Tworzenie definicji czworokątów przez przyszłych nauczycieli – podejście poprzez odkrywanie

Abstract

This research addresses elementary school prospective teachers’ education in geometry and takes place in Portugal, in the context of a geometry course taught by the first author of this paper. We aim to characterize the activity of prospective teachers in defining quadrilaterals as they work on exploratory tasks in a design-based research study for a geometry course at the 2nd year of their teacher education program. The teaching-learning conjecture is based on the relevance of exploratory work, also valuing prospective teachers’ reflection on their learning. We also assume that the activity of defining quadrilaterals plays an important role in developing reasoning, deepens the knowledge of its properties and supports a more complete view of mathematics. Data was gathered from the participants’ reports and portfolios, as well as audio and video records of classroom discussions. The results show that initially the participants had misconceptions about what a definition is, just focusing on the necessary properties of the figures. However, at the end of the sequence of tasks, they were able to take into account that the set of properties has to be sufficient to identify the defined figure, revealing the development of their knowledge. At the end of the sequence, most participants presented correct definitions, using properties that they previously ignored and showing comprehension of the underlying concepts. They produced economical definitions in few cases, and performed better in inductive than deductive reasoning. We conclude that the exploratory work allowed participants to construct their knowledge in a meaningful way and reflection played an important role in making them aware of personal preconceptions and knowledge. In the beginning, the participants analysed definitions strictly focusing on the necessary properties of the figure, which seemed to be simple when the properties related to visible elements or were well known facts. In the final stage of the sequence of tasks, the participants also took into account that the set of properties has to be sufficient to identify the defined figure and, in the construction of definitions, they presented mostly correct definitions using properties that they previously ignored, showing an understanding of the underlying concepts and properties. Doing this process after investigating and classifying quadrilaterals encouraged the prospective teachers to mobilize different kinds of properties and to reason according to the established classifications, which was a very challenging activity.