# A dilemma that underlies an existence proof in geometry

## Abstract

Proving an existence theorem is less intuitive than proving other theorems. This article presents a semiotic analysis of significant fragments of classroom meaning-making which took place during the class-session in which the existence of the midpoint of a line-segment was proven. The purpose of the analysis is twofold. First follow the evolution of students’ conceptualization when constructing a geometric object that has to satisfy two conditions to guarantee its existence within the Euclidean geometric system. An object must be created satisfying one condition that should lead to the fulfillment of the other. Since the construction is not intuitive it generates a dilemma as to which condition can be validly assigned initially. Usually, the students’ spontaneous procedure is to force the conditions on a randomly chosen object. Thus, the second goal is to highlight the need for the teacher’s mediation so the students understand the strategy to prove existence theorems. In the analysis, we use a model of conceptualization and interpretation based on the Peircean triadic SIGN.

### Keywords

Existence proofs Students’ procedure to prove existence theorems Meaning-making Teacher semiotic-mediation## Notes

### Acknowledgments

Our long-term project was financed by the Colombian national research institute, Colciencias, and Centro de Investigaciones, Universidad Pedagógica Nacional (CIUP). The views and conclusions expressed here are those of the authors and not of the institutions.

We want to thank the reviewers and editor for their suggestions which definitely improved this article.

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