Conceptual Structure of the Accumulation Function in an Interactive and Multiple-Linked Representational Environment

Abstract

In light of the recent growing interest in conceptual learning and teaching of calculus, and especially with the focus on using technological environments, our study was designed to explore the learning processes and the role played by multiple-linked representations and by interactive technological environment in objectifying the accumulation function. The study focuses on 13 pairs of 17-year-old students familiar with the concept of differentiation but not of integration. The study was inspired and guided by semiotic mediation theory, which considers artefacts to be fundamental to cognition and views learning as a process of becoming aware of the knowledge that exists within a cultural context. Students were asked to explain the possible connection between multiple-linked representations: the function graph, the accumulation function graph, and the table of values of the accumulation function. We include three types of analysis representing the mathematical elements involved in the students' learning path of the accumulation function, and the students' interactions with the artefact. The data analysis identified four phases in the processes of objectification. We describe a conceptual structure that typifies the learning of the accumulation function concept in an interactive, multiple-linked representation environment.

Adherencia al discurso matemático escolar: el caso de la integral definida en la formación inicial docente

“…A su vez, destacan las investigaciones que favorecen el uso de la tecnología en la enseñanza de la matemática, por ejemplo, para promover los conceptos del Cálculo Integral a partir del uso de las nuevas herramientas digitales (Swidan & Yerushalmy, 2016, Moreno-Armella, 2014. Particularmente, Jácome, Fiallo & Parada (2018) proponen el diseño, implementación y evaluación de una secuencia de tareas mediante el uso de la tecnología.…”

Disclosure of mathematical relationships with a digital tool: a three layer-model of meaning

“…In the context of learning calculus, research has focused mainly on how student interaction with digital tools helps them conceptualize calculus ideas (Arzarello & Robutti, 2010;Swidan & Yerushalmy, 2016). In particular, studies have emphasized two aspects: the use of digital tools as facilitative resources for solving problems; and the use of symbolic, graphic, and numeric digital tools to illustrate the concepts of co-variation, derivation and integration (Tall, 2010;Thompson & Carlson, 2017).…”