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Keeping meaning in proportion: The multiplication table as a case of pedagogical bridging tools. Unpublished doctoral dissertation
, 2004
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Collaborative Interpretive Argumentation as a PhenomenologicalMathematical Negotiation: A Case of Statistical Analysis of a Computer Simulation of Complex Probability
"... "To swallow something in the hope that it may be wholesome is clearly a commitment, and so is every act of seeing things in one particular way.” Polanyi (1958, p. 363) “A full understanding of the power and pitfalls of visual representations is no doubt a long way off. But lack of understanding shou ..."
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"To swallow something in the hope that it may be wholesome is clearly a commitment, and so is every act of seeing things in one particular way.” Polanyi (1958, p. 363) “A full understanding of the power and pitfalls of visual representations is no doubt a long way off. But lack of understanding should not block their use in cases where it is clearly legitimate.” Barwise and Etchemenday (1991, p. 23) Computers are powerful tools for modeling mathematical concepts, but models are not selfevident phenomena. We wish to query the use of mathematical computer modeling as a form of argumentation and discourse. We presume that mathematical knowledge should be grounded in contextualized activity (Piaget, 1952; Papert; 1991). Computer simulations afford opportunities for such contextualized activity (Wilensky; e.g., 1993) in learning environments where students can ‘connect ’ their qualitative intuitions to formal quantitative articulation, e.g., graphs and formulae. We believe in the advantage of collaboration over exclusivelyindividual learning as a catalyst of argumentative rhetoric, through which individuals articulate hitherto implicit interpretive models (HaroutunianGordon & Tartakoof, 1996; Cobb & Bauersfeld, 1995; Krummheuer, 2000). Also, we see great heuristicdidactic value in shifting between different interpretive models for making sense of observed phenomena, and between isomorphic mathematical representations (diagrams, graphs, and equations; Post, Cramer, Behr, Lesh, & Harel, 1993). A collaborative phenomenologicalcummathematical negotiation affords opportunities for formulating and bartering interpretive
Gödel's Incompleteness Theorems: A Revolutionary View of the Nature of Mathematical Pursuits
"... The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicia ..."
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The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicians studies mathematics. This paper provides a history of the mathematical developments that laid the foundation for Gödel's work, describes the unique method used by Gödel to prove his famous incompleteness theorem, and discusses the farreaching mathematical implications thereof. 2 I.
Random Generator
, 2009
"... This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan or sublicensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express ..."
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This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan or sublicensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. COGNITION AND INSTRUCTION, 27(3), 175–224, 2009
Probabilistic thinking: presenting plural perspectives (PT: PPP). Rethinking Probability Education: Perceptual Judgment as Epistemic Resource
"... ABSTRACT: The mathematics subject matter of probability is notoriously challenging, and in particular the content of random compound events. When students analyze experiments, they often omit to discern variations as distinct outcomes, e.g., HT and TH in the case of flipping a pair of coins, and thu ..."
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ABSTRACT: The mathematics subject matter of probability is notoriously challenging, and in particular the content of random compound events. When students analyze experiments, they often omit to discern variations as distinct outcomes, e.g., HT and TH in the case of flipping a pair of coins, and thus infer erroneous predictions. Educators have addressed this conceptual difficulty by engaging students in actual experiments whose outcomes contradict the erroneous predictions. Yet whereas empirical activities per se are crucial for any probability design, because they introduce the pivotal contents of randomness, variance, sample size, and relations among them, empirical activities may not be the unique or best means for students to accept the logic of combinatorial analysis. Instead, learners may avail of their own preanalytic perceptual judgments of the random generator itself so as to arrive at predictions that agree rather than conflict with mathematical analysis. I support this view first by detailing its philosophical, theoretical, and pedagogical foundations and then presenting empirical findings from a designbased research project. Twenty eight students aged 9–11 participated in tutorial, taskbased clinical interviews that utilized an innovative random generator. Their predictions were mathematically correct even though initially they did not discern variations. Students were then led to recognize the formal