Making sense of the Fundamental Theorem of Calculus
This dissertation provides an elaborated examination of how a group of students, four years after first studying calculus, create an explanation of the Fundamental Theorem of Calculus for use by a figurative "novice student". The group of students studied AP Calculus in 1999, and had participated for 15 years in a longitudinal study at Rutgers University. In that study, students were invited to participate on a yearly basis in mathematical activities that asked them to justify their conclusions and make their thinking public.
The students were videotaped as they worked cooperatively on developing an explanation during two sessions one month apart. The students structured their own activity, using their own recollections, printed resource materials, and Geometer's Sketchpad sketches provided for the students' use. The students' discussions are examined with specific attention to the mathematical representations the students used, the meanings students attached to those representations, how the students decided upon those meanings.
The AP Calculus course description emphasizes "understanding the meaning" of the Fundamental Theorem from multiple perspectives, but students' understandings of the ideas of calculus have been found to be "poorly coordinated" (Judson & Nishimori, 2005; Thompson, 1994). Flexible use of representations has been suggested as a route to bring meaning to these ideas (Tall & Thomas, 1991). This dissertation illustrates in detail how students can coordinate graphical, numerical, verbal, and numerical representations to build meanings for the Fundamental Theorem of Calculus. The students' discussions provide evidence that students can develop meanings through engaging in non-evaluative dialogue where they negotiate and refine the meanings of words, symbols, and graphs.
The students, who had grown accustomed to participating in mathematical discourse (Maher, 2002), negotiated meanings in ways that may provide models for the teaching and learning of calculus in the classroom. The students' use of representations also has implications for those interested in how students might come to understand the ideas of calculus and the Fundamental Theorem in particular. The ways in which the students organized ideas serves as an example of how learners can build powerful understandings in multiple ways.