Making sense of accumulation functions in a technological environment

The present case study was designed to analyze the process of making sense of the
accumulation function when the integral concept is being studied in the geometry
track. This study is guided by the socio-cultural learning theory and the assumption
that mathematical concepts are learned by means of constructions in three worlds:
embodied, symbolic, and formal. We concentrate on the first. The learning
environment designed for this study includes activities focusing on the use of
computer applications that support direct manipulations of graphs and parameters
related to the visual representations of accumulation and integrals. The case study
focuses on two 17 year old students who have studied differentiation but not
integration. In the course of the discourse micro-analysis we identified four
elaborations of the meaning of the accumulation function: (1) noticing the presence
of a lower limit, (2) awareness of the negative and positive areas of bounded
rectangles, (3) paying attention to zeros in the integral as indications of
accumulation of negative and positive areas, and (4) explaining the global effect of
the change of the lower limit as vertically transforming the integral function.