Handbook of psychology volume 7 educational psychology
Inscriptions Transform Mathematical Thinking and Learning
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- Studies of Inscription in Classrooms Designed to Support Invention and Appropriation
- Notation: A Privileged Inscription
- Geometry and Measurement 373
- Dynamic Notations
- GEOMETRY AND MEASUREMENT
- THE MEASURE OF SPACE
Inscriptions Transform Mathematical Thinking and Learning 371 inscription is more accurately described as a fusion between signifiers and signified. In their view, coming to interpret an inscription mathematically often involves treating the signi- fiers and the signified as undifferentiated, even though one knows very well that they can be treated distinctly (the roots of these capabilities are likely found in pretense and possibil- ity, as we described previously). In their studies of students’ attempts to interpret graphical displays of physical motion, they recounted an instance of teacher scaffolding by using “these” to refer simultaneously to lines on a graph, objects (toy bears), and a narrative in which the bears were nearing the finish of a race. This referential ambiguity helped the student create an interpretation of the inscription that was more consistent with disciplinary practice as she sorted out the relations among inscription, object, and the ongoing nar- rative that anchored use of the inscription to a time course of events.
According to Stevens and Hall (1998), mathematical learning mediated by inscription is tantamount to disciplining one’s perception: coming to see the inscription as a mathe- matical marking consistent with disciplinary interpretations, rather than as a material object consistent with everyday in- terpretations. That such a specialized form of perception is required is evident in the confusions that even older students have about forms of notation like the graph of a linear func- tion. For example, a student’s interpretation of slope in a case study conducted by Schoenfeld, Smith, and Arcavi (1993) included a conception of the line as varying with slope, y- intercept, and x-intercept. The result was that the student’s conception of slope was not stable across contexts of use. Stevens and Hall (1998) traced the interventions of a tutor who helped an eighth-grade student working on similar problems of interpretation of graphical displays. Their analy- sis focused on the tutoring moves that helped reduce the student’s dependence on a literal grid representing Cartesian coordinates. Some of the teacher’s assistance included literal occlusion of grid, a move designed to promote disciplinary understanding by literally short-circuiting the student’s reliance on the grid in order to promote a disciplinary focus on ratio of change to describe the line. Moschkovich (1996) examined how pairs of ninth-grade students came to disci- pline their own perceptions by coordinating talk, gestures, and inscriptions of slope and intercept. Inscriptions helped orient students toward a shared object of reference, and the use of everyday metaphors such as hills and steepness grounded this joint focus of conversation. Ultimately, how- ever, the relative ambiguity of these everyday metaphors instigated (for some pairs) a more disciplined interpretation because meanings for these terms proved ambiguous in the context of conversation. However, not all pairs of students evolved toward disciplinary-centered interpretation, again suggesting the need for instructional support.
Some research provides glimpses of invention and use of inscription in classrooms where the design of instruction supports students’ invention and appropriation of varying forms of mathematical inscription. These studies are oriented toward a collective level of analysis (i.e., treating the class as a unit of analysis) because the premise is that, following Latour (1990), inscriptions mobilize arguments in particular commu- nities. In these studies the community is the mathematics cul- ture of the classroom. Moreover, “a focus on inscriptions requires traditional learning environments to be redesigned in such a way that students can appropriate inscription-related practices and discourses” (Roth & McGinn, 1998, p. 52). Cobb, Gravemeijer, Yackel, McClain, and Whitenack (1997) traced children’s coordination of units of 10 and 1 in a first grade class. Instruction was designed to situate investi- gation of these units and unit collections in a context of pack- aging candies. Arithmetic reasoning was constituted as a “chain of signification” (Walkerdine, 1988) in which unifix cubes first signified a quantity of candies packed in the shop and then this sign (the unifix cubes–candies relation) was incorporated as a signified of various partitions of candies inscribed as pictured collections. At this point the structure of the collection, rather than the original packaging of candy, became the object of thinking. The structure of the collection, in turn, served as the signified of yet another signifier, a notational rendering of collections as, for instance, 3r13c (3 rolls, 13 candies). Cobb