Classification and Concept Consistency

Abstract

This paper investigates the extent to which undergraduates consistently use a single mechanism as a basis for classifying mathematical objects. We argue that the concept image/concept definition distinction focuses on whether students use an accepted definition, but does not necessarily capture the more basic notion that there should be a fixed basis for classification. We examine students' classification of real sequences before and after exposure to definitions of 'increasing' and 'decreasing'; we develop an abductive 'plausible explanations' method to estimate the consistency within the participants' responses and suggest that this provides evidence that many students may lack what we call 'concept consistency'.Running head: Classification and Concept Consistency

Interactions between defining, explaining and classifying: the case of increasing and decreasing sequences

“…First, Alcock and Simpson (2011) found that in relation to these concepts, students classified sequences in a wide variety of ways. Second, their formal definitions are relatively straightforward compared with many definitions at this level (each has only one quantifier and one relation), so poor classifications are less likely to result from failing to understand the definitions (Dubinsky & Yiparaki, 2000).…”

“…Moreover, defining tasks have varied considerably: some studies provided participants with a formal definition (Alcock & Simpson, 2011;Inglis & Simpson, 2008), some asked for a formal definition (Heinze & Kwak, 2002), some asked for something more akin to an explanation ('try to explain, in simple words. .…”

“…Alcock and Simpson (2011), for instance, began by asking new undergraduates to complete a classifying task for which the concepts, increasing and decreasing for sequences, had not been formally introduced. They then presented the standard definitions and asked participants to repeat the classifying task with these definitions clearly visible.…”

“…Alcock & Simpson, 2004;Moore, 1994). At a local level, it means that students' ideas about category membership might be skewed towards common types of examples, so that their answers are inconsistent with defined concepts (Fujita, 2012;Vinner, 1991) and even, in some cases, inconsistent with one another (Alcock & Simpson, 2011). Advanced mathematics thus requires that students learn to classify and argue using standard concept definitions (Vinner, 1991) and to modify pre-existing concept images so that they are consistent with defined concepts.…”

Teaching and Learning of Calculus

“…The degree of legitimacy of this ultimate accomplishment is largely discussed in Vinner (1991). For Alcock and Simpson (2011), students do not necessarily feel the need for classifying a mathematical object as a member of a coherent set in which elements obey the same formal definition. They argue that before deciding on the use of a definition by students, it is important to investigate what they call concept consistency in order to consider student performance in the judgement of classification.…”