Professores, recursos curriculares e o ensino de combinatória no contexto das reformas curriculares e do ensino médio: o caso da rede estadual paulista de ensino

Abstract

The investigation reported in this document focuses on curricular resources produced for the teaching of combinatorics, within the scope of the São Paulo State Education Network, with a view to investigating their adequacy to research results. Since the approval of the São Paulo Curriculum between 2019 and 2020, the São Paulo Education Department (SEDUC-SP) has produced and made available to teachers and students a series of physical and digital resources that constitute the Curriculo em Ação program, aiming at the implementation of this curricular document in public schools. It is in this context that the investigation, of a qualitative nature, aims to evaluate in which aspects the material provided by SEDUC-SP approaches or moves away from the elements identified in research, as essential, for the development of combinatorial reasoning, and, as these resources, along with other possibilities, are articulated by teachers in the teaching of combinatorics. As guidance, two questions were proposed: a) what situations, operational invariants and symbolic representations can be inferred from the proposed approach for teaching combinatorics in the physical and digital resources of the Currículo em Ação program? b) what operational invariants are mobilized by teachers when using these curricular resources and how are they articulated in the context of other resources that they eventually use to teach combinatorics? To answer the first, combinatorial situations and combinatorial teaching situations were analyzed, present in the physical materials (teacher's notebook) and digital materials (slides and online tasks) of the Currículo em Ação program. At this stage, analyzes were carried out based on Bardin's Content Analysis and considering Vergnaud's Conceptual Field Theory, for whom a concept must encompass three fundamental sets: situations, operational invariants (object, relations, properties, theorems) and symbolic representations. Through the analysis of physical and digital materials, the following were observed: 1. scarce and little diverse combinatorial situations; 2. little exploration of representations such as possibility trees, lists and tables; 3. prioritization of the use of formulas, especially in digital material; 4. combinatorics teaching situations based on definition, presentation of formulas, examples, and application exercises. Regarding the second question, we followed two mathematics teachers, both teaching in 2nd grade classes of high school from schools in the São Paulo State Education Network, and a third who teaches in a private school to assist with the analyses. The data was collected and analyzed based on the theoretical assumptions of the Documentational Approach to Didactics by Trouche, Gueudet and Pepin, who proposed what they called documents, that is, what the teacher prepares, creates, or recreates based on the resources available to them. This approach goes hand in hand with Vergnaud's constructs, such as schema and conceptual fields. These authors conceive that the teacher's action in the production of documents (documentary genesis) is always mediated by resource use schemes, thus constituting their documentary work. Monitoring the teacher in his documentary work required the construction of his own methodological contribution: the Reflective Investigation of the teacher's documentary work. Based on the principles of this contribution, we sought to accompany the three participants in this research in the preparation and execution of their classes on combinatorics. Data collection was carried out remotely through semi-structured interviews, through participation and in-person observation of classes by the first two participating teachers, and by monitoring the preparation of classes by the third. The integration of resources from the Currículo em Ação program into the participants' documentary work took place in different ways. In the case of the first teacher, we did not observe integration of these resources with others aimed at teaching combinatorics, having opted for full use of the information contained on an educational website. The second participant used the digital material, integrating it into the textbook, but without adjusting or criticisms regarding the way in which combinatorics is approached in this material. The third, when preparing his classes, established clippings in the teacher's notebook to adapt it to the way he conceives the teaching of combinatorics, a way that is based, above all, on the use of multiplicative and additive principles as main strategies in resolving situations of score. Due to these observations, we consider it important to strengthen the relationship between researchers and teachers, with a view to the collaborative creation of schemes for the use of resources, whether curricular or of another nature, based on what studies on the constitution and development of combinatorial reasoning say