et al. (1997) noted that this ren- dering served as the vehicle by means of which the pictured collections became models of arithmetic reasoning (also see Gravemeijer, Cobb, Bowers, & Whitenack, 2000). Kemeny (2001) examined the collective dialogic pro- cesses during a lesson in which a third-grade teacher helped students construct the mathematical object referred to by the inscription of the Cartesian system. Her analysis underscores the interplay between collective argument and inscription. It also highlights the role of the teacher’s orchestration of con- versation and inscription. First, the teacher introduced a new signifier, drawing the axes of the coordinate system on the blackboard, and invited students to consider whether it might be a good tool for thinking about relationships between the sides of similar rectangles. Because these students had a prior history of investigating concepts of ratio via the study of geometric similarity (Lehrer, Strom, & Confrey, in press), the introduction of the signifier (the inscription) created an
372 Mathematical Learning opportunity for students to create the signified—the Carte- sian grid (see Sfard, 2000). Children’s first attempts to gen- erate a signified were based on projecting metaphors of measure. They decided, for example, that the lengths of the axes should be subdivided into equal measures and that this subdivision implied an origin labeled numerically as zero because movement along the axis was a distance, not a count. They debated where this origin should be placed and generated several valid alternatives. At this point, the teacher stepped in to introduce a convention, which students ac- cepted as sensible. Some students then transported a practice they had gener- ated in previous investigations, superimposing paper models of similar rectangles to observe their growth, to the axes on the blackboard, drawing rectangles that mimicked the paper material. This invited consideration of the axes as a literal support (and raised questions about what to label them), but it also inspired one student to notice a stunning possibility— a rectangle might be represented by one of its vertices. Per- haps there was no need to draw the whole thing! Their teacher promptly seized upon this suggestion, and the stu- dents went on to explore its implications. Eventually, they concluded that there could be as many rectangles as they liked, not just the cases initially considered, and that all sim- ilar rectangles could be represented and generated as a line through the origin. Inscription (Cartesian coordinates) and argument (a gen- eralization about similar figures) were co-originated. The inscription did not spring out of thin air, but it became a tar- get of metaphoric projection and extension and was ulti- mately treated as an object in its own right. The construction of this object invited a format for generalization, the line rep- resenting all rectangles, and also an epistemology of pattern. What was true for three or four cases was accepted as true for infinitely many. Over the course of several lessons, students’ inscriptions of similarity as numeric ratio, as algebraic pattern (e.g., the class of similar rectangles described by LS ϭ 3 ϫ SS, where LS and SS refer to “long side” and “short side,” respectively), and as a line in the Cartesian sys- tem introduced a resonance among inscriptional forms. For example, the sense of pattern generalization could be ex- pressed in three distinctive forms of inscription, yet the equivalence of these forms invited construction of a signified that spanned all three (Lehrer et al., in press). The lesson analyzed by Kemeny (2001) was anchored in a history of inscription in the classroom (Lehrer, Jacobson, Kemeny, & Strom, 1999; Lehrer & Pritchard, in press). The norms in the classroom included a stance toward adopting inscriptions as tools for thinking and, further, toward assum- ing that no inscription would be wasted; that is, if students de- veloped a stable (and public) system of mathematical inscription, they could reasonably expect to use it again. One such opportunity was presented to students later in the year when they conducted investigations about the growth of plants. Lehrer, Schauble, Carpenter, and Penner (2000) tracked students’ inscriptions of plant growth during succes- sive phases of inquiry over the course of approximately three months. The investigators found a reflexive relationship be- tween children’s inscriptions of growth and their ideas about growth. Over time, children either invented or appropriated inscriptions that increasingly drew things together by increas- ing the dimensionality of their models of growth. For exam- ple, initial inscriptions were one-dimensional records of height, but these were later supplanted by models of plant vol- ume that incorporated variables of height, width, and depth and that were sequenced chronologically to facilitate test of the conjecture that plant growth was an analogue of geometric growth (which it was not). Inscription and conception of growth were co-originated in Rotman’s (1993) sense.
