UC San Diego

The study of student representation use and specifically the distinction between analytic and visual representations has fueled a long line of mathematics education literature that began more than 35 years ago. This literature can be partitioned into two bodies of work, one that is primarily cognitive and one that is primarily social. In spite of the large overlap in the results and foci of these two bodies of work they have tended to not inform one another. I bridge these two bodies of work by creating and implementing an analysis tool, referred to as the VAP-model, which can be used within both group-work and individual interview settings. This model allows focusing on how students in both settings transition between representations during problem-solving and how these transitions fuel students' mathematical advancement. The VAP-model considers physical representations in addition to the analytic/visual distinction, which has driven much of the research on representation use. Physical representations are references to realistic or imaginable scenarios in mathematical problem-solving. Fluency with multiple representations has been long considered an important aspect of deep understanding of mathematics in general and calculus in particular. This study examines the representation use of three calculus students from a technologically enriched calculus course in both in-class group-work settings and individual interview settings. This analysis reveals that students incorporate representations not required by given tasks and that, in spite of their common classroom experiences, the representations used in problem-solving vary widely across students. Comparing representation usage in the two settings showed similarities in how frequently each participant transitioned between modes of representation, which modes appeared within their reasoning and the types of questions in which particular shifts between representations occurred. This analysis also highlighted cross-setting similarities in how social roles facilitated shifts in representation use and in what ways these transitions occurred. Within both settings physical representations played an important role in students' thinking and were often introduced to reason about non- physical problems. Since in past studies this type of representation use was commonly subsumed under other categories this phenomenon was often overlooked