Handbook of psychology volume 7 educational psychology
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- References 355
- Mathematical Learning
- Mathematical Argument Emerges in Classrooms That Support It 363 Reprise of Mathematical Argument 367
- Inscriptions as Mediators of Mathematical Activity and Reasoning 369
- Bridging Epistemologies 382 Cycles of Modeling 383
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Yuill, N. M., & Oakhill, J. V. (1991). Children’s problems in text comprehension: An experimental investigation. Cambridge, England: Cambridge University Press. CHAPTER 15 Mathematical Learning RICHARD LEHRER AND RICHARD LESH 357 MATHEMATICAL LEARNING 357 THE GROWTH OF ARGUMENT 358 Conversational Structure as a Resource for Argument 359 From Pretense to Proof 359 Mathematical Argument Emerges in Classrooms That Support It 363 Reprise of Mathematical Argument 367 INSCRIPTIONS TRANSFORM MATHEMATICAL THINKING AND LEARNING 367
Disciplinary Practices of Inscription and Notation 368 The Development of Inscriptions as Tools for Thought 369 Inscriptions as Mediators of Mathematical Activity and Reasoning 369 GEOMETRY AND MEASUREMENT 373 THE MEASURE OF SPACE 374 Mental Representation of Distance 375 Developing Conceptions of Unit 375 Developing Conceptions of Scale 376 Design Studies 376 Estimation and Error 378 STRUCTURING SPACE 380
MODELING PERSPECTIVES 382
IMPLICATIONS 384 REFERENCES 385 MATHEMATICAL LEARNING Does beauty have structure? How does a hinge work? What happens if zero divides a number? Do the symmetries of a triangle and the set of integers under addition have any struc- ture in common? How many distinct patterns of wallpaper design are possible? What are Nature’s numbers? How do nurses determine the dosage of drugs (e.g., Pozzi, Noss, & Hoyles, 1998) or entomologists quantify relations among ter- mites (e.g., Hall, Stevens, & Torralba, in press)? What forms of mathematical activity are found in automotive production (Smith, 1999)? Questions like these suggest the enormous imaginative scope and practical reach of mathematics and demonstrate that mathematicians are jugglers not of num- bers, but of concepts (e.g., Stewart, 1975). Mathematical practice spans a universe of human endeavor, ranging from art and craft to engineering design, and its products extend over much of recorded history. Despite this long history of mathematics, systematic study of mathematical learning oc- cupies only a brief slice in time. Nevertheless, research in mathematics education and in the psychology of mathemati- cal learning continues to grow, so that any review of this research is necessarily incomplete and highly selective. Our choices for this review stem from a genetic view of knowledge (Piaget, 1970), a “commitment that the structures, forms, and possibly the content of knowledge is determined in major respects by its developmental history” (diSessa, 1995, p. 23). Mathematics develops within a collective his- tory of argument and inscription (Davis & Hersh, 1981; Devlin, 2000; Kline, 1980; Lakatos, 1976; Nunes, 1999; Polya, 1945), so a genetic account of mathematical learning describes potential origins and developmental landscapes of these modes of thought. Accordingly, we first examine the nature of mathematical argument, tracing a path between everyday forms of argument and those that are widely recog- nized as distinctly mathematical. In this first section we focus on the epistemology (the grounds for knowing) and skills of argument, rather than on the more familiar heuristics and processes of mathematical reasoning (see, e.g., Haverty, The authors appreciate the thoughtful comments and suggestions of Leona Schauble, David Williamson Shaffer, Kathy Metz, Ellice Forman, and the editors, William Reynolds and Gloria Miller. This work was supported by the National Science Foundation (REC 9903409). The opinions expressed do not necessarily reflect the position, policy, or endorsement of the NSF. 358 Mathematical Learning Koedinger, Klahr, & Alibali, 2000; Leinhardt & Schwarz, 1997; Schoenfeld, 1992). We suggest that developmental roots of mathematical argument reside in the structure of nar- rative and pretend play but note how these roots must be nurtured to promote epistemic appreciation of proof and related forms of mathematical argument. We next turn to the role that inscriptions (e.g., markings in a medium such as paper) and notations play in the growth and development of mathematical ideas. Our intention once again is to illuminate the developmental relationship between infor- mal scratches on paper and the kinds of symbol systems em- ployed in mathematical practice. In concert with the core role assigned to argument, we suggest that mathematical thinking emerges as refinement of everyday claims about pattern and possibility yet departs from these everyday roots as these claims are progressively inscribed and otherwise symbolized. Inscription and mathematical thinking co-originate (Rotman, 1993), so that mathematics emerges as a distinct form of lit- eracy, much in the manner in which writing distinguishes itself from speech. From these starting points we examine how these general qualities of mathematical thinking play out in two realms: geometry measurement and mathematical modeling. We chose the former because spatial mathematics typically receives short shrift in reviews of this kind, yet it encom- passes a tradition that spans two millennia. Furthermore, spa- tial visualization is increasingly relevant to scientific inquiry and is undergoing a renaissance in contemporary computa- tional mathematics. Modeling was selected as the second strand because modeling emphasizes the need for a broad mathematical education that includes several forms of math- ematical inquiry. Moreover, modeling underscores the need to develop accounts of mathematical learning at the bound- aries of professional practices and conventionally recognized mathematical activity (e.g., Moschkovich, 2002). The studies selected for this review reflect both cognitive (e.g., Anderson & Schunn, 2000) and sociocultural perspec- tives (e.g., Forman, in press; Greeno, 1998) on learning. Stud- ies of cognitive development typically shed light on individual cognitive processes, for example, how young students might think about units of measure and how their un- derstandings might evolve. In contrast, sociocultural perspec- tives typically underscore thinking as mediated activity (e.g., Mead, 1910; Wertsch, 1998). For example, one might con- sider the history of cultural artifacts, such as rulers, in chil- dren’s developing conceptions of units. We believe that both forms of analysis are indispensable and that, in fact, these per- spectives are interwoven for learners, regardless of re- searchers’ proclivities to consider them as distinct enterprises. Consider, for example, the idea of learning to construct a geometric proof. On the one hand, a cognitive analysis char- acterizes the kinds of skills required to develop a proof and describes how those skills must be orchestrated (e.g., Koedinger & Anderson, 1990). These forms of char- acterization seem indispensable to instructional design (Anderson & Schunn, 2000). On the other hand, the need for proof is cultural, arising from an epistemology that values proof as explanation (Harel & Sowder, 1998; Hersh, 1993). Accordingly, this perspective poses the challenge not just of accounting for the understanding of proof, but also of how one might inculcate a classroom culture that values proof. In the sections that follow, we attempt to strike a balance be- tween these two levels of explanation because both supply important accounts of mathematical learning. Because we as- sume that readers are familiar with the general nature of these two kinds of analysis, we will not flesh out the assumptions of each perspective in this chapter. Download 9.82 Mb. Do'stlaringiz bilan baham: |
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