mathematics Article Norms That Regulate the Theorem Construction Process in an Inquiry Classroom of 3D Geometry: Teacher’s Management to Promote Them Oscar Molina 1, * , Vicenç Font 2, * and Luis Pino-Fan 3   Citation: Molina, O.; Font, V.; Pino-Fan, L. Norms That Regulate the Theorem Construction Process in an Inquiry Classroom of 3D Geometry: Teacher’s Management to Promote Them. Mathematics 2021, 9, 2296. https://doi.org/10.3390/math9182296 Academic Editor: Jay Jahangiri Received: 21 July 2021 Accepted: 14 September 2021 Published: 17 September 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad Pedagógica Nacional, Bogotá 110221, Colombia 2 Departament d’Educació Lingüística i Literària i de Didàctica de les Ciències Experimentals i de la Matemàtica, Facultat d’Educació, Campus Mundet, Universitat de Barcelona, 08035 Barcelona, Spain 3 Departamento de Ciencias Exactas, Universidad de Los Lagos, Osorno 5312574, Chile; luis.pino@ulagos.cl * Correspondence: ojmolina@pedagogica.edu.co (O.M.); vfont@ub.edu (V.F.); Tel.: +57-311-240-9819 (O.M.); +34-93-403-5035 (V.F.) Abstract: This paper aims to illustrate how a teacher instilled norms that regulate the theorem construction process in a three-dimensional geometry course. The course was part of a preservice mathematics teacher program, and it was characterized by promoting inquiry and argumentation. We analyze class excerpts in which students address tasks that require formulating conjectures, that emerge as a solution to a problem and proving such conjectures, and the teacher leads whole-class activities where students’ productions are exposed. For this, we used elements of the didactical analysis proposed by the onto-semiotic approach and Toulmin’s model for argumentation. The teacher’s professional actions that promoted reiterative actions in students’ mathematical practices were identified; we illustrate how these professional actions impelled students’ actions to become norms concerning issues about the legitimacy of different types of arguments (e.g., analogical and abductive) in the theorem construction process. Keywords: norms; professional actions; theorem construction process; three-dimensional geometry; abductive and analogical arguments 1. Introduction In an inquiry classroom, an atmosphere of intellectual challenge is generated in which students are expected to: (i) propose and defend mathematical ideas and conjectures and (ii) respond thoughtfully to the mathematical arguments of their peers. In our study, mathematical inquiry begins when a task is proposed that requires solving an open- ended problem using a Dynamic Geometry Software (DGS), formulating a conjecture that encapsulates the solution of the problem, and proving the conjecture. The mathematical practice of a classroom with these characteristics requires focusing on students’ production and teacher and students collectively building norms (social and socio-mathematical) that regulate and support these practices [1]. Examples of social norms are: (i) active listening, intellectual risk-taking (sharing incomplete ideas), and building on the ideas of others [2]; and (ii) assuming the responsibility of solving the given task [3]. Examples of socio-mathematical norms are: research in mathematics involves creatively solving problems; valid arguments should be based on properties of mathematical ob- jects [1,4,5]; open-ended problems require exploration, formulation of conjecture, and argumentation of conjecture [3]. The cited authors have illustrated how classroom norms that tend to promote ar- gumentation in the classroom are generated. In this research line, the complexity of the negotiation norms process has been illustrated, particularly when the production of con- vincing arguments by students is elicited. Additionally, teachers’ professional actions Mathematics 2021, 9, 2296. https://doi.org/10.3390/math9182296 https://www.mdpi.com/journal/mathematics