ABSTRACT OF THE DISSERTATION.......................................ii ACKNOWLEDGEMENTS.............................................................iv DEDICATIONS................................................................................vi
LIST OF FIGURES...........................................................................x
CHAPTER 1: INTRODUCTION....................................................1
        1.1    Nature of the Study
        1.2    The Setting and the Participants
        1.3    The problem Task
        1.4    The Taxicab problem and its significance
        1.5    Research Purpose and Guiding Question
        1.6    Mathematical Microcultural Norms of the Participants
CHAPTER 2:  Review of literature.................................................13
       2.1    Introduction
       2.2    Learning
       2.3    Discourse
              2.3.1    Teacher-to-student discursive interactions
                2.3.2    Student-to-student discursive interactions
        2.4    Intended Contribution of the Study
CHAPTER 3:  Theoretical Framework and Analytic Method....32
       3.1    Introduction
       3.2   Theoretical Framework
                3.2.1    Maher's notion of critical events
                3.2.2    Inscriptions
                3.2.3    Gattegno's contents of mathematical experience
                3.2.4   Interlocution
        3.3    Analytic Method
                3.3.1    Data collection
                3.3.2    Data analysis
                          3.3.2.1   Viewing attentively the video data
                          3.3.2.2   Describing consecutive time intervals
                          3.3.2.3   Identifying critical events
                          3.3.2.4   Transcribing the video record
                          3.3.2.5   Coding synchronously the transcripts, videotapes, and
                                         inscriptions in an inductive and deductive manner
                                  3.3.2.5.1   Coding transcript
                                  3.3.2.5.2   Coding inscriptions
                         3.3.2.6     Writing analytical memoranda
                                  3.3.2.6.1    Categorize codes, identifying properties, and
                                                     dimensionalizing properties within categories
                          3.3.2.7   Constructing visual representations of codes and categories
                          3.3.2.8   Constructing a storyline
                          3.3.2.9   Composing narrative
        3.4    Researcher as participant interlocutor 
CHAPTER 4:   DETAIL DESCRIPTION OF DATA......................63
        4.1    Notation and nomenclature of the Problem
        4.2    Narrative Description of Video Data
CHAPTER 5: RESULTS...................................................................103  
        5.1   Introduction
        5.2   Structuring the investigation
        5.3   Understanding the problem
        5.4   Developing problem-solving strategies, overcoming heuristic hurdles
        5.5   Implying, hypothesizing, and articulating combinatorial algorithms
        5.6   Building conjectures, counterexamples, and justifications
        5.7   Building and clarifying isomorphisms
CHAPTER 6: CONCLUSION..........................................................169  
        6.1   Introduction
        6.2   Limitations of the study
        6.3   Interpretations: Responding to research questions
                6.3.1    The first question
                6.3.2    The second question
                6.3.3    The third question
                6.3.4    The fourth question
        6.4   Implications for Research and pedagogy
                6.4.1    Research question
                6.4.2    Pedagogical implications
        6.5   Summary
APPENDIX A: The problem task....................................................192  
APPENDIX B:  Combinatorics problem sequence........................193
 
APPENDIX C:  Transcript of problem-solving session................196
 
APPENDIX D:  Transcription convention......................................283
 
APPENDIX E:  Participants' inscriptions........................................284
 
APPENDIX F:  Layered building of inscription (SK)...................309
 
APPENDIX G:  Layered building of inscription (LS)...................316
 
APPENDIX H:  Student-researcher interaction episodes.............323
 
REFERENCES:  References..............................................................326