External vs. Internal Didactic Transposition of the Definition of Limit in Calculus
DOI:
https://doi.org/10.23917/varidika.v39i1.17312Keywords:
External Didactic Transposition, Internal Didactic Transposition, Limit of Function, Calculus Learning, Knowledge TransformationAbstract
The limit of a function is a foundational concept in calculus, yet its transition from scholarly knowledge into knowledge to be taught, taught knowledge, and learned knowledge may produce shifts in meaning that affect students’ conceptual understanding. Limited studies have examined how these shifts occur through external and internal didactical transposition. This study examines how the meaning of the limit definition is transformed through these stages and how such transformations are reflected in students’ understanding. A qualitative approach was employed through document analysis of calculus textbooks, analysis of students’ lecture notes to reconstruct the lecturer’s taught knowledge, and analysis of students’ responses to a diagnostic task. Using purposive sampling, 25 undergraduate students enrolled in a Calculus course were selected as participants. The findings show that, through external didactical transposition, the formal ε–δ definition is simplified into intuitive and procedural forms in textbooks, thereby reducing its epistemological scope. Through internal didactical transposition, this simplification is further reinforced in the taught knowledge reconstructed from students’ lecture notes, which predominantly emphasize algebraic manipulation and direct substitution. Consequently, students’ understanding is expressed through three main response patterns: interpreting the limit as approaching a value, applying procedural techniques, and equating the limit with direct substitution. These findings reveal a shift from the formal mathematical meaning of the limit toward intuitive and procedural interpretations. The study highlights the need for didactical designs that explicitly connect intuitive representations and procedural techniques with the formal ε–δ definition.
Keywords: External didactical transposition, internal didactical transposition, limit of function, calculus learning, knowledge transformation
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