Developmental studies of children’s symbolization, microge- netic studies of individuals’ efforts to appropriate inscription, and collective studies of classrooms where inscriptions are recruited to argument describe a complementary genetic path- way for the development of mathematical reasoning: the in- teractive constitution of inscription and mathematical objects. These studies also reveal the cognitive and social virtues of privileging notations among inscriptions. Goodman (1976) suggested heuristic principles to dis- tinguish notational systems from other systems of inscrip- tion. The principles govern relations among inscriptions (signifiers–literal markings), objects (signified), character classes (equivalent inscriptions, such as different renderings of the numeral 7), and compliance classes (equivalent ob- jects, such as dense materials or emotional people). Two prin- ciples govern qualities of inscriptions that qualify as notation: (a) syntactic disjointedness, meaning that each inscription be- longs to only one character class (e.g., the marking 7 is rec- ognized as a member of a class of numeral 7s, but not numeral 1s), and (b) syntactic differentiation, meaning that one can readily determine the intended referent of each mark (e.g., if one marked quantity with length, then the differences in length corresponding to differences in quantity should be per- ceived readily). Two other principles regulate mappings between charac- ter classes and compliance classes. The first is that all inscrip- tions of a character class should have the same compliance class, which Goodman (1976) referred to as a principle of unambiguity. For example, all numeral 7s should refer to the same quantity, even though the quantity might be comprised
Geometry and Measurement 373 of seven dogs or seven cats. It follows, then, that character classes should not have overlapping fields of compliance classes—the principle of semantic disjointedness. For exam- ple, the numeral 7 and the numeral 8 should refer to different quantities. This requirement rules out natural language’s inter- secting categories, such as whale and mammal. Finally, a prin- ciple of semantic differentiation indicates that every object represented in the notational scheme should be able to be clas- sified discretely (assigned to a compliance class)—a principle of digitalization of even analog qualities. For example, the quantities 6.999 and 7.001 might be assigned to the quantity 7, either as a matter of practicality or as a matter of necessity before the advent of a decimal notation. These features of notational systems afford the capacity to treat symbolic expressions as things in themselves, and thus to perform operations on the symbols without regarding what they might refer to. This capacity for symbolically mediated generalization creates a new faculty for mathemat- ical reasoning and argument (Kaput, 1991, 1992; Kaput & Schaffer, in press). For example, the well-formedness of no- tations makes algorithms possible, transforming ideas into computations (Berlinski, 2000). Notational systems simulta- neously provide systematic opportunity for student expres- sion of mathematical ideas, but the same systematicity places fruitful constraints on that expression (Thompson, 1992). We have seen, too, how notations transform mathematical experiences genetically, both over the life span (from early childhood to adulthood) and over the span of growing expertise (from novices to professional practitioners of math- ematics and science). Consider, for example, the van Oers (2000, in press) account of parental scaffolding to notate children’s counting. This marking objectifies counting activity so that it becomes more visible and entity-like. The use of a symbolic system for number foregrounds the quantity that results from the activity of counting and backgrounds the counting act itself. This separation of activity (counting) from its product (quantity) sets the stage for making quantity a sub- strate for further mathematical activities, such as counts of quantities as exemplified in the Cobb et al. (1997) study of first graders. Microgenetic studies like those of Hall (1990) and Meira (1995) suggest that inscriptions tend to drift over time and use toward notations that stabilize interactions among par- ticipants. The classroom studies by Kemeny (2001) and Lehrer et al. (2000) also suggest a press toward notation as a means of fixing, selecting, and composing mathematical objects as tools for argument. These studies, however, concen- trate largely on the world on paper, so in the next section we address the implications of electronic technologies for bootstrapping the reflexive relation between conception and inscription. Dynamic Notations The chief effect of electronic technologies is the correspond- ing development of new kinds of notational systems, often described as dynamic (Kaput, 1992). The manifestations of electronically mediated notations are diverse, but what they share in common is an expression of mathematics as compu- tation (Noss & Hoyles, 1996). DiSessa (2000) suggested that computation is a new form of mathematical literacy, conclud- ing that computation, especially programming, “turns analy- sis into experience and allows a connection between analytic forms and their experiential implications” (p. 34). Moreover, simulating experience is a pathway for building students’ un- derstanding, yet it is also integral to the professional practices of scientists and engineers. Sherin (2001) explored the implications of replacing alge- braic notation with programming for physics instruction. Here again, notations did not simply describe experience for students, but rather reflexively constituted it. Programming expressions of motion afforded more ready expression of time-varying situations. This instigated a corresponding shift in conception from an algebraically guided physics of bal- ance and equilibrium to a physics of process and cause. Resnick (1994) pointed out that introducing students to parallel programming (e.g., multiple screen “turtles”) pro- vides an opportunity to develop mathematical descriptions at multiple levels and to understand how levels interact. The programming language provides an avenue for decentralized thinking. Wilensky and Resnick (1999) noted the difficulties that people have in comprehending levels of phenomena such as traffic jams. At one level, traffic jams result from cars mov- ing forward, but the interactions among cars create jams that proliferate backward. This effect seems at first glance to violate common sense, so it is hard for people to compre- hend, but dynamic notations such as parallel programming place new tools in the hands of students for thinking about re- lations between local agents and aggregate levels of descrip- tion. Our (much) abbreviated tour of dynamic notations clearly indicates that this form of inscription affords new op- portunities to coconstitute mathematical thought and writing. In the sections that follow, we revisit this theme in the realms of geometry measurement and mathematical modeling.
