The Spirit of Four

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Droujkova, M. (2004). The spirit of four: Metaphors and models of number construction. Paper presented at the 26th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Toronto, Ontario, Canada.
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  The spirit of four: Metaphors and models of number construction Maria A. DroujkovaObjectives Researchers have noted the need for investigation of relationships between different types of reasoning and number construction models (Confrey & Smith, 1995; Olive, 2001; Pepper &Hunting, 1998; Steffe, 1994). The goal of this study was to look at the development of number construction through the lens of metaphor.In particular, the study investigated the interplay between a largely multiplicative environment andthe development of reasoning within this environment that was significantly different from scenariosfrom other studies. Conceptual framework  Observing young children makes a strong case for viewing mathematical thinking as fundamentallymetaphoric (R. Davis, 1984). Metaphor is the recursive movement between a source and a targetthat are structurally similar, both changing in the dynamic process of learning (B. Davis, 1996; R.Davis, 1984; English, 1997; Lakoff & Johnson, 1980; Lakoff & Nunez, 1997, 2000; Pimm, 1987;Presmeg, 1997; Sfard, 1997).For analyzing number construction, I used the counting scheme (Olive, 2001; Steffe, 1994) and thesplitting conjecture (Confrey & Smith, 1995; Lehrer, Strom, & Confrey, 2002). The metaphor thatconnects sources of sharing, folding or similarity, and the target of multiplicative one-to-manyactions can be considered the basis of splitting as a cognitive scheme. The metaphor that connects- 1 -  the source of counting and the target of the number sequence is the basis of the counting scheme. Inthe splitting world multiplicative reasoning develops via grounding metaphors with sources such assharing. In the counting world multiplicative reasoning is based on the linking metaphor whichconnects interiorized, reversible counting with iterable units (Olive, 2001; Steffe, 1994). Modes of inquiry This paper presents a longitudinal case study of reasoning in a child up to the age of five, whosehome environment was restructured to incorporate more multiplicative activities. Researchers oftenconsider metaphor to be private, unformulated and difficult to study (Presmeg, 1997). Additionalaccess issues came from the need for a very young subject necessary to trace the beginnings of number concept development, and from the longitudinal nature of the study. These considerations pointed to the necessity of a close relationship between the subject of the study and the researcher,and I invited my daughter “Katya” to be the subject of the study. As a parent, I was in a privileged position of access to the majority of the details of Katya’s day-to-day life, as well as to the meaningof her utterances and gestures. Data sources and evidence Data for the study came from fieldnotes of observations as a participant-observer; videotapes andaudiotapes of unstructured and semi-structured interviews; photographs of activity settings; and acollection of artifacts used in activities. Results The non-sequential order in which conventional number names first appeared in Katya’s speechcorresponded to multiplicative, rather than counting, actions. For example, the utterance “two twos” - 2 -  appeared about eight months earlier than the word “four,” and also earlier than the word “three.”Appearance of “two threes” in games preceded the use of the words “four,” “five” and “six,” andappearance of “two fours” preceded the use of numbers greater than four. In constructing numbers from one to four, Katya used individual (Presmeg, 1997) metaphors basedon instant recognition of the quantity. In these metaphors, the source was an image with a quantityintrinsically embedded in it, such as “dog’s legs” for “four.” Katya mostly used mixed references formultiplicative situations, for example, “two dogs” to signify “two times four.” This availability of two systems of signifiers provided a language necessary to address the asymmetrical nature of themultiplication models Katya used. For example, in the case of “two dogs” the words underlined thedistinction between sets and set members in the set model of multiplication. Lack of signifiers forthis asymmetry of multiplication models may be problematic and may hinder development of multiplicative reasoning. Confrey and Smith (1995) note that “a counting number is typically usedto name the result or outcome of a split” (p.75, italics mine). If learners see the splitting and counting worlds as isomorphic (Confrey & Smith, 1995), they canunderstand structures of one world by making parallels with the corresponding structures of theother world. Children’s structure transfer attempts become especially visible when they differ fromaccepted standards. For example, researchers often focus on children inappropriately applyingadditive strategies to multiplicative situations (Post, Behr, & Lesh, 1986). Katya frequently tried touse multiplicative relationships instead of additive . For example, when asked to continue a patternof arrays made out of circles: 2 by 1, 2 by 2, 2 by 3, ___ she attempted to iterate the previous arraytwice, drawing a 2 by 6 array instead of the expected “2 by 4”. Upon my explanation that a pair of  - 3 -  circles is added  to the array in each step, Katya said, somewhat angrily, that these pictures “are notreal.” Multiplicative relationships were more “real” to her.In another unexpected example, a square was split into four equal squares, and then each of thesmall squares was split into four tiny squares. Katya used words signifying size gradients, such as“large, small, and tiny,” and “babies and adults,” consistently across different multiplicative worlds.This metaphor of “growth” united different multiplicative worlds and allowed Katya to comparetheir structures, working on what a mathematician would call “powers” or “base systems.” Katyaused the word “spirit” to denote the action in each world, for example, talking about “the spirit of four” in the split square above. She claimed that if we cut the 4-square piece in four, the resultwould be zero. Upon cutting, she was surprised that the result was one square. However, in repeatedactivities with the same picture, or with pictures based on other powers from other split worlds,Katya consistently said that the result of splitting the power base picture would be “zero”, or“nothing,” even after observing again and again that it turned out to be one.I hypothesized that these names were expressions of metaphors for the srcin, and I told Katya thatresearchers call the entity in question “the srcin.” We compared the srcins of additive and power-based structures, and Katya felt validated to discover a “real” zero at least at some srcin. Thisinstance of isomorphism between additive and multiplicative worlds helped Katya to build her ideaof the srcin as a “superordinate construct” (Confrey & Smith, 1995), whereas the idea wasproblematic while she stayed within the multiplicative world. - 4 -
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