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The informal logic of mathematical proof
 ASPIC2 Argumentation Service Platform with Integrated Components http://www.argumentation.org
, 2007
"... Paul Erdős famously remarked that ‘a mathematician is a machine for turning coffee into theorems ’ [9, p. 7]. The proof of mathematical theorems is central to mathematical practice and to much recent debate about the nature of mathematics. This paper is an attempt to introduce a new perspective on t ..."
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Paul Erdős famously remarked that ‘a mathematician is a machine for turning coffee into theorems ’ [9, p. 7]. The proof of mathematical theorems is central to mathematical practice and to much recent debate about the nature of mathematics. This paper is an attempt to introduce a new perspective on the argumentation characteristic of mathematical proof. I shall argue that this account, an application of informal logic to mathematics, helps to clarify and resolve several important philosophical difficulties. It might be objected that formal, deductive logic tells us everything we need to know about mathematical argumentation. I shall leave it to others [14, for example] to address this concern in detail. However, even the protagonists of explicit reductionist programmes—such as logicists in the philosophy of mathematics and the formal theorem proving community in computer science—would readily concede that their work is not an attempt to capture actual mathematical practice. Having said that, mathematical argumentation is certainly not inductive either. Mathematical proofs do not involve inference from particular
Advanced MathematicalThinking at Any Age: Its Nature and Its Development
"... This article argues that advanced mathematical thinking, usually conceived as thinking in advanced mathematics, might profitably be viewed as advanced thinking in mathematics (advanced mathematicalthinking). Hence, advanced mathematicalthinking can properly be viewed as potentially starting in ele ..."
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This article argues that advanced mathematical thinking, usually conceived as thinking in advanced mathematics, might profitably be viewed as advanced thinking in mathematics (advanced mathematicalthinking). Hence, advanced mathematicalthinking can properly be viewed as potentially starting in elementary school. The definition of mathematical thinking entails considering the epistemological and didactical obstacles to a particular way of thinking. The interplay between ways of thinking and ways of understanding gives a contrast between the two, to make clearer the broader view of mathematical thinking and to suggest implications for instructional practices. The latter are summarized with a description of the DNR system (Duality, Necessity, and Repeated Reasoning). Certain common assumptions about instruction are criticized (in an effort to be provocative) by suggesting that they can interfere with growth in mathematical thinking. The reader may have noticed the unusual location of the hyphen in the title of this article. We relocated the hyphen in “advancedmathematical thinking ” (i.e., thinking in advanced mathematics) so that the phrase reads, “advanced mathematicalthinking” (i.e., mathematical thinking of an advanced nature). This change in emphasis is to argue that a student’s growth in mathematical thinking is an evolving process, and that the nature of mathematical thinking should be studied so as to
Towards a theory of mathematical argument
"... Abstract. In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, an ..."
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Abstract. In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a fullfledged mathematical proof or merely some nondeductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in nonmathematical contexts. I demonstrate this claim by considering the assessment of proofs, probabilistic evidence, computeraided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator ’ view of proofs because it places derivations – which may be thought to invoke formal logic – at the center of mathematical justificatory practice. However, when the notion of ‘derivation ’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.
An assessment model for proof comprehension in undergraduate mathematics.
"... Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimens ..."
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Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics. Building on Yang and Lin’s (2008) model of reading comprehension of proofs in school geometry, we contend that in undergraduate mathematics a proof is not only understood in terms of the meaning, logical status, and logical chaining of its statements, but also in terms of the proof’s highlevel ideas, its main components or modules, the methods it employs, and how it relates to specific examples. We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory.
Developing Understanding for Different Roles of Proof in Dynamic Geometry. Paper presented at ProfMat 2002
, 2002
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Proof as method: A new case for proof in mathematics curricula
, 2003
"... In recent years there has been a call to reform mathematics education to produce what the NCTM calls “mathematical literacy ” for all students. One of the NCTM’s Standards involves the use of problem solving as a method of learning mathematics. In this thesis I put forward the hypothesis that proof ..."
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In recent years there has been a call to reform mathematics education to produce what the NCTM calls “mathematical literacy ” for all students. One of the NCTM’s Standards involves the use of problem solving as a method of learning mathematics. In this thesis I put forward the hypothesis that proof is valuable in the school curriculum because it is instrumental in the cognitive processes required for successful problem solving. My view of proof does not supersede, but rather supplements, the traditional arguments for teaching proof. The evidence I present here draws on those traditional arguments as well as evidence from cognitive psychology concerning the role of metacognition in learning. The picture of proof that emerges emphasizes a role in mathematical discovery which mathematicians have noted but which
Contributions to a science of contemporary mathematics, preprint; current draft at http:// www.math.vt.edu/people/quinn
"... Abstract. This essay provides a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century, and in other sciences. Roughly, modern practice is well adapted to the structure of the subject and, within this constraint, much better ad ..."
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Abstract. This essay provides a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century, and in other sciences. Roughly, modern practice is well adapted to the structure of the subject and, within this constraint, much better adapted to the strengths and weaknesses of human cognition. These adaptations greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. The subject is approached in a bottomup ‘scientific ’ way, finding patterns in concrete microlevel observations and being eventually lead by these to understanding at macro levels. The complex and intenselydisciplined technical details of modern practice are fully represented. Finding accurate commonalities that transcend technical detail is certainly a challenge, but any account that shies away from this cannot be complete. As in all sciences, the final result is complex, highly nuanced, and has many surprises. A particular objective is to provide a resource for mathematics education. Elementary education remains modeled on the mathematics of the nineteenth century and before, and outcomes have not changed much either. Modern methodologies might lead to educational gains similar to those seen in professional practice. This draft is about 90 % complete, and comments are welcome. 1.
AN INVESTIGATION OF 10TH GRADE STUDENTS ’ PROOF SCHEMES IN GEOMETRY WITH RESPECT TO THEIR COGNITIVE STYLES AND
, 2007
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mathematics professor is trying to convey
"... Abstract. We describe a case study in which we investigate the effectiveness of a lecture in advanced mathematics. We first video recorded a lecture delivered by an experienced professor who had a reputation for being an outstanding instructor. Using video recall, we then interviewed the professor t ..."
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Abstract. We describe a case study in which we investigate the effectiveness of a lecture in advanced mathematics. We first video recorded a lecture delivered by an experienced professor who had a reputation for being an outstanding instructor. Using video recall, we then interviewed the professor to determine the ideas that he intended to convey and how he tried to convey these ideas in this lecture. We also interviewed six students to see what they understood from this lecture. The students did not comprehend the ideas that the professor thought were central to his lecture. Based on our analyses, we propose two factors to account for why students failed to understand these ideas.