Dissertation Defense Announcement Ph.D. in Education Program: Rasha Abadir, “A Teaching Experiment for Calculus I Using Tasks Designed for Learning to Solve Applied Optimization Problems”

This qualitative study investigates the process by which two students build knowledge structures for solving applied optimization problems in the Calculus I context. Applied optimization is a topic that has historically been documented as one of the most challenging for Calculus I students to navigate. Grounded in Davis’s (1984) theoretical framework, this research had two main goals: (1) to identify the knowledge structures students exhibited during their problem-solving sessions, and (2) to investigate how a teaching experiment contributed to the evolution of these structures.

A teaching experiment consisting of five task-based research sessions was designed and implemented, where students could negotiate ideas, develop sense-making, and build on prior knowledge. Targeted tasks were designed for students to work on collaboratively during the series of sessions with the intent of facilitating students’ learning of solving calculus applied optimization problems. Video recordings of five task-based research sessions, student work, and recorded exit interviews provided the data for this study. Data were analyzed using the Cognitive Framework for Mathematical Inquiry (CFMI), a framework adapted from Davis’s (1984) “skeleton outline of typical steps” of problem-solving, to identify how students worked to build a solution to a series of tasks. More specifically, the data were coded to illuminate in detail what procedures they used, how their speech and actions aligned thematically with the eight cognitive processes defined by the CFMI, and how the students’ mathematical understanding changed over time. The findings show that, through their participation in this teaching experiment, a pair of students recognized recurring patterns among problem-solving tasks and built knowledge structures for solving optimization problems using calculus techniques. They demonstrated ability to connect their prior knowledge (e.g., assigning variables, applying derivative techniques, maximum and minimum values, and horizontal tangent lines) and use different representations to build a more flexible understanding of the topic. These results offer useful insights for designing effective calculus lessons in an area that has historically been challenging for students. The study offers pedagogical strategies to support students in building a strong and rich understanding of applied optimization problems and related concepts.