Handbook of psychology volume 7 educational psychology
Inscriptions Transform Mathematical Thinking and Learning
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- Reprise of Mathematical Argument
- INSCRIPTIONS TRANSFORM MATHEMATICAL THINKING AND LEARNING
- Inscription and Argument
- Inscriptions Transform Mathematical Thinking and Learning 369
- Inscriptions as Mediators of Mathematical Activity and Reasoning
- Microgenetic Studies of Appropriation of Inscription
Inscriptions Transform Mathematical Thinking and Learning 367 teachers who were schooled traditionally. At first, students were satisfied by induction over several cases to “prove” that the area of a rectangle could be constituted by multiplication of its width and length. Rather than challenging something that the students all knew to be true, the teacher (Simon) directed the conversation toward explanation, subtly reori- enting the grounds of argument from the particular to the general (e.g., whether this would work all the time). Simon’s emphasis on the general was further illustrated in another episode in which students attempted to determine the area of an irregular blob by transforming its contour to a more famil- iar form. Although students could see in a case that their strat- egy in fact also transformed the area, they were not bothered by this refutation (see also Schauble, 1996), a manifestation of an everyday sense of the general, rather than a mathemati- cal sense. Simon and Blume (1996) also encountered the lim- its of persuasion when students considered justifications of their predictions about the taste of mixtures that were in different ratios. Here students talked past one another, appar- ently because some thought of the situation as additive and others as multiplicative. Such studies of teaching and learn- ing again emphasize both the role of the teacher in establish- ing formats of argument consistent with the discipline and the need for enculturation so that students can see the functions of proof, not simple exposure to proof practices. Reprise of Mathematical Argument Mathematical argument emphasizes generality and certainty about patterns and is supported by cognitive capacities to represent possibility and to reason counterfactually about pos- sible patterns. These capacities seem to be robustly supported by cultural practices such as pretense and storytelling. Never- theless, dispositions to construct mathematically sound argu- ments apparently do not arise spontaneously in traditional schooling or in everyday cultural practices. Mathematical forms such as proof have their genesis in mathematics classrooms that emphasize conjecture, justification, and explanation. These forms of thinking demand high standards of teaching practice because the evidence suggests that although students may be the primary authors of these arguments, it is the teachers who orchestrate them. Classroom dialogue can spawn overlapping epistemologies, as students are oriented toward mathematics as structure and pattern while they simultaneously examine the grounds of knowl- edge. Ideally, pattern and proof epistemologies co-originate in classrooms because pattern provides the grounds for proof and proof the rationale for pattern. Thus, classroom conversation and dialogue constitute one possible genetic pathway toward the development of proof reasoning skills and an appreciation of the epistemology of generalization. Yet even as we empha- size proposition and language, we are struck with the role played by symbolization and tools in the development of argu- ments in classrooms and in various guises of mathematical practice. This is not surprising when one considers the central historical role of such symbolizations in the development of mathematics. We turn next to considering a complementary genetic pathway to mathematical knowledge, that of students as writers of mathematics. INSCRIPTIONS TRANSFORM MATHEMATICAL THINKING AND LEARNING In this section we explore the invention and appropriation of inscriptions (literal marks on paper or other media, following Latour, 1990) as mediational tools that can transform mathe- matical activity. This view follows from our emphasis in the previous section on mathematics as a discursive practice in which everyday resources, such as conversation and pre- tense, provide a genetic pathway for the development of an epistemology of mathematical argument, of literally talking mathematics into being (Sfard, 2000; Sfard & Kieran, 2001). Here we focus on the flip side of the coin, portraying mathe- matics as a particular kind of written discourse— “a business of making and remaking permanent inscriptions . . . operated upon, transformed, indexed, amalgamated” (Rotman, 1993, p. 25). Rotman distinguished this view from a dualist view of symbol and referent as having independent existence, proposing instead that signifier (inscription) and signified (mathematical idea) are “co-creative and mutually origina- tive” (p. 33). Accordingly, we first describe perspectives that frame inscriptions as mediators of mathematical and scien- tific activity, with attention to sociocultural accounts of inscription and argument. These accounts of inscription but- tress the semiotic approach taken by Rotman (1988, 1993) and set the stage for cognitive studies of inscription. We go on to describe children’s efforts to invent or appropriate in- scriptions in everyday contexts such as drawing or problem solving. Collectively, these studies suggest that the growth of representational competence, as reviewed in the previous section, is mirrored by a corresponding competence in the uses of inscription and notation. In other words, the having of ideas and the inscribing of ideas coevolve. Studies of inscrip- tionally mediated thinking in mathematics indicate that math- ematical objects are created as they are inscribed. This perspective calls into question typical accounts in cognitive science, where inscriptions are regarded as simply referring to mathematical objects, rather than constituting them. We conclude this section with the implications of these findings
