Hooks and Shifts 1 Running head: HOOKS AND SHIFTS Hooks and Shifts in Instrumented Mathematics Learning

This paper presents and analyzes vignettes selected from empirical data gathered in a designbased research study investigating the microgenesis of proportional reasoning through technology-mediated embodied interaction. 22 Grades 4-6 students participated in individual or paired semi-structured clinical interviews in which they engaged in problem-solving activities involving remote manipulation of virtual objects on a computer screen. At different points throughout these interviews, the interviewer introduced various symbolic artifacts into the problem space, such as a Cartesian grid and y-axis numerals. In our on-going microgenetic analyses, we have observed that participants appropriated the artifacts as strategic or discursive means of accomplishing their goals; yet in the course of enacting these goal-oriented tasks, the participants recognized certain other embedded affordances in these symbolic artifacts as supporting the development of a more sophisticated strategy. We have characterized this twostep mediated-discovery process as: (a) hooks—appropriating an artifact as enactive, explanatory, or evaluative means; and (b) shifts—spontaneously reconfiguring current or emerging strategy to avail of the artifact’s newly discovered affordances. This paper stems from this on-going work and extends the findings. Specifically, I here provide quantitative data about the frequency of participants’ hooking/shifting the grid and numerals and elaborate on interaction parameters enabling or hindering hooks and shifts in artifact-mediated discovery-based learning.