GENERATIVE ADOLESCENT MATHEMATICAL LEARNERS: THE FABRICATION OF KNOWLEDGE by BRIAN R. LAWLER (Under the Direction of Leslie P. Steffe) ABSTRACT This dissertation is embedded in a deconstruction of the field of Mathematics Education in order to reconstitute the mathematics student as a generative mathematical learner. The purpose of the dissertation is to understand how generative adolescent mathematical learners (GAMLs) maneuver through their mathematics courses while maintaining such a disposition. As a result, an initial model of constitutive characteristics of the GAML emerged. A generative disposition is meant to characterize the learner as someone who operates mathematically in ways that reflect an internal sense of authority for knowing and a constructive orientation to the knowledge they come to know. Drawing upon the radical constructivist teaching experiment methodology, I conducted a poststructural qualitative inquiry of students’ experiences that may have informed a generative disposition, their orientation toward mathematical knowledge, and the role of schooling in the interactions of these students. I co-taught the mathematics classes of the studies’ three students during April 2005 and interacted with the students as both teacher and classroom researcher. Field notes served as a primary data source, with some classroom episodes and student interviews videotaped in order to aid in the retrospective analysis that followed. Case studies were first developed, and then used to enrich an initial definition of the GAML. The GAMLs of this study relied on their own thinking to come to know. Confirmation of their teachers’ judgments that they were competent thinkers was strongly evident. The three GAMLs demonstrated confidence in what they knew and in their potential to come to know. They considered that mathematics is a human activity and were mathematically interactive among their classroom peers. Further, they enjoyed a high social and academic status among their classroom peers. A disconnection was observed in the three GAMLs between observed classroom behavior and their observed generativity. In particular, they rejected schooling in various ways and engaged in deviant classroom behavior. INDEX WORDS: Authority, Critical Theory, Critical Postmodern Theory, Deconstruction, Disposition, Equity, Knowledge, Mathematics Education, Mathematics of Children, Personal Epistemology, PostStructuralism, Power/Knowledge, Radical Constructivism, Relational Equity, Social Justice GENERATIVE ADOLESCENT MATHEMATICAL LEARNERS: THE FABRICATION OF KNOWLEDGE by BRIAN R. LAWLER B.S., Colorado State University, 1992 M.A., California State University – Dominguez Hills, 1999 M.A., The University of Georgia, 2006 A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY ATHENS, GEORGIA 2008 © 2008 Brian R. Lawler All Rights Reserved GENERATIVE ADOLESCENT MATHEMATICAL LEARNERS: THE FABRICATION OF KNOWLEDGE by BRIAN R. LAWLER Major Professor: Leslie P. Steffe Committee: George M. A. Stanic Dorothy Y. White Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia August 2008 iv TABLE OF CONTENTS Page LIST OF TABLES........................................................................................................................ vii LIST OF FIGURES ..................................................................................................................... viii FORWARD .....................................................................................................................................1 CHAPTER 1 THE PROBLEM AND ITS BACKGROUND ..............................................................2 The Study ..................................................................................................................5 The Place of This Study in Equity Research in Mathematics Education................17 Why Thinking about the GAML is Work for Social Justice...................................35 2 RELEVANT LITERATURE.......................................................................................44 Relevant Work in Mathematics Education..............................................................45 Contributions of Other Sciences to Mathematics Education...................................71 The Postmodern Influence on Knowledge Construction ........................................85 A Postmodern Epistemology?.................................................................................96 3 THEORETICAL FRAMEWORK...............................................................................98 Epistemology / Learning Theory...........................................................................100 Ontology / Worldview / Macro-Theory / Midlevel Theory ..................................115 Agency, Identity, Authority...................................................................................130 The Generative Learner, Constructivism, Poststructuralism, and Social Justice..135 v 4 EPISTEMOLOGY OF RESEARCH.........................................................................145 The Politics of Methodology.................................................................................147 Researcher Subjectivity.........................................................................................159 Methods for Data Collection and Analysis ...........................................................171 Researcher Ethics ..................................................................................................179 5 METHODOLOGY ....................................................................................................183 Research Setting – Background to Phase I............................................................184 Subject Selection – Data Collection and Analysis in Phase I ...............................191 Research Design – Co-Teaching, Observation & Interviews in Phase II .............201 Research Design – Member Check during Phase III ............................................215 Methods of Data Analysis – Phase IV...................................................................217 6 DATA REPORTING AND ANALYSIS...................................................................224 Fisk ........................................................................................................................228 Kate .......................................................................................................................260 Jack........................................................................................................................293 7 SUMMARY AND CONCLUSION...........................................................................333 Enriched Understanding of the Theoretical GAML..............................................333 The Abstracted GAML..........................................................................................344 8 THE FABRICATION OF KNOWLEDGE ...............................................................354 The Study ..............................................................................................................354 Limitations.............................................................................................................361 Implications...........................................................................................................368 AFTERWARD.............................................................................................................................376 vi REFERENCES ............................................................................................................................377 APPENDICES A STUDENT IDENTIFICATION REQUEST .............................................................408 B SUBJECT SURVEY..................................................................................................412 C SUBJECT MATHOGRAPHY ..................................................................................414 D SUBJECT INTERVIEW 1 PROTOCOL ..................................................................415 E SUBJECT INTERVIEW 2 PROTOCOL ..................................................................416 F SUBJECT VIDEO MATHOGRAPHY .....................................................................417 G SUBJECT INTERVIEW 3 PROTOCOL ..................................................................419 H FORMER TEACHER INTERVIEW PROTOCOL ..................................................420 vii LIST OF TABLES Page Table 1: Classrooms Observed During Phase I: Subject Selection, Labeled by Teacher........197 Table 2: Data Collection/Analysis During Phase II.................................................................202 Table 3: Phase II Attendance Summary and Notations of Particular Data Types Collected...215 Table 4: Data Collection/Analysis During Phase III................................................................216 Table 5: Data Collection/Analysis During Phase IV ...............................................................217 Table 6: Boaler and Greeno’s (2000) Ways of Knowing ........................................................219 viii LIST OF FIGURES Page Figure 1: Various ways to theorize mathematical learning...........................................................7 Figure 2: Fisk’s model to factor x2 – 7x + 10 ............................................................................251