Characterizing Thematized Derivative Schema by the Underlying Emergent Structures

Abstract

This paper reports on different underlying structures of the derivative schema of three undergraduate students that were considered to be at the trans level of development of the derivative schema (action-process-object-schema). The derivative schema is characterized in terms of the students' ability to explicitly transfer the relationship between a function and its first derivative to the derivative function and the second derivative. This conscious shift of properties of derivatives is differently manifested by the students in the trans level of development of the derivative schema and can be considered evidence of the different characteristics of the thematization of derivative schema. From here we suggest that there are different underlying structures in the constructed schema due to the consciousness in which students use the relations between a function and its derivative.

Students’ understanding of the function-derivative relationship when learning economic concepts

“…This suggests that integration of the different systems of representation may be a key aspect in understanding the relationship between the notion of functions and derivative functions in economic concepts. García, Llinares and Sánchez-Matamoros (2011) have emphasised the importance of the relationship between the derivative at a point (local perspective) and the first and second derivatives of a function (global perspective) in order to acquire a proper understanding of the relationship between a function and its derivative. In our research, the last two tasks proposed constituted an example of how to analyse the process of obtaining the first and second derivative from the item of data used to obtain the derivative at a point.…”

“…In this study, we proposed a genetic decomposition of the relationship between a function and its derivative in the use of economic concepts (Table 1), based on the previous research on how students develop an understanding of this relationship (García, Llinares and Sánchez-Matamoros, 2011;Habre and Abboud, 2006;Sánchez-Matamoros, García and Llinares, 2013;Haciomeroglu, Aspinwall and Presmerg, 2010;Zandieh, 2000). In particular we considered the following:…”

Una Propuesta para el Análisis de las Prácticas Matemáticas de Futuros Profesores sobre Derivadas

“…Las características de la comprensión del concepto de derivada de los estudiantes de Bachillerato y primeros cursos de universidad es un tema que ha sido ampliamente investigado por diversos autores, desde distintas aproximaciones teóricas -particularmente las cuestiones de índole cognitiva e instruccional - (ASIALA et al, 1997;BAKER;COOLEY;TRIGUEROS, 2000;SÁNCHEZ-MATAMOROS et al, 2006;ARTIGUE; BATANERO; KENT, 2007;SÁNCHEZ-MATAMOROS, 2011). No obstante, aunque también ha sido de interés caracterizar los conocimientos de los profesores para una enseñanza efectiva de la derivada (GAVILÁN; GARCÍA; LLINARES, 2007; BADILLO; SÁNCHEZ-MATAMOROS et al, 2012), pocos han sido los estudios centrados en los profesores y su relación con la gestión de los aprendizajes sobre derivada.…”

Developing Pre-Service Teachers’ Noticing of Students’ Understanding of the Derivative Concept

“…Finally, students have also difficulties transferring the relationship between the function and the derived function to the relationship between the first and second derivative of a function (Baker et al, 2000;García, Llinares & Sánchez-Matamoros, 2011) Taking into account previous research regarding students' understanding of the derivative concept and considering the Spanish curricula, an intended learning trajectory (Clements & Sarama, 2004) of the derivative concept was described (Figure 1). Limit, continuity and derivability concepts are part of the Spanish curricula.…”

“…The aim of adding excerpts from students' interviews was to give more insight into their responses and to provide more information about their understanding of derivative. Students' responses in the questionnaires real and were chosen from a previous study (Sánchez-matamoros, García & Llinares, 2006;García, Llinares, & Sánchez-matamoros, 2011). PMTs had to answer the next two questions for each one of the high school students' responses:…”