Fostering Hooks and Shifts: Tutorial Tactics for Guided Mathematical Discovery

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.