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3:05 – 3:35 pm

Session 3 – Contributed Reports

Marquis A

A new perspective to analyze argumentation and knowledge construction in undergraduate classrooms

Karen Keene, Derek Williams and Celethia McNeil

Using argumentation to help understand how learning in a classroom occurs is a compelling and complex task. We show how education researchers can use an argumentation knowledge construction framework (Weinberger & Fischer, 2006) from research in online instruction to make sense of the learning in an inquiry oriented differential equations classroom. The long term goal is see if there are relationships among classroom participation and student outcomes. The research reported here is the first step: analyzing the discourse in terms of epistemic, social, and argumentative dimensions. The results show that the epistemic dimension can be better understood by identifying how students verbalize understanding about a problem, the conceptual space around the problem, the connections between the two and the connections to prior knowledge. In the social dimension, we can identify if students are building on their learning partners’ ideas, or using their own ideas, and or both.



Marquis B

Prototype images of the definite integral

Steven Jones

Research on student understanding of definite integrals has revealed an apparent preference among students for graphical representations of the definite integral. Since graphical representations can potentially be both beneficial and problematic, it is important to understand the kinds of graphical images students use in thinking about definite integrals. This report uses the construct of “prototypes” to investigate how a large sample of students depicted definite integrals through the graphical representation. A clear “prototype” group of images appeared in the data, as well as related “almost prototype” image groups.




Marquis C

The graphical representation of an optimizing function

Renee Larue and Nicole Infante

Optimization problems in first semester calculus present many challenges for students. In particular, students are required to draw on previously learned content and integrate it with new calculus concepts and techniques. While this can be done correctly without considering the graphical representation of such an optimizing function, we argue that consistently considering the graphical representation provides the students with tools for better understanding and developing their optimization problem-solving process. We examine seven students’ concept images of the optimizing function, specifically focusing on the graphical representation, and consider how this influences their problem-solving activities.



Grand Ballroom 5

Support for proof as a cluster concept: An empirical investigation into mathematicians’ practice

Keith Weber

In a previous RUME paper, I argued that proof in mathematical practice can profitably be viewed as a cluster concept in mathematical practice. I also outlined several predictions that we would expect to hold if proof were a cluster concept In this paper, I empirically investigate the viability of some of these predictions. The results of the studies confirmed these predictions. In particular, prototypical proofs satisfy all criteria of the cluster concept and their validity is agreed upon by most mathematicians. Arguments that satisfy only some of the criteria of the cluster concept generate disagreement amongst mathematicians with many believing their validity depends upon context. Finally, mathematicians do not agree on what the essence of proof is.