Students’ struggles with temporal order in the limit definition: uncovering resources using knowledge in pieces

Abstract

A few case studies have suggested students’ struggles with the temporal order of epsilon and delta in the formal limit definition. This study problematizes this hypothesis by exploring students’ claims in different contexts and uncovering productive resources from students to make sense of the critical relationship between epsilon and delta. A three-step analysis supports these aims. The analysis starts by investigating the generalizability and specificity of the struggle with the temporal order. Then, analysing students’ justifications reveals dominant ideas supporting students’ claims. Finally, attending to the foci of the justifications reveals the potential resources to make sense of the temporal order. This study illustrates the productivity of the principles context sensitivity, cueing priority, and reliability priority from Knowledge in Pieces in understanding students’ struggles. The study offers the three-step analysis as a method to approach students’ understanding from an anti-deficit perspective.

KEYWORDS:

• Cognition
• constructivism
• context sensitivity
• limit
• knowledge in pieces

Acknowledgements

I thank the anonymous reviewers and the editor for critical comments on revising this article. I also thank Alan Schoenfeld, Andy diSessa, Joe Wagner, Tom Dick, Elise Lockwood, Kendrice James, and Allison Dorko for their help during all phases of this research.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The notation AE describes the quantification for every, or for all epsilon there exists a delta.

2 Naïve conceptualizations can also include misremembered facts, which can easily fade from use. However, mistakes are not always assumed to be misremembered facts. 

3 The protocol was also piloted with two Real Analysis students. Marco was one of the Real Analysis students. The current study limits its population to students who were exposed to the formal definition in their calculus class, prior to Real Analysis.

4 Instead of focusing on the quantification in the AE statement, many students (like Erin) focused more on the sign of epsilon and delta, that they are both positive. Erin’s taking the sign of epsilon to be assumed suggests that she may have understood that epsilon is a given quantity.

5 There is not a particular number epsilon that is given, but the existence of epsilon is stipulated. The AE statement can be interpreted as for any given positive number epsilon, there exists a number delta.