Functions and their representations in mathematics education
A function is a specific relation between two nonempty sets which associates each element of set A (the domain) to exactly one element of set B (range) (Büchter & Henn, 2010; Niss, 2014). Because of the curricular focus in the teaching–learning context of this study, in the following, we will relate particularly to functions from real numbers to real numbers. As can be seen from the above definition, functions are abstract and difficult to grasp, like other mathematical objects. However, learners can access functions and communicate about them using different representations (Duval, 2006; Vogel et al., 2007; see also Rolfes et al., 2021). Function representations that are commonly used in school contexts are situational representations (e.g., descriptions, pictures), tables, graphs, and equations (KMK, 2004; Land BadenWuerttemberg, 2004a, 2004b, 2012, 2016; see also Büchter & Henn, 2010). In the following, we will illustrate such representations related to the linear function with the equation^{Footnote 3}\(y=2x+3\). The graph of this function is a straight line with the Slope 2, which intersects the vertical axis at the yIntercept 3 (see Fig. 1a). Both parameters, the slope and yintercept, can be detected in the corresponding function equation \(y=2x+3\) as well as in the table (see Fig. 1b). If we relate the positive part of this function’s domain to the context price of a taxi ride (see Fig. 1c), these parameters also express relevant situational features.
Changes between representations of functions
Changes between different representations are crucial for concept formation as well as for cognitive flexibility and adaptivity when solving (mathematicsrelated) problems (Duval, 2006; Heinze et al., 2009). This general consideration also applies to the domain of functions, as different representations highlight specific characteristics of a function and might be more or less adequate for solving a concrete problem. Moreover, a function should not be limited to a single representation, as the interplay between several representations can be meaningful for students to grasp all relevant characteristics of a real function (Barzel et al., 2005; Niss, 2014; Vogel, 2006). Learners might struggle when they consider a representation in isolation (Duval, 2006) or when they identify a function with a single representation (Niss, 2014). In this context, Leinhard, Zaslavsky, and Stein (1990, p. 3) point out that: “… algebraic and graphical representations are two very different symbol systems that articulate in such a way as to jointly construct and define the mathematical concept of function.” If this citation is extended to apply to function representations in general, the extent to which an individual is able to (I) read/interpret different representations, (II) recognize and use the advantages of specific representations, and (III) change between representations reveals his or her understanding of functions (AduGyamfi, 2007; Barzel et al., 2005; Büchter & Henn, 2010; Vollrath, 1989).
A closer look at concrete representational changes reveals that they require a learner to engage in specific (mental) activities (Barzel & Ganter, 2010; Hußmann & Laakmann, 2011; Leuders & Prediger, 2005; Lichti, 2019; see also Vogel, 2006) in addition to mastering the source and target representation. On top of other factors, such as how often and in what manner the corresponding representational changes are treated in the mathematics classroom (some subtleties might be treated and acknowledged differently by different mathematics teachers), these specific (mental) activities might influence the empirical difficulty of these representational changes (Bossé et al., 2011a). In the following, we will describe the representational changes in the context of linear functions which are investigated in this study:

When changing from a situational representation to a graph, learners have to understand the underlying situation in order to determine concrete points or the yintercept and slope that they need in order to draw the graph.

When changing from a situation to an equation, students need to read the characteristics of the situation that directly determine the slope and yintercept or to identify pairs of associated values in the situation which enable them to calculate the slope and yintercept. Moreover, these parameters need to be arranged meaningfully in an equation.

When changing from a graph to a situation—depending on the concrete task—learners need to interpret the graph itself (e.g., increasing, decreasing, type of function) or relevant points or parameters thereof. Then, they have to illustrate or verbalize these aspects in light of the given situation.

When changing from an equation to a graph, students can use conceptual knowledge about the parameters; i.e., they identify the slope and yintercept in the equation and mark them directly or use a gradient triangle in the coordinate system. Alternatively, students can also calculate the coordinates of two or more points (recorded, e.g., in a table) and draw a graph using them.

