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Adolescent Reasoning in Mathematics: Exploring Middle School Students’ Strategic Approaches in Empirical Justifications

2011; Wiley; Volume: 33; Issue: 33 Linguagem: Inglês

1551-6709

Jennifer Cooper, Candace Walkington, Caroline Williams, Olubukola Akinsiku, Charles W. Kalish, Amy B. Ellis, Eric Knuth,

Mathematics Education and Teaching Techniques

Adolescent Reasoning in Mathematics: Exploring Middle School Students’ Strategic Approaches in Empirical Justifications Jennifer L. Cooper (jcooper4@wisc.edu) A Candace A. Walkington (cwalkington@wisc.edu) A Caroline C. Williams (ccwilliams3@wisc.edu) C Olubukola A. Akinsiku (akinsiku@wisc.edu) C Charles W. Kalish (cwkalish@wisc.edu) B Amy B. Ellis (aellis1@wisc.edu) C Eric J. Knuth (knuth@education.wisc.edu) C A Wisconsin Center for Education Research, 1025 W. Johnson Street B Department of Educational Psychology, 1025 W. Johnson Street C Department of Curriculum and Instruction, 225 N. Mills Street Madison, WI 53706 USA Abstract Twenty middle-school students participated in semi- structured interviews in which they were asked to assess the validity of two mathematical conjectures. In addition to being free to develop a valid proof as a justification, students were also asked to generate numeric examples to test the conjecture. Students demonstrated strategic reasoning in their empirical approaches by varying the quantity, parity, magnitude, and typicality of the numbers selected. These strategies were more developed in students who initially believed in the truth of the conjecture as well as in students who generated a valid, deductive proof. Emphasizing students’ strategic selection of diverse examples parallels inductive reasoning in other domains. Strategic use of examples in justifying conjectures has the potential to assist students’ development of deductive proof strategies. Keywords: inductive reasoning; middle mathematics; proof; empirical-based reasoning school Background Many consider proof to be central to the discipline and practice of mathematics. Yet surprisingly, the role of proof in school mathematics has traditionally been peripheral at best, usually limited to high school geometry. More recently, however, mathematics educators and researchers are advocating that proof should play a central role in mathematics education. Reasoning about the properties, relationships, and patterns in math, as one does with proofs, supports the development of mathematical expertise. Yet, despite the growing emphasis on justifying and proving in school mathematics, students rely overwhelmingly on examples to justify the truth of statements rather than using deductive proofs (e.g., Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009; Koedinger, 1998; Porteous, 1990). Many students fail to understand the nature of what counts as evidence and justification (Kloosterman & Lester, 2004). In mathematics, testing examples is not sufficient for proof – a deductive argument is necessary to cover all possible cases. The preceding discussion regarding students’ reliance on examples to “prove” the truth of statements (i.e., provide empirical-based justifications) is not meant to imply that examples do not play an important role in mathematical activity. Indeed, mathematicians often utilize examples to gain insight, develop an argument, and verify that an argument works (Alcock, 2004). The challenge remains, however, to help students learn to differentiate these appropriate uses of examples from their use as a primary means of justification. Although reasoning inductively 1 features prominently in students’ math justifications, the strategies underlying such reasoning are typically treated by mathematics educators as stumbling blocks to overcome rather than as objects of study in their own right or as starting points from which to foster the development of more sophisticated (deductive) ways of reasoning. The research has focused primarily on distinctions between the inductive, empirical approach and deductive justifications. Questions such as what might make one example or empirical justification stronger than another have not been well addressed. In contrast, inductive strategies have been an ongoing focus of research in other domains such as biology where children and adults reason competently using inductive reasoning (e.g., Gelman & Kalish, 2006; Gopnik et al., 2004; Rhodes, Brickman, & Gelman, 2008). Inductive approaches and predictive inferences are appropriate in this domain, and they are supported by category knowledge. In particular, empirical justifications are rated as stronger when based on typical examples with high similarity to the category (Osherson et al., 1990). Having a diverse set of examples increases the coverage of the category. People’s knowledge about the underlying category structure supports successful inferential reasoning (Osherson et al., 1990). Effectively employing strategies to select informative examples depends, at least in part, on intuitions about similarity and typicality relations. It is unclear to what degree students have robust intuitions about the relations and category structure of mathematical objects and the Here we refer to making generalizations about a class of numbers based on observing or testing particular instances of that class, not mathematical induction, which is a valid method of proof.