STUDENTS\u27 DEVELOPMENT IN PROOF: A LONGITUDINAL STUDY

Despite importance of teaching proof in any undergraduate mathematics program, many students have difficulties with proof (Dreyfus, 1999; Harel & Sowder, 2003; Selden & Selden, 2003; Weber, 2004). In this qualitative case study, nine undergraduate students were each interviewed once every two weeks over the course of an academic year. During each interview, the students were asked to complete, evaluate or discuss mathematical proofs. The results of these interviews were then analyzed using two different frameworks. The first focused on proof type, which refers to what kind of proof is created and how it came about. The second framework addressed identifying each student\u27s proof scheme, which constitutes ascertaining and persuading for that person (Harel & Sowder, 1998). Using these structures as a guide, the question I sought to answer is: What, if any, identifiable paths do students go through while learning to prove? Unfortunately, the data from this study failed to demonstrate any identifiable path that was common to all participants. In fact, only a single student made clear progress as judged by the criteria laid out at the beginning of this study. Specifically, the way she attempted proofs changed which was reflected in a greater tendency to use a particular proof type as time passed: semantic. Of the other students, six entered the study with a fairly mature view of proof that remained unchanged and thus had little progress to make relative to the frameworks used in the study. These students were also generally successful with the proofs they attempted and were more likely to use semantic proofs. The remaining two students were generally less successful and used semantic proofs rarely. This seems to imply that as students become more comfortable with proof, they become inclined toward the semantic proof type and this coincides with becoming more successful with proof in general

Proof and Proving in Mathematics Education

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Universal Logic and the Geography of Thought - Reflections on logical pluralism in the light of culture

The aim of this dissertation is to provide an analysis for those involved and interested in the interdisciplinary study of logic, particularly Universal Logic. While continuing to remain aware of the importance of the central issues of logic, we hope that the factor of culture is also given serious consideration. Universal Logic provides a general theory of logic to study the most general and abstract properties of the various possible logics. As well as elucidating the basic knowledge and necessary definitions, we would especially like to address the problems of motivation concerning logical investigations in different cultures. First of all, I begin by considering Universal Logic as understood by Jean-Yves Béziau, and examine the basic ideas underlying the Universal Logic project. The basic approach, as originally employed by Universal Logicians, is introduced, after which the relationship between algebras and logics at an abstract level is discussed, i.e., Universal Algebra and Universal Logic. Secondly,I focus on a discussion of the translation paradox , which will enable readers to become more familiar with the new subject of logical translation, and subsequently comprehensively summarize its development in the literature. Besides helping readers to become more acquainted with the concept of logical translation, the discussion here will also attempt to formulate a new direction in support of logical pluralism as identified by Ruldof Carnap (1934), JC Beall and Greg Restall (2005), respectively. Thirdly, I provide a discussion of logical pluralism. Logical pluralism can be traced back to the principle of tolerance raised by Ruldof Carnap (1934), and readers will gain a comprehensive understanding of this concept from the discussion. Moreover,an attempt will be made to clarify the real and important issues in the contemporary debate between pluralism and monism within the field of logic in general. Fourthly, I study the phenomena of cultural-difference as related to the geography of thought. Two general systems in the geography of thought are distinguished, which we here call thought-analytic and thought-holistic. They are proposed to analyze and challenge the universality assumption regarding cognitive processes. People from different cultures and backgrounds have many differences in diverse areas, and these differences, if taken for granted, have proven particularly problematic in understanding logical thinking across cultures. Interestingly, the universality of cognitive processes has been challenged, especially by Richard Nisbett s research in cultural psychology. With respect to these concepts, C-UniLog can also be considered in relation to empirical evidence obtained by Richard Nisbett et al. In the final stage of this dissertation, I will propose an interpretation of the concept of logical translation, i.e., translations between formal logical mode (as cognitive processes in the case of westerners) and dialectical logical mode (as cognitive processes in the case of Asians). From this, I will formulate a new interpretation of the principle of tolerance, as well as of logical pluralism

The Role of Inversion in the Genesis, Development and the Structure of Scientific Knowledge

The main thrust of the argument of this thesis is to show the possibility of articulating a method of construction or of synthesis--as against the most common method of analysis or division--which has always been (so we shall argue) a necessary component of scientific theorization. This method will be shown to be based on a fundamental synthetic logical relation of thought, that we shall call inversion--to be understood as a species of logical opposition, and as one of the basic monadic logical operators. Thus the major objective of this thesis is to This thesis can be viewed as a response to Larry Laudan's challenge, which is based on the claim that ``the case has yet to be made that the rules governing the techniques whereby theories are invented (if any such rules there be) are the sorts of things that philosophers should claim any interest in or competence at.'' The challenge itself would be to show that the logic of discovery (if at all formulatable) performs the epistemological role of the justification of scientific theories. We propose to meet this challenge head on: a) by suggesting precisely how such a logic would be formulated; b) by demonstrating its epistemological relevance (in the context of justification) and c) by showing that a) and b) can be carried out without sacrificing the fallibilist view of scientific knowledge. OBJECTIVES: We have set three successive objectives: one general, one specific, and one sub-specific, each one related to the other in that very order. (A) The general objective is to indicate the clear possibility of renovating the traditional analytico-synthetic epistemology. By realizing this objective, we attempt to widen the scope of scientific reason or rationality, which for some time now has perniciously been dominated by pure analytic reason alone. In order to achieve this end we need to show specifically that there exists the possibility of articulating a synthetic (constructive) logic/reason, which has been considered by most mainstream thinkers either as not articulatable, or simply non-existent. (B) The second (specific) task is to respond to the challenge of Larry Laudan by demonstrating the possibility of an epistemologically significant generativism. In this context we will argue that this generativism, which is our suggested alternative, and the simplified structuralist and semantic view of scientific theories, mutually reinforce each other to form a single coherent foundation for the renovated analytico-synthetic methodological framework. (C) The third (sub-specific) objective, accordingly, is to show the possibility of articulating a synthetic logic that could guide us in understanding the process of theorization. This is realized by proposing the foundations for developing a logic of inversion, which represents the pattern of synthetic reason in the process of constructing scientific definitions

Mathematics and Its Applications, A Transcendental-Idealist Perspective

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies

Tartu Ülikooli toimetised. Tööd semiootika alalt. 1964-1992. 0259-4668

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