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Try to See It My Way: The Discursive Function of Idiosyncratic Mathematical Metaphor

2012; Taylor & Francis; Volume: 14; Issue: 1 Linguagem: Inglês

10.1080/10986065.2012.625076

1532-7833

Dor Abrahamson, J.F. Gutiérrez, Anna K. Baddorf,

Creativity in Education and Neuroscience

Abstract What are the nature, forms, and roles of metaphors in mathematics instruction? We present and closely analyze three examples of idiosyncratic metaphors produced during one-to-one tutorial clinical interviews with 11-year-old participants as they attempted to use unfamiliar artifacts and procedures to reason about realistic probability problems. Our interpretations of these episodes suggest that metaphor is both spurred by and transformative of joint engagement in situated activities: metaphor serves individuals as semiotic means of objectifying and communicating their own evolving understanding of disciplinary representations and procedures, and its multimodal instantiation immediately modifies interlocutors' attention to and interaction with the artifacts. Instructors steer this process toward normative mathematical views by initiating, modifying, or elaborating metaphorical constructions. We speculate on situation parameters affecting students' utilization of idiosyncratic resources as well as how socio-mathematical license for metaphor may contribute to effective instructional discourse. ACKNOWLEDGEMENTS This article draws on empirical data collected in a study supported by a National Academy of Education/Spencer Postdoctoral Fellowship 2005–2006 (Seeing Chance, Abrahamson). A synopsis of the paper focusing on different data was presented at SRTL-6 (CitationAbrahamson, 2009d). An elaborated treatment of the Li episode has appeared in CitationAbrahamson (2010). We are grateful to all members of the Embodied Design Research Laboratory at the University of California, Berkeley (Abrahamson, Director), for generative contention during the microgenetic analysis of our data. Thanks to Katie Makar, Dani Ben-Zvi, and the MTL editors and anonymous reviewers of earlier drafts who helped us realize how we see things. The second and third authors, a graduate student and an undergraduate research apprentice, respectively, made equivalent contributions to this article—their names are listed in reverse-alphabetical order. Notes 1Lennon, J., & McCartney, P. ("The Beatles") (1965). We can work it out. London: Abbey Road Studios. 2Viewed in "slow motion," the experiment is revealed to be not binomial but hypergeometric. That is, when the first marble is scooped, the urn has one less of that marble's color to be scooped into the remaining three concavities, so that the experiment is strictly a case of "without replacement." Nevertheless, the experiment can be viewed as functionally approximating the binomial due to the minute ratio of the sample size (four marbles) to the content of the urn (hundreds of marbles). FIGURE 1 Materials used in the study—experimental and analytic embodiments of the 2-by-2 mathematical object: (a) the marbles scooper; (b) a template for performing combinatorial analysis; (c) the combinations tower—a distributed sample space of the marbles-scooping experiment; and (d) an actual experimental outcome distribution produced by a computer-based simulation of this probability experiment. (Color figures available online.) Display full size 3For discussions of research on students' challenges and possible trajectories in learning to perform combinational analysis, see CitationBatanero, Navarro–Pelayo, and Godino (1997), CitationEnglish (2005), and CitationSriraman and English (2004). FIGURE 2 Selected phases in Jake's guided combinatorial analysis of the three-blue combination (Figures 2a–2d) and the two-blue combination (Figure 2e, 2f). The darker line on top of each card was designed as a means of supporting the analysis by making rotations distinct. Display full size 4Jake's modest discovery faintly echoes Friedrich August Kekulé von Stradonitz's momentous metaphor-based discovery of benzene's ring-shaped structure. Kekulé apparently dreamed of a snake swallowing its own tail (CitationBenfey, 1958). In CitationAbrahamson, Wilensky, and Janusz (2006) we discuss a case of middle-school students inventing similar figural strategies to analyze a 3-by-3 green/blue table. Although they discovered many of the 512 possibilities, the students ultimately abandoned the strategy, because they had no clear criteria for determining whether they had exhausted the sample space. 5It is as though Sima and Dor are both staring at Jastrow's proverbial duck–rabbit ambiguous image and arguing not over what it is but whether this thing should be fed carrots or fish (CitationAbrahamson et al., 2009; CitationRowland, 1999). See also CitationNewman, Griffin, and Cole (1989) on the enabling "looseness" in a novice and expert's joint attention to objects used for the enactment of disciplinary cultural practice. 6A "batting average" is the ratio of hits (successful batting) to at-bats (opportunities to do so) over a given period of time. This "average" is usually expressed as a decimal quotient, and it can be updated by adding appropriately to the numerator and denominator. At the beginning of a season, the "average" changes erratically, but as the season progresses, it stabilizes.