Problem-solving Abilities Among First-Year Undergraduate Students with Qualifying AP Calculus Exam Scores

Open Access
Author:
McMahon, Jeffery Thomas
Area of Honors:
Curriculum and Instruction
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
  • Rose Mary Zbiek, Thesis Supervisor
  • Rose Mary Zbiek, Honors Advisor
  • Mary Kathleen Heid, Faculty Reader
Keywords:
  • mathematical proficiency
  • problem solving
  • AP Calculus
Abstract:
During the past few decades, substantial reforms within the mathematics education community have radically redefined the notion of what it means for a student to be mathematically proficient to include, for example, adaptive reasoning and strategic competence (Kilpatrick, Swafford, & Findell, 2001). A review of relevant literature informs definitions of “problem” and “problem solving” as the terms appear in current interpretations of mathematical proficiency. Additionally, literature-based criteria for characterizing problem-solving performance arise in four areas: resources, heuristics, control, and belief systems. Guided by these criteria, this study investigates the problem-solving abilities of first-year undergraduate students who achieved qualifying scores on the AP Calculus AB Exam in order to determine to what extent and in what combination the problem-solving influences of adaptive reasoning, strategic competence, and established experience are employed in task solving. In this exploratory study, first-year undergraduate students with qualifying AP Calculus Exam scores participated in a series of task-based interviews, which were videotaped and analyzed for evidence of the problem-solving influences. The interview tasks are modifications of common calculus tasks and were designed to be challenging problems for which students did not have preexisting solution strategies. The results indicate that students did not exhibit adaptive reasoning to draw appropriately on their resources when forced experience—the recollection and forced unproductive use of ideas recalled from past work on mathematical tasks—inhibited the use of strategic competence to develop additional representations of the problems. Displaying limited strategic competence, students who fixated on a single representation of the problem did not explore alternative strategies and draw on relevant resources. Students who did not initially employ rich adaptive reasoning did show evidence of the capacity for adaptive reasoning in response to scaffolding strategies that provided information about a missing piece of strategic competence. This combination of observations suggests that forced experience, with its unproductive use of ideas from earlier work with mathematical tasks, occurs in combination with an absence of strategic competence and subsequent lack of adaptive reasoning to inhibit students’ success in solving novel problems although students show the capacity for adaptive reasoning in these problem situations. Although more research must be done to generalize this result, the implications of the study for practice suggest the need to implement task modifications and scaffolding strategies in all mathematics classrooms in order to give students the opportunity to develop and exhibit strategic competence.