Relational Understanding of the Derivative Concept through Mathematical Modeling: A Case Study

Abstract

The purpose of this study was to investigate three second-year graduate students' awareness and understanding of the relationships among the "big ideas" that underlie the concept of derivative through modeling tasks and Skemp's distinction between relational and instrumental understanding. The modeling tasks consisting of warm-up, modeleliciting, and model-exploration activities were used to stimulate participants to reflect and construct ideas about the concept of change. The data indicated that the participants' understanding of derivative was rather instrumental. Their explanations couldn't reveal the role of the big ideas regarding the concept of derivative with respect to what they mean, why and how they are related to the derivative and to each other. The results highlighted the fact that even if one of big ideas is ignored, the concept of derivative may not be fully understood relationally due to the compartmentalization of these big ideas in students' conceptual systems. If this happens to be the case, even though students can solve differentiation tasks/problems correctly, which implies procedural understanding, they may not be actually making sense of what the concept of derivative conceptually means.

Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative

“…Derivative was chosen because the majority of students can solve a routine differentiation problem easily, but not developed a correct conceptual understanding of derivative's graphical representation, and the relationship between the algebraic and graphical representation (Asiala et al, 1997;Baker et al, 2000;Clark et al, 1997;Sahin, Erbas, Yenmez, 2015). According to these studies as well as our personal teaching experiences, several students have not developed a deep conceptual understanding of the derivative's graphical representation.…”

Application of the Complementarities of Two Theories, APOS and OSA, for the Analysis of the University Students’ Understanding on the Graph of the Function and its Derivative

“…For this reason, most of the students can solve a routine differentiation problem easily but have no correct conceptual understanding of derivative's graphical representation and the correlation between the algebraic and graphical representation (Baker, Cooley, & Trigueros, 2000). For instance, some of the students or even teachers do not know the correlation between the slope of the tangent line and difference quotient limit (lim (Badillo, Azcárate, & Font, 2011;Özmantar, Akkoç, Bingölbali, Demir, & Ergene, 2010;Sahin, Erbas, & Yenmez, 2015;Sánchez-Matamoros, Fernández, & Llinares, 2015).…”

Characteristics of the process of solving mathematical tasks

“…На пример, Ормонд (Ormond, 2012) користи термин погрешно разумевање, наглашавајући да се појављује у ситуацији решавања алгебарских задатака у нижим разредима основне школе, што се у случају употребе знака једнакости јавља као погрешно разумевање значења знака једнакости. Сахин и сарадници (Sahin, Yemez, & Erbas, 2015) разматрају потешкоће које се јављају при решавању задатака из извода једначина (средњошколска настава математике), које се означавају као тешкоће у разумевању. У разматрању одређених тешкоћа које се јављају у разумевању математичких садржаја, Шахрил (Shahrill, 2013) користи појам погрешно разумевање (incorrect understanding).…”