DIDACTIC ENGINEERING MEDIATED BY A HYBRID LEARNING ENVIRONMENT FOR TEACHING VECTOR ALGEBRA
Anthropological Theory of the Didactic – ATD; T4TEL; Teaching and Learning; Hybrid Learning Environment – HLE.
This research discusses Vector Algebra teaching and learning in the Mathematics Teaching Course at Universidade Estadual de Feira de Santana (Feira de Santana Federal University, in free translation), which will be called CLM – UEFS in this text. It is based on the pillars provided by the Anthropological Theory of the Didactic (ATD) and on studies developed in the scope of Didactics of Mathematics, which have shown peculiarities that categorize vector notion as a concept that makes learning harder. Starting from those preliminary ideas, our research followed a path that led us to vectors as a mathematical object and to their institutional structure in CLM – UEFS, according to two aspects: the first one refers to the application and institutional relevance of vectors, while the second aspect is related to the conceptual multiplicity of vectors, which can be linked to the obstacles detected in the didactical-mathematical context observed in the teaching and learning processes of such mathematical element. In this setting, we admit the existence and functioning of Didactical Systems (DS), coordinated by a set of conditions and restrictions. Therefore, there was the need to analyze students’ productions and interactions regarding vector knowledge in terms of Personal Practice Paths (PPP) in the scope of vector tasks, which were based on techniques produced in a Praxeological Reference Model (PRM), materialized through a Hybrid Learning Environment (HLE) – a form of development centered on the didactical reference framework T4TEL. In other words, there is the question “How to harness praxeological recombinations – guided by conditions and restrictions – which can mediate personal and institutional interactions in the DS related to vector knowledge in CLM – UEFS?” With regard to methodological support, Didactical Engineering (DE) was implemented in order to test hypotheses and outline possible praxeological recombinations – path mapping and the reach of the techniques in each Didactical Device (DD) that “lives” in the HLE –, in addition to enabling an immersion into the descriptions of the personal praxeological complex produced by the players in this setting. Regarding results, we point to the susceptibility of dialogues that might contribute to removing obstacles related to choices that are dominant and widely crystalized in the functioning of DS in CLM – UEFS. Moreover, there is the emergence of a Didactical Device repertoire aimed at vector teaching and learning.