematical representations (progression from Appendix B, Figure 3, to Figure 4, sections 1.3-1.9 and accompanying endnotes), however, not without scaffolding from the researcher. It must be said that the main work in representation beyond the pictorial levels was accomplished by Ed with minimal or no scaffolding at all—just a tenacity to represent and re-represent his refined mathematical conceptualizations—considered precissive as they moved from explanatory conversation or written explanation to solid, concise, propositionally oriented mathematical exposition. I interpret this reasoning as moving from abductive inference refinement to precissive abstraction into clear perceptual judgments of the tersely stated meaning of the Markov Property (Appendix B, Figure 4, section 1.2—the textbook writer’s abstract representation), to explicated mathematical representations of Markov chain processes and steady state properties presented from the student’s understanding (Figure 4, sections 1.3-1.9). The abductive explanations were refined to the point of precissively abstracted perceptual judgments (mathematical insight or “intuitive” grasp), towards progressions of representations of those representations—so common to mathematics, a product of discriminatory abstraction (Table 1, 16). This is noted in one instance (or others similar) where Ed and I moved from the diagrammatic representation in Appendix C, Figure 7, to the abstracted transition matrix representation in Figure 6. In conclusive summary, Ed’s work described above were reasoning acts which intellectually moved him ahead. These acts were in no small way facilitated by the interactive constructional tractability (Table 1, 19) offered by the NMI. When Ed began, he was not able to externally represent based on internal constructs, using the NMI or in
a representational system) is represented as mathematical representations, or the mathematical representations are represented by logical representations (Table 1, 16). 109 Facilitated in the sense that the tedious eigen calculations were done automatically—so that one only need know the theoretical approach without having to perform the tedious computations that can be left to a machine. Most (if not all) natural mathematical calculations done by machines are “typical” deductive paths, thus named “calculational” in that they are so generally of use that they take the name of algorithm.
165 conversation, what Markov chains meant other than copying down a text book author’s already abstracted definition (Appendix B, Figure 3). However, following his successive expository drafts, and intermittent exploration and experimentation using the NMI (and diagramming by hand and using other programs), Ed’s work evidenced representations in forms such as in Appendix C, Figure 7, along with his expository mathematical representations (Appendix B, Figure 4).110 These showed strong indications of his forming many varied constructs concerning concepts regarding Markov chains. The primary role I saw in the use of NMI was a) his motivation to re-think and re-write his drafts where the NMI served as a mathematical word processor, b) his abductive participation in our exploratory and experimental (inductive verification) work evidenced in Appendix C, Figures 5-6, and c) Ed wanted to move forwards from the plausible reasoning activity characteristic of NMI use, towards the conclusive, or formally demonstrative multiple-chaining reasoning activities available to students of mathematics—to engage in precissively abstracted perceptual judgment and conceptualization. I conclude that Ed’s NMI oriented plausible reasoning work did prove pedagogically useful for aiding his precissively abstracted conceptualizations involved in personal perceptual judgment. It further seemed to challenge Ed to show things he was capable of that the NMI could not do. For example, Ed worked to exhibit his own mathematical explanations (abduction), and, framing (Table 1, 12) of propositions so they may serve as deductive premises (see sections 0.6-1.5 of Figure 4, Appendix B). I finally conclude that this higher level re-representational (or theoretically explicit) framing
110 Here is a point where it may be considered explicitly, if mathematics is a language, or something more, or less—or something altogether different—and only analogous to language. It would be difficult to think of a “natural language interface” in the sense of a NMI. NMI’s are interfaces to machines that respond, through computations, in a dialogical way. Word processors, for example, have no “computational” backgrounds, other than grammer and spell check. Machines cannot hold conversations (have dialogue) in natural language (as opposed to NMI interaction). It is interesting to note, on the other hand, that machines are approaching abilities to translate from one language to another. This is an area von Glasersfeld worked in. He came to the conclusion the interfacing of language to language had too high an error rate to be considered viable—leading to his “internalist” construtivist positions (minds are the interface, and they cannot be “externally” represented). Point being though, generally speaking, mathematical dialogue is meaningful with a machine (across say an NMI interface, or a simulation)—this is not the case with natural language (this is one of the enormous projects and problems in the field of artificial intelligence). Peircean abstractions lead one to reognize mathematics as having a good balance of dissociative, precissive and discriminatory abstraction, while natural language is weak in the precissive abstraction area. Mathematics has precise objects while natural language has tremendous discriminatory power—perhaps necessitated in that its domain has few “precise” objects. Natural language is “about” things, not being a thing (i.e., a circle, square, eigen-value, statistic, etc.).
166 (formal-axiomatic ‘natural mathematics’ thinking according to Tall, Table 3) facilitated by Ed’s perceptual judgmental insights, and recursive re-writes using NMI, exhibited discriminatory abstraction. These kinds of abstractions I conclude are useful for distinguishing what a student might (or might not) be understanding in mathematical subtleties when using technology such as NMI—as NMI does not demand, in itself, work on proof (as opposed to calculation--see footnotes 110-111). NMI may automate for the student beyond what they have a precissively abstracted concept with regards to its proceptually related activity (see Table 3).
Empirically Tagged Theoretical Description: Ann Ann, differing from Ed’s style of learning, seemed to enjoy the exploratory abduction and inductive experimentation offered by the NMI—with minimal and eventually no scaffolding at all (in fact, for the NMI exploratory work she did in Appendix J, Figures 27-28, she scaffolded me on what she had accomplished). I conclude that Ann picked up strong IPE’s right from the start, and began her studies from this base, taking them forwards to NMCE insights (see Table 1, 34 and 36). She was able to jettison the concreteness of the IPE embodiments111 and move forward building concepts through precissive abstraction, (see Table 1, 15) a construct formation aided by external interactive dynamic diagram NMCE’s (e.g., her dragging around of “pictures,” but, soon facilitating her perceptual judgment of related mathematical object constructs (Appendix F, Figure 15). Ann remarked, in the researcher’s words, that her pulling together the concepts (i.e., colligating a copula facilitating a precissively abstracted perceptual judgment) to latter prove theorems was made easier through our NMI facilitated work. Much of this was done in an efficient manner due to the implicit, or intersubjective meanings she and I had negotiated in using the NMI together, as a kind of interacting intermediary which, quite usefully, could not be biased or swayed to one person’s side or the other.
111 As discussed, IPE’s involve iconic, or picture-like embodiments, as opposed to embodiments of representations. Picture-like in the sense of memory intensive, reasoning weak. I interpret algorithmic thinking to be analogous to picture-like. This differs from “visualization”—as internal visualizations are mentalesque representations, rather than mere iconic “things” in themselves—non-representative of anything slightly more general. I can imagine a boulder as a big rock, or, as a feeling of heaviness and instability, as its near the edge of a cliff, and one might be “teetering” under a heavy load—such being a representation aside from a mere “big rock.”
167 I conclude her first mathematical conceptualizations can be interpreted as results of precissive abstractions lifted from the IPE interactions with which she was so comfortable and familiar. This kind of work is evidenced in Appendix E with its pictorial-like IPE expressions inseparable from an easy going memorized algorithmic automaticity. However, as Ann’s interactions with the NMI became more deliberate, i.e., having the purpose of learning geometric concepts; the NMI was observed to function as a scaffold for her to build NMCE oriented concepts. I conclude that NMCE building follows from inductively reasoning through lower level (partly understood, partly cluttered) mathematical concepts (with the goal to fully, or precissively understand them). Her higher level diagrammatic conceptualizations were facilitated through interactive NMI use--dragging around of IPE facilitated literal “pictures,” as well as non- thinkingly following algorithms (Table 1, 31)—a “picture-seeing-like” memorization intensive activity. The ‘pictures’ moved from external, recognized from memory icons, to precissively abstracted NMCE perceptual judgments—these insights taking her to the mathematical representation, construction, and construct stage (Table 1, 25, 19, 20, respectively). Here, she became inductively confident in her interpretation of the NMCE represented perceptual judgment of the three straight angles, recognizing them as alternate interior angle formations (an inductive inference over successively dragging- represented image frames). Ann at this point might then be described to have moved from this plausibly oriented NMCE to a precissive GE of purely general mathematical insight at the heart of perceptually judging there to be all possible triangles (to similarity) represented in the geometric construction in Appendix F, Figure 15 (see Table 1, 19). Note, interestingly enough, that geometric constructions are in fact technologically derived representations, i.e., compass and rule. Its important here to see that NMI presents a more complex virtual manipulative than a compass and rule—it can be ‘un-natural’. It seems clear that educators need to be weary with younger children as to not cognitively overload them to the point where they can do nothing but abduce, in that there is no way for them to make mathematical (precissively abstracted) sense out of all the “moving diagrams.” This can force the mathematical diagrammatic nature of the representation to be a) lost in one’s interest in
168 the pretty pictures (e.g., Bob from the pilot study), or b) unaware of how intellectually overwhelmed they really are—taken in by how easy it was to merely kinesthetically interact with the “picture.” Here it seems, the kinesthetic control can lull one into a false sense of intellectual security (i.e., not being perturbed in that they have a false sense of confidence in being able to move the diagram “like the teacher did,” while the teacher’s intended lesson was not comprehended). There come opportunities where IPE’s can progress towards GE (i.e., student manageable perturbations). How can teacher’s help facilitate these opportunities? These are conclusions and questions I bring forward from as far back as the pilot study, and see them as an area that might be addressed in future research. Reviewing above then, Ann’s first alleged GE (per her previous NMCE induction, and that inferred per IPE abduction) was explicitly evidenced in Appendix F, figure 14 (see the far right statement about “four lines”). I interpreted her reasoning act being of a precissive abstraction, from IPE to NMCE--moving to the inductively prefaced NMCE relationship noted by Ann through interactive visual-dynamic external representation using the NMI. Finally, Ann makes mathematically central middle-out reasoning moves (Table 1, p. -10) during our scaffolding in putting together her first proof here (Appendix F, Figure 16). She was colligating a copula for a new kind of reasoning, a chaining of inferences, a multiple-chain deduction which was nontrivial for one who had not proven a theorem before (Ann indicated she did not have this opportunity in high school growing up in the projects in Detroit).112 This copula guided her to explicitly state what premises were necessary to set up the proof of the stated theorem (see Figure 14). This reasoning act of well-forming mathematical propositions I observed to be strongly related to NMI use. Figure 14 also came about through Ann’s drive not only to reason through the proof, but also in consideration of her future grade school students, in wanting to figure out what they needed to get a handle on as they moved ahead in their studies.
