The aim of this paper is to present and discuss some of the evidence regarding the resources that students use when they establish relationships between a contextual situation and an ordinary differential equation (ODE). We present research results obtained from work by seven students in a graduate level course in mathematics education, where they were involved in solving tasks based on the study of ODEs. The students' mastery of digital tools allowed them to use and articulate different mathematical representations to comprehend how phenomena develop. The use of digital tools helped to enhance the students' interpretation of the relationships between the context and the mathematical model associated with it. We also found that difficulties related to interpretation are grounded in the literal relationship that students make between their mental model of how a phenomenon develops and its mathematical representations.
Resumen. En este trabajo se analizan algunas de las dificultades que surgen en el momento de interpretar las soluciones de una ecuación Diferencial Ordinaria (EDO) cuando se expresan en forma gráfica y algebraica. La investigación se llevó a cabo con estudiantes de ingeniería que estaban al final de un primer curso de EDO, los cuales realizaron varias actividades en un ambiente dinámico incluyendo algunas relacionadas con la interpretación de las soluciones representadas en forma algebraica y gráfica de una EDO que modela el comportamiento de una población. Se encontró que, aun cuando los estudiantes dominaban algunas herramientas algebraicas para resolver una EDO, no las utilizaban para interpretar el comportamiento de las soluciones dentro del registro gráfico. Además, al trabajar con el campo de direcciones para obtener información sobre las soluciones de la EDO su uso fue sólo local, esto es, centraron su atención en regiones relativamente pequeñas del campo de direcciones. Palabras clave. Ecuaciones diferenciales ordinarias, interpretación, registros de representación, dificultades. Difficulties experienced by students in the interpretation of the solutions of ordinary differential equations that models a problem Summary. In this work we analyzed some of the difficulties that arise when the student interpret the solutions of Ordinary Differential Equation (ODE) when they are expressed in graphical and algebraic form. The investigation was carried out with Engineering students who were at the end of a first course of ODE, who realized some activities in a graphical dynamic environment that included the interpretation of the solutions represented in algebraic and graphic form of a ODE that models the behavior of a population. One the one hand, even though the students dominated some algebraic tools to solve a ODE, they did not use them to interpret the behavior of the solutions within the graphical registry. On the other hand, when they working with the field of directions to obtain data on the solutions of the ODE its use was only local, that is to say, focused its attention on relatively small regions of the field of directions.
In this paper, we develop a review of the research focused on the teaching and learning of ordinary differential equations with the following three purposes: to get an overview of the existing literature of the topic, to contribute to the integration of the actual knowledge, and to define some possible challenges and perspectives for the further research in the topic. The methodology we followed is a combination of a systematic literature review and a bibliometric analysis. The contributions of the paper are given by the following: shed light on the latest research in this area, present a characterization of the actual research lines regarding the teaching and learning of ordinary differential equations, present some topics to be addressed in the next years and define a starting point for researchers who are interested in developing research in this field.
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