Geometry is a spatial mathematics that has its roots in antiq- uity yet continues to evolve in the present, as witnessed by continuing concern with computer-generated experiments in visualization. Although common school experiences of geometry emphasize the construction and proof schemes 374 Mathematical Learning of the ancient Greeks, the scope of geometry is far wider, ranging from consideration of fundamental qualities of space such as shape and dimension (e.g., Banchoff, 1990; Senechal, 1990) to the very fabric of artistic design, commercial craft, and models of natural processes (e.g., Stewart, 1998). Con- sider, for example, the designs displayed in Figure 15.1. Both were created from the same primary cell (unit) but with dif- ferent symmetries (the left by a translation, the right by a ro- tation). Systematic analyses of symmetries of design stimulate both mathematical inquiry (e.g., Schattschneider, 1997; Washburn & Crowe, 1988) and the ongoing practice of crafts such as quilting (e.g. Beyer, 1999). Geometry’s versatility and scope have oriented us to sur- vey a range of studies that demonstrate the potential role of geometry in a general mathematics education (Goldenberg, Cuoco, & Mark, 1998; Gravemeijer, 1998). Our chief em- phasis is on studies of the growth and development of spatial reasoning in contexts designed to support development (prin- cipally, schools). We first consider studies of children’s unfolding understanding of the measure of space. Although measurement is (now) traditionally separated from geometry education, we argue for its reinstatement on two grounds. First, measuring a quality of a space invokes consideration of its nature. For example, although measure of dimension seems transparent, the dimension of fractal images in not ob- vious, and consideration of their measure leads one toward more fundamental ideas about their construction (e.g., Devaney, 1998). Second, measurement is inherently approx- imate so that it constitutes a bridge to related forms of math- ematics, such as distribution and reasoning about variation. Third, practices of measurement span multiple realms of en- deavor, especially the quantification of physical reality (Crosby, 1997). Even apparently simple acts, such as match- ing the color of a sample of dirt to an existing classification scheme, are in fact embedded within systems of inscription and practice, so that measurement is a window to the inter- play between imagined qualities of the world and the practi- cal grasp of these qualities (Goodwin, 2000). Consequently, our review focuses on research that helps us understand the kinds of thinking at the heart of the interplay between this imaginative leap (i.e., an imagined quality of space) and practical grasp (e.g., its measure). After completing our review of measure, we consider how inquiry about shape and form frames developing types of arguments, especially proof and related “habits of mind” (Goldenberg et al., 1998). Here we focus on the role of dynamic notational systems, embodied (currently) as soft- ware tools such as Logo (Papert, 1980) and the Geometer’s Sketchpad (Jackiw, 1995), because these spotlight the role of dynamic notation in the development of mathematical reasoning and argument about space. THE MEASURE OF SPACE In the sections that follow, we review investigations of chil- dren’s reasoning about measure. We focus primarily on stud- ies of linear measure to illuminate the interactive roles of inscription and developing conceptions of space because these studies encapsulate many of the findings, issues, and approaches that emerge in investigations of other dimensions and qualities of space, such as area, volume, and angle (see Lehrer, 2002; Lehrer, Jaslow & Curtis, in press, for more extensive review of the latter). We include studies from mul- tiple perspectives. Studies of cognitive development typically compare children at different ages (cross-sectional) or follow the same children for a period of time (longitudinal) to ob- serve transitions in thinking, typically about units of mea- sure. These studies provide glimpses of children’s thinking under conditions of activity and learning that are typically found in the culture. They follow from the tradition first es- tablished by Piaget and his colleagues (e.g., Piaget, Inhelder, & Szeminska, 1960). In contrast, design studies modify the learning environment and then investigate the effects of these modifications (Brown, 1992; Cobb, 2001). These stud- ies are often conducted from sociocultural perspectives with attendant attention to forms of inscription and notation and to forms of classroom talk that seem important to help Download 9.82 Mb. Do'stlaringiz bilan baham: |
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