Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition might best serve in articulating implications of Enactivism for mathematics education. We offer a blend of Dynamical Systems Theory and Sociocultural Theory as an analytic lens on micro-processes of action-to-concept evolution. We also illustrate the methodological utility of design-research as an approach to such theory development. Building on constructs from ecological psychology, cultural anthropology, studies of motor-skill acquisition, and somatic awareness practices, we develop the notion of an ''instrumented field of promoted action''. Children operating in this field first develop environmentally coupled motoraction coordinations. Next, we introduce into the field new artifacts. The children adopt the artifacts as frames of action and reference, yet in so doing they shift into disciplinary semiotic systems. We exemplify our thesis with two selected excerpts from our videography of Grade 4-6 volunteers participating in task-based clinical interviews centered on the Mathematical Imagery Trainer for Proportion. In particular, we present and analyze cases of either smooth or abrupt transformation in learners' operatory schemes. We situate our design framework vis-à-vis seminal contributions to mathematics education research. All doing is knowing, and all knowing is doing. (Maturana and Varela 1992, p. 26).
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...
Unscaffolded problem-solving before receiving instruction can give students opportunities to entertain their exploratory hypotheses at the expense of experiencing initial failures. Prior literature has argued for the efficacy of such preparatory activities in preparing students to learn from instruction. Despite growing understanding of the underlying cognitive mechanisms, the pedagogical value of success or failure in initial problem-solving attempts is still unclear. We do not know yet whether some ways of succeeding or failing are more efficacious than others. We report empirical evidence from a classroom intervention (N = 221), where we designed scaffolds to explicitly push student problem-solving toward success via structuring, but also toward failure via problematizing. Our rationale for explicit failure scaffolding was rooted in facilitating problem-space exploration. We subsequently compared the differential preparatory effects of success-driven and failure-driven problem-solving on learning from follow-up instruction. Results suggested that failure-driven scaffolding (nudging students to generate suboptimal solutions) and success-driven scaffolding (nudging students to generate optimal solutions by giving them heuristics with low specificity) had similar outcomes on posttest assessments of conceptual understanding. Students exposed to failure-driven scaffolding, however, demonstrated higher quality of constructive reasoning. These trends were more salient for the learning concept with greater difficulty.
Recent, empirically supported theories of cognition indicate that human reasoning, including mathematical problem solving, is based in tacit spatial-temporal simulated action. Implications of these findings for the philosophy and design of instruction may be momentous. Here, we build on design-based research efforts centered on exploring the potential of embodied interaction (EI) for mathematics learning. We sketch two emerging, reciprocal contributions: (1) a sociocognitive view on the role of automated feedback in building the perceptuomotor schemes that undergird conceptual development; and (2) a heuristic EI design framework. We ground these ideas in vignettes of children engaging an EI design for proportion. Increasing ubiquity and access to mobile devices geared to avail of EI principles suggests the feasibility of mass-disseminating materials evolving from this line of research.
This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion that mathematics should be taught like swimming. The general claim being made is that "direct instruction" and "discovery" are not opposite but complementary, linked by repetitive yet explorative practice. This claim is elaborated through an empirical case of martial arts instruction. That repetitive practice can nonetheless be a fountainhead of discovery is explained using Bernstein's notion of repetition-withoutrepetition. Finally, we attend to parallels in canonical mathematics practice. Implications are discussed, with a focus on reconceptualizing direct instruction, repetition, and discovery as complementary and synergistic.
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