On the different ways that mathematicians use diagrams in proof construction

Samkoff, Aron; Lai, Yvonne; Weber, Keith

doi:10.1080/14794802.2012.657438

Cited by 20 publications

(12 citation statements)

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“…To highlight this difficulty, Samkoff, Lai, and Weber (2012) asked eight research mathematicians at prestigious universities to prove the following claim: BProve that f(x) = sin(x) is not injective on any interval of length greater than π.^All eight mathematicians drew a graph of the sine function and used this graph to convince themselves that the statement was true. Also, all participants felt that their graphical arguments did not constitute proofs and attempted to transform these arguments into verbal-symbolic proofs.…”

Section: Limitations Of Basing Proofs On Graphical Argumentsmentioning

confidence: 98%

Bridging the gap between graphical arguments and verbal-symbolic proofs in a real analysis context

2016

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We examine a commonly suggested proof construction strategy from the mathematics education literature-that students first produce a graphical argument and then work to construct a verbal-symbolic proof based on that graphical argument. The work of students who produce such graphical arguments when solving proof construction tasks was analyzed to distill three activities that contribute to students' successful translation of graphical arguments into verbal-symbolic proofs. These activities are called elaborating, syntactifying, and rewarranting. We analyze how engaging in these activities relates to students' success in proof construction tasks. Additionally, we discuss how each individual activity contributes to the translation of a graphical argument into a verbal-symbolic proof.

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Section: Limitations Of Basing Proofs On Graphical Argumentsmentioning

confidence: 98%

Bridging the gap between graphical arguments and verbal-symbolic proofs in a real analysis context

2016

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“…In the same vein, translating an informal argument into a proof can be a time-consuming process, even for those who are experienced proof writers (cf., Samkoff et al, 2012). The time constraints in this study, and in most undergraduate mathematics assessments, might push students away from a potentially time-consuming targeted strategy and toward a shotgun strategy, which has the potential to pay quicker dividends.…”

Section: Limitations Of Our Findings and Future Research Questionsmentioning

confidence: 95%

“…Lockwood, Ellis, Dogan, Williams, and Knuth (2012) studied the ways in which mathematicians used examples to solve problems and write proofs. Samkoff, Lai, and Weber (2012) explored how mathematicians used diagrams when writing a proof. This focus on mathematicians provides valuable insight into how proofs may be successfully written, but it has an important limitation.…”

Section: Introductionmentioning

confidence: 99%

Two proving strategies of highly successful mathematics majors

Zazkis

Weber

Mejía-Ramos

2015

The Journal of Mathematical Behavior

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a b s t r a c tWe examined the proof-writing behaviors of six highly successful mathematics majors on novel proving tasks in calculus. We found two approaches that these students used to write proofs, which we termed the targeted strategy and the shotgun strategy. When using a targeted strategy students would develop a strong understanding of the statement they were proving, choose a plan based on this understanding, develop a graphical argument for why the statement is true, and formalize this graphical argument into a proof. When using a shotgun strategy, students would begin trying different proof plans immediately after reading the statement and would abandon a plan at the first sign of difficulty. The identification of these two strategies adds to the literature on proving by informing how elements of existing problem-solving models interrelate.Published by Elsevier Inc.

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“…The study of diagrams has become a growing field in mathematics education (de Freitas & Sinclair, 2012;Gibson, 1998;Samkoff, Lai, & Weber, 2012). The findings of Gibson and Samkoff indicate students and mathematicians use diagrams to help them comprehend the information available to them, to reason about the truthfulness of an assertion, to discover new ideas, to make conjectures, and to help them communicate their own ideas.…”

Section: Diagrammatic Reasoningmentioning

confidence: 98%