Posing inverse modeling problems for task enrichment in a Secondary Mathematics teachers training program

Abstract

In this presentation the structure of the thesis is briefly explained. The thesis fieldwork was carried out with prospective teachers, focusing on the reformulation of given problems in an inverse way, in order to enrich tasks for secondary school students. As a consequence, in the first chapter, important mathematics education topics, such as tasks, problems, modeling, problem posing, problem solving and task enrichment –among other less frequent ones, like inverse problems– are introduced. Thus, the introduction finishes with the research questions and objectives. Taking into account the above considerations, the different sections of the thesis theoretical framework, were chosen. The theoretical framework –which constitutes the second chapter of the thesis–, includes problem posing, inverse problems, mathematical modeling and task enrichment. Besides, it should be pointed out that all the results obtained in the fieldwork are analyzed using the Didactic Analysis, developed by Rico and collaborators (Rico & Fernández-Cano, 2013; Rico, Lupiáñez, & Molina, 2013; Rico & Ruiz-Hidalgo, 2018). In particular, our work focuses on three of the dimensions of Didactic Analysis: analysis of meanings, cognitive analysis, and instructional analysis. For this reason, Didactic Analysis is a fundamental part of the theoretical framework of the thesis, developed in the second chapter. The third chapter is devoted to the methodological framework and it describes the characteristics of the sample and the instruments utilized to analyze the results. It is important to mention that these instruments used to analyze the productions of the subjects (i.e., the prospective teachers), have been transformed during the research, and these modifications are also described in this chapter. Additionally, it is worth mentioning that the fieldwork was carried out in two well-differentiated parts: a pilot study that took place in 2017 and a definitive study, done in 2019. Between both experiments, there was a research design in which modifications were proposed, in order to correct or at least attenuate the difficulties observed during the pilot study; this is also part of the third chapter. The fourth chapter is devoted to the description and analysis of the results obtained during the first part of the fieldwork, that is, the pilot study, done in 2017. In that opportunity, two direct problems were provided to the prospective teachers and they were asked to reformulate in an inverse form the second one. Furthermore, the participants were requested to design the tasks associated with their own reformulated problem and their responses are the inputs for the corresponding cognitive and instruction analysis. In the fifth chapter, the results of the second fieldwork, designed throughout 2018 and carried out in 2019, are described and analyzed. In this second experience, the direct problems provided to the participants were the same as in the pilot study and the prospective teachers were asked to reformulate both in an inverse form. Once again, the participants were asked to propose the corresponding tasks, associated with their reformulation, so the productions included the cognitive and instructional analysis. Finally, the sixth and last chapter is devoted to expound the conclusions of the research and the limitations of this work as well as the possible continuations of the research line.