School of Mathematical Sciences Educational Mathematics
University of Northern Colorado
Type of Resources
Place of Publication
University of Northern Colorado
Although undergraduate complex variables courses often do not emphasize formal proofs, many widely-used integration theorems contain nuanced hypotheses. Accordingly, students invoking such theorems must verify and attend to these hypotheses via a blend of symbolic, embodied, and formal reasoning. Using Tall’s three worlds of mathematics as a theoretical lens, this research explores undergraduate student pairs’ collective argumentation about integration of complex functions, with emphasis placed on students’ attention to the hypotheses of integration theorems. Data consisted of videotaped, semistructured interviews with two pairs of undergraduates, during which they collectively reasoned about thirteen integration tasks. Videotaped classroom observations were also conducted during the integration unit of the course in which these students were enrolled. Interview data were analyzed by categorizing participants’ responses according to Toulmin’s argumentation scheme, as well as classifying each statement as embodied, symbolic, formal, or blends of the three worlds. The student pairs’ responses were further coded according to Levinson’s four speaker roles in order to document how individuals contributed socially to the collective arguments, and backing statements were identified as either supporting a warrant’s validity, correctness, or field. Findings revealed that participants’ nonverbal modal qualifiers and explicit challenges to each other’s assertions catalyzed new arguments allowing students to reach consensus, verify conjectures, or revisit prior assertions. Hence, while existing frameworks identify two types of participation in collective argumentation, the aforementioned challenges suggest an important third type of participation. Although participants occasionally conflated certain formal hypotheses from the integration theorems, their arguments married traditional integral symbolism with dynamic gestures and clever embodied diagrams. Participants also attended to a phenomenon, referred to in the literature as thinking real, doing complex, in three distinct manners. First, they took care to avoid invoking attributes of real numbers that no longer apply to the complex setting. Second, they intermittently extended their real intuition to the complex setting erroneously. Third, they deliberately called upon attributes of the real numbers that were productive in describing analogous complex number operations. This three-tiered attention to the thinking real, doing complex phenomenon is notable because only the second type is currently documented in existing literature. Collectively, the findings suggest that instructors of complex analysis courses might wish to heavily underscore the importance of geometric interpretations of complex arithmetic early in the course and avoid utilizing acronyms that de-emphasize individual theorem hypotheses. The results also indicate that a more multimodal stance is needed when studying collective argumentation in order to capture covert aspects of students’ communication.
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