The continuous technological progress of the past few decades affords a different orientation to the teaching-learning process, which is useful and advantageous when mathematical concepts are transmitted. This article pursues two different aims. The first one is to examine the present state of the traditional process in relation to the concepts of convergence of sequences and functional series in first-year university mathematics courses, as well as the short-term effects of this process. The second one is to present, from a global and integrative perspective, a curricular variant on the teaching of mathematics, in which MAPLE V software and visualization play an essential role. In later work, the effects of this proposal on first-year students of mathematics at the authors' university will be analysed.