Abstract This research aims to explain a part of the modeling path of the Fundamental Theorem of Calculus for the characterization of a microworld system, which was conceived in our doctoral thesis (OLIVEIRA, 2022) as a computer support for mathematical activity in the computational environment. Didactic-Informatics Engineering (EDI) was chosen as a methodological contribution, with an adaptation of the process model, structured as a research praxeology to describe the investigation stages, and with the analysis of the didactic, cognitive, epistemological and informatics dimensions. In this process, the theorem modeling was carried out with the connection between the Anthropological Didactic Theory (ADT) and the T4TEL approach to study the fundamental theorem through praxeologies and be able to characterize the microworld. The analysis of dimensions allowed specifying a set of requirements and functionalities to describe the microworld. The system of variables that was created for the generators of types of tasks showed the role of the ostensible dimension in the development of techniques and allowed us to conjecture about the existence of other objects that can be mobilized with the intermediation of a software and, consequently, for the production new techniques or solving new tasks. OLIVEIRA, OLIVEIRA (OLIVEIRA 2022 environment DidacticInformatics Informatics EDI (EDI contribution model stages didactic cognitive ADT (ADT TTEL T TEL consequently 202 20 2

Resumo Esta pesquisa tem como objetivo explicitar uma parte do percurso de modelização do Teorema Fundamental do Cálculo para a caracterização de um sistema micromundo, que foi concebido em nossa tese de doutorado (OLIVEIRA, 2022) como um suporte informático à atividade matemática no ambiente computacional. A Engenharia Didático-Informática (EDI) foi escolhida como aporte metodológico, com uma adaptação do modelo de processo, estruturada como uma praxeologia de pesquisa para descrever as etapas de investigação, e com a análise das dimensões didática, cognitiva, epistemológica e informática. Nesse processo, a modelização do teorema foi realizada com a conexão entre a Teoria Antropológica do Didático (TAD) e a abordagem T4TEL para estudar o teorema fundamental por meio das praxeologias e poder caracterizar o micromundo. A análise das dimensões permitiu especificar um conjunto de requisitos e funcionalidades para descrever o micromundo. O sistema de variáveis que foi constituído para os geradores de tipos de tarefas evidenciou o papel da dimensão ostensiva no desenvolvimento das técnicas e permitiu conjecturar sobre a existência de outros objetos que podem ser mobilizados com a intermediação de um software e, consequentemente, para a produção de novas técnicas ou a resolução de novas tarefas. micromundo OLIVEIRA, OLIVEIRA (OLIVEIRA 2022 computacional DidáticoInformática Informática EDI (EDI metodológico processo investigação didática cognitiva informática TAD (TAD TTEL T TEL consequentemente 202 20 2

Abstract: Here are reports the design, implementation and evaluation of an activity conformed by two realistic mathematical situations framed in the phenomenon of free fall, in the framework of an investigation supported by the theory of Realistic Mathematical Education, whose objective was to characterize the levels of mathematization achieved by students of a course of Integral Calculus in the study of the Fundamental Theorem of Calculus with the use of digital technologies. In the analysis of the models created by the students in the first three levels of mathematization described by the theory, it is evident that, from the tasks designed, the students manage to represent algebraically the existing relationship between variable quantity, accumulation of change and rate of change of accumulation, referring to the physical context presented. They identify the relationship between derivative and integral, explicitly stating that they are inversely related, since each one undoes what the other does, a relationship that is evidenced in the Theorem.

Resumen: Aquí se reporta el diseño, implementación y evaluación de una actividad conformada por dos situaciones matemáticas realistas enmarcadas en el fenómeno de caída libre, en el marco de una investigación sustentada en la teoría de la Educación Matemática Realista, cuyo objetivo fue caracterizar los niveles de matematización logrados por estudiantes de un curso de Cálculo Integral en el estudio del Teorema Fundamental del Cálculo con el uso de tecnologías digitales. En el análisis de los modelos creados por los estudiantes en los tres primeros niveles de matematización descritos por la teoría, se evidencia que, a partir de las tareas diseñadas, los estudiantes logran representar algebraicamente la relación existente entre cantidad variable, acumulación de cambio y razón de cambio de acumulación, haciendo referencia al contexto físico presentado. Identifican la relación existente entre derivada e integral, manifestando explícitamente que están relacionadas de forma inversa, puesto que cada una deshace lo que hace la otra, relación que se evidencia en el Teorema.

Abstract This paper reported a research which aims to identify connections that a university student group established to solve problems in a context. These problems can be solved using the relationship between the derivative and integral established in the Fundamental Theorem of Calculus. To identify such connections, we adopted the theoretical framework proposed by Businskas (2008). As a research method, we used the case study and applied a questionnaire which included five problems, the solutions of which we obtained the data. The data indicate that students establish extra-mathematics and intra-mathematics connections when they solve problems in a context, but mainly used the procedural type and the different representations. We also noticed a strong connection with their prior knowledge learned in previous levels of education rather than at the university level.

Resumen En el presente artículo se reporta una investigación cuyo objetivo fue identificar las conexiones que un grupo de estudiantes universitarios establecen al resolver problemas en contexto. Estos problemas pueden resolverse utilizando la relación entre la derivada y la integral establecida en el Teorema Fundamental del Cálculo. Para identificar las conexiones adoptamos el marco teórico que al respecto plantea Businskas (2008). Como método de investigación utilizamos el estudio de casos y aplicamos un cuestionario integrado por cinco problemas, de cuyas soluciones obtuvimos los datos. Los datos indican que los estudiantes establecen conexiones extramatemáticas e intramatemáticas al resolver problemas en contexto, pero principalmente utilizan las de tipo procedimental y las representaciones diferentes, además notamos una fuerte conexión con sus conocimientos previos aprendidos en los niveles de educación precedentes al nivel superior.