112 Ann and her sister both had full scholarships to Harvard, in large measure, Ann said, due to her nearly perfect SAT scores. Ann told me if she missed a spelling word on a quiz at school her grandfather had her write it 100 times, no questions asked. Ann’s choice of universities, she said, was determined by her family housing considerations as she was a mother. Ann was also a founder and director of a campus club for women to help them further their future careers, and a part time model, having had the opportunity to appear several times on the Music Television Video (MTV) show.
169 As Ann became better at learning to deductively reason through, and doing proofs (Appendices F, G, H, I, and J), she spoke about her having used NMI for what she considered necessary background learning, which I interpreted to result from her plausible reasoning using the NMI. I interpreted her then, in preparing for proofs, to progress to colligation of copula of premises and this plausible background knowledge (this knowledge helping to form mathematical propositions).113 There then was a general progressing forwards again to middle-out, plausible reasoning (e.g., conjecturing, or abducing; internal visualization, concomitant to perceptual judgment). These reasoning acts I interpreted through student observation and member checking conversation. When Ann came to understand a proof path (i.e., succeeded in completing a deductive multiple-chain; premises forcing the conclusion, all constructed using clean propositions), I sometimes noticed that she felt it (the proof) “end to end,” it was apparent in her affect.114 Sometimes she (the teacher in her) was disappointed in my scaffolding, happily though there was a sense of serenity in comprehending the entire proof, in sharp contrast at times to our scaffolding conversations at the NMI. Although I sometimes scaffolded in ways she didn’t like, her own sense of accomplishment was far greater in that she (in large part) did the reasoning—plausible and deductive. Ann also did here own dissociative abstraction--separating among MUI driven procedures, even when there was no definitively reasonable sense for why these procedures were implemented in just how they were (differing drastically from the NMI motivated interface procedures). I conclude this is dissociative abstraction in that initially, before any mathematics enters in—a key stroke is a key stroke. Ann, over time began to separate out which key-strokes performed mathematical (i.e., NMI) functions, and which were simply artifacts of the program implementation. It was also interesting to note how excited she was when that something I had taught her on a blackboard in a previous mathematics course could be NMI- dynamically depicted, i.e., her first GE discussed above concerning the alternate interior angles theorem.
113 I reflectively observed that learning mathematics is difficult in part because students have trouble with believing in their own black and white precissive abstractions—believing that something (their own mathematical mentalesque construct, or concept) is absolutely true, and if not, is absolutely false (a logical statement or proposition)—not something common in everyday plausible reasoning. 114 The proofs we did had few steps, and it was possible to have a sense of the whole deduction “at once.”
170 Finally, I noted Ann’s fulfillment in successfully deducing theorems— independent (at that time) of any NMI use. I interpret that her previous GE’s (compact insights) were needed to facilitate the discriminatory abstraction (Table 1, 16) it takes to hold together a correctly ordered admittance of mathematical objects in logical adaptation to force a sought result. Ann did not simply precissively abstract a proof, but instead had to chain together several precissive concepts (propositions), discriminating which were needed or not, and which logical order to present her argument so that the theorem’s conclusion was inevitable. These reasoning acts were subtle but sharp, discriminatory abstractions. I conclude that Ann both gained confidence and competence in her plausible reasoning and abstraction capabilities using the NMI, as well as was able to transfer this knowledge (through middle-out connections) into the deductive chaining; reaching forced conclusions drawn from propositionally framed premises. This is how I interpreted Ann’s deductive reasoning approach. She knew what the propositions she needed to start with meant (via intensive plausible reasoning acts, much of it using the NMI), and she knew (understood) where she needed to take them (again, in large measure through NMI intensive plausible reasoning). Concluding then, my interpretation of a plurality of observations was that a knowledge of geometric postulates (assumptions), properties, definitions, and constructions (clear propositions), were robustly learned through plausible reasoning use of the NMI. Concerning proof, it was then a middle-out reasoning task for her to back out a demonstrative argument, applying the geometric propositions at hand, such that they be in conformance to the rigorous demands of a logical deduction of a certain theorem’s proposition (see Appendix F-J).
171 Exploitation of Mind-independent Representations It is of theoretical interest that what is external to Ann—or “mind-independent”— is something she takes information from with greater confidence in knowing that she is being shown something “real.” This stands opposed to a purely mentalesque construct with which she may feel to be misled in her internal visualization. The external, or “real” experiment verifies the construct she is, has, or will form. It very much involves mentalesque constructs in that there is substantial interactivity with the mind-independent reality—her forming and reforming it to her sense of how it would be if her construct were correct (roughly speaking, a process of attempted disconfirmation). Theoretically, the student works with mind-independent reality much like getting an “objective” answer from an instructor she trusts—although the NMI removes certain doubt concerning her a) perhaps misrepresenting her question to the instructor, b) the instructor misinterpreting her questions, or c) whether or not the instructor is competent here. Ann has free reign to experiment by all manner of means—and there are many (precissive) means available to her on the NMI that are not available to her with verbal, or compass, rule construction situations, due to possibly impatient instructors with large class sizes, or demands of tedious drawing activities. The theoretical point is that the student sees their work in light of a mind-independent reality, while the researcher, rightly so, makes consideration of her internal understanding (or constructs). I have concluded that NMI provides much implicit information to instructors in observing their student’s work, as well as much useful work done by the NMI under the student’s explicit control.
Kinds of Abstractions: Ann and Ed I discussed with Ann her making of dissociative abstractions. For example, she was making distinctions while working within the “picture,” or the interface procedure level. I conclude this to be a level of concreteness, with emphasis on memorization involved, and the windows layout of iconic pictures on the interface that led me here. Icons on computer interfaces are spatially oriented pictures (e.g., a printer to print, a hand with the index finger extended, is a picture of the practical action itself; of pointing with
172 the mouse).115 Further, there was dissociative abstraction, or separation between the iconic interface procedural use, and the pictorial (not seen as diagrammatic) representations displayed, interestingly enough, as a result of the iconic interface procedural work. Finally, there was the plausibly reasoned (perhaps implicitly) distinction between different memorized procedures regarding particularly the MUI implementation details. These dissociative abstractions (picture level from picture level, or arbitrary procedure from arbitrary procedure—see Table 1, 14) built the concepts facilitating the IPE (see Appendix E). Later, using these separations of the iconic memory intensive, she would begin to associate, and precissively abstract from, various iconic level commands to do mathematics proceptual work (see Table 3). This I see as NMCE formation as it involved with Ann, the real-time, or relatively instantaneous, interactively sought computation of seen to be mathematically relevant graphics. The computations behind the interface can result in an NMCE, or, an NMCE perceptual judgment can be facilitated by the computational power behind the interface. This separation of a mathematical representation I conclude to be a precissive abstraction as it moved Ann from dynamic pictures involving memorized (but mathematically oriented) procedures to mentalesque representations or constructs. Just as students generalize the embodiments of “apples”--Ann generalized from the pictorial or iconic procedural IPE level of perceptual judgment to a mathematical (NMCE) construct. I conclude these can be a beginning point of mathematical concept formation (e.g., a circle are all the equidistant points from its center, see her maturation of this concept via Appendix E, Figure 13, to Appendix G, Figures 18-20, to Appendix I). Ed’s NMI use helped him with his discriminatory abstraction reasoning acts (Table 1, 16). This was noted in his careful iteration through eleven drafts of work, carefully refining his chosen notation to best fit his own construct of the Markovian theory’s intertwining of probabilistic, statistical, and linear algebraic representations.
115 These iconic windows operating system screen layouts are designed to be (nonmathematical) embodiments, and so ideally speaking, do not require reasoning beyond abduction for their use. The interface designer chooses iconic, windows screen layouts to be images which are so familiar and practical (action orientations) as to not seem to be abstract representations, let alone mathematical abstractions.
173 Concrete to Abstract Perceptual Judgments At the high school geometry (including proof) and calculus level mathematics courses, there are great levels of abstractions occurring for student’s who have a genuine understanding of the material (e.g., are not just manipulating symbols from memorized formulae). If these students were using NMI, I would hypothesize they might construct embodiments at various levels of abstraction. Recall that Peirce orders least to highest levels of abstraction as dissociative, precissive, to discriminatory (where discriminatory can do anything the other two can and more). Following Peirce, abstraction is a separation of certain types of things. Dissociative abstraction separates picture level icons from other icons, or just the image of something from another image of something related to it (e.g., a branch from a whole tree). Precissive abstraction separates out a representation of something from the (iconic) thing (e.g., the concept of addition from looking at apples—this is typically the level of mathematical embodiment when using non-computer manipulatives like Deine’s (1961) blocks). Finally, there is the separation of a representation from a representation which I conclude to be common in NMI use near the level of high school mathematics just mentioned. This is because the NMI already is presenting mathematical representations, so indeed, forming abstract embodiments from NMI interaction is likely to involve discriminatory abstraction.116 IPE typically requires not much more than dissociative abstraction, while the other embodiment types, the way I see it, may involve all three types of abstraction. Below is an idealized sketch of how I see the embodiments in relation to one another as regards abstraction in general. Table 4, from top to bottom, is an ordered list of the most concrete perceptual judgments to the most mathematically abstract.
