Characterizing the content on tangent lines of six analytical geometry textbooks

Abstract

We developed a qualitative content analysis of six Analytical Geometry textbooks with the aim of identifying learning opportunities offered by the content on tangent lines. The theoretical concepts used to organize the study involve a qualitative content analysis from a mathematical perspective, considering four different epistemological planes on tangent lines as a priori categories: (a) geometric, (b) algebraic, (c) analytical and (d) mechanical or kinematic. The categories construction was carried out in a deductive-inductive way, considering as analysis units the narrative content blocks. The analytical approach was identified as the most frequent when addressing the topic of tangents, followed by the algebraic and the geometrical approximation. In only one book was found the mechanical approximation. Two emerging categories were identified, related to the type of problems solved and graphic representations used in the texts. About the learning opportunities, the emphasis is on the notion of the tangent as a limit of secants. This leaves other important approximations aside that could be useful for mathematical thinking development. This is due to the fact that analytical geometry, in the revised texts, is conceived as a preparatory subject for calculus courses.