Abstract
Most teachers agree that if a student understands a particular mathematical topic well, he/she will probably be able to do problems correctly. The converse, however, frequently fails: students who do problems correctly sometimes do not actually have robust understandings of the topic in question. In this paper we explore this phenomenon in the context of power series, one of the most challenging topics in the calculus curriculum. We report on a small pilot study of calculus students working in pairs on power series problems in an interview setting. We first describe general tendencies in the students' work and then, concentrating on one pair of students, we document how they arrived at correct answers, explaining their answers in appropriate terms, despite having significantly flawed ideas about those series. Implications for further research and the teaching of power series are discussed.
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