Fostering Hooks and Shifts: Tutorial Tactics for Guided Mathematical Discovery

Abstract

How do instructors guide students to discover mathematical content? Are current explanatory models of pedagogical practice suitable to capture pragmatic essentials of discovery-based instruction? We examined videographed data from the implementation of a natural user interface design for proportions, so as to determine one constructivist tutor’s methodology for fostering expert visualization of learning materials. Our analysis applied professional-perception cognitive–anthropological frameworks. However, several types of tutorial tactics we observed appeared to “fall between the cracks” of these frameworks, due to the discovery-based, physical, and semantically complex nature of our design. We tabulate and exemplify an expanded framework that accommodates the observed tactics. The study complements our earlier focus on students’ agency in discovery (in Abrahamson et al., Technol Knowl Learn 16(1):55–85, 2011) by offering an empirically validated resource for researchers, instructors, and professional developers interested in preparing future teaching for future technology.

Appendix

1.1 Novice Tutorial Tactics

Charlie (pseudonym), a novice tutor, was facilitating his first session ever, as part of his training as a graduate student. The elementary-school study participant had been manipulating the two cursors on the screen and had determined a demonstrably effective strategy for moving her hands while keeping the screen green. Charlie had thus reached the point along the protocol where he was to overlay a virtual grid upon the screen. He said to the student that he was about to bring up something on the screen and that she should see whether that changes anything. He then lit up the grid. The child picked up the tracker devices that had been lying on the desk. Just as before, she located a “green spot” and then lifted her hands further up, maintaining a green screen in accord with her existing strategy. No, she reported, nothing has changed. She laid down the trackers and did not proceed to avail of the grid.

Of the total of a near two-dozen students who participated in this study, this student was the only one who responded thus. Other students tended to appropriate the grid as a means of better enacting, explaining, or evaluating their strategy. In retrospective analysis, we realized that how the tutor frames the introduction of a new artifact partially predicts whether or not the student engages it as a useful instrument (Gutiérrez et al. 2011). Thereafter, Charlie learned to frame the introduction of new symbolic artifacts as potentially promoting the interaction, and the research group amended the protocol to provide the clinical interviewers with appropriate guidance.

Incidences such as this, which we have been archiving for training purposes, are essential in the preparation of interviewers, because they occasion opportunities for supervisors to flesh out implicit dimensions of their own practice. As such, for a PI charged with training graduate students as much as with conducting research, videotaped documentations of “interview bloopers” are vital for building a laboratory’s organizational knowledge and capacity. Yet for the particular methodological needs of the current study, these incidences accentuate the rationale of our approach. Namely, professional tutors exercise a repertory of specific tactics that affect the nature (if not quality) of students’ engagement in learning activities.