An alternative framework to evaluate proof productions: A reply to Alcock and Inglis

Abstract

In a recent paper, Alcock and Inglis (2009) noted a distinction between the way that Weber (2009) and they defined syntactic and semantic proof productions. Weber argued that “a syntactic proof production occurs when one works predominantly within the representation system of proof [. . .] Alternatively, a semantic proof production occurs when much of the prover’s work takes place [outside] the representation system of proof.” Alternatively, Alcock and Inglis cited their definition that “during a syntactic proof production, the prover works entirely . . .within the representation system of proof” whereas “during a semantic proof production, the proverworks, at least in part, in a different representation system” (Alcock & Inglis, 2008). The authors then illustrated how using different definitions can lead to different classifications of some proof productions and argue that their characterization is preferable. Alcock and Inglis (2009) contended that by focusing on students’ ability to translate between representation systems, their terminology captures the fundamental aspect of this distinction, while at the same time remaining “sufficiently wellspecified for classifying all existing published case-studies on undergraduate proof production” and “useful for analyzing large-scale datasets” (a test that they believed should be pivotal in determiningwhich framework to adopt). Alcock and Inglis further argued that, in contrast, Weber’s (in press) set of definitions relied upon “an as yet unspecified metric for gauging the amount of work conducted in each representation system” (making it hard to employ in the analysis of large datasets) and failed to recognize an important asymmetry between the status of representation systems involved in undergraduate proof productions.