116 There is the interesting case of abstracting an AE from a GE—say interpreting a deeply understood mathematical model (to the point of it being an embodiment) to some kind of image of a physical situation. I suspect Peirce would call this a discrimination in that one is not merely “viewing” the physical situation from simple memory, but viewing it in terms of theoretical properties predicted by the model—this abstraction involving deductive inference from the premises one assumes regarding the model and its parameters.
174 TABLE 4. Reasoning Acts Leading from Concrete to Abstracted Embodiments. Kind of Typical Some NMI independent Some NMI dependent Embod Orientation of reasoning acts leading to reasoning acts leading to iment Embodiments these embodiments these embodiments IPE NMI dependent Fluent, compact Fluent, compact memorization of step-by-step memorization of individual (by hand) algorithms—the computer program specific “interface” can merely be a idiosyncrasies, however, piece of paper. Involves these idiosyncrasies have nothing more than logical correlation with dissociative abstraction, e.g., computer command algorithmically summing structures that are based on columns of numbers requires natural mathematics. nothing more than: a separation of numbers in certain columns, numbers which are carried, numbers which are added, etc.
NMCE NMI dependent Interactive use of paper, Same abstractions. Using pencil and erasure. Typically results of computing power, involves discriminatory user friendly interaction abstraction (when with great range of natural computation is proceptually mathematical understood);117 separating representations, e.g., computed representations symbols and diagrams. from internal. AE Can be either Curious wondering and Same, but admitting NMI depend- pursuing knowledge of a broader, deeper ent or independ- real-world situation mathematical situations and ent, e.g., real- concomitant with their explanations which can world problems mathematical modeling or be pursued using interactive of less, or, model interpretation. NMI as a tool. NMI allow greater Mathematics is used strictly certain model uses (e.g., complexity.118 as a tool. All three with fine-grained data) not abstractions may occur. available before.
117 If the computing process is not understood, than the abstraction is precissive, since it is a moving not from representation to representation, but from a concrete IPE (a memory orientation termed here to be iconic in Peirce’s semiotic sense), to a precise mathematical concept or, representation. 118 This research does not conclude that “more complex” necessarily leads to more accurate and robust AE’s. Robust here is in the sense of “gracefully degrading.” For example, in artificial intelligence (AI) pattern recognition, a pattern which can still be recognized when it has been rotated speaks to there being a more “gracefully degrading” AI recognition algorithm. This can be analogized in a sense to a perceptual judgment (i.e., the recognition of the pattern).
175 TABLE 4. Continued GE NMI indepen- Involving a mathematical Same, but admitting greater dent. situation with which a freedom for a prefacing student is intimately familiar spontaneous interactive (e.g., understood computational and computational procedures), diagrammatic exploration leading to its forming a more and experimentation in general, compacted, and achievement of a familiarity precissively abstracted which may serve as this (a construct. A pinnacle pure GE). mathematical concept. Invites Discrimination.
Table 4 provides a summary of certain reasoning acts which might aide students forming mathematical perceptual judgments (mathematical embodiments). Top to bottom idealizes a progression of embodiments from the more concrete to the abstract.119 “Same” refers to previous box to the left.
Future Research Considerations Several questions come to mind in light of the conclusions above. How are colligations drawn? We mentioned hypothesis formation above. Yet how does a student know where to go, either intellectually or through NMI use, to gather the copula, or perform the action of colligation? We mentioned several kinds of abduction. However, what may facilitate this hypothesizing? Roughly, one may say, “The student’s conceptualizations of the situation led him through various perspectives, contexts, etc.” The researcher suggests that the pragmatic forms of abstraction are, at the least, usefully central to describe how students mathematically conceptualize (Chapter 4, Table 1, 14- 16). But notice that pragmatic reasoning act discussion is not for the purposes of describing the perspectives experienced by the student, or the context in which he or she found them self. Rather, it is to devise descriptions for discussing how the student investigated, navigated, and arrived at their inferences—context come what may. Pragmatic reasoning methods facilitate the student to be sensitive to their problem context, but the reasoning act descriptions—their definitions as a theoretical
119 One might interpret AE, at times, to be more abstract than GE—this being the case for the second kind of AE described in Table 1, 37.
176 terminology—are not at all context-dependent. This is why it is difficult and lengthy to try to describe what the pragmatic reasoning act terms mean. However, once they are understood, they make description of student’s reasoning with NMI fairly well disciplined, but open to most any relevant context in which the learner finds them self. Future research would be necessary for testing this hypothesis. Useful research questions might involve distinguishing student’s use of the different types of abstraction. Very young students might interact with various geometric shapes and learn to dissociatively abstract the triangles from the other common, memorized shapes.120 Precissive abstraction in students can be studied using NMI— observing their being able to derive inner mathematical understanding (create internal representational constructs) from external interactions initiating at a more concrete embodiment level. For example, moves from IPE to NMCE involve conceptualization or internal representation of what previously involved the blind operational use of procedures, missing the mathematical sense of (precissively abstracted) “objects,” or concepts, being manipulated. These procedures involved merely icons. Students found them so obvious in their dissociated (Table 1, 14) meanings as to not require explicit consideration—it would be as if they were looking at different “pictures.” These pictures were concrete, self contained perceptual judgments—closed-ended or complete—they motivated no further curiosity. However, when opened up (or recognized to be an entrée, rather than an IPE completed situation) they gradually facilitated pursuit of NMCE’s, via precissive abstractions, stimulating new, mathematical questions for the students. As a future research, all three types of abstraction (Table 1, 14-16) can be considered during certain instructor guided, pedagogical NMI learning tasks. Then one can see if these distinctions (in some repeatable, and predictive sense) are helpful in the description of how student’s conceptualizations in interacting with NMI relate. The task described above involved dissociative abstraction within IPE, but progressing to precissive abstraction facilitating NMCE embodiments. Other embodiment transitions due to other (or the same) kinds of abstractions can also be the subject of future research.
120 This relates to van Heile’s first level in learning geometry.
177 Explicitness, Four Embodiments, and Mathematical Manipulatives A significant conclusion I have noticed regards the explicitness to which the students do their mathematics using the NMI. Ed was observed to work hard to represent his mathematical ideas in a highly explicit manner using the NMI. Part of this I interpret to be a) because the stochastic, probabilistic nature of Markov chains theory requires careful exposition, b) because of his interest in compiling a mathematical document in his own words expressed in a natural mathematics language (see Appendix B, Figure 4), and c) because the NMI forces the use of certain natural mathematics syntax, in part, so that it might perform calculations directly from the written text. Aside from forcing mathematical form of exposition, the NMI facilitated a focusing of their reasoning to follow in a deliberate manner. For example, Ed and my conjecturing and experimentation in Appendix C, Figure 5, were distinct—the “guesses and checks” were explicitly deliberate. Ann’s explicit work is noted, for example, in her breaking her proofs down into separate cases, each expressing a conscientious use of diagrammatic constructions. This was shown in Appendices G, and I. Ann’s NMI, however, not only aided in explicitness due to its concrete geometric diagram display, but also aided her in coming to work with higher, or more abstract levels of mathematics (e.g., her work with vector and rotational transformations, Appendix J, Figures 27-28). This leads to my third conclusion--the NMI generally facilitates students working at higher, more abstract levels in their mathematics. Ed was observed to be able to work at higher theoretical levels in using eigen-equation solution techniques on large matrices, as well as the NMI facilitating other matrix manipulation activities (e.g., powers of matrices, and vector-matrix multiplication, see Appendix C, Figure 5). Finally then we see, following from Table 4 above, that the NMI can aide students a) in forming various distinct types of mathematical embodiments; b) both forces and enables explicit, as well as deliberate mathematical expression; and finally, c) helps students work at higher mathematical levels. I would generally suggest that a poignant description of how and why the NMI gets this capability goes to seeing it as a vastly flexible, yet mathematically pointed virtual manipulative.
