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Architectures of School Mathematics: Vernaculars of the Function Concept

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Architectures of School Mathematics: Vernaculars of the Function Concept Jacob Frias Koehler Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2016 ©2016 Jacob Frias Koehler All Rights Reserved ABSTRACT Architectures of School Mathematics: Vernaculars of the Function Concept Jacob Frias Koehler This study focuses on the history of school mathematics through the discourse surrounding the function concept. The function concept has remained the central theme of school math- ematics from the emergence of both obligatory schooling and the science of mathematics education. By understanding the scientific discourse of mathematics education as directly connected to larger issues of governance, technology, and industry, particular visions for students are described to highlight these connections. Descriptions from school mathemat- ics focusing on expert curricular documents, developmental psychology, and district reform strategies, are meant to explain these different visions. Despite continued historical inquiry in mathematics education, few studies have offered connections between the specific style of mathematics idealized in schools, the learning theories that accompanied these, and larger societal and cultural shifts. In exploring new theoretical tools from the history of science and technology this study seeks to connect shifting logic from efforts towards rational organization of capitalist society with the logic of school mathematics across the discursive space. This study seeks to understand this relationship by examining the ideals evinced in the protocols of educational science. In order to explore these architectures, the science of mathematics education and psychology are examined alongside the practices in the New York City public schools{the largest school system in the nation. To do so, the discourse of the function concept was viewed as a set of connections between mathematical content, psychology, and larger district reform projects. Four architectures{the mechanical, thermodynamic, cybernetic, and network models{are examined. Outline List of Figures . iv List of Tables . vi 1 Introduction 1 1.1 Need for the Study . .1 1.2 Purpose of the Study . .3 1.3 Procedures of the Study . .8 1.3.1 Methodological Needs . .9 1.3.2 Socio-Political Problems . 10 1.3.3 Mathematical Background . 12 1.3.4 Mathematics Education and the Function Concept . 14 1.4 Resources for the Study . 15 2 Historical Background 17 2.1 Manhattan Architectures . 17 2.1.1 Schooling Background . 19 2.2 Scientific Technological Images . 25 2.2.1 Clockwork Imagery . 26 2.2.2 Thermodynamics . 33 2.2.3 Cybernetic Science . 38 3 Mathematical Background 45 3.1 Introduction . 45 3.1.1 Mechanical Functions . 49 i 3.1.2 Thermodynamic Functions . 53 3.1.3 Cybernetic Functions . 56 3.1.4 Network Functions . 59 4 Mechanical Mathematics 64 4.1 The New Army . 65 4.1.1 Technology and the Emerging School . 66 4.1.2 The Cartesian Ideal . 71 4.1.3 A Mechanical Function . 73 4.1.4 Mechanical Function in Text . 82 4.2 Mechanism Embodied . 91 4.2.1 Physiology of the Nervous System . 93 4.2.2 Intelligence . 98 4.2.3 Psychology of Algebra . 100 4.3 Conclusion . 105 5 The Thermodynamic Model 106 5.1 Introduction . 106 5.1.1 Shifts in Resistance . 107 5.1.2 Scientific Consequences . 111 5.1.3 The New School Science . 117 5.2 Individuation . 119 5.2.1 Function and Functionality . 121 5.2.2 Functional Textbooks . 125 5.2.3 Examinations . 129 5.3 Thermodynamic Management . 132 5.3.1 Psychological Conversions . 137 6 Cybernetic Mathematics 149 6.1 Centralization . 150 6.2 Institutional Change . 154 ii 6.2.1 Integration . 155 6.2.2 Federal Scientific Research . 157 6.3 The Structural Function . 160 6.3.1 Structure in Texts . 164 6.3.2 Examinations . 176 6.4 Structures, Psychology, Cyborgs . 179 6.4.1 Woods Hole and Bruner . 180 6.4.2 Piaget and Structure . 184 6.4.3 A Genetic Function . 189 6.5 Piaget and Skinner's Teaching Machine . 193 7 Networks and Conclusion 196 7.1 Network Models . 197 7.2 A Network Function . 201 7.2.1 Common Core Reforms . 201 7.2.2 Network Texts . 203 7.2.3 Autopoietic Cognition . 206 7.3 Conclusion . 214 Bibliography 217 iii List of Figures 2.1 Early Organization of the City of New York . 17 2.2 Map of Common Schools in New York City 1850 . 19 2.3 Skeleton of the Crystal Palace . 20 2.4 Tenement House Life in New York 1881. 21 2.5 Jacob Riis' Images of Tenement Schooling . 22 2.6 New York City Board of Education Employee Repairing Clocks . 25 2.7 Emergence and Endurance of the Graph . 27 2.8 Turner's Temeraire ................................. 37 3.1 Descartes Figure for Inverse Tangent Problem . 50 3.2 Descartes Transformation . 51 4.1 Mechanical Addition. 76 4.2 Linearizing Addition. 76 4.3 A linkage diagram. 77 4.4 A Mechanical Parabola. 78 4.5 A Mechanical Hyperbola. 78 4.6 Descartes Mechanism. 79 4.7 A Nerve . 95 5.1 Two Different Kinds of Soldier and Army . 110 5.2 LaGuardia's program . 111 5.3 Peace ....................................... 116 5.4 Fite's Function . 128 iv 5.5 Rapid Calculation ................................ 130 5.6 Brownell's Equipment . 141 5.7 Number Arrangements . 141 5.8 Spectrum of Meaning . 147 6.1 Walter Harrison's P.S. 34 . 150 6.2 Whitey's on the Moon . 153 6.3 Map of Segregated Schooling prior to Brown v. Board . 155 6.4 Al's Arithmetic . 166 6.5 Justice? ...................................... 167 6.6 UICSM Tree Chart . 168 6.7 Two different kinds of relations . 169 6.8 Objects to Manipulate . 183 6.9 Piaget's Composition Games . 190 6.10 Apparatus for Wheel Experiment . 192 6.11 Pressy's Machine . 194 7.1 The Modeling Cycle . 206 7.2 Simulating Nature . 207 7.3 A realization tree of the signifier \solution of the equation 7x + 4 = 5x + 8 . 211 7.4 Probing Initial Conditions . 213 7.5 Kitten Life . 214 v List of Tables 3.1 Significant Contrasts In Classical And Western Mathematics . 46 vi Acknowledgments The oversight of Dr. Bruce Vogeli made this study possible. It is doubtful that I would have been able to produce this work in a different setting than that of Teachers College. The encouragement and suggestions from committee members Dr. Erica Walker and Dr. Daniel Friedrich were central to understanding the shortcomings and limitations of this work. Also, Dr. Peter Gallagher and Dr. Lalitha Vasudevan provided important feedback to improve the final draft. Early on in the process, Dr. Joe Malkevitch provided significant support and suggestions, and Dr. Orit Halpern at The New School who provided me great support in understanding the literature around technology and.
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