368 Mathematical Learning for an emerging arena of dynamic inscriptions, namely, com- putational media.
Studies in the sociology of science demonstrate that scientists invent and appropriate inscriptions as part of their everyday practice (Latour, 1987, 1990; Lynch, 1990). Historically, sys- tems of inscription and notation have played important roles in the quantification of natural reality (Crosby, 1997) and are tools for modeling the world on paper (Olson, 1994). DiSessa (2000, p. 19) noted, Not only can new inscription systems and literacies ease learn- ing, as algebra simplified the proofs of Galileo’s theorems, but they may also rearrange the entire terrain. New principles become fundamental and old ones become obvious. Entirely new terrain becomes accessible, and some old terrain becomes boring.
Visualizing Nature One implication of this view of scientific practice as the in- vention and manipulation of the world on paper (or electronic screen) is that even apparently individual acts of perceiving the world, such as classifying colors or trees, are mediated by layers of inscription and anchored to the practices of discipli- nary communities (Goodwin, 1994, 1996; Latour, 1986). Goodwin (1994) suggested that inscriptions do not mirror discourse in a discipline but complement it, so that profes- sional practices in mathematics and science use “the distinc- tive characteristics of the material world to organize phenomena in ways that spoken language cannot—for exam- ple, by collecting records of a range of disparate events onto a single visible surface” (p. 611). For example, archaeolo- gists classify a soil sample by layering inscriptions, field practices, and particular forms of talk to render a professional judgment (Goodwin, 2000). Instead of merely looking, archaeologists juxtapose the soil sample with an inscription (the Munsell color chart) that arranges color gradations into an ordered grid, and they spray water on the soil to create a consistent viewing environment. These practices format dis- cussion of the appropriate classification and illustrate the moment-to-moment embedding of inscription within particu- lar practices. Repurposing Inscription Inscriptions in scientific practice are not necessarily stable. Kaiser (2000) examined the long-term history of physicists’ use of Feynman diagrams. Initially, these diagrams were invented to streamline, and make visible, computationally intensive components of quantum field theory. They drew heavily on a previous inscription, Minkowski’s space-time diagrams, which lent an interpretation of Feynman diagrams as literal trajectories of particles through space and time. Of course, physicists knew perfectly well that the trajectories so described did not correspond to reality, but that interpretation was a convenient fiction, much in the manner in which physi- cists often talk about subatomic particles as if they were macroscopic objects (e.g., Ochs, Jacoby, & Gonzales, 1994; Ochs, Gonzales, & Jacoby, 1996). Over time, the theory for which Feynman developed his diagrams was displaced, and a competing inscription tuned to the new theory, dual dia-
tages, the new inscription (dual diagrams) never replaced the Feynman diagram. Kaiser (2000) suggested that the reason was that the Fenyman diagrams had visual elements in com- mon with the inscriptions of paths in bubble chambers, and this correspondence again had an appeal to realism: Unlike the dual diagrams, Feynman diagrams could evoke, in an unspoken way, the scatterings and propagation of real particles, with “realist” associations for those physicists already awash in a steady stream of bubble chamber photographs, in ways that the dual diagrams simply did not encourage. (Kaiser, 2000, pp. 76–77) Hence, scientific practices of inscription are saturated in some ways with epistemic stances toward the world and thus cannot be understood outside of these views.
