Contents lists available at ScienceDirect Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb The basis step in the construction of the principle of mathematical induction based on APOS theory Isabel García-Martínez a, ⁎ , Marcela Parraguez b a Departamento de Matemáticas, Universidad Católica del Norte (UCN), Avenida Angamos 0610, Antofagasta, CP 1270709, Chile b Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso (PUCV), Blanco Viel 596, Cerro Barón, Valparaíso, CP 2350026, Chile ARTICLE INFO Keywords: APOS Genetic decomposition Principle of mathematical induction University education ABSTRACT Using APOS theory as the framework along with a case study from a perspective within the methodological design of APOS theory, this study presents a cognitive model of the Principle of Mathematical Induction (PMI) in higher education. Based on evidence from university classrooms and the result of an initial measurement, the genetic decomposition designed by Dubinsky and Lewin for this concept was reformulated, introducing and defining the basis step in the PMI as a mental process. Using this reformulated genetic decomposition, the productions of four university students are analysed in order to support or refute the constructions it proposes. The results show that the reformulated genetic decomposition is viable and that the inclusion of the basis step as a mental process was seen in the cognitive model of the PMI shown by the students. The instruments used provide activities for a teaching sequence for the PMI at university level. 1. Introduction This research looks at the Principle of Mathematical Induction (PMI) from a cognitive perspective in order to show how university students construct this principle –as a method of proof–. As PMI is also used in mathematics for proving, this research also refers to PMI as a method of proof. The following are different versions of the origin of PMI. According to Bussey (1917), PMI was first used in 1575 by Maurolycus in the book Arithmeticorum Libri Duo. In it, Maurolycus describes and uses PMI as a way to proof the properties of the relations between different types of numbers (integers, even, odd, triangular, polygonal, etc.). Later, Pascal (1623–1662) carried out proofs with PMI in what is today known as Pascal’s Triangle. However, for Bourbaki (1972) PMI was conceived and used for the first time by Pascal in the 17th century and has been used often by mathematicians since the first half of that century. It was only in 1888 that the principle was formulated precisely when Dedekind presented a complete system of axioms for arithmetic (a system that was later reproduced by Peano and which carries his name today), one of which is the PMI. In Mathematics literature in different study programs at Chilean universities, there are different statements regarding PMI, though one of the most explicit definitions of the propositional functions of the principle is in the text by Grimaldi (1997), and therefore in the present study, the following definition of PMI is considered: If Pn ( ) is a propositional function, where n represents a natural number, such that: a) Pn ( ) 0 is true for an initial natural value n 0 (basis step) and http://dx.doi.org/10.1016/j.jmathb.2017.04.001 Received 2 September 2016; Received in revised form 6 April 2017; Accepted 11 April 2017 ⁎ Corresponding author. E-mail addresses: igarcia@ucn.cl (I. García-Martínez), marcela.parraguez@pucv.cl (M. Parraguez). Journal of Mathematical Behavior 46 (2017) 128–143 0732-3123/ © 2017 Elsevier Inc. All rights reserved. MARK