ADHEME

For years now, the research group has developed a series of studies on didactic, historical and epistemological analysis of school mathematics, on arithmetic, algebra, problem solving and modelling. Its intention is to improve the teaching and learning of mathematics in school systems, not only by understanding the phenomena that occur in them, but also by providing well-founded information on the processes of teaching and learning mathematics, and by providing teaching models for use by practising teachers and curriculum designers. Part of these studies have been carried out within the framework of the agreement between the Universitat de València and the Centro de Investigación y de Estudios Avanzados de México, signed in 1988 and renewed in 2004. The theoretical and methodological framework that organises the studies is that of Local Theoretical Models (Filloy, E., Puig, L., and Rojano, T. (2008). Educational Algebra. A Theoretical and Empirical Approach. New York: Springer), in which the processes of teaching and learning mathematics are considered as processes of communication and production of meaning. More recent studies combine three aspects: The first is the study of problem solving, both, on the one hand, from the general point of view of heuristics or heuristic methods for classes of problems, and, on the other hand, in specific arithmetic-algebraic domains. But it also takes into account both aspects simultaneously, considering the role of heuristic elements such as control, management and decision mechanisms (which are usually described as metacognitve) in the resolution of arithmetic-algebraic problems, or in the modelling of phenomena by means of families of functions expressed algebraically. The second is the use of interactive learning environments, be it the spreadsheet, the symbolic graphing calculator, GeoGebra, the intelligent tutorial system HBPS, or, more recently, apps for tablets. The third is the historical analysis of mathematical ideas, from the point of view of research in didactics of mathematics. In particular, we have carried out or are carrying out studies on: The modelling process and the concepts of function family and parameter. Several studies for which we developed and tested teaching materials. One in Bachillerato, another in a first course of the Faculty of Mathematics of the Universitat de València and another in a first course of the degree of Economics and Administration of a university in Colombia. Some with a symbolic graphing calculator, others with GeoGebra, others with iPad apps and one with Maple. Some with experiments using real data, and others using real data from official sources or based on a proposed model. The teaching and learning of the algebraic resolution of verbal arithmetic-algebraic problems in secondary school, using a computer environment as a didactic artefact. In some cases the environment has been the Excel spreadsheet, in others an intelligent tutorial system developed for this purpose, which we have called Hypergraph Based Problem Solver (HBPS). The study of the forms of demonstration of algorithms for solving second-degree equations in medieval Arabic texts and in the first algebra texts printed in Spanish, and the study of the representation of functions in historical texts. The study of the history of algebra and its teaching in Spain in the 16th, 17th and 18th centuries. The design and implementation of a proposal for teaching the concepts of reaso