Nevertheless, Latour (1990) suggested that systems of inscription, whether they are about archaeology or particle physics, share properties that make them especially well suited for mobilizing cognitive and social resources in service of argument. His candidates include (a) the literal mobility and immutability of inscriptions, which tend to obliterate barriers of space and time and fix change, effectively freezing and preserving it so that it can serve as the object of reflection; (b) the scalability and reproducibility of inscriptions, which guarantee economy even as they preserve the configuration of relations among elements of the system represented by the inscription; and (c) the potential for recombination and superimposition of inscriptions, which generate structures and patterns that might not otherwise be visible or even conceivable. Lynch (1990) reminded us, too, that inscriptions not only preserve change, but edit it as well: Inscriptions reduce and enhance information. In the next section we turn toward studies of the development of children
Inscriptions Transform Mathematical Thinking and Learning 369 as inscribers, with an eye toward continuities (and some discontinuities) between inscriptions in scientific and every- day activity. The Development of Inscriptions as Tools for Thought Children’s inscriptions range from commonplace drawings (e.g., Goodnow, 1977) to symbolic relations among maps, scale models, and pictures and their referents (e.g., DeLoache, 1987) to notational systems for music (e.g., Cohen, 1985), number (e.g., Munn, 1998), and the shape of space (Newcombe & Huttenlocher, 2000). These inscriptional skills influence each other so that collectively children develop an ensemble of inscriptional forms (Lee & Karmiloff-Smith, 1996). As a con- sequence, by the age of 4 years children typically appreciate distinctions among alphabetical, numerical, and other forms of inscription (Karmiloff-Smith, 1992). Somewhat surprisingly, children invent inscriptions as tools for a comparatively wide range of circumstances and goals. Cohen (1985) examined how children ranging in age from 5 to 11 years created inscriptions of musical tunes they first heard, and then attempted to play with their invented scores. She found that children produced a remarkable diver- sity of inscriptions that did the job. Moreover, a substantial majority of the 8- to 11-year-olds created the same inscrip- tions for encoding and decoding. Their inscriptions adhered to one-to-one mapping rules so that, for example, symbols consistently had one meaning (e.g., a triangle might denote a brief duration) and each meaning (e.g., a particular note) was represented by only one symbol. Both of these properties are hallmarks of conventional systems of notation (e.g., Good- man, 1976). Other studies of cognitive development focus on children’s developing understandings and uses of inscription for solving puzzle-like problems. Karmiloff-Smith (1979) had children (7–12 years) create an inscriptional system that could be used as an external memory for driving (with a toy ambulance) a route with a series of bifurcations. Children invented a wide range of ade- quate mnemonic marks, including maps, routes (e.g., R and L to indicate directions), arrows, weighted lines, and the like. Often, children changed their inscriptions during the course of the task, suggesting that children transform inscriptions in response to local variation in problem solving. All of their re- visions in this task involved making information that was im- plicit, albeit economically rendered, explicit (e.g., adding an additional mark to indicate an acceptable or unacceptable branch), even though the less redundant systems appeared adequate to the task. Karmiloff-Smith (1992) suggested that these inscriptional changes reflected change in internal repre- sentations of the task. An alternative interpretation is that children became increasingly aware of the functions of inscription, so that in this task with large memory demands, changes to a more redundant system of encoding provided multiple cues and so lightened the burden of decoding—a tradeoff between encoding and decoding demands. Communicative considerations are paramount in other studies of children’s revisions of inscriptions. For example, both younger (8–9 years) and older (10–11 years) children adjusted inscriptions designed as aides for others (a peer or a younger child) to solve a puzzle problem in light of the age of the addressee (Lee, Karmiloff-Smith, Cameron, & Dodsworth, 1998). Compared with adults, younger children were more likely to choose minimal over redundant inscrip- tions for the younger addressee, whereas the older children were equally likely to chose either inscription. Overall, there was a trend for older children to assume that younger addressees might benefit from redundancy. In a series of studies with older children (sixth grade through high school), diSessa and his colleagues (diSessa, in press; diSessa, Hammer, Sherin, & Kolpakowski, 1991) in- vestigated what students know about inscriptions in a general sense. They found that like younger children, older children and adolescents invented rich arrays of inscriptions tuned to particular goals and purposes. Furthermore, participants’ inventions were guided by criteria such as parsimony, econ- omy, compactness (spatially compact inscriptions were pre- ferred), and objectivity (inscriptions sensitive to audience, so that personal and idiosyncratic features were often suppressed). Collectively, studies of children’s development suggest an emerging sense of the uses and skills of inscription across a comparatively wide range of phenomena. Invented inscrip- tions are generative and responsive to aspects of situation. They are also effective: They work to achieve the goal at hand. Both younger and older children adapt features of in- scriptions in light of the intended audience, suggesting an early distinction between idiosyncratic and public functions of inscription. Children’s invention and use of inscriptions are increasingly governed by an emerging meta-knowledge about inscriptions, which diSessa et al. (1991) termed
the deployment of inscriptions for mathematical activity, although we shall suggest (much as we did for argument) that if mathematics and inscription are to emerge in coordination, careful attention must be paid to the design of mathematics education. Inscriptions as Mediators of Mathematical Activity and Reasoning Mathematical inscriptions mediate mathematical activity and reasoning. This position contrasts with inscriptions as mere
370 Mathematical Learning records of previous thought or as simple conveniences for syn- tactic manipulation. In this section we trace the ontogenesis of this form of mediated activity, beginning with children’s early experiences with parents and culminating with class- rooms where inscriptions are recruited to create and sustain mathematical arguments.