178 I conclude there is a possible pedagogy associated here that maximizes resolvable perturbations in minimal time on task. The student is facilitated to take very small steps in their learning, allowing for many small perturbations within their grasp of resolving— assuming the student learns to use the NMI in an exploratory and experimental way. An awareness of the four distinct embodiments facilitates instructors to be aware of how they might present material in a balanced way. For example, some students may have a proclivity to want to attack the MUI (Table 1, 2) aspects of the machine, shredding and poking the software-specific command and windows structures, enjoying finding solutions for the puzzle of the software allowances and disallowances themselves. This is a means for them to grasp IPE, which has been seen with respect to Ann, can lead them to mathematical aspects of the NMI, as the NMI supervenes on these MUI, software specific aspects, but in a manner where once the IPE has formed, the supervenience of NMI on MUI dissolves. The connection between MUI and NMI can progressively become apparent to the student. MUI then serves as embodiment of direct logical facilitation of NMI, and in so doing, creates a transparency (a concrete relationship) between MUI and NMI. Other students may enjoy NMCE oriented work, say, exploring and experimenting with real, or student figured-out and put together data (e.g., large Markov matrices) in a computationally manipulative way. These numerical approaches are computing intensive and students may enjoy the power in using the computational engine behind the interface—learning from their mistakes along the way. This was true of Ed and my exploration (Appendix C, Figure 5) and problem solving activity (Appendix C, Figures 6-7). NMI certainly gives science and engineering oriented students (the great majority of the students in upper level calculus courses at the college level) opportunity to work on applied mathematics problems—taking advantage of AE’s facilitated by interaction with the NMI. The capacity for using real-world data, and the consideration of problems of greater complexity (e.g., integrating infrared heat signatures over more interestingly symmetrical surfaces; useful in pattern recognition of tanks or aircraft). NMI opens the way for studying complex problems in the biological, social and psychological sciences
179 as well. Long term projects students can work on over a semester become opportunities to explore areas of genuine interest. Finally, and most instructionally intensive, students can be guided (or work on their own) to discovery of generic patterns and relationships in the mathematics itself— facilitating their internalizing (e.g., discriminately abstracting) of mentalesque constructs (e.g., mathematical visualization; e.g., Flake, McClintock, & Turner (1990); Presmeg (1986); Presmeg, (1997b); Wheatley (1997)). They may not be totally correct generalizations, but the personal generics they discover and construct gives them footholds to move ahead in their mathematical concept formation through further abstraction and generalization. These student-personal conceptualizations might stem from the Generic Embodiments (GE) students can glean from a kind of iterated, or recursive feedback (doing the same general interaction but shaping it as one goes): sculpting, refining and polishing of their mathematical ideas which were interactively (e.g., experimentally) aided by use of an NMI. Here, an instructor can provide well planned problems to lead students into these pathways—not wholly unlike the lectures that have been put together in the past—only considering use of NMI interaction rather than chalk talk and question answering. The bottle-neck funneling student’s access to scaffolding, available typically through lecture questions and face to face work with the professor or a tutor can be significantly widened using NMI tools.
Final Conclusions and Research Questions The main conclusion I see overall from this research involves computer interactivity—in other words, the NMI can be seen to act as a (highly sophisticated) virtual manipulative. Student-NMI interactivity facilitated many cogent, fruitful pragmatic reasoning approaches for Ed and Ann’s learning, however, required considerable background in their mathematical knowledge before they were able to use the NMI at the level of mathematics portended by that particular NMI. That is, before they could use it in any way that they could make sense of the manipulations open to them. However, when students are familiar with the level of NMI mathematics, they are not held back in their learning, as how in some classroom situations this might occur. The scaffolding mathematical dialogue held between the instructor, student, and NMI helped
180 to focus pedagogical interaction, and facilitated an implicit means of communication providing for maximal resolvable perturbations in minimal time on task. Further, in response to student interactions, the mind-independent logical presentations of the NMI forced students to be particularly explicit in their NMI dialogue. Finally, the mind- independence of NMI representations can be exploited by students initially looking for a foothold. Then, a mind-dependent construct or conceptualization can be sought with which they can draw mathematical inference—abductive, inductive, or deductive.121 Several more questions I conclude with are: 1) what definable, general ways are there to usurp MUI aspects and make certain NMI more user-friendly? By definable and general, I mean in the sense that they can be passed on to the writers of the NMI software. 1) What are ways to prompt faculty to use NMI in their instruction? 2) How are college courses already using NMI122 different, and non-different from typical lecture oriented courses? 3) Does NMI use consistently facilitate students to work at higher mathematical levels? Are students found to have a greater explicit understanding in using NMI? 4) Relatedly, might NMI cause a greater implicit dependence on the computer? Or, on the other hand, might there be cogent and fruitful NMI independent mentalesque
121 I would suggest developing a philosophical discussion topic for prospective teachers that may enliven their interest in their chosen profession. Consider: a) the mathematical efficacy of notions of embodiments in learning mathematics, b) Peirce’s “essential elements of reasoning,” colligation, observation, and judgment (EP2, p. 24), c) Peirce’s sense that precissive abstraction (although generally less powerful than discriminatory) is (at least) equally powerful in mathematics, and d) “Unscientific people have a very imperfect sense of the shades of assurance that attach to scientific propositions. (EP2, 25).” NMI (and some other computer programs, see pp. 26-40 above) works as a manipulative, allowing externally aided colligational and observational activity, prompting the perceptual judgments of embodiments. How might one consider the art (EP2, p. 11) of their reasoning activities w.r.t. the deepest of hypotheticals; axioms explained as embodiments. This, in light of (d) above, can perhaps inspire students to consider the sense of perfection, and pragmatic use, of the critically chosen axioms in the maths, analogous to theories in science. And analogous to being taught F = ma in science, only to refute it later, we may intrigue students with the progress of axiomatizations (analogous to scientific theorizing)—but the major difference being, mathematician’s creatively choose their suppositions via pure reasoning—to Wigner’s (1960) amazement. So for future research, consider inducing perspective teacher students towards philosophical considerations by developing the richly debated notion of synthetic a priori (i.e. roughly, observed truth and a truth of pure reasoning), in the context of computer interaction-observation and creative reasoning in learning mathematics. Finally, consider: what might be the human cognitive constraints and capabilities that may back up this or that axiomatization scheme (see Wolfram, 2002)? Consider the meaning of a mathematical embodiment in this context. Consider the argument’s of Quine in this area, and the differences in precissive and discriminatory abstraction. 122 I am aware of coursework going on at the University of Colorado, University of Texas, the Naval Academy, and Kenesaw State University (in Georgia) using the NMI Scientific Notebook.
181 constructs resulting for students using NMI that aide in their building their own implicit (or tacit) knowledge of mathematics. 5) Might the framework on reasoning discussed here be used to discuss other mathematics education research that may have been done with respect to a detailed psychological basis?
Disclaimer Due to the nature of applying pre-existing theory to frame and discipline a case study, it must be explicitly stated that reflective interpretations were made by the researcher that, at times, required natural generalization on the part of the researcher. 123 However, these researcher interpretations apart from the case study data are clear from the context in which they are presented. Further, as described in the summary section, and research methodology (Chapter III), an emergent recasting of the operational descriptions of theoretical terminology was necessary to articulate a reasonable specificity to the case study and the researcher’s related reflections. Therefore, the researcher does not claim that the conclusions are nothing more than a direct condensation of observations from the case study data; only that interpretational reflections described make qualitatively general sense within the framework of the accompanying theory driven, emergent operational descriptions (i.e., literature reviewed in Chapter II, tabulated theory in Chapter IV, and focusing interpretations in Chapter V). The conclusions of the study are hypotheses, similar to other case studies which do not include such a disciplining theoretical framework. That is, case studies are just that, they consider individual cases (as in abductive inference), however, this study differs only in that its operational terminology is based on a fit between pre-existing theory and emergent case study descriptive demands. The conclusions are interpretations of empirical evidence. It is only that the interpretations were in part prompted by the operational theory. However, the conclusions are not taken far from the data in that the
123 ‘Natural generalization’ is a term for how a reader of qualitative research might interpret the research findings for their own purposes. Typically, natural generalizations are left to reader’s interpretations, not the researcher’s.
182 operational terminology emerged in concert with the observed data’s analysis and reflection. The researcher’s reflections drew from multiple experiences. They are, a) the case study observations, b) immersion in the theories of Peirce and Tall, c) readings in mathematics education literature, d) constructs learned in mathematics education research courses, e) teaching experience, and f) engineering and scientific research experience. By in large however, there are no conclusions drawn in the dissertation which the researcher does not feel confident that can be drawn through careful interpretation, directly from the case study—as there were triangulations among various parts of the research, and sufficient member checking to validate research conclusions. Such is a disclaimer for any generalizations that might be drawn from the researcher’s interpretations. I do believe though that some theoretical aspects of mathematics learning and reasoning, made clear to the researcher through reflection on Peirce’s and Tall’s theories (e.g., distinctions between dissociative, precissive, and discriminatory abstraction), are useful in considering student’s reasoning when using NMI. The understanding the researcher takes from the work is that there are multiple levels of generality and specificity (e.g., descriptions of reasoning compared with psychologically based descriptions, per various technology use orientations) that are helpful in the enormous endeavor of describing ways we can better educate students in mathematics in light of the rapid growth of technology use.
183 APPENDIX A
SCREEN SHOTS: SCIENTIFIC NOTEBOOK NMI
Appendix A contains interface screen shots representing some of the representation capabilities and functionalities of the NMI Scientific Notebook (SNB) are shown.124
Figure 1. Calculations using the NMI Scientific Notebook Screen shot showing “Matrices” selection on “Compute” menu.
124 These screen shots were actually taken from a sister program called Scientific Workplace—but the SNB presentations, capabilities and functionalities are identically a portion of Scientific Workplace.
184
Figure 2. Plotting using the NMI Scientific Notebook Screen shot showing plot of sin(x)sin(y), and some of the plotting features. This plot is a default setting, demonstrating ideal simplicity and fruitfulness of NMI graphing capabilities.
185 APPENDIX B
ED’S EXPOSITORY WRITING AND RELEVANT PROBLEMS
Appendix B contains four Figures (numbered 3 – 6) which represent the span of Ed’s work. The Figures following contain Scientific Notebook (SNB) representations. Figure 3, entitled, “Markov Chains,” contain a labeling of pp. 1-7, and Figure 4 entitled, “Part I, Markov Chains: Applied and Theoretical Study,” contains a labeling of pp. 1-17. Figure 3 is the initial draft of work done by Ed at the start of the research, while Figure 4 is the final NMI document of Ed’s exposition. Note that Figure 3 is 7 pages in length, Figure 4 is 17 pages, Figure 5 is 8 pages, and Figure 6 is 8 pages.
