On the density of ℚ in ℝ: Imaginary dialogues scripted by undergraduate students

Abstract

This study aims to explore the notion of the density of the set of rational numbers in the set of real numbers, as interpreted by undergraduate mathematics students. The data comprise 95 responses to a scripting task, in which participants were asked to extend a hypothetical dialog between two student characters, who argue about the existence of one or infinitely many rational numbers in a real number interval. The analysis leans on the framework of reducing abstraction to provide explanations for the participants’ mathematical behaviour when coping with the task. The findings point to students’ informal ideas related to density that can be mapped to formal proofs, as well as to unconventional understandings of related concepts and ideas, including rational and irrational numbers, infinity, and mathematical justification. Implications are drawn.

KEYWORDS:

• Undergraduate mathematics
• density
• rational numbers
• scripting tasks
• reducing abstraction
• technology

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The applet can be found in https://www.geogebra.org/m/nruYwQAd

2 Despite inappropriate writing, it is clear that the intended expression was (1+2)/2.

3 The Portuguese title in the image, ‘Conjuntos Numéricos’, translates to ‘Number Sets’.