Instructional Tasks for Introducing Derivatives in College Calculus with Inquiry

Abstract

In this dissertation I explore instructors’ work with instructional tasks as they plan to introduce various representations of derivatives. Drawing on the notion of framing from sociology, I adopt a situative lens to study calculus instructors’ planning of instructional tasks. I rely on Herbst and Chazan’s (2012) notion of instructional situations and Doyle’s (1984) notion of activity structures to describe how instructors frame their instructional tasks in terms of both content and social interactions. I purposefully selected eight instructors from a pool of 48 who reported using three different patterns of inquiry in their teaching of Calculus I. Subsequently, I conducted individual interviews with each instructor during which I prompted them to propose instructional tasks for introducing derivatives. The prompts, designed based on Zandieh’s (2000) framework of conceptions of derivatives, provided the instructors with students’ assumed pre- and post-conceptions of derivatives before and after working on the tasks within four representations of derivatives. Within each representation, instructors were asked to propose two instructional tasks: one targeting derivatives at the limit layer and another targeting the function layer. Collectively, the instructors proposed a total of 56 instructional tasks throughout the interviews. As each instructional task often comprised multiple parts, I deconstructed them into their smallest calculus-specific problems, which I call calculus-specific task units or CTUs. In total, I identified 37 CTUs within the proposed instructional tasks. I then used deductive/inductive hybrid thematic analyses (Proudfoot, 2022) to qualitatively identify the ways the instructors framed their tasks with calculus-specific instructional situations and activity structures. The CTUs were categorized into 11 instructional situations: four calculating, two exploring/conjecturing, two graphing, two installing, one proving, and one solving equation. Additionally, six activity structures were proposed by instructors to socially frame their instructional tasks: individual seatwork, whole-class instruction or lecturing, recitation or whole-class presentation, discussion, group work, and prep work. After identifying the instructional situations and activity structures, I used thematic analysis (Saldaña, 2021) to synthesize the findings regarding each generic type of situations and to identify usage patterns for each activity structure across all instructional tasks. The findings describe the kinds of work the identified calculus-specific instructional situations are expected to engage students with together with their roles within instructional tasks, while also exploring the interactions between different instructional situations. I demonstrate how these instructional situations were manifested or realized through the calculus-specific task units (CTUs) within the proposed tasks, providing insight into how instructors utilized them within the context of their instructional tasks to frame students’ mathematical work. Additionally, I describe how instructors conceptualized each activity structure based on the nature of tasks, their instructional goals, and their relationship to other activity structures. The instructors in this study mainly relied on calculating, graphing, and exploring situations to frame their instructional tasks, while also heavily scaffolding students’ work on the tasks. Collectively, the findings underscore the complex nature of the calculus instructors’ framing of instructional tasks for teaching derivatives, as they proposed a wide range of tasks with various combinations of CTUs, incorporating multiple representations of derivatives and instructional situations. Nonetheless, within this complexity, the study suggests that content-related and social patterns emerge when we investigate calculus teaching at the task level across instructors, as almost all instructors exhibited consistent patterns of activity structure across their instructional tasks with rare deviations from their patterns.