Van Oers (2000, in press) claimed that early parent-child in- teractions and play in preschool with counting games set the stage for fixing and selecting portions of counting via in- scription. In his account, when a child counts, parents have the opportunity to interpret that activity as referring to cardi- nality instead of mere succession. For example, as a child completes his or her count, perhaps a parent holds up fingers to signify the quantity and repeats the last word in the count- ing sequence (e.g., 3 of 1, 2, 3). This act of inscription, although perhaps crudely expressed as finger tallies, curtails the activity of counting and signifies its cardinality. As suggested by Latour (1990), the word or tally (or numeral) can be transported across different situations, such as three candies or three cars, so number becomes mobile as it is recruited to situations of “how many.” Pursuing the role of inscription in developing early num- ber sense, Munn (1998) investigated how preschool children’s use of numeric notation might transform their understanding of number. She asked young children to par- ticipate in a “secret addition” task. First children saw blocks in containers, and then they wrote a label for the quantity (e.g., with tallies) on the cover of each of four containers. The quantity in one container was covertly increased, and children were asked to discover which of the containers had been incremented. The critical behavior was the child’s search strategy. Some children guessed, and others thought that they had to look in each container and try to recall its previous state. However, many used the numerical labels they had written to check the quantity of a container against its previous state. Munn found that over time, preschoolers were more likely to use their numeric inscriptions in their search for the added block, using inscriptions of quantity to compare past and current quantities. In her view, children’s notations transformed the nature of their activity, signaling an early integration of inscriptions and conceptions of number. Coconstitution of conceptions of number and inscription may also rely on children’s capacity for analogy. Brizuela (1997) described how a child in kindergarten came to under- stand positional notation of number by analogy to the use of capital letters in writing. For this child, the 3 in 34 was a “capital number,” signifying by position in a manner reminiscent of signaling the beginning of a sentence with a capital letter. Microgenetic Studies of Appropriation of Inscription The cocreation of mathematical thought and inscription is elaborated by microgenetic examination of mathematical ac- tivity of individuals in a diverse range of settings. Hall (1990, 1996) investigated the inscriptions generated by algebra prob- lem solvers (ranging from middle school to adult participants, including teachers) during the course of solution. He sug- gested that the quantitative inferences made by solvers were obtained within representational niches defined by interaction among varied forms of inscription (e.g., algebraic expres- sions, diagrams, tables) and narratives, not as a simple result of parsing strings of expressions. These niches or material de- signs helped participants visualize relations among quantities and stabilized otherwise shifting frames of reference. Coevolution of inscription and thinking was also promi- nent in Meira’s (1995, in press) investigations of (middle school) student thinking about linear functions that describe physical devices, such as winches or springs. His analysis focused on student construction and use of a table of values to describe relations among variables such as the turns of a winch and the distance an object travels. As pairs of students solved problems, Meira (1995) noted shifting signification, reminiscent of the role of the Feynman diagrams, in that marks initially representing weight shifted to represent distance. He also observed several different representational niches (e.g., transforming a group of inscriptions into a single unit and then using that unit in subsequent calculation), a clear dependence of problem-solving strategies on qualities of the number tables, and a lifting away from the physical devices to operations in the world of the inscriptions—a way of learning to see the world through inscriptions. Izsak (2000) found that pairs of eighth-grade students experimented with different possibilities for algebraic expressions as they explored the alignment between computa- tions on paper and the behavior of the winch featured in the Meira (1995) study. Pairs also negotiated shifting signification between symbols and aspects of device behavior, suggesting that interplay between mathematical expression and qualities of the world may constitute one genetic pathway for mediat- ing mathematical thinking via inscriptions. (We pick this theme up again in the section on mathematical modeling.) In their studies of student appropriation of graphical displays, Nemirovsky and his colleagues (Nemirovsky & Monk, 2000; Nemirovsky, Tierney, & Wright, 1998) sug- gested that learning to see the world through systems of
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