Abstract
This is a report of a study of students’ understanding of infinite series. It has a three-fold purpose: to show that students may construct two essentially different notions of infinite series, to show that one of the constructions is particularly difficult for students, and to examine the way in which these two different constructions may be built so that we may uncover ways to help students improve their understanding. The theoretical framework consists of action–process–object–schema theory and the specific model of conceptions in Balacheff’s theory of conception, knowing, and concept. Approaching the problem from these two different theoretical perspectives allows us to provide different and at the same time complementary explanations of observed phenomena. The two different infinite series constructions are, briefly stated, series as an infinite unending process of addition and series as a sequence of partial sums. Students are found to have difficulty building an understanding of series as a sequence of partial sums and thus tend to have difficulty in problem situations that require this interpretation. The study uses semi-structured interviews with 10 graduate students. The interviews explore situations that might give insight into students’ notion of the sequence of partial sums.
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Notes
In this question we first defined S to be the "sum" of the series, then gave two arguments giving different values for S, and referred to these two different "results" for S just prior to asking the question.
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Acknowledgments
The authors wish to thank Professor Ed Dubinsky for his guidance and many insightful comments. His collaboration in the project was made possible by a RUME mentoring grant of the Exxon/Mobil Education Foundation. The authors also thank the referees of previous versions of this paper for their many valuable comments.
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Martínez-Planell, R., Gonzalez, A.C., DiCristina, G. et al. Students’ conception of infinite series. Educ Stud Math 81, 235–249 (2012). https://doi.org/10.1007/s10649-012-9401-2
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DOI: https://doi.org/10.1007/s10649-012-9401-2