Signifying the accumulation graph in a dynamic and multi-representation environment

Abstract

The present study focuses on the accumulation process involved in the integration of a single-variable function. Observing the work of two high-school calculus students who had not yet learned any other integral-related ideas, we analyze the emergence of the semiotic relationship between personal and mathematical meanings, as expressed through the understanding of mathematical signs in integration tasks. Adopting Radford's educational perspective whereby learning is defined as a process of objectification, we identify a three-stage evolution of the double semiotic meaning of the lower boundary and of its role in the definition of the accumulation graph: (1) objectifying a zero accumulation in relation to the lower limit, (2) objectifying zero as marking the zero sum of accumulated areas, and (3) objectifying the accumulation graph as dependent on the lower-limit value. This evolution is marked by semiotic changes related to the pivotal role of the "zeros" in the accumulation graph.

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

“…For those students who were not able to objectify the lower limit value as a relative zero, their personal meanings did not evolve into mathematical meanings. Therefore, based on our data analysis and on another study (Yerushalmy and Swidan 2012), it was reasonable to conclude that objectifying the reflection of the lower limit as a relative zero is an essential element in objectifying the overview of the accumulation function.…”

“…In view of the recent growing interest in conceptual learning and teaching of calculus, especially learning in technological environments, we designed a learning environment that enables students to learn the concept of integration based on the idea of accumulation in a multi-representational and interactive setting. In a previous study (Yerushalmy and Swidan 2012) we examined the ways used by one pair of students to conceptualize the accumulation function while learning with an artefact designed to support exploration through experimentation with interactive multiple-linked representations of the function and of its accumulation function. The artefact supports interactive changes of parameters, providing immediate feedback, and direct manipulation of mathematical objects in a graphic presentation.…”

“…The artefact supports interactive changes of parameters, providing immediate feedback, and direct manipulation of mathematical objects in a graphic presentation. In that study, we identified the role played by the exploration of the functions of the lower limit in the definition of the accumulation graph (Yerushalmy and Swidan 2012). To deepen our understanding of the ways in which students use interactive and multiple-linked representation artefacts to conceptualize the accumulation function, we broadened our research sample and observed how students objectify the accumulation function when they vary the upper limit but keep the other parameters invariant.…”

Math majors' visual proofs in a dynamic environment: the case of limit of a function and the ϵ–δ approach

“…They explained that the horizontal line y = 0.5 would represent the horizontal asymptote in each case as x tends to plus and minus infinity, indicative of a perceptual proving scheme. [29,33] Anna had a unique description as demonstrated in her explanations: 'When I keep going like this [by moving the graphics view always to the left along the left part of the function graph (Figure 13 Yerushalmy and Swidan [47] postulated that 'becoming aware of an existing mathematical object requires engagement in mathematical activity to grant meaning to the object' (pp.288-289). Anna's visualization could be explained with the dynamic conception of limit [5] during which she utilized a variety of semiotic tools such as dynamic keywords and phrases along with motions on the graphs and creative hand-gestures via which she objectified [48] the mathematical content.…”

Fostering Dialogue in the Calculus Classroom Using Dynamic Digital Technology

“…The introduction of digital technology allows different approaches, which look to technology as a means to an end (e.g. Thompson et al 2013;Hong and Thomas 2013;Yerushalmy and Swidan 2012). The meaning of rate of change or of accumulation is illustrated by technology as a process that simulates the sophisticated understanding of the limit concept.…”