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Figure 3. Markov Chains Showing first draft of Ed’s work in Scientific Notebook (SNB). Figure 3 is continued on the following 7 pages.
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Figure 4. Markov Chains: Applied and theoretical study Shows Ed’s final paper. Figure 4 is continued on the following 17 pages.
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211 APPENDIX C
MATHEMATICS EXPLORATION USING THE NMI: SCIENTIFIC NOTEBOOK
This Appendix contains an “Exploration,” and a “Serial Production” problem. The last Figure 7 shows a mathematical diagram of the Serial Production problem typical of a Markov process.
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Figure 5. An NMI facilitated exploration: Conjecture on finding Steady State A conjecture which would not be feasible to explore without the NMI is considered. The conjecture arose from noticing a pattern during indirectly related work. A means to finding steady state vectors by taking Markov matrices to very high powers (order 1,000, to 10,000) was concluded to be extremely likely. As it turns out, this is a theorem. Figure 5 continues for 7 pages.
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Figure 6. An NMI facilitated exploration: “Serial Production” An Operations Analysis problem here requires computation. Ed and I explored modeling this problem via an 8 by 8 Markov transition matrix. The problem was designed to be solved using a “black box,” input-output MUI on the internet. We approached the problem’s solution in purely natural mathematical terms using the NMI. Figure 6 continues for 7 pages.
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Figure 7. “Serial Production”: State Space Flow Graph Depicts the probabilities of transitioning from one state to the next as described in the transition matrix above in Figure 6, the Serial Production problem.
228 APPENDIX D
SCREEN SHOTS: GEOMETER’S SKETCHPAD NMI
Here is presented interface screen shots representing some of the presentation, capabilities and functionalities of the NMI Geometer’s SketchPad (GSP).
Figure 8. Constructions using the NMI GSP Showing NMI interface, and geometric construction menu.
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Figure 9. Transformations using the NMI GSP Showing NMI interface, transformation menu, and a mirror reflection across a segment.
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Figure 10. Rotation using the NMI GSP Showing NMI interface, rotation sub-menu, and several 90 degree rotations of the smaller triangle around point A.
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Figure 11. Analytic Geometry using the NMI GSP Showing NMI interface, the measure menu, and the graph of a circle. The circle was created first by Euclidean means--specifying a center (A) and radius (A to B). Then an arbitrarily placed coordinate system was drawn, within which the NMI provided an equation for the circle.
232 APPENDIX E
ANN’S INITIAL LEARNING OF INTERFACE PROCEDURES
This appendix shows some of Ann’s initial work learning the NMI’s Mathematics User Interface (MUI) procedures—those procedures which are not natural mathematics, but computer program specific. These are the procedures referred to concerning NMI Interface-Procedural Embodiments (IPE) or perceptual judgments.
Figure 12. Triangles within triangles Showing Ann’s first activity in learning the NMI. The segment midpoint command was primarily used.
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Figure 13. Equilateral triangle by radii Shows a pattern Ann created when first working with the NMI.
234 APPENDIX F
ANN’S FIRST PROOF
The Measures of the Interior Angles of All Triangles Sum to 180°
CONSTRUCTION: 2 lines, 2 intersecting transversals Can you find a combination of What are "alternate interior angles"? four lines and What are "transversal lines"? the concept of "transversals of parallel lines" to What is a "straight angle" and how many degrees is it? represent all What does "m(∠ABC)" mean? possible shapes of triangle s? We will prove that m(∠A)+m(∠B)+m(∠C)=180°.
Figure 14. Pedagogical motivation: Angle sums Shown are pedagogical introductions to proving the theorem on the sum of the measures of the internal angles of a triangle. Ann (who was an education major) thought about the kinds of things that might prepare a student to look at the proof she did below in Figure 16.
235 E D A
B C
Figure 15. Construction: Sum of the inner angles of a triangle Shows parallel lines with two transversals, which have the property that upon dragging around, depict all possible triangle (to within similarity). Ann studied this construction in a previous liberal arts major freshman mathematics course I taught. Ann said she learned more about this construction due to her ability to drag it around. I interpreted this learning to be due to inductive reasoning because she was looking at a sampled sequence of cases which had identical relevant properties.
236 Theorem: The measures of all interior angles of a triangle sum to 180°.
GIVEN: DE is parallel to BC and AB and AC are transversals. Assume we know that alternate interior angles are congruent (Alt. Int. ∠'s Theorem.). Assume we know what a straight angle is.
Prove that the m(∠ABC)+m(∠BCA)+m(∠CAB)=180°.
PROOF ______(JUSTIFICATION)______1) m(∠DAC) + m(∠EAB) + m(∠CAB) = 180°. (by Definition of straight angles). 2) m(∠EAB) = m(∠ABC) (by Alt. Int. ∠'s Theorem.) 3) m(∠DAC) = m(∠ACB) (by Alt. Int. ∠'s Theorem.) 4) m(∠ABC)+m(∠ACB)+m(∠CAB)=180°. (by algebraicly substituting (2) and(3) into (1)). QED
Figure 16. Formal proof concerning the interior angles of a triangle This was the first proof Ann worked on. All proofs Ann worked on were scaffolded, rather than presented so they might have been simply memorized. Once Ann saw the straight angles involved, she saw a (middle-out) reason for the theorem’s validity. Later, we negotiated how to formally represent a logical path (multiple-chain) of reasoning moves leading to the deduced conclusion or theorem.
237 APPENDIX G
POINTS EQUDISTANT FROM TWO POINTS: CONSTRUCTIONS AND PROOF
2 point Equidistant Theorem Theorem: Points are equidistant from a segment's endpoints iff (<-->) they lay on the ⊥ bisector of the segment.
We must prove in 2 cases, forward and reverse: IF "equid" THEN "on ⊥ bi" AND IF "on ⊥ bi" THEN "equid"
CASE 1. Forward Direction. IF "equid" then "⊥ bi" C' M B CONSTRUCTION of ANTECEDENT of IF "equid" then "⊥ bi" C 1. Mark a vector from A to B. A D 2. Label arbitrary pt. C on AB 3. Vector Translation of AC length along direction AB towards B. This makes C'. 4. Construction of an "arbitrary" equidistant point D now exists! 5. Join AD and C'D (equal radii), therefore D is equidistant from endpoints! 6. Construct midpoint M of AC'. Connect MD.
Figure 17. Construction for case 1 of 2pt Shows Ann’s construction, and sketch of an explanation for the Two Point Equidistant Theorem (2pt) in the forward direction of the biconditional statement. The construction provides points for a perpendicular bisector of segment AC’. See Figures 18-20. Euclidean plane geometry supposes the existence of the concepts of line and circle. These concepts have typically been concretely represented by using the straight edge and compass. These same basic concepts are applied by Ann in a creative way that demonstrates an understanding of her construction.
238 B and C' B and C'
A and C A and C
C' C' B B C C A A
C' C' B C B C A A
Figure 18. 2pt construction using vector translation. Shown on the right are typical diagrams for constructing the perpendicular bisector of segment AC’. The left diagrams show how vector C to C’ (initially overlaying segment AB) can be used by vector translation to maintain constant radii centered at A and C’. The vector translation displayed on the left shows a generalizing construction of multiple lengths of segment AC’, as with having a constant “compass” width. I noted that Ann seemed motivated by the NMI capability to create these kinds of drag-able diagrams. They also provided a concrete, creative demonstration to me of her mathematical understanding of the generality the diagrams show. Appendices E through J each contain drag-able diagrams.
239 M C' B C A D
Figure 19. Construction of a perpendicular bisector Showing construction for the perpendicular bisector of segment AC’ through points D and M. Point M, the midpoint construction was known by Ann to be a result of connecting the circle intersections. Ann was interested in using the “construct midpoint” command of Geometer’s SketchPad as the construction is specifically used in the proof she was anticipating below.
240 M C' B A C D
Figure 20. Triangle constructions facilitating proof. Ann created this generalizable (i.e., drag-able) diagram construction (see Figure 18 and 19) to prove the forward conditional statement of the 2pt bi-conditional as stated above (see top of Figure 17). All this then led to the reasoning below resulting from the simultaneous use of the NMI, and our scaffolding discussions over several days. Figure 20 shows how the points M and D can be used to show, by SSS, that triangle AMD is congruent to triangle C’MD. Assume segment AC’ has a midpoint called M, so AM equals C’M. D is equidistant from A and C’ by the congruent radii A to D and C’ to D. MD equals MD. The proof of the first case of the 2pt bi-conditional uses this construction and this proof sketch (above) fills in the reasoning assumed in the “proof” of case 1 below. Ann and I negotiated further for the proof of case 2 in Figure 22 below. This proof is stated completely, minimally, and precisely.
241 CASE 1. PROOF: IF "equid" THEN "on ⊥ bi"
1. AD =C'D by construction (given)
2. A midpoint between A and C' called M will allow us forming two triangles with a shared side of measure: MD.
3. But note that by SSS, we must have two congruent triangles! But now, how do I know that D is on the ⊥ bisector of AC' ? This is the CONSEQUENT we wish to prove!
4. ∠AMD must equal it's corresponding ∠C'MD (by congruent triangles).
