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• W2992509895 abstract "This article is motivated by a commitment to the ideas underlying discovery learning, namely the epistemological notion of grounded, meaningful, generative knowledge. It is also motivated by concern that these ideas have been implicitly misinterpreted in curriculum and instruction, ulti mately to the detriment of students. Accordingly, I discuss an alternative, empirically based, theoretical articulation of discovery pedagogy that addresses the criticisms it has faced. The research question framing this alternative approach is, What exactly about a mathematical concept should students discover via discovery learning? I will pursue this question by reflecting on two case stud ies of children who participated in activities of my own design. Empirical data from these and other studies have served me over the past decade as contexts for inquiry into the cognition and instruction of mathematical concepts, an inquiry that, in turn, keeps feeding back into further design and articulation of design principles. In this essay, I will use these data to offer an empirically grounded centrist answer to the question of what students should discover, at least with respect to a particular class of mathematical concepts (intensive quantities) as embodied in a particular type of design (perception-based learning). First, though, some fur ther clarification of terms is due. The rationale of my proposal hinges on a common dis tinction between and in mathematical learning activities. By process, I am referring to a general problem-solving sequence: (a) construing, parsing, and modeling a realistic situation along dimensions relevant to goal information; (b) determining targeted values within these dimensions (by enumeration and/or measurement); (c) manipulating these extracted values algorithmically with the aid of further mathematical instruments, tools, forms, and media, such as inscribing and developing an algebraic formula; and (d) reinterpreting obtained values or inferences in light of the source situation (e.g., Verschaffel, Greer & De Corte, 2000). By product, I refer primarily to any of the milestone mathematical displays created through engaging in the process, including inscriptions, such as diagrams, and multimodal utterance, such as speech or gesture. An exam ple of a would be the event space of a probability experiment, which is created through combinatorial analysis of a random generator—a novice can be guided to build this product, yet only an expert can infer from it an anticipated outcome distribution. I do no cas my articulations of the terms process and product as offering in-and-of-themselves any unique insight i t the nature of mathematical cognition. I am not referring to epistemological debates about arithmetical oper ations as pro esses or objects (e.g., Confrey & Costa, 1996; Sfard, 1991). Similarly, by process, I am not referring to s hematic mental activity with operational closure (Varela, Thompson & Rosch, 1991, p. 139) but to the cultural rote algorithmic practice that can be enacted piecemeal and imi tatively with littl to no grounding or clear goal-based orientation. By product, I am not referring to an interim state of self-adapted chemes resulting from situated enactment, but to external or externalized information, such as inscrip tional artifacts or vocalized or gestured expressions that come to exist in the world, in forms that Norman (1991) might call cogni ive rtifacts or what Hutchins (1995) might view as mean of instrumenting and distributing col lective cognitive activity. Finally, my objective is not so much to survey or evaluate the variety of pedagogical philosophies in dialogue with con structiv sm (see Cobb, 2005; Freudenthal, 1991; Norton, 2009; Radford, 2005). Nor will I comment on whether math ematical knowledg is formed as a cognitive schema, social practice, semiotic system, etc. Rather, I am exploring whether the exacting tenets of the more radical forms of constructivist mathematical pedagogy may be upheld, yet adapted, so as to include instructional situations in which the locus of reinven" @default.
• W2992509895 created "2019-12-13" @default.
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• W2992509895 date "2012-01-01" @default.
• W2992509895 modified "2023-09-28" @default.
• W2992509895 title "Discovery Reconceived: Product before Process." @default.
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