# Fostering Hooks and Shifts: Tutorial Tactics for Guided Mathematical Discovery

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## 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.

## Keywords

Cognition Design-based research Discovery Education Embodied interaction Guided reinvention Insight Mathematics education Natural user interface (NUI) Proportion Proportional reasoning Remote control Semiotic-cultural Sociocultural Symbolic artifact Teaching Technology Virtual object## Notes

### Acknowledgments

This manuscript builds on the authors’ AERA 2012 paper. The research reported here was supported by a University of California at Berkeley Committee on Research Faculty Research Grant (Abrahamson) and an Institute of Education Sciences, U.S. Department of Education predoctoral training grant R305B090026 (Gutiérrez, Charoenying). The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education. We wish to thank Lucie Vosicka and Brian Waismeyer for ongoing conversations and for their formative comments on earlier drafts. We are grateful to the journal Editor in Charge Richard Noss as well as anonymous TKL reviewers for their very constructive tutorial tactics.

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