5. ∠AMD + ∠C'MD = 180 degrees (by supplementarity)
6. So, if X + X = 180 (the two equal angles above in 5 and 4), X = 90 degrees.
7. If X equals 90 deg., we have proven the CONSEQUENT: that MD is BOTH ⊥ to AC' and bisecting it (since it intersects at M the midpoint). QED
Figure 21. Forward case proof sketch. 2pt proof as negotiated with Ann (found in a typical geometry text) of the forward conditional statement: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. See diagram above in Figure 20. As these diagrams are taken from work done with the student, errors such as denoting a measure of an angle are left as they were written. This explication of Case 1 was not written until Ann and I found that she was having difficulty with the biconditional aspects of the theorem. Ann worked with the initial diagrammatic idea in case 1 after working on the proofs which follow in Appendices H-I. She re-emphasized her understanding of the biconditional nature of this theorem. Concluding discussion on this is included at the end of Appendix H.
242 Construction: Case 2. IF "on ⊥ bi" THEN "equid"
1) Construct two points (A and B), and adjoining segment (AB), and contruct the Numerical Measure. midpoint of the segment. Label all points.
2) Recall: perpendicular bisector- a line that A forms a 90° angle with and passes through the midpoint of a segment. Construct AC = 5.01 cm perpendicular bisector to segment (AB). BC = 5.01 cm Place an arbitrary point (C) on the perpendicular bisector. M 3) Connect endpoints of original segment C arbitrary point on perpendicular bisector (AC and BC). THEOREM: If points lay on the ⊥ bisector of a segment, then are equidistant from that segment's endpoints ( ⊥ bi → equid ). GIVEN: MC is the ⊥ bisector of AB.
PROOF: B Claim Justification
1) AM ≅ BM Defn. of ⊥ bisector (midpoint).
2) MC ≅ MC Identity
3) ∠AMC ≅ ∠BMC Defn. of ⊥ Bisector.
4) AMC ≅ BMC SAS
5) AC ≅ BC Corresponding sides of congruent triangle. QED
Figure 22. Case two construction and formal proof. Shown is the 2pt construction and proof; case 2—the reverse of the biconditional. The statement of the case to be proven is labeled above as “theorem”. Note the use of the exploitation of the NMI numerical measure capability. This is not used in the proof, but provides a plausible reason for the proof’s result. Ann tended to neglect case 1 in interpreting later proofs as discussed in Figure 22.
243 APPENDIX H
THEOREM SB: THE SIDE BISECTORS OF A TRIANGLE ARE CONCURRENT
BM = 6.98 cm Circumcenter, M MA = 6.98 cm Side bisector CM = 6.98 cm concurrency point
M CY = 2.96 cm m∠BXM = 90.00° A YA = 2.96 cm m∠AYM = 90.00° BX = 4.10 cm m∠AZM = 90.00° XC = 4.10 cm Z Y BZ = 6.11 cm ZA = 6.11 cm
B X C
Figure 23. Side bisector concurrency construction Note the GSP NMI’s capacity for displaying numerical measures. I interpreted these features helped Ann reach an NMCE concerning Theorem SB—seen generally by virtue of the fact that this figure could be dragged around, yet maintain the same properties (e.g., the numerical measures and circumscribing circle). Ann made this construction for several different triangles, re-confirming her NMCE by induction.
Discussion and Proof
The proof implicit in Figure 23 is similar to the proof Ann did of Theorem ABC displayed below in Appendix I. The similarities hinge on the use of the transitive property of equality—a GE Ann had from grade school arithmetic (if A = B, and B = C, then A = C). Below is the proof we studied together for triangle side bisector concurrency (Theorem SB).
244
Given: Triangle ABC with perpendicular side bisectors XM, ZM and YM. The 2 point Equidistant Theorem (2pt): Points are equidistant from a segment’s endpoints if and only if they lie on that segment’s perpendicular bisector. Prove: Theorem SB; The side bisectors of a triangle are concurrent.
Proof: XM and YM are nonparallel because XM and YM are given as perpendicular bisectors of each segment BC and CA respectively, the sides of the triangle. Therefore, they intersect at one point M. By 2 pt in the reverse biconditional direction: if points lie on a segment’s perpendicular bisector, then they are equidistant from its endpoints. Therefore, point M is equidistant from both B and C (via its lying on XM), and likewise, C and A (via YM). However, since distances BM=CM, and CM=AM, then BM=AM (equality is transitive). It is tempting here especially given the dynamic NMI construction to believe one has the completely understood the proof. However, just because we know we have a point M which is equidistant from certain end points, how do we know it lay on the perpendicular bisector ZM?
So I asked Ann, “How do you know if M lie on ZM?” Ann had trouble explicitly answering this question. Through discussion, I realized Ann took it as implicit that we were using the entire Two Point Equidistant theorem (2 pt) in the argument just above. Ann indicated that since transitivity went in both directions, the question was answered. I could tell Ann was searching for a biconditional type of explanation, as we had spoken of this much earlier in my scaffolding this proof. After a time, Ann realized, and explained that given the initially proven fact that BM=AM (by transitivity), we then satisfied the conditional for the forward direction of 2 pt (If equid., then on perp. bi.), and so then we knew that M must lay on ZM. Ann’s first distractor involved a memory conflation. Ann was in taking the apparent “reversibility” of the transitive property (a mistaken “reversal” for mere substitution), and conflating that (through a memory distractor) with the forward direction of 2 pt (i.e., remembering the need for a biconditional, as a type symmetric reversal, transitivity was mis-associated in memory rather than following a path of reasoning at this new point in time—long after the initial proof of 2 pt was accomplished). Finally, Ann seemed to believe her constructed diagram to be a demonstration itself, circumventing a need to explicitly answer the question by citing the forward direction of 2pt. This is a case where an NMCE (her dynamic construction) may have interfered with perceiving a GE (the two sided biconditional).
This then was considered a “mistaken GE” concerning the Two Point Equidistant theorem, in consideration of the biderectionality in the Two Point Theorem (Appendix G). The transitive property GE alone was a crystal clear embodiment for Ann. This tended though, it seemed, to cloud her sense of need for the other biconditional case.
245 APPENDIX I
THEOREM ABC: ANGLE BISECTOR CONCURRENCY
Angle bisectors are m∠ABM = 26.47° are concurrent. A m∠CBM = 26.47° m∠BAM = 18.60° m∠CAM = 18.60° m∠ACM = 44.94° m∠MZA = 90.00° m∠BCM = 44.94° m∠MXC = 90.00° m∠MYA = 90.00° C' Z B' Incenter, M, M Equidistant from each side of the Y triangle.
B A' X C m ZM = 2.73 cm m YM = 2.73 cm m ZM = 2.73 cm
Figure 24. Triangle angle bisector concurrency diagram Shows Ann’s construction used for proof work on the theorem that a triangle’s angle bisectors are concurrent. NMI displays numerical measures of interest.
246 Theorem: A triangle's angle bisectors are concurrent. They intersect at a single point.
Lemma: A point lay on an ∠ bisector iff. it is equidistant from each side of the angle.
Sketch of Lemma Proof:
Case 1. If a point lay on an ∠ bisector, then it is equidistant from the sides of the angle ("on ∠ bi. → equid"). Construction: Drop ⊥ from arbitrary point on bisector. Show: Each length from the ∠ bisector to the side is equal. Proof Sketch: Use the sum of angles in a to apply AAA, including an identical side, to show congruency.
Case 2 ("equid → on ∠ bi."). Proof Sketch: Assuming each length to the angle bisector is equal, apply the "hypotenuse-leg theorem" to show congruency.
Hypotenuse-leg theorem: Recall that a theorem similar to what would be called "SSA" holds in particular if the involved has a 90° ∠. This holds because the fixed 90° ∠ rules out the possibility of two cases occuring in the general "SSA" relationship.
Figure 25. Angle bisector equidistance: Lemma AB proven by two cases Ann’s proof sketch for the Lemma AB: A point lay on an angle bisector iff it is equidistant to each side of the triangle. This lemma is used in proving Theorem ABC below.
Discussion and Proof of Lemma AB
Since the proofs Ann worked on were not part of a predefined course, we applied middle-out reasoning to find which lemmas we would need (and settle for as a first principle) to prove Theorem ABC. This led to the Lemma AB above in Figure 25, which in turn, led us to needing the “hypotenuse-leg theorem,” as a second lemma, to prove case 2 above in Figure 25. The proof sketch stands as written in Figure 25. The construction necessary for proving Lemma AB is very simple and so not included. Note in case 1 that the angle bisected forms a congruent pair of adjacent angles.
Discussion and Proof of Theorem ABC
Proof of the claim diagrammed in Figure 24, Theorem ABC, “Triangle Angle Bisectors are Concurrent,” is discussed. Given triangle ABC and angle bisectors from A to A’, B to B’ and C to C’, show that these angle bisectors must be concurrent for any triangle ABC.
247 Label ‘M’ at the intersection of the angle bisectors of B and C (since lines interest at a single point). Using bisector B to B’, drop equal perpendicular distances MZ and MX (forward direction of Lemma AB). Now from bisector C to C’, drop equal perpendicular distances MX and MY (forward direction of Lemma AB). If distances MZ = MX and MX=MY, then MZ=MY (transitivity of equality). This part of the proof was most challenging to Ann, and was discussed in Appendix H. Now, since MZ=MY--an equidistance from the sides of angle A--it must lay on the bisector of A (reverse direction of Lemma A). Since M is a single point where bisectors of angles B and C intersected, and this is same point involved in showing MZ=MY (i.e., M lies on the angle bisector of angle A), there is that single point M where all bisectors intersect—or are concurrent. QED.
