Development of a Notion of Limit as Proximity From Discourse Analysis in Secondary-School Function Classes

Mathematical teaching and learning of an introductory calculus conception of limit in secondary school before students learn the formal definition of limit in higher education are discussed. In this study, applications of the theory of instruction based on “Formalizing Introductory Notions (FIN)” (Nagle, 2013), in which a theory incorporating new pedagogical approaches was introduced to describe static notions of limit, without ignoring dynamic notions of limit, are presented to help foster an informal limit conception better aligned with the formal definition. A qualitative discourse analysis based on students’ utterances, including students’ drawing pictures on graphs, was done. In the results of the investigation, it was found that the students’ utterances drawings on a graph of the secants (segments) used static notions of limit supported by dynamic notions of limit according to the operating activities. There were learning activities in which students developed a notion of limit as the proximity of the predicted tangent line of a function. Consequently, students’ discussions changed focus to the validation of limit candidates with static notions of limit. To overcome the contradiction of their explanations of the operating activities with dynamic notions of limit, the students changed to an explanation with static notions of limit. In light of the findings, this study suggests adapting the pedagogical approach used by the Nagle (2013).