# Generalizing Single Variable Functions to Two-Variable Functions, Function Machine and APOS

## Article excerpt

Abstract

The focus of this study in which the theoretical framework of APOS was used is students' generalizing function notion from single variable to two-variable function concepts in Analysis II course in the elementary mathematics education program. In the teaching process, teaching activities that support generalizing the function notion with multiple representations and relations between them and the function machine were used. For data collection, two tests on each of single and two-variable function concepts and interviews with six students were used. It was concluded that the students' understanding level of function concept and students' schema of threedimensional space is fundamental for their construction of two-variable function. Moreover, significant results that can support prospective studies for students' conceptual levels of two-variable functions were obtained.

Key Words

Function Notion, Two-variable Function Concept, APOS, Function Machine.

(ProQuest: ... denotes formulae omitted.)

Mathematics educators have paid attention to the function concept, which in fact deserves this attention, by supporting the literature with their studies on this concept. There are many different studies on the concept of function; however, all of them are on the notion of general function or the concept of single variable function. The studies on the concept of two-variable function, which requires transferring the notion of general function and many aspects of single variable function are few. Nonetheless, promising development is that this concept has started to be studied recently. Two-variable function concept, which has quite importance for science, engineering, mathematics and mathematics education students, is one of the fundamental concepts of advanced mathematics. Since understanding two-variable functions requires transferring not only general function notion but also key aspects of single variable functions, this causes difficulty for most of the students (Montiel, Vidakovich, & Kabael, 2008). Moreover, generalizing geometric properties of functions to two-variable functions also requires sufficient knowledge of three-dimensional geometry and visualization, which makes difficult to understand geometric properties of two-variable functions for students. Trigueros and Martinez-Planell (2009) emphasized the necessity of relating different representations of functions to understand geometric properties of function related to visualization. Yerushalmy (1997) stated the complexity of generalizing single variable functions to two-variable function and emphasized that this complexity depends on both the concept and its representation. He insisted on the importance of multiple representations and relation between them in the process of generalizing function notion from single variable to two-variables. The significance of multiple representations and relations between them has been highlighted in most of the studies (Bower, & Lobato, 2000; Breidenbach, Hawks, Nichol, & Dubinsky, 1992; Carlson, Oehrtman, Thompson, 2007; Christou, Elia, & Gagatsis, 2004; Ferrini-Mundy & Graham, 1990; Janvier, 1987; National Council of Teachers of Mathematics [NCTM], 1989; Sierpinska, 1992; Yerushalmy, 1997). Trigueros and Planell (2009) who have studied on geometrical aspects related to students' notion of the two-variable function emphasized that relating treatments of representations with converting different representations should be paid attention before abstract concepts related to functions are given. Meanwhile, they stated that the table representations of functions were not analyzed in their study, the reason of this was not that table representation is less important. Conversely, they insisted that different representations of functions should also include table representation.

This study is on students' generalizations about the concept of function from single variable to twovariables through two-variable calculus course in which function machine was used as a cognitive root. …

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