A Calculus Graphing Schema

Abstract

In this study, we analyzed students' understanding of a complex calculus graphing problem. Students were asked to sketch the graph of a function, given its analytic properties (1st and 2nd derivatives, limits, and continuity) on specific intervals of the domain. The triad of schema development in the context of APOS theory was utilized to study students' responses. Two dimensions of understanding emerged, 1 involving properties and the other involving intervals. A student's coordination of the 2 dimensions is referred to as that student's overall calculus graphing schema. Additionally, a number of conceptual problems were consistently demonstrated by students throughout the study, and these difficulties are discussed in some detail.

Analysis of STEM Majors’ Calculus Knowledge by Using APOS Theory on a Quotient Function Graphing Problem

“…APOS theory applicable is not necessarily a good method to analyze data as experienced by Clark, Cordero, Cottrill, Czarnocha, DeVries, St. John, Tolias, and Vidakovic (1997) in their research. Piaget et al (1983) introduced the triad stages intra, inter, and trans, used by Baker et al (2000), to introduce the property and interval schemas to analyze undergraduate students conceptual function knowledge on a calculus graphing problem. In 2007, Cooley, Trigueros, and Baker built on their work from 2000 (Baker et al (2000)) by focusing on the thematization of the schema with the intent to expose those possible structures acquired at the most sophisticated stages of schema development.…”

“…Piaget et al (1983) introduced the triad stages intra, inter, and trans, used by Baker et al (2000), to introduce the property and interval schemas to analyze undergraduate students conceptual function knowledge on a calculus graphing problem. In 2007, Cooley, Trigueros, and Baker built on their work from 2000 (Baker et al (2000)) by focusing on the thematization of the schema with the intent to expose those possible structures acquired at the most sophisticated stages of schema development. In their study, the problems were structured in a way that participants were required to respond to the first eight questions and continue with the ninth question only if they succeeded in answering the first eight questions (please see Cooley et al (2007) and the appendix, pg.…”

“…Therefore, understanding STEM majors' ways of viewing and recalling calculus concepts has been an important task of researchers interested in STEM education. (see for example Baker, Cooley, and Trigueros (2000) and Cooley, Trigueros, and Baker (2007)) This is due to the fact that students need to recall and apply calculus concepts in advanced courses. Numerical Methods/Analysis is an example of such advanced level courses in which calculus concepts are frequently used.…”

“…The motivation behind this study is to analyze conceptual calculus knowledge of the undergraduate and graduate students' who were enrolled in a Numerical Methods or Analysis course by using a theoretical framework similar to that of Baker et al (2000) and Cooley et al (2007) In this study a quotient function is given in its algebraic form and participants are asked to employ their calculus knowledge to determine the graph of the corresponding function. Written and video recorded oral interview responses of participants are analyzed by using the APOS theory.…”

“…Baker, Cooley, and Trigueros (2000) analysis of students' conceptual calculus understanding was based on a calculus graphing problem in which several conditions are required to be satisfied. In 2007, Cooley, Trigueros, and Baker conducted a more detailed study than in their research in 2000 to observe calculus concept knowledge of students who were considered to be successful by their professors in diverse disciplines, and by using the same theoretical framework of their study in 2000.…”