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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 partnersÕ ideas, or using their own
ideas, and or both. 42 
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 ÒprototypesÓ to investigate how a large sample of students
depicted definite integrals through the graphical representation. A clear
ÒprototypeÓ group of images appeared in the data, as well as related Òalmost
prototypeÓ image groups. 15 
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 problemsolving process. We examine seven
studentsÕ concept images of the optimizing function, specifically focusing on
the graphical representation, and consider how this influences their
problemsolving activities. 93 
Grand Ballroom 5 
Support for proof as a cluster concept: An empirical
investigation into mathematiciansÕ 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. 116 