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This paper describes our work to determine the naturalistic images that first-time second-semester university calculus students possess for series convergence. We found that the students we interviewed most frequently determined whether a series converged by imagining a process of appending summands into a running total and examining whether this running total appeared to approach an asymptotic value. We provide examples and three corresponding implications of this "asymptotic running total" that informed students' actions while determining series convergence or the value of convergence. Our paper adds to the research literature by confirming students' meanings for limits reported for other topics (e.g., limit of sequence, function, Taylor series) apply to infinite series and proposing relationships between previously reported meanings for series convergence. [For the complete proceedings, see ED630210.]

The purpose of this paper is to highlight issues related to students' personal inferences that arise when students verbally explain their justification for calculus statements. We conducted clinical interviews with three undergraduate students who had taken first-semester calculus but had not yet been exposed to formal proof writing activities through undergraduate mathematics courses. We analyzed these students' verbal justification of four statements, each of which described a different relationship between two quantified variables in calculus contexts. In this paper, we document students' personal inferences that were evoked in their justifications and discuss how they are similar to or different from the logical inferences that have been accepted and practiced by the mathematics community for mathematical arguments. [For the complete proceedings, see ED630210.]

This report documents how one undergraduate student used set-based reasoning to reinvent logical principles related to conditional statements and their proofs. This learning occurred in a teaching experiment intended to foster abstraction of these logical relationships by comparing the predicate and inference structures among various proofs (in number theory and geometry). We document the progression of Theo's emergent set-based model from a model-of the truth of statements to a model-for logical relationships. This constitutes some of the first evidence for how such logical concepts can be abstracted in this way and provides evidence for the viability of the learning progression that guided the instructional design. [For the complete proceedings, see ED630060.]

This study investigates one Calculus student's meanings for quantifiers in Calculus statements involving multiple quantifiers. The student was asked in a two-hour long clinical interview to evaluate and interpret the Intermediate Value Theorem (IVT) and three other statements whose logical structure was similar to the IVT except for the order of both the quantifiers and their attached variables. Four different meanings for variables attached to universal and existential quantifiers emerged from his responses at various moments of the interview. In this paper we detail these four meanings with empirical evidence and discuss implications of our findings to research and teaching of quantified variables in Calculus statements. [For the complete proceedings, see ED606531.]

We explore ways that university students handle proving statements that have the overall structure of a conditional implies a conditional, i.e., (p [right arrow] q) [implies] (r [right arrow] s). We structure our analysis using the theory of conceptual blending. We find conceptual blending useful for describing the creation of powerful new ideas necessary for proof construction as well as for describing the creation of blends that slow or hinder student efforts at proof construction. [For the complete proceedings, see ED585874.]