Explaining Infinite Series: An Exploration of Students' Images
This study uses self-generated representations (SGR) - images produced in the act of explaining - as a means of uncovering what university calculus students understand about infinite series convergence. It makes use of student teaching episodes, in which students were asked to explain to a peer what that student might have missed had they been absent from class on the day(s) when infinite series were introduced and discussed. These student teaching episodes typically resulted in the spontaneous generation of several SGR, which provided physical referents with which both the student and an interviewer were able to interact. Students' explanations, via their SGR, included many more aspects of what they found important about that content than did the standard research technique of asking students to answer specific mathematics tasks.
This study was specifically designed to address how students construct an understanding of infinite series. It also speaks to the broader goal of examining how students use SGR as a tool for explaining concepts, rather than simply as tools for solving specific problems. The main analysis indicates that both students and their professors/textbook, when introducing the topic of infinite series, make use of the following five different image types: plots of terms, plots of partial sums, areas under curves, geometric shapes, and number lines. However, the aspects of the mathematical concepts that the students and the professors/textbooks highlight in their explanations and modes of use for those image types are different, and at times conflicting. In particular, differences emerged along three dimensions of competence - limiting processes (Tall, 1980), language, and connections.
While students using SGR generated many of the images that had been used by their professors, the limiting processes that they discussed via those images contrasted sharply. The professors and textbook chapter prioritized the limiting processes represented in particular image types to support mathematically sound conclusions. In contrast, many student explanations focused on limiting processes that did not lead to valid arguments about series convergence. There were also differences in use of language, in that students often assumed much more meaning than was intended in their professors' language choices, leading to problems with their explanations. Finally, while the experts connected their representations in meaningful ways, using other images to clarify or exemplify those that were used to define, students connected their understanding in different ways that were not always supportive of the convergence arguments that they were trying to make.
This study expands the literature on students' understanding of infinite series topics, pointing to gaps in student understanding and ways in which students mis-applied what teachers had presented. In doing so, it suggests many avenues for improving infinite series instruction. In addition, the methods employed in this study are general, and open up ways of looking at student thinking that can be applied to many problematic areas in the curriculum. Typical studies ask students to address tasks and issues framed by a researcher. This study instead asked students to explain the content, thereby providing a much larger window into what counts, from the student perspective.