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Problem Solving for Today's Social Context Holly K. Brewster Submitted in Partial Fulfillment O The Teacher as Mathematician: Problem Solving for Today’s Social Context Holly K. Brewster 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 2014 © 2014 Holly K. Brewster All rights reserved ABSTRACT The Teacher as Mathematician: Problem Solving for Today’s Social Context Holly K. Brewster A current trend in social justice oriented education research is the promotion of certain intellectual virtues that support epistemic responsibility, or differently put, the dispositions necessary to be a good knower. On the surface, the proposition of epistemically responsible teaching, or teaching students to be responsible knowers is innocuous, even banal. In the mathematics classroom, however, it is patently at odds with current practice and with the stated goals of mathematics education. This dissertation begins by detailing the extant paradigm in mathematics education, which characterizes mathematics as a body of skills to be mastered, and which rewards ways of thinking that are highly procedural and mechanistic. It then argues, relying on a wide range of educational thinkers including John Dewey, Maxine Greene, Miranda Fricker, and a collection of scholars of white privilege, that an important element in social justice education is the eradication of such process-oriented thinking, and the promotion of such intellectual virtues as courage and humility. Because the dominant paradigm is supported by an ideology and mythology of mathematics, however, changing that paradigm necessitates engaging with the underlying conceptions of mathematics that support it. The dissertation turns to naturalist philosophers of education make clear that the nature of mathematics practice and the growth of mathematical knowledge are not characterized by mechanistic and procedural thinking at all. In these accounts, we can see that good mathematical thinking relies on many of the same habits and dispositions that the social justice educators recommend. In articulating an isomorphism between good mathematical thinking and socially responsive thinking, the dissertation aims to offer a framework for thinking about mathematics education in and for a democratic society. It aims to cast the goals of mathematically rigorous education and socially responsible teaching not only as not in conflict, but also overlapping in meaningful ways. Table of Contents List of Figures ......................................................................................................................................... v Acknowledgments .............................................................................................................................. vi Chapter 1: Introduction ..................................................................................................................... 1 1.1 The question .......................................................................................................................................... 1 1.2 Objectives ............................................................................................................................................... 6 1.3 Existing Lines of Inquiry: Mathematics ......................................................................................... 7 1.4 Existing lines of inquiry: Education .............................................................................................. 9 1.4.1 Pedagogy of the Oppressed ........................................................................................................................ 12 1.4.2 Critical Mathematics Education .............................................................................................................. 14 1.4.3 Critical mathematics education on the ground ................................................................................ 16 1.5 Methodology ....................................................................................................................................... 19 1.6 Theoretical Framework .................................................................................................................. 19 1.6.1 Focus on the Privileged .............................................................................................................................. 19 1.6.2 Deweyan Democratic Education ............................................................................................................... 24 1.7 Overview & Chapter Summaries ................................................................................................. 26 1.8 On Radicality, Positionality, and the Role of the Philosopher .......................................... 27 Chapter 2: ProBlem Solving Skills Paradigm ......................................................................... 30 2.1 Distinction among mathematicians ............................................................................................ 31 2.2 Problem Solving in Mathematics Education: Historical Context ....................................... 33 2.3 Problem Solving Skills Paradigm ................................................................................................ 37 2.3.1 Problem solving skills in action: Khan Academy ........................................................................... 39 i 2.3.2 Further discussion ........................................................................................................................................ 44 2.4 Animating Ideology ........................................................................................................................... 45 2.4.0 There are no myths in mathematics ..................................................................................................... 46 2.4.1 Mathematics is universal, unified, and transcendental ................................................................ 46 2.4.2 Mathematics is objective and impersonal .......................................................................................... 51 2.4.3 Mathematics is provable and certain ................................................................................................... 54 2.4.4 Mathematics is a solitary achievement ............................................................................................... 55 2.5 Counter Currents ............................................................................................................................................. 56 2.5 Generalizing Mathematical Thinking ......................................................................................... 58 Chapter 3: Epistemic Aspects of Social Justice ...................................................................... 63 3.1 Introduction ........................................................................................................................................ 63 3.2 Thinking for Freedom ...................................................................................................................... 65 3.3 Focus on the powerful ...................................................................................................................... 71 3.3.0 A Note on Terminology .............................................................................................................................. 72 3.3.1 Unconscious prejudice: Institutional Knowledge ........................................................................... 72 3.3.2 Unconscious Prejudice: Individual Level (Credibility) ................................................................. 76 3.3.3 Objectification and Direction of Fit ....................................................................................................... 78 3.3.4 A Second Approach: Ways of being and phenomenology of privilege. ................................. 81 3.4 Conclusion ........................................................................................................................................... 89 Chapter 4: Intellectual Courage and Humility ...................................................................... 93 4.1 Introduction ........................................................................................................................................ 93 4.2 Uncertainty, Open-mindedness, and Humility ....................................................................... 96 4.2.1 Unertainty and Open-mindedness ............................................................................................................. 96 4.2.2 Humility ................................................................................................................................................................. 97 ii 4.3 Courage and Surprise .......................................................................................................................... 99 4.4 Virtues vs. Skills ............................................................................................................................... 104 4.4.1 Judgment and the Will ................................................................................................................................. 104 4.4.2 Learning virtues .......................................................................................................................................... 107 Chapter 5. Recasting Mathematics ........................................................................................
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