Paper presented at the annual meeting of the Jean Piaget Society, Berkeley, June 2-4.

Responding to recent calls to develop cognitivist-cum-sociocultural theoretical models of mathematical learning, we present case analyses of students’ creative appropriation of forms emerging from interaction with symbolic artifacts. In this bootstrapping procedure, problem solvers: (1) *hook *– when a symbolic artifact is first introduced into the interaction space, they engage it because they recognize its contextual utility for enhancing the enactment, explanation, or evaluation of their current solution strategy; then (2) *shift *– in the course of implementing these new affordances, they notice in these artifacts additional embedded properties as affording a new or reconfigured strategy that better meets domain-general criteria of conciseness, precision, prediction, and – in the case of co-production with a peer – communication, coordination, and collaboration. These new strategies are then sanctioned by the instructor, who views the shift as advancing the child’s process of mathematization closer to disciplinary structures and procedures, in accord with the intervention’s pedagogical objectives. We support and elaborate the proposed constructs with a set of selected episodes from videographed empirical data gathered in a design-based research study that investigated the emergence of mathematical concepts from guided embodied-interaction activity (n=22; ages 9-11). We list and explain critical interaction dimensions enabling such learning.

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**Summary:**

diSessa has recently called for “dialectical” approaches to the study of cognition and instruction in mathematics and science, the principle being that neither cognitivist nor sociocultural approaches *per se*appear to have presented satisfactory narratives of learning processes in the disciplines, so that the field should develop integrated theoretical models that leverage and interweave the unique contributions of these historically parallel approaches, thus offering more complete, more nuanced descriptions (diSessa, 2008). Such conciliatory efforts bespeak an earlier call from Cole and Wertsch (1996)to go “beyond the individual–social antinomy in discussions of Piaget and Vygotsky.” Some researchers have attempted to create dialogues between these dominant perspectives (e.g., Abrahamson et al., 2007; diSessa, Philip, Saxe, Cole, & Cobb, 2010; Halldén, Scheja, & Haglund, 2008). Yet, in order to advance from symposium to synthesis—or, if you will, from mixture to solution—we must go beyond patchwork models that aggregate individual and social factors as co-present cumulative effects to synergistic models that represent this co-presence as situated, emergent, microgenetic reciprocities (cf. Hall, 1996).

One possible reason for the apparent challenges of developing dialectical narratives of mathematical learning is that the empirical data sources for these theoretical modeling efforts are often pre-biased by epistemological and pedagogical perspectives that side either with the individual or the social interpretation of learning, such that the observed phenomena make salient one, but not both of these perspectives. Our current efforts to develop dialectical models of mathematical learning draw on a corpus of data from an empirical study whose methodology was informed *a priori*by a dialectical perspective on the microgenesis of mathematical knowledge, so that resulting observed phenomena are perhaps more conducive for dialectical modeling. Designing for dialectical analysis as an intellectual pursuit, we conjecture, is perforce designing for powerful learning as a pedagogical epiphenomenon, because the methodology encourages students to ground cultural forms in their intuitive perceptions and conceptualizations.

In particular, the objective of our presentation is to share and discuss several brief videographed episodes that we view as cases of study participants bootstrapping mathematical theorems-in-action (Vergnaud, 2009)through engaging in problem-solving activities (cf. Hall, 2001; Neuman, 2001). In this bootstrapping process, the participants rely on domain-general sensory-perception mechanisms, problem-solving heuristics, and evaluation criteria (Gelman & Williams, 1998; Gigerenzer & Brighton, 2009; Tversky & Kahneman, 1974; Xu & Vashti, 2008), yet their insights are mediated through appropriating semiotic potential inherent in the symbolic artifacts they encounter in their interaction spaces (Bartolini Bussi & Mariotti, 1999; Radford, 2009; Vygotsky, 1930/1978). Importantly, the instructor never demonstrates or even insinuates these new situated practices that the students discover as the symbolic artifacts’ interactive potential, such that the students experience “guided reinvention” of logico–linguistic structures familiar to experts as mathematical or proto-mathematical (Gravemeijer, 1999; Resnick, 1992).

This bootstrapping procedure, as we analyze it, is characteristically two-stepped, and we have named it “hook and shift,” as follows: (1) *hook*– when a symbolic artifact is first introduced into the interaction space, problem-solvers engage it because they recognize its contextual utility for enhancing the enactment, explanation, or evaluation of their current solution strategy; then (2) *shift*– in the course of implementing these new affordances, problem-solvers notice in these artifacts additional embedded properties as affording a new or reconfigured strategy that better meets domain-general criteria of conciseness, precision, prediction, and – in the case of co-production with a peer – communication, coordination, and collaboration. These new strategies are then sanctioned by the instructor, who views the shift as advancing the child’s process of mathematization closer to disciplinary structures and procedures, in accord with the intervention’s pedagogical objectives (Streefland, 1993). We delineate critical interaction dimensions enabling this process.

Our hook-and-shift model is similar to the “intimation and implementation” model (Sfard, 2002, 2007), only that we wish to underscore that much of a child’s “theory building is endogenously provoked rather than socially mediated” (Karmiloff-Smith, 1988, p. 184). The hook-and-shift model is also similar to “form and function” (Saxe & Esmonde, 2005), only that we are looking specifically at mathematical learning in contexts where a community’s overall goals are stable – the mediation of targeted subject content matter – and the instructional processes are, accordingly, deliberately designed and facilitated toward desirable end-states. Also, our work generally resonates with research on semiotic mediation (Bartolini Bussi & Mariotti, 2008; Mariotti, 2009), only that we foreground the child’s agency in developing one “artifact–sign” into another, toward building mathematical signs (see also Engeström, 2008, on the Vygotskian principle of double stimulation in formative interventions). Finally, we are inspired by French didactique work on “instrumented activity situations” (Artigue, 2002; Vérillon & Rabardel, 1995; White, 2008), only that we investigate how students semi-spontaneously initiate the *re*-instrumentalization of artifacts even as they are instrumenting themselves toward these artifacts.

The data source of this study was a design-based research project examining the emergence of mathematical concepts from embodied-interaction activity (Hommel, Müsseler, Aschersleben, & Prinz, 2001; Nemirovsky, 2003; Núñez, Edwards, & Matos, 1999). We worked with 22 Grade 4-6 (age 9-11) volunteer students. The intervention was in the form of semi-structured, task-based individual or paired clinical tutorial interviews (diSessa, 2007; Ginsburg, 1997; Goldin, 2000; Piaget, 1952). The task consisted of remote-controlling a pair of virtual objects on a computer display in an attempt to make the display color turn green and then stay green even as one moves the hands. This “mystery device” was set up to display a green screen whenever the manipulated virtual objects were located on the screen at heights that corresponded to an unknown ratio controlled on the interviewer’s console. For example, a 1:2 ratio setting would require the hands to be at 1’ and 2’ up along the screen or at 2’ and 4’, 2.5’ and 5’, etc. By strategizing for green, we argue, students discovered a new ontology anticipating an understanding of proportional equivalence that is differentiated from non-proportional classes, for example articulating the recursive theorem-in-action, “For every one unit that you go up on the left, I go up two units on the right.”

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