Building the Concept of Integral Function and Covariational Reasoning: two Case Studies

Abstract

The aim of this paper is to characterize how high school students build the integral function concept in the context of a teaching experiment using applets, and designed according to a hypothetical learning trajectory. Five out of fifteen students were chosen, which have participated in the teaching experiment and shown different moments of reflective abstraction process, to carry out a study of cases. Our results indicated three characteristics for building the integral function concept: (i) not identifying the relationship between a bound of an interval and the value of the area under a straight line belonging to this interval; (ii) identifying this relationship as a way of simple covariation; and (iii) identifying the complex covariation among the variable x, x ∈ [a, b], a function f(t), t ∈ [a, x] and the integral function $F\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)dt}$, in linear functions. These results provide information about the learning trajectory of the integral function concept; this trajectory can be used for designing curricular materials that support students' covariational reasoning.

Keywords:
Reflexive abstraction; Teaching Experiment; Hypothetical Learning Trajectory; Covariational Reasoning; Integrating applets