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 is historically documented to be 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 what kinds of knowledge structures students develop, and (2) to investigate how a specific teaching approach influences the evolution of these structures. A teaching experiment consisting of five task-based research sessions was designed and implemented to foster a space where students could negotiate ideas, develop sense-making, and build on prior knowledge. Targeted tasks were specifically designed for students to work on during the research session to facilitate students' learning of calculus applied optimization. Video recordings of five task-based research sessions, student work, and recorded exit interviews provided the data for this study, which were analyzed using the Cognitive Framework for Mathematical Inquiry (CFMI) to identify how students reasoned, what procedures they used, and how their understanding changed.The findings show that with the help of the teaching experiment, a pair of students among a larger sample of eight participants recognized recurring patterns and built lasting and transferable knowledge structures. They were able to connect their prior knowledge and use different representations to build a more flexible understanding of the topic. These results offer useful ideas for designing effective learning environments in mathematics education. The study also contributes to training educators and teaching assistants by suggesting strategies that can help students build a strong, rich understanding of calculus and other mathematical concepts.