Conocimiento geométrico especializado en estudiantes para profesor de matemáticas de secundaria. Un estudio entorno a los polígonos

Abstract

This thesis presents a case study of three prospective secondary mathematics teachers (PTs). The aim of the study is to characterise the specialised knowledge implicitly or explicitly deployed by the teachers with respect to their conceptualisation of a polygon and the hierarchical classification of quadrilaterals. The theoretical foundations and methodological approach are those of the Mathematics Teachers’ Specialised Knowledge (MTSK) model, developed by the SIDM1 research group at the University of Huelva, Spain. This framework is complemented by further theoretical analysis and research concerning geometric thinking and the use of definition and classification in mathematics, which supply the context for the descriptors emerging from the preliminary analysis of the data. Hence the analytical tools employed in the study are provided by the subdomains and categories of the MTSK model alongside the emergent descriptors. The research follows an interpretive paradigm and was carried out in the context of a Professional Practice course forming part of the Degree in Education at a private university in Peru, during which participants conducted simulated teaching. The research data was collected from several sources: a survey via an open-ended questionnaire, a record of the prospective teachers’ intentions in the form of their lesson plans, and observation of performance through video recording. The questionnaire presented respondents with four hypothetical classroom situations for which they needed to consider: an analysis of the definition of polygons; the construction of a working definition of a polygon with illustrations; an analysis of the typology of quadrilaterals and the underlying concepts involved; and the concept of the hierarchy of quadrilaterals and the rationale for an inclusive classification of them. The lesson plan outlined each PT’s proposed teaching-learning activities for developing either the topic of polygons or quadrilaterals, as previously directed by the teacher trainer. The plan was then carried out and the resulting session was recorded on video and transcribed. This enabled information units to be identified and presented alongside the associated descriptions of specialised knowledge deployed by the PTs. The results show that the PTs conceive of a polygon as flat, geometric shape delimiting an interior region, composed of sides, angles and vertices. One of the PTs restricted the set of polygons to solely convex polygons, a second amplified the set to include convex plus concave, and the third additionally recognised that polygons can have crossed sides (complex polygons). With respect to quadrilaterals, the key elements which determined their conceptualisation were found to be: the length of the sides and their parallelism, the size of the angles, and, in one case, the intersection of the diagonals. The PTs recognised that graphic illustration plays an important role in the teaching and learning of quadrilaterals, but did not consider its conceptual basis, although they showed awareness that the standard, or prototypical, positions used by teachers to graphically represent quadrilaterals can lead to erroneous learning on the part of the students. It was also observed that the PTs had difficulties in five specific areas: 1) identifying the most generalised quadrilateral, or the one with least additional properties; 2) using this (the most generalised) to create a hierarchy of the remaining quadrilaterals; 3) specifying the necessary (and sufficient) characteristics for defining a quadrilateral; 4) constructing a definition for each quadrilateral, in accordance with the features of a mathematical definition; and 5) devising an inclusive classification of quadrilaterals. With regard to this latter, it was observed that they favoured disjoint classifications. This study highlights the complexity of teachers’ knowledge, specifically prospective teachers, and the need for this to be studied across a range of topics (as it is very topic-specific) and across a range of mathematical practices, such as definition and classification.