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Calculus has long been known for its gatekeeping role in postsecondary students' pursuit of STEM careers. In addressing this pressing issue, researchers at Montclair State University developed a model of peer-led complementary instruction to engage Calculus I students in small-group, collaborative problem solving on inquiry-oriented, groupworthy tasks. This work comes from a multiple-case study that sought to address the question, “How do undergraduate students experience and navigate their calculus learning in the parallel spaces of coursework and inquiry-oriented complementary instruction?” The analytic representations that were constructed to represent the findings of that study are presented here. Those findings include characterizations of the different forms of Calculus I students’ agentive participation and the figured worlds of class and complementary instruction. The analytic representations depict those findings in the form of word clouds and Venn diagrams. The analytical representations of “Victor’s” participation are presented and discussed here, and an argument is made for their particular representational power and efficiency. As such, this work seeks to make a methodological contribution to education research that seeks to characterize the nature of participation by the actors in figured worlds. 

We describe an experience within mathematics teacher preparation that engages pre-service teachers of mathematics (PMTs) in Making and design practices that we hypothesized would inform their conceptual, curricular, and pedagogical thinking. With a focus on the design of new tools that can generate new possibilities for mathematics teaching and learning, this Learning by Design experience has PMTs exploring at the intersection of content, pedagogy, and Making. We describe the forms of knowledge brought to bear on their experiences through a case study analysis of one pair of PMTs’ Making experience. As the advancement of these forms of knowledge is essential to effective mathematics teaching, these findings suggest the promise of a Making experience within mathematics teacher preparation