Abstract
This paper reports a teaching experiment in which two students engaged in tasks that challenged them to describe a final state for a variety of infinite iterative processes. The results from the study indicate that the students used multiple reasoning strategies for addressing these tasks. Refinements in the students’ reasoning occurred as students constructed relationships between the problems they were solving and problems they had solved previously, applying some of the reasoning strategies that they used for one problem to make sense of or solve another problem. We discuss how these findings relate to the existing body of research on infinite iterative processes.
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Notes
We decided to work with mathematics undergraduate students to ensure that the participants had a certain maturity regarding proof techniques and because a mathematics student would have already been exposed to a wide variety of mathematical concepts, which gave us more mathematical contexts to choose from for our task variations.
We worked only with two students at a time in order to be able to closely follow each student’s reasoning, both during each session and later during data analysis.
All student names are pseudonyms.
All tasks used in this study (and proposed solutions) can be found in the Appendix, which is part of the electronic version of the article and can also be accessed at http://rci.rutgers.edu/~khweber/radu_esm_tasks.pdf.
In all the geometrical construction tasks, the process was defined in such a way that any intermediate state was included in the subsequent one (so \( {S_0} \subseteq {S_1} \subseteq ... \subseteq {S_n} \subseteq ... \)).
An interesting question that can be considered for future research is whether these students’ refinements in their reasoning on infinite iteration constitute procedural or conceptual changes in their knowledge. We note that before being able to address such a question, it will be necessary to further develop the theoretical ground with respect to what type of reasoning constitutes a procedural or conceptual approach, respectively, when reasoning about infinite iteration.
For a more detailed discussion of APOS with respect to constructing infinite iteration processes, see Radu (2009).
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An erratum to this article can be found at http://dx.doi.org/10.1007/s10649-011-9332-3
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Radu, I., Weber, K. Refinements in mathematics undergraduate students’ reasoning on completed infinite iterative processes. Educ Stud Math 78, 165–180 (2011). https://doi.org/10.1007/s10649-011-9318-1
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DOI: https://doi.org/10.1007/s10649-011-9318-1