When changing from a graph to an equation, learners can also use conceptual knowledge about the parameters and directly read the slope (using a gradient triangle) and the yintercept from the graph. Then, both parameters have to be arranged in a meaningful way in a function equation. As an alternative, students can read specific points of the graph and use formulas/a linear equation system to calculate the slope and yintercept.
The procedures outlined for changing between graphs and equations using conceptual knowledge about the parameters appear to be commonly included in German textbooks (e.g., Freudigmann et al., 2016; Maroska et al., 2006), in particular in the teaching–learning contexts of this study. The ability to identify the parameters in these specific representations is a prerequisite for changing between them.
As described, when changing between particular representations of functions, students should draw on conceptual knowledge about these representations; i.e., they should understand the meaning of the source and target representations as well as know how particular characteristics of the function can be translated to make the representational change. Unfortunately, previous studies on functions report that students only learn “rules without understanding the underlying concepts to which they refer, and this often results in mathematics becoming a formal, dull, and virtually unusable subject” (Swan, 1985, p. 6; see also Bossé et al., 2011a; Sajka, 2003). Hence, students are able to perform a representational change without having any underlying conceptual understanding. This is in line with our prior research (2020), which indicates that changes between graphs and equations are partly taught and learned in an algorithmic way. In other words, students learn to follow rules without understanding them. This means that students do not draw on conceptual knowledge, but that they follow an algorithm such as “doing XY with the first number of the equation and YZ with the second number” and vice versa. Following such an algorithm without understanding it can be considered purely procedural knowledge, which is expected to be prone to error and likely to be forgotten (RittleJohnson et al., 2015).
An algorithmic approach to functions might be reduced by referring to situational representations. Zindel (2019) suggests using representational changes that include situations, as students can only master these changes if they understand the corresponding situational representation and its mathematical equivalent. Nathan and Koedinger (2000) empirically show that learners often work more successfully on tasks with a situational context than on purely mathematical tasks (see also Sproesser et al., 2020) as they can draw on informal and less abstract solution strategies. However, these studies also provide empirical evidence that teachers often hold a socalled symbolprecedence view; i.e., teachers consider solving purely mathematical tasks as a prerequisite for solving tasks with a situational context. Moreover, in our previous study (Sproesser et al., 2020), we reported that some teachers tend to neglect tasks with situational representations and focus on purely mathematical tasks in their lessons on functions (see also Bossé et al., 2011a). Despite this focus, their students do not systematically perform better when making purely mathematical representational changes. Such a disregard of situational representations is not conducive to developing mathematical literacy in general (cf. OECD, 1999). It also fails to comply with the German Educational Standards (KMK, 2004) and, in particular, with the educational aim that students should learn to connect mathematical and situational representations of functions in order to use functions “for describing and analyzing many aspects of our economic, physical and social environment” (Swan, 1985, p. 6).
Although tasks involving a situational representation might reduce the risk of algorithmic computation and result in learners using informal solution strategies, changing between situational and mathematical representations can also pose specific difficulties for learners. With regard to the modeling cycle, Vogel (2006) explains that the situational representation involved addresses another abstraction level than a purely mathematical representation (see also Geiger, 2019). Nitsch (2015) deduces from this consideration that students’ everyday concepts might interfere and cause specific errors—in particular in the field of functions. Similarly, Bossé et al., (2011a) argue that representational changes involving a situational representation constitute the most difficult changes for learners.
The preceding paragraphs illustrate that different representational changes require particular (mental) activities; moreover, the representations involved have distinct characteristics which might make such a representational change more or less difficult, from a theoretical point of view (cf. Bossé et al., 2011a). It stands to reason that these theoretically driven considerations result in learners encountering empirical difficulties. Furthermore, it should be noted that—beyond task characteristics—classroom and learner characteristics might also affect whether an individual successfully performs a task such as a representational change or if s/he encounters problems (ibid.). Such a characteristic might be, e.g., affiliation to a specific school track. Therefore, the next section gives an overview of the curricular requirements related to elementary functions, with details for the different school tracks involved in the study.
Curricular requirements related to functions
The German Educational Standards for mathematics apply to Grades 5 to 10 of all German middle schools. Accordingly, in the area of elementary functions, the curriculum focuses on their characteristics, different representations, and representational changes as well as, in particular, on situational representations in order to solve realworld problems (KMK, 2004, pp. 11). In most regions of Germany and in particular in the teaching–learning contexts of this study, a threetier school system is established for middle schools. This means that students can choose one of three different tracks (the academic, medium, and most basic track^{Footnote 4}) for middle school depending on the schoolleaving qualification they wish to obtain. There are distinct curricular requirements for these tracks, which are based on and in line with the German Educational Standards. A comparison of the curricular requirements related to elementary functions for the academic and nonacademic tracks reveals that there are