248 APPENDIX J
PYTHAGOREAN CONSTRUCTION AND PROOF
A E B
F
H
C D G
Figure 26. Square within a square Shows a construction to be used to prove Pythagorean’s theorem. Ann devised two ways on her own to construct the inner square using geometric transformations (See Figures 27 and 28).
Ongoing Discussion of Inner Square Constructions
Ann used higher level mathematical ideas to create the diagrams in this appendix. They are not compass and straight edge, but they are natural—ideas that are learned proceptually (e.g., a geometric transformation involves the thought of the thing in itself— the concept, as well as the process involved in carrying out the transformation).125 Ann first created the outer square by rotating a side segment 90 degrees about a corner point, this point marked as the center of the rotation. This she did three times to get all four sides. I scaffolded Ann on the first rotation, but she did the last two herself— and found it interesting how this transformation worked to create a perfect square. Ann commented, “This is a good way to teach the definition of a square… using the 90
125 The use of geometric transformations to construct inner squares was conveyed to both Ann and myself in private conversation with Janice Flake.
249 degrees and the same [or congruent] line segment four times.” Next, I drew a picture of a square within a square, and labeled the sides as they are in Figure 30 below. Ann was then shown the construction shown in Figure 30 below.126 Noticing how this inner square was constructed on the NMI, Ann invented the following two methods of construction shown in Figures 28 and 29, completely on her own.
V1
A
D E
H1 H2
C F
B G
V2
Figure 27. Inner square construction: Reflections and rotations Ann’s original, and creative construction of a quadrilateral (AEGC) inside an outer square (not labeled) using the geometric transformation of reflection and rotation.
Figure 27 Construction Discussion
1) A was reflected across the horizontal to B. 2) B was rotated around the corner of the square + 90 degrees to C. 3) C was reflected across the horizontal (H1 to H2) to D. 4) D was reflected across the vertical (V1 to V2) to E. 5) E was reflected across the horizontal to F. 6) B was returned to, and reflected across the vertical to G.
126 This was shown to Ann by Janice Flake during a course she was teaching which concerned the use of GSP.
250 7) She then connected the segments of the inscribed square by sight, beginning with A, to form quadrilateral AEGC. The proof that AEGC is square is similar to the proof in Figure 30.
I want to note that this construction involved both seeing the chosen reflections and rotations (through a concept), and performing the actions via a “mark mirror” command, and a “reflection” command. The rotations involved a “mark center” command and a “rotate” command, which asked for the degree measure (+90 degrees in both cases). These actions were a process and thus the natural mathematics notion of the proceptual. All her concepts and actions followed from mathematical conventions, or mathematical language, not this or that computer programming language operations. Thus Ann demonstrated the employment of natural mathematics, so that it was apparent that she comprehended the mathematics of the symmetries involved by using the explicit geometric transformations. Again, the constructions in Figures 28 and 29 are completely original creations of Ann. The only scaffolding by me was previous to the fact, helping her learn these transformations in arbitrary situations. Further, each “square within a square” construction demonstrated a perceptual judgment formed from lower level concepts for at least two initiation of reasoning points. Ann spoke about how these constructions helped her to conceptualize how transformations could be applied. I consider this a GE as it is mathematics applied to itself. Secondly, these inner square constructions served to emphasize for Ann, the generalized properties to all right triangles facilitating the proof. Ann commented that the flexible attributes in building the construction (the translations, rotations, etc.) gave her a deeper sense of how generality can be achieved in multiple ways. I felt good about Ann finding these multiple constructions as it helped her, I believe, to see my point all along in urging her to explore and experiment using the NMI, rather than I come in and “tell” her a proof to be simply memorized. Appendix G provides an example of how the NMI directly applies to the reasoning of a proof. Ann’s constructions there were quite different than one might expect—a different creative perspective is found and explicitly demonstrated (Figures 17). Ann and I discussed this construction and found that it could be explicated as in Figure 18. The notion of lengthening the distance between the segment’s endpoints served to emphasize the method of construction of the perpendicular bisector. It is a kind of inductive demonstration of many cases (viewable display frames) that serve, to my interpretation of Ann’s comments, “The “equidistance” from the endpoints is plain here.” This was her own created external representation, reinforcing what she understood verbally. The NMI reinforced her own conceptualization, the “conceptual-embodiment” (according to Tall), is an assertion of a perceived intuition—a perceptual judgment. Please see Chapter V. The relevance to education is her creative demonstration of understanding through the flexibility of the interactive virtual manipulative (see Chapter IX). Ann told me she had a “general picture in mind” and aimed at “showing it”—as her interest was in teaching. She also had fun doing these original constructions—as well as I did in seeing her creations. These explicit creations were noted to provide insight. One day Ann and I would be verbalizing and sketching on paper—and I could tell she wasn’t confident in her understanding of the difference in the cases in Appendix G. However, this challenged her to go home and come up with a representation that made it plain as day—and dispelled
251 my own lack of confidence that she genuinely “saw” (could perceptually assert) what the distinctions between cases looked like—as she would say, “anyone can flip symbols, I want to see this plainly.” I know she wanted at times for me to show her construtions— but the perturbations of my resisting at times paid off. Further, there was the consideration as well that my not knowing exactly what would help her make her own sense out of the situation—its easy to provide many differing kinds of examples, but its another thing to have the student do this exercise, e.g., the several “squares within squares” she did in Appendix J. My conclusion is that Ann enjoyed using the computer in that she could a) see it being of interest to future students, b) externally (mind-independently) demonstrate (to herself as well as others) her sometimes not so confident, or meaningful verbalizations and “symbol flipping”, and c) take the opportunity to show me ideas, rather than it (as is typical in a mathematics course) visa-versa. As for (b), this bridging of “picture” to logical symbolization is a precissive abstraction—serving for a more fruitful, cogent concept than the discriminatory verbalizations—piling one representation upon another— without a concrete basis. Or, precissively, or tightly encapsulated, to the point of a general concept with “plain” properties, one that could serve as clear initiation points of reasoning (GE’s) for further work, e.g., serve to colligate meanings surrounding premises of deductive arguments—often necessary in finding a multiple-chain to logically connect premises to theorem. This GE then, stood opposed to symbolic representations built on verbal quick sand for Ann. I interpreted Ann to not have as much intellectual respect for the verbal-symbolic as she did for her own NMI facilitated, mind-independent constructions. These were my interpretations of how Ann viewed our discussions involving “symbol flipping.”
Construction of the outer square for Figure 27-29.
1) Construct segment AB.
2) Mark B as a center of rotation, mark (i.e., mouse click on it) the desired segment from the new corner to point A, and rotate + 90 degrees to form the side of the square designated by segment BC.
3) Mark C as a center of rotation, mark the desired segment from the new corner to point B, and rotate + 90 degrees to form the side of the square designated by segment CD.
4) Mark D as a center of rotation, mark the desired segment from the new corner to the point C, and rotate + 90 degrees to form the second side of the square designated by segment DA.
252
A E B
F K
G J
D C H I
Figure 28. Square within square: All reflections Ann’s original, creative construction of an inner quadrilateral (EGIK, not drawn) by nothing more than reflections of a single point. She created an inner quadrilateral (proven to be square in a similar manner to the proof in Figure 30) by choosing the arbitrary point E, and reflecting across the nearest diagonal, then across the horizontal, point E to F to G, etc.
253
b a
a C D A 1) Construct a segment (CA) and an b arbitrary point (D). Construct a segment from arbitrary point to first endpoint. 2) Mark vector from one endpoint to the other, and translate the new segment c (CD). Rotate resultant projection 90°. Repeat 3 times, until a square, with an c inscribed square is constructed. Label all points. K 3) Label resultant segments around the square (a,b), noting congruency of four resulting triangles. c 4) Measure distances between endpoints L and arbitrary points, and calculate addition of each resultant segment. 5) Drag original arbitrary point (D) along c original segment (CA), and note the relationships among measurements. b a
Ea M b B CD = 4.87 cm DA = 3.31 cm CD+DA = 8.18 cm m∠DLM = 90.00° AL = 4.87 cm LB = 3.31 cm AL+LB = 8.18 cm m∠LMK = 90.00° ME = 3.31 cm BM = 4.87 cm BM+ME = 8.18 cm m∠MKD = 90.00° EK = 4.87 cm KC = 3.31 cm EK+KC = 8.18 cm m∠KDL = 90.00°
Figure 29. Construction used (and animated) for proof of Pythagorean’s Theorem The construction here is similar to Figure 27.127 Segment CD is vector translated to have tail at A, and rotated – 90 degrees to determine point L, then segment AL is vector translated, etc. The construction is labeled so that lengths a + b, and c, are the sides of the outer and inner square respectively.128 The proof begins with showing the inscribed quadrilateral MLDK to be square.
127 The construction technique of the inner square in Figure 29 was communicated to me in private conversation, and taught to Ann with/by Janice Flake. 128 The square within square onstruction above was shown to Ann and myself in private conversation with Janice Flake.
