Abstract
Certain terms in mathematics were created according to conventions that are not obvious to students who will use the term. When this is the case, investigating the choice of a name can reveal interesting and unforeseen connections among mathematical topics. In this study, we tasked prospective and practicing teachers to consider: What is geometric about geometric sequences? Participants embedded their answers to this question within a scripted dialogue between teacher- and student-characters in a mathematics classroom, provided commentary on this dialogue, and expanded on its mathematical content. In analyzing their submissions, we identified four different abductive arguments leveraged by participants to explain why geometric sequences are named as such. To analyze these informal arguments, we constructed and analyzed composite Toulmin models (Toulmin, 1958/2003) corresponding to each of the four arguments. We also compared the warrants and rebuttals that arose in response to each explanation and discuss the reappearance of similar rebuttals across arguments.
Résumé
On a créé certains termes du domaine des mathématiques selon des conventions qui ne sont pas toujours évidentes pour les étudiants qui ont à utiliser ces termes. Lorsque c’est le cas, l’étude du choix d’un nom peut révéler des liens intéressants et imprévus entre les sujets mathématiques. Dans cette étude, nous avons demandé à de futurs enseignants et à des enseignants déjà en exercice de réfléchir à la question suivante. Qu’y a-t-il de géométrique dans les suites géométriques ? Les participants ont intégré leurs réponses à cette question dans un dialogue scénarisé entre des personnages d’« enseignant» et d’« élève» dans une classe de mathématiques, ils ont ensuite commenté ce dialogue et ont étoffé leur propos sur son contenu mathématique. En analysant leurs contributions, nous avons identifié quatre arguments abductifs différents soulevés par les participants pour expliquer pourquoi les suites géométriques sont nommées ainsi. Afin d’étudier ces arguments informels, nous avons créé et analysé des schémas composites de Toulmin (Toulmin, 1958/2003) qui correspondent à chacun des quatre arguments. Nous avons également comparé les lois de passage et les réfutations qui sont apparues en réponse à chaque explication et discuté de la réapparition de réfutations similaires d’un argument à l’autre.
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Data Availability
The datasets generated and analyzed during the current study are not publicly available due the fact that they constitute an excerpt of research in progress but are available from the corresponding author on reasonable request.
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This research was supported by a grant from the Social Sciences and Humanities Research Council of Canada.
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Kercher, A., Bergman, A.M. & Zazkis, R. What is Geometric About Geometric Sequences? Informal Justifications for Mathematical Terminology. Can. J. Sci. Math. Techn. Educ. 23, 48–65 (2023). https://doi.org/10.1007/s42330-023-00271-4
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DOI: https://doi.org/10.1007/s42330-023-00271-4