Given a perpetually new natural and historical situation to control, the perceiving subject undergoes a continued birth; at each instant it is something new. (Merleau-Ponty 1964, p. 6)
Abstract
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 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) hooking—engaging 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 that are disclosed only through actively engaging the artifact. Looking closely at two cases and surveying others, we delineate mediated interaction factors enabling or impeding hook-and-shift learning. The apparent cognitive–pedagogical utility of these behaviors suggests that this ontological innovation could inform the development of a heuristic design principle for deliberately fostering similar learning experiences.
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Notes
We recognize that this distinction between carpenters’ and mathematicians’ practice is not clear-cut. Carpenters can internalize some instruments to a degree (e.g., measuring tape), and many mathematical forms still must be inscribed to provide optimal use.
We view the semi-structured nature of our task-based clinical interview as well as the iterative modification of the protocol from one interview to the next as affording viable responses to Engeström's thoughtful critique of some genres of design-based research as being non-responsive. In fact, we view students’ shifts as manifest evidence that our interventions are formative, not linear; and that our practice accords with the Vygotskian principle of “double stimulation” (Engeström 2008, p. 5).
Interviews consisted primarily of working with the MIT. In designing the MIT, we leveraged the high-resolution infrared camera available in the Nintendo Wii remote to perform motion tracking of students’ hands, similar to work by Johnny Lee (2008). The Wii remote is a standard Bluetooth device, with several open-source libraries available to access it through Java or C#. In an earlier version of the MIT, an array of 84 infrared (940 nm) LEDs aligned with the camera provides out light (source), and 3 M 3000X high-gain reflective tape attached to a tennis ball enables effective motion capture at distances as far as 12 feet. In use, infrared rays emanate from the MIT, reflect off tape covering tennis balls held by the student, and are then sensed, interpreted, and visually represented on a large display in the form of two crosshair symbols (trackers; see Fig. 2a). In the current version of the MIT, students point LED beams directly at the special camera. The display is calibrated so as to continuously position the crosshairs at the actual physical height of its controlling hand in an attempt to enhance the embodied experience of remote manipulation.
The mathematical concept of proportion is one of several interrelated concepts in the multiplicative conceptual field, which also includes rational numbers and other intensive quantities (Vergnaud 1983, 2009). In order to enable students to construct proportion as a new equivalence class, we needed a technological contrivance that associated pairs of ontologically independent left and right hand locations—a token of what Vergnaud (1983) named isomorphism of measure—in the form of a single epistemic entity, a product of measure, wherein x and y are ontically integrated as a single intensive quantity x/y. In so doing, we hoped, students would experience the ordered pairs [2 3] and [4 6] as “the same.” Thus, the MIT links isomorphism-of-measure input with product-of-measure sensory feedback. As such, the MIT resembles designs used by Yerushalmy and collaborators (Botzer and Yerushalmy 2008; Yerushalmy 1997).
RT = Right-Tracker device; LT = Left-Tracker device; Rc = Right-hand crosshair; Lc = Left-hand crosshair; //= utterance overlapped by next speaker. We mark spoken utterance with bold characters for readability.
In passing, we note different types of pronouns employed to designate action or measurement. Action is attributed to individuals (I and you), whereas measurement is about absolute magnitudes (it). These linguistic marks suggest subtle conceptualization of action as pragmatic or epistemic (cf. Kirsh 2006).
Note that at this point in the interview, these independent locations of the two hands or crosshairs are not yet absolute or explicit—they do not designate the hand/crosshair’s unitized distance from the desktop or screen bottom. Rather, the students will conceptualize the locations as self-referential, with each hand motion recursively departing from the previous location toward the new location. Study participants typically referred to locations in absolute terms only once the numerals appeared on the screen.
Close examination of Eden’s actions reveals that rather than directly going up by half a unit each time that Uri went up by a whole unit, Eden was going up by a whole unit but then lowering by a half, as though correcting each time.
See Trninic et al. (in press). Also note that the five motivation dimensions are not orthogonal but overlap in ways that are difficult to tease apart. This list should not be taken as offering an exhaustive blueprint for the development of a precision coding system as much as to lay out in general the scope of motivations we have witnessed and characterized in our data analyses (see also diSessa 1995; Sarama and Clements 2009, for the affordances of computer-based "concrete" interactive objects).
We reflect that whereas study participants who generated and connected more strategies and meanings were typically the students who had been characterized by their teachers as higher achieving, the other participants were impeded as much by our experimental methodology as by their own knowledge. Namely, at times we marched on through the interview just to ensure that we “cover” all protocol items for subsequent analysis, regardless of whether or not participants were optimally prepared to work with the new symbolic artifacts. Only during analysis did we fully appreciate the detrimental cumulative effect of our facilitation practice on the quality of the experimental implementation as a learning experience for some participants. Whereas the dual role of research interviewers in both agitating and measuring behavior is universal and familiar, the hook-and-shift construct underscores a reliability problematic of data gathered in multi-item interviews requiring cumulative feedforward reasoning.
A recent large empirical dissertation study implemented our Mathematical Imagery Trainer as part of a 2-week curriculum in a controlled experiment in which fourteen classrooms participated (Petrick and Martin 2011). Students involved in actual embodied manipulation sessions later outperformed on relevant conceptual tasks other students who watched screenings of the embodied sessions but did not have any agency in controlling the virtual objects nor saw a person actually manipulate them. The researchers interpret their data as supporting our own team’s conjecture pertaining to the learning gains inherent in physical solution procedures that enact kinesthetic image schemas underlying target curricular concepts (Abrahamson and Howison 2008; Abrahamson and Trninic in press).
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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. The Kinemathics project was sponsored by a UC Berkeley Committee on Research Junior Faculty Research Grant (Abrahamson, PI) and IES pre-doctoral training grant R305B090026 (Gutiérrez, Trninic). Project findings were first presented to the Gesture Study Group at University of California, Berkeley (Abrahamson and Howison 2008). This paper builds on AERA presentations (Abrahamson and Howison 2010a; Abrahamson et al. 2011; Gutiérrez et al. 2011; Trninic et al. 2011) and conference proceedings of CHI (Howison et al. 2011) and PME-NA (Reinholz et al. 2010; Trninic et al. 2010). We wish to thank Andy diSessa, Maria Alessandra Mariotti, Luis Radford, Geoff Saxe, Jeanne Bamberger, and Katharine G. Young for illuminating conversations around our data analyses and theoretical modeling and Tobin White for his comments on a draft. We further thank Lulu Healy and TKL anonymous reviewers who offered very constructive critiques. Live and learn.
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Abrahamson, D., Trninic, D., Gutiérrez, J.F. et al. Hooks and Shifts: A Dialectical Study of Mediated Discovery. Tech Know Learn 16, 55–85 (2011). https://doi.org/10.1007/s10758-011-9177-y
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DOI: https://doi.org/10.1007/s10758-011-9177-y