In the early 1980s, Dreyfus and Eisenberg (1982, 1984) proposed a framework for a systematic approach to curriculum development pertaining to the concept of function. The framework includes three dimensions: settings or representations (e.g., tables, equations, graphs, etc.), concepts (e.g., image, zeros, extrema, domain and range, injectiveness, surjectiveness, etc.), and generalization and abstraction (extending function concepts to multivariable functions, complex-valued functions, etc.). The authors summarized this framework by what they called the function block with three mutually perpendicular axes. Along the x-axis are the representations of the function concept; along the y-axis are the formal concepts related the notion of function; and along thez-axis are the different levels of abstraction of the function concept. According to the framework, at any level of abstraction (on the z-axis), movement can be along the x-axis, to consider different representations of that function, and/or along the y-axis, to consider the formal concepts related to that function. Dreyfus and Eisenberg (1984) referred to movement along the x-axis as horizontal transfer and movement along the z-axis asvertical transfer. The authors asserted, “ultimately, ways have to be found to extend the discussion of selected concepts to functions of several variables, complex valued functions, implicitly defined functions, etc.” (Dreyfus & Eisenberg, 1984, p. 78).
A considerable amount of research over the past three decades has investigated students’ understanding of many of the elements described in Dreyfus and Eisenberg’s (1982, 1984) function block. This section provides an overview of such research.
A significant amount of literature from the 1980s and 1990s focused on students’ un- derstanding of different representations of functions (Even, 1998; Hitt, 1998; Janvier, 1987; Moschkovich, Schoenfeld, & Arcavi, 1993; Schwingendorf et al., 1992). However, Thomp- son (1994) reflected on the multiple representation perspective, reconsidering that the “core concept of function” is not represented by the common representations of function. He stated:
Tables, graphs, and expressions might be multiple representations of functions to us, but I have seen no evidence that they are multiple representations ofanything to students. In fact, I am now unconvinced that they are multiple representations even to us, but instead may be areas of representational activity among which, as Moschkovich, Schoenfeld, and Arcavi have said, we have built rich and varied connections. (Thompson, 1994, p. 39)
Thompson went on to assert, “we should focus on them as representations of something that, from the students’ perspective is representable, such as aspects of a specific situation” (p. 39).
David Tall and colleagues investigated the “core concept of function” and the repre- sentable “something” referred to by Thompson (Akko¸c & Tall, 2002, 2003, 2005; McGowen, DeMarois, & Tall, 2000; Tall, McGowen, & DeMarois, 2000). Tall, McGowen, and DeMarois (2000) suggested that the “something” referred to by Thompson could be an embodied image of the function concept, such as the function machine. Meanwhile, Akko¸c and Tall (2003) describe the “core concept of function” as a mathematically simplistic idea that generates ideas that are often too complicated for students who do not make connections among the different representations of function. Arnon et al. (2014) claimed that the reason students have difficulty transitioning from one representation of function to another is that instruc- tion focuses on the direct translation without passing through the cognitive meaning of the function concept. From the perspective of APOS, the authors suggested that the student should determine the process of the function represented and use that process conception to transfer to a new representation.
Oehrtman et al. (2008) recommended that students be given more opportunities to experience multiple representations functions in different coordinate systems. While there
is a substantial amount of literature on students’ understanding of Cartesian graphs of functions, only recently have researchers begun to investigate students’ transferring their knowledge of the function concept other coordinate systems (Montiel et al., 2008, 2009, 2011; Moore et al. 2013).
Montiel et al. (2008) investigated the relationship between students’ understanding of functions in the Cartesian and polar coordinate systems in a single-variable context. The authors stated that when standard textbooks introduce the polar coordinate system, “very little emphasis is placed on the actual function concept, as it is supposed that by this state the student has developed the concept and that it will transfer” (p. 58). However, the authors found that the majority of students in the study were not able to transfer the notion of function to the polar coordinate system. Many of them took the vertical line test as the definition of function instead of a definition in terms of input and output. As the authors stated, “the generic definition of function [. . .] seems to often be lost amongst the different representations students are exposed to, without recognizing any implicit hierarchy” (p. 64). Moore et al. (2013) investigated two pre-service secondary mathematics teachers’ ways of thinking when graphing in the polar coordinate system. They found that by engaging in covariational reasoning, the students were able to conceive graphs in different coordinate systems as representative of the same relationship. As a result, the authors asserted that pro- viding students with opportunities to graph relationships in multiple coordinate systems can establish a need for covariational reasoning, which can promote less problematic conceptions of functions and their graphs.
Several studies have shown that students have difficulty generalizing their conception of function. For example, Mamona (1990) observed that students had misconceptions about sequences. She found that many students treated sequences as series, as they felt the need to sum the list of numbers generated by a sequence in order to find its limit. Mamona conjectured that this misconception might be due to students’ lack of acclimation to ab- stract concepts. That is, she argued that students feel the need to “do” something to the numbers, such as addition. Moreover, Mamona (1990) found that students resisted to con-
sider sequences as functions, due to the discrete nature of the domain. She conjectured that this is a consequence of students’ previous exposure to only “well-behaved” functions that correspond to a continuous graph.
Similar to Mamona’s (1990) findings, McDonald et al. (2000) reported that students tend to construct two notions of sequence, one as a list—SEQLIST—and the other as a function—SEQFUNC—with domainN. While the notion of SEQLIST was found to be easy for students to construct, the notion of SEQLIST was much more difficult, especially for students who had a weak function schema. For these students, it was inconceivable for a sequence to have a domain.
Mart´ınez-Planell, Gonzalez, DiCristina, and Acevedo (2012) built on the study by Mc- Donald et al. (2000) by investigating students’ conceptions of infinite series. The authors argued that students construct two notions for infinite series, one as a never-ending process of addition—SERLIST—and the other as sequence of partial sums—SERFUNC—each of which corresponds to a positive integer. They observed that weak schemas for function and limit caused students difficulty in constructing a strong SERFUNC conception.
A pair of studies by Trigueros and Mart´ınez-Planell (Mart´ınez-Planell & Trigueros Gais- man, 2012; Trigueros & Mart´ınez-Planell, 2010) investigated students’ reasoning about two- variable functions. In these studies, the authors proposed a preliminary genetic decomposi- tion (in the sense of APOS theory) for how students might generalize their understanding of one-variable function to two-variable functions. The genetic decomposition involved the essential constructions of a schema for R3, set, and function, that should be coordinated through the actions and processes of assigning a unique height to each element of a given subset of R2. The first study (Trigueros & Mart´ınez-Planell, 2010) focused on students’ development of R3 and their understanding of graphs of two-variable functions, while the second study (Mart´ınez-Planell & Trigueros Gaisman, 2012) investigated students’ general understanding of two-variable function, specifically the notions of domain, range, uniqueness of function value, and the arbitrary nature of function. These studies revealed that students have difficulty generalizing their notion of function to include relations between R2 and
They reported that students have difficulty recognizing that the domain of a two-variable function is a set of points in the plane. Furthermore, they reported that students have difficulty finding the range of two-variable functions when the domain is restricted, which is linked to their inability to imagine a graph of a function of two-variable functions given algebraically.
Weber and Thompson (2014) described a way of thinking about graphs of two-variable functions defined by z =f(x, y). They said that by thinking of x ory as a fixed parameter, an individual can imagine “sweeping out” a point to create a graph in the plane. Then by varying the parameter, the individual can imagine sweeping the graph through space to create a surface. The authors found that covariational reasoning can support the visualization of graphs of two-variable functions, rather than relying on the memorization of prototypes.