Abstract
A teaching experiment—using Mathematica to investigate the convergence of sequence of functions visually as a sequence of objects (graphs) converging onto a fixed object (the graph of the limit function)—is here used to analyze how the approach can support the dynamic blending of visual and symbolic representations that has the potential to lead to the formal definition of the concept of limit. The study is placed in a broad context that links the historical development with cognitive development and has implications in the use of technology to blend dynamic perception and symbolic operation as a natural basis for formal mathematical reasoning. The approach offered in this study stimulated explicit discussion not only of the relationship between the potential infinity of the process and the actual infinity of the limit but also of the transition from the Taylor polynomials as approximations to a desired accuracy towards the formal definition of limit. At the end of the study, a wide spectrum of conceptions remained. Some students only allowed finite computations as approximations and denied actual infinity, but for half of the students involved in the study, the infinite sum of functions was perceived as a legitimate “object” and was not perceived as a dynamic “process” that passes through a potentially infinite number of terms. For some students, the legitimate object was vague or generic, but we also observed other students developing a sense of the formal limit concept.
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Notes
The translation given by Child omitted the term “continuous” that was in the original Latin.
In the original question of Davis, Porta and Uhl, and in our study, the sequence of responses in the question was given with Euler first and Zeno second. Here, we place Zeno first and Euler second, to follow the same order both in history and in cognitive development.
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This research was supported by the Israel Science Foundation (grant no. 843/09).
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Kidron, I., Tall, D. The roles of visualization and symbolism in the potential and actual infinity of the limit process. Educ Stud Math 88, 183–199 (2015). https://doi.org/10.1007/s10649-014-9567-x
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DOI: https://doi.org/10.1007/s10649-014-9567-x