254 Given: A square with inscribed square and points at all vertices (A, B, C, D, E, K, L, M). Assume: Sum of the interior angles of all triangles is 180°. Sum of any angles that form a straight line is 180°. Prove: The resulting side segments a, b, and c contribute to the formula a2+b2=c2. First we must prove ∠KML, ∠MLD, ∠LDK, and ∠DKM are each 90°. 1) ∠KME+∠KML+∠LMB=180°, by supplementarity 2) ∠LMB=∠EKM, by corresponding angles of congruent triangles. 3)∠MEK=90°, by construction. 4)∠EKM+∠KME+∠MEK=180 °, by triangles=180°. 5)If ∠MEK=90°, then ∠EKM+∠KME=90 °, by algebra 6) If LMB=EKM, the ∠LMB+∠KME=90 °. 7)Recall, ∠KME+∠KML+∠LMB=180°, so 8) So, KML=90°, for any KML. 9) The sums of the ares of the parts of the square add to the area of the whole square, much like the area of a parking lot is equal to the sum of the areas of all the parking spaces therein, so the area of the whole square (a+b)2, is equal to the area of it's parts, the four congruent triangles plus the inscribed square, or 4(1/2ab) + c2. 10) So, (a+b)2=4(1/2ab) + c2. 11) a2+b2=c2, by algebra. QED
Figure 30. Ann’s (pre-scaffolded) proof of Pythagorean’s Theorem Refer Figure 29. I scaffolded Ann on this proof early on in the research. Weeks later she presented me this write-up of the proof. Clarification of justification on step 2: the triangles correspond by the typically stated theorem ‘SAS’. Clarification on typo and justification on step 6: [If angle LMB = angle EKM, then angle LMB + angle KME = 90 degrees] is justified by the fact that Triangle EMK is congruent to Triangle BLM, and, the sum of the angles of any triangle must sum to 180 degrees. Note that by SSS; a, b, & c are self-same, therefore step (2) does indeed refer to “congruent triangles” as claimed.
255 APPENDIX K
HUMAN SUBJECTS COMMITTEE FORMS
256
257 CONSENT FORM. Instructional and Student Uses of Mathematics User Interfaced Computing: A Case Study
Dear ______,
Described here is the research project we have discussed for which I have prepared this formal request your voluntary participation (see box below).
I am a graduate student under the direction of Professor Janice Flake for the Mathematics Education Program of the Department of Middle and Secondary Education at Florida State University. All participants in this research must be at least 18 years of age. I am conducting a research study to determine how effective certain “Natural Mathematics Representational, Mathematics User Interfaced” computer programs might be in helping teachers-in-training (and others) better learn their mathematics.
The research procedure involves you and me working with specific computer programs on certain mathematics topics. There will be some audio-taping of our discussion for the purposes of reviewing our sessions. The minimum time required for your participation will be approximately 20 hours total, which includes 1) an initial interview about your background, 2) computer work and discussion with me as we go along, 3) computer work done on your own, and 4) exit interviewing about the work we accomplished. We may wish to work for longer than this minimum time frame, but you are free to withdraw from the study at any time with no prejudice. All audio-tapes will be kept confidential, and your name will not be used in any publications of this research that may follow after its completion to the extent allowed by law.
There are no foreseeable risks or discomforts for you as a participant in this research. Although there may be no direct benefit to you, the possible benefit of your participation is 1) getting some help learning certain mathematics related to your course work, 2) finding out about some computer programs that might be useful to you, and, if your interest is deep enough, 3) being advised of professional opportunities during the course of this research where you yourself might be able to present your own experiences on the educational uses of these new kinds of software tools.
If you have any questions concerning this research study, please call me at 850-574-4946 ([email protected]), or Dr. Janice Flake at 850-644-8481 ([email protected]). If you have any questions about your rights as a subject/participant in this research, or if you feel you have been placed at risk, you can contact the Chair of the Human Subjects Committee, Institutional Review Board, through the Vice President for the Office of Research at (850) 644-8633.
Sincerely,
______Erich G. Nold
I give my consent to participate in the above study. I understand that I will be tape recorded by the researcher. These tapes will be kept by the researcher in a locked filing cabinet. I understand that only the researcher will have access to these tapes and that they will be destroyed by August 15, 2007.
______(Signature) (Date).
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278 BIOGRAPHICAL SKETCH
George Erich Paul Charles Nold was born on May 20, 1959, in Patchogue, N.Y., on Long Island. His mother worked for a time as a ballerina in Hannover, Germany. His father was a Naval Academy graduate, retired a Commander, and retired physics professor from the State University of New York, Farmingdale, where there is an indoor sports arena dedicated in his name. Erich was his mother’s only child, born when she was age 50, and his father, 60. Erich grew up involved in music from a young age, beginning with the violin, then the piano, trombone, guitar, and drums—subsequently playing in several bands at school, and later in (quite awfully) smoky bars. He gave piano lessons part time for a number of years. He presently enjoys Sunday worship services singing bass in the choir at St. Peter’s Anglican Church in Tallahassee. Erich graduated from Oak Ridge High School in Orlando, a school excelling in math and science. The school as well had first place state rankings of their symphonic band, stage band, and several ensembles, within which he was sometimes, first chair trombonist. Erich attended the University of Florida (U of F) and graduated with honors in Electrical Engineering. He worked two cooperative education jobs through U of F as an undergraduate; a position at the Naval Research Laboratory’s Underwater Sound Reference Detachment, and one at Lockheed-Martin Aerospace, both in Orlando. Erich began work in Apopka, Florida for SAWTEK Inc., doing surface acoustic micro-wave band-pass filter design. He then was hired at Lockheed-Martin Orlando Aerospace Division where he worked 8 years, leaving as a senior engineer—returning later as a private contractor. Erich had the opportunity to work in five departments at the company: (i) Analog circuit design and production, (ii) Electro-optical design and production, (iii) Electro-optical research and development, (iv) The Digital image and signal processing lab, and (v) The Digital-Analog Hybrid computing lab. Lockheed-Martin awarded Erich with the Industrial Associates Research Fellowship which paid for his first Master’s degree in Electrical Engineering at the University of Central Florida (UCF), doing his thesis research on a neural network analog circuit implementation, with the help of Leslie Canney. Les and he had the opportunity to
279 work together using the largest digitally controlled analog computer facility in the free world. There he applied for a patent on a digital neural network pattern recognizor, and received a monetary award for subsequent publications in the area. Erich participated in several grant writing proposals concerning various state of the art technologies, taking him on fact-finding travels to places such as Denver, and Michigan’s Upper Peninsula. One example of his research concerning a proposed grant was on statistically representing the infrared signatures of various natural background textures such as forest, tundra, grassland, slushy snow, etc. The question was in determining the veracity of the simulation of computer generated scenery to be used in testing pattern recognition algorithms. Erich received several awards for research publications. He received two first place student paper awards from the Florida Artificial Intelligence Research Symposia (FLAIRS), and a monetary award for a first place student paper from an Aerospace Simulations Conference. He was invited to sit on a panel at the first joint Auburn University--NASA Neural Networks Symposia. Erich was awarded the 1997 student researcher of the year award for his (second) Master’s degree work at the UCF’s department of mathematics. He has presented papers at several conferences in the fields of engineering, mathematics, and technology use in mathematics education. Erich began his teaching career as a graduate student in mathematics at UCF, and in the mathematics and engineering departments at Valencia Community College, in Orlando. Also in Orlando, he taught 8th grade at Lee Middle, an inner city school that was interestingly enough fed by 35 different elementary schools. At Lee, he received an award for teaching, and other activities on campus. He has also taught 4th – 8th grade at COAST charter school in St. Marks, Florida (where he taught mathematics, science, and social studies), and worked short-term for the Florida State University Lab School finishing an 8th grade mathematics semester. At St. Marks he submitted a grant for student research on Monarch butterfly migration, worked setting up their marine biology touch tank, and helped their music program by writing and accompanying music for student talent shows. This activity energized several students to bring their own instruments to school and learn to play together at recess. He had the opportunity to work for a private tutoring firm for grade
280 school children, as well as volunteer tutoring at a church mission for “at risk” students. All these grade school experiences guided his attentional focus towards ways of teaching children, and later encouraged his graduate work in the field of mathematics education. He has taught as a graduate teaching assistant at Florida State (FSU), and as an adjunct mathematics professor at Florida Agricultural and Mechanical University (FAMU), Tallahassee Community College, and North Florida Community College—all in the Tallahassee area. Erich worked on research projects at FSU during his doctoral program, and UCF during his two Master’s programs. He worked for the Department of Computational and Engineering Sciences at FSU on the development of a statewide website for teaching grade school science. He also worked in the Program of Elementary Education on two separate curriculum development projects, each concerning lower grade mathematics education. Through a UCF affiliation, he researched an infrared full- face holographic projection system for individual soldiers in the field. He also researched ways of implementing international parallel-distributed “war-games” using the internet and was invited to write a book chapter on this in a subsequent publication. Through FAMU, where he worked over a period of five years as an adjunct mathematics professor, he worked on an alliance project for “bridging” minority students to doctoral programs. Through FAMU, he also trained with EDUCO Co. on using their computer-aided mathematics curriculum for lower division college mathematics teaching. Erich’s college course teaching experience spans Liberal Arts mathematics, College Algebra, Geometry, Trigonometry, and Electrical Circuits courses; while his grade school experience spanned 4th-8th grade mathematics. Erich is finishing his doctoral program at FSU in the department of Secondary and Middle Education. Erich left engineering at the close of the cold war to pursue mathematics teaching for the following reasons. He believed that learning mathematics is important in that it (a) teaches reasoning, (b) illuminates, as well as facilitates, knowledge in science and engineering, and (c) provides opportunity to engage an intellectual art which spans millennia. A mathematics education lends of understanding hard-learned and deep-felt ideas accreted over a virtually continuous time span from pre-school through college. And significantly here; all that was genuinely figured out by the student, over all these many years, can be found to be: entirely consistent.
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