We contribute to research on visualization as an epistemic learning tool by inquiring into the didactical potential of having students visualize one phenomenon in accord with two different partial meanings of the same concept. 22 Grade 4-6 students participated in a design study that investigated the emergence of proportionalequivalence notions from mediated perceptuomotor schemas. Working as individuals or pairs in tutorial clinical interviews, students solved non-symbolic interaction problems that utilized remote-sensing technology. Next, they used symbolic artifacts interpolated into the problem space as semiotic means to objectify in mathematical register a variety of both additive and multiplicative solution strategies. Finally, they reflected on tensions between these competing visualizations of the space. Micro-ethnographic analyses of episodes from three paradigmatic case studies suggest that students reconciled semiotic conflicts by generating heuristic logico-mathematical inferences that integrated competing meanings into cohesive conceptual networks. These inferences hinged on revisualizing additive elements multiplicatively. Implications are drawn for rethinking didactical design for proportions. I didn't pay enough attention to change of perspective. …. The subject deserves a more systematic treatment, which I do not dare undertake. …. Learning processes are marked by a succession of changes in perspective which should be provoked and reinforced by those who are expected to guide them. (Freudenthal, 1991, as cited in Streefland, 1993 This article builds on a paper presented to the Special Interest Group on Research in Mathematics Education (SIG RME) at the 2012 annual meeting of the American Educational Research Association (AERA).

Radical constructivists advocate discovery-based pedagogical regimes that enable students to incrementally and continuously adapt their cognitive structures to the instrumented cultural environment. Some sociocultural theorists, however, maintain that learning implies discontinuity in conceptual development, because novices must appropriate expert analyses that are schematically incommensurate with their naive views. Adopting a conciliatory, dialectical perspective, we concur that naive and analytic schemes are operationally distinct and that cultural-historical artifacts are instrumental in schematic reconfiguration yet argue that students can be steered to bootstrap this reconfiguration in situ; moreover, students can do so without any direct modeling from persons fluent in the situated use of the artifacts. To support the plausibility of this mediated-discovery hypothesis, we present and analyze vignettes selected from empirical data gathered in a conjecture-driven design-based research study investigating the microgenesis of proportional reasoning through guided engagement in technology-based embodied interaction. 22 Grade 4-6 students participated in individual or paired semi-structured tutorial clinical Abrahamson's coauthors are all members of the Embodied Design Research Laboratory in the Graduate School of Education at the University of California, Berkeley. The construct of ''hook and shift'' first emerged during a debriefing interaction between Abrahamson and Trninic. Thanks to Mark Howison for his technological development work on the MIT, Daniel Reinholz for earlier contributions to the project and the art in the figures, and Brian Waismeyer and Lucie Vosicka for thinking with us and reviewing earlier drafts.interviews, in which they were tasked to remote-control the location of virtual objects on a computer display monitor so as to elicit a target feedback of making the screen green. The screen would be green only when the objects were manipulated on the screen in accord with a ''mystery'' rule. Once the participants had developed and articulated a successful manipulation strategy, we interpolated various symbolic artifacts onto the problem space, such as a Cartesian grid. Participants appropriated the artifacts as strategic or discursive means of accomplishing their goals. Yet, so doing, they found themselves attending to and engaging certain other embedded affordances in these artifacts that they had not initially noticed yet were supporting performance subgoals. Consequently, their operation schemas were surreptitiously modulated or reconfigured-they saw the situation anew and, moreover, acknowledged their emergent strategies as enabling advantageous interaction. We propose to characterize this two-step guided re-invention process as: (a) hookingengaging an artifact as an enabling, enactive, enhancing, evaluative, or explanatory means of effecting and elaborating a current strategy; and (b) shifting-tacitly reconfiguring current strategy in response to the hooked artifact's emergent affordances t...