In this article we present the results of a qualitative investigation into the teaching and learning of Taylor series and local approximations. In order to perform a comparative analysis, two investigations are conducted: the first is historical and epistemological, concerned with the pedagogical evolution of semantics, syntax and semiotics; the second is a contemporary institutional investigation, devoted to the results of a review of curricula, textbooks and course handouts in Tunisia and the United States. Our results show that contemporary curricula emphasize the algebraic and analytic approaches to the theory but omit the graphical and numerical registers, which both played a large role in the historical development. We contend that by presenting a more historically accurate didactic approach, student learning can be enriched to give a deeper and more conceptually accurate understanding of the topic.