Imagens conceituais de combinatória no ensino superior de matemática

Abstract

This qualitative doctoral research as case study is inserted in the field of mathematical education and is linked to the Postgraduate Program in Education of the Education Center of the Federal University of Espírito Santo - PPGE/CE/UFES. Their objectives were to identify conceptual images evoked by university students when solving combinatorial tasks, and to investigate how the conceptual images of university students about simple groupings of arrangement, permutation and combination reconstruct when we work with specific combinatorial tasks. In this sense, this research brings theoretical perspectives from Tall and Vinner (1981) abour concept image and concept definition, Hershkowitz (1994), when dealing with prototypes and associations with relevant and irrelevant attributes, and Skemp (1976) about instrumental and relational understanding. For the combinatorial analysis, we used the studies of Morgado, Carvalho, Carvalho and Fernandez (1991), Hazzan (1993), Batanero, Godino and Navarro-Pelayo (1996), Guzmán (2000) and Paiva (2009). Therefore, based on the study object located from the perspective of the initial formation, the research was carried out in the Mathematics Teacher Education Degree Course of the IFES - Federal Institute of Education, Science and Technology of the Espírito Santo, Cachoeiro de Itapemirim campus. In this study participated seven undergraduate students of the sixth period who studied the discipline of Combinatorial Analysis and Probability in 2017/2. The analysis of the data pointed to the fragility of the conceptual images of the undergraduate students, showing that the prior knowledge of combinatorics and the simple groupings of arrangement, permutation and combination of them were not all coherent. They presented some conceptual images of the definition of the concept, oscillating between coherent, fragmented, incoherent and mistaken in relation to the formal concept described by the mathematical literature. Therefore, from the scientific point of view, they have a conceptual image that is more intuitive than formal. With the literature review and the studies about the mathematics teacher education degree courses of Espírito Santo state it was possible to see that, from basic education up to higher education, it is given a brief and superficial treatment to combinatorial. In addition, the discipline of Combinatorial Analysis and Probability occupies little space in the curriculum and a small workload of the actual work is destined to the combinatorial content. In this context, the collaborative planning of research and classroom lessons and/or lecture and research has proved potent in the reconstruction of conceptual images by the undergraduate students in mathematics. The integrated and simultaneous work involving simple combinatorial groupings through the understanding of the mathematical structure underlying the statement from classes based on problem solving and discursive practices in mathematics allowed the enrichment of the conceptual images. In addition, the research shows the potential of collaborative work between teachers and researchers.