some common requirements, but also several differences (Land BadenWürttemberg, 2016, pp. 39; see also Land BadenWuerttemberg, 2004a, 2004b, 2012). Students of all tracks have to learn to represent functional relationships (in particular proportional, antiproportional, and linear relationships) using texts, tables, equations, and graphs. Learners in the most basic track, however, do not have to learn to flexibly change between representations. Moreover, all students should be able to read situationrelated characteristics from different function representations (e.g., time points, increasing/decreasing). Additionally, the curriculum requires all students, except for those in the most basic track, to be able to draw graphs of linear functions with a slope and a gradient triangle and to identify an equation for a given graph. Students should also be able to calculate the slope and yintercept from the coordinates of two points and, hence, be able to determine the corresponding linear function equation. Furthermore, they should be able to interpret changes in situational contexts, although only students from the academic track are taught to use the notion of change rate in this context. Only students in the academic track learn formal characteristics of functions. Thus, although function representations and representational changes are taught in all school tracks, there are differences in the levels of proficiency targeted in the particular tracks.
Textbooks that are commonly used within our teaching–learning contexts (e.g., Backhaus et al., 2017; Freudigmann et al., 2016; Maroska et al., 2006) meet these curricular requirements. They cover all of the mentioned representational changes (between graphs, equations, situations, tables). However, tables play a minor role in the teaching unit on linear functions and are often only used as intermediate representations (see also Nitsch, 2015). As mentioned above, our previous study (Sproesser et al., 2020) revealed that several teachers strongly focused on changes between graphs and equations, in particular in the nonacademic tracks, and tended to neglect tasks with situational representations (see also Bossé et al., 2011a; Cunningham, 2005).
The aim of this study is to analyze students’ competencies in changing between representations of elementary functions and, in particular, to identify which changes (specific groups of) students can(not) perform easily. Beyond investigating different school tracks as a possible reason for differences in competency, we will also evaluate gender effects in this field, which are well documented for the subject of mathematics in general. Therefore, the next section provides an overview of gender differences in mathematics and, especially, gender differences in dealing with functions.
Gender differences in the fields of mathematics and functions
Various national and international studies have revealed significant advantages for boys in terms of general mathematical competency, with mostly small effect sizes (Hyde et al., 1990; Köller & Klieme, 2000; Lindberg et al., 2010; OECD, 2001, 2004, ; Schroeders et al., 2013). Despite this widespread evidence, there are empirical findings indicating that gender differences do not exist per se. Differences are rare or small at elementary school level and increase as children grow older, in particular at secondary school level (Beller & Gafni, 1996; Hyde et al., 1990; see also Winkelmann et al., 2008). Moreover, there is variation between countries regarding gender differences; i.e., the magnitude of gender effects differs from one country to another, and in some cases there is empirical evidence in favor of girls (Blum et al., 2004; ElseQuest et al., 2010; Guiso et al., 2008; OECD, 2010).
Looking at these findings, one might seek an underlying reason for these gender differences in mathematics. Cognitive and (neuro)biological models attribute differences in mathematical competency mainly to varying spatial abilities between boys and girls (Maier, 1999; see also Büchter, 2010). However, this does not explain the age or countryrelated variation in gender differences outlined above. In contrast, psychosocial models explain the differences based on gender stereotypes in children’s domestic environment, such as family members considering mathematics to be a typically male domain. This might result in girls having a less positive attitude towards mathematics (Eccles et al., 1990), more anxiety (Chipman et al., 1992), and lower interest and selfconcept (Wigfield & Eccles, 1992). Consequently, girls perform worse in mathematics for affective or motivational reasons. Models of schoolrelated socialization imply that such gender stereotypes related to mathematics might also be common among teachers who therefore treat boys and girls differently—at least unintentionally—and increase gender differences (Fennema et al., 1990; Muntoni et al., 2020). In this context, Chipman et al., (1991) report that mathematics curricula and textbooks are more geared to boys than to girls. As reducing gender differences in mathematics is a declared objective of mathematics education (Budde, 2009; Leder & Forgasz, 2008), teachers should be aware of genderrelated socialization mechanisms and, in particular, of the fact that they often hold mathematicsrelated gender stereotypes (Keller, 1998; Muntoni et al., 2020).
Empirical research on gender differences in mathematics indicates that the processes and content areas underlying tasks make a difference. At the process level, gender differences in favor of boys are particularly frequent for complex problemsolving and modeling tasks which require nonstandard strategies; girls, on the other hand, perform better on calculusoriented tasks requiring standard solution strategies (Gallagher et al., 2000; Hyde et al., 1990, 2008; Köller & Klieme, 2000). At the content level, boys have been found to have the biggest advantage in geometry and analysis, compared to fewer or no differences or even advantages for girls in algebra and arithmetics (Hyde et al., 1990; Kaiser & Steisel, 2000; OECD, 2009). Studies focusing on the domain of functions also report that boys achieve higher competency scores on average (Klinger, 2018; Lichti & Roth, 2019; Nitsch, 2015; Schroeders et al., 2013). Moreover, the findings reveal a pattern that is largely consistent with the findings of general mathematicsrelated research. Boys perform better on tasks that include a situational context and graphs, whereas girls are more proficient in executing purely mathematical tasks requiring procedural or calculusoriented knowledge and tasks in the verbal form (BayrhuberHabeck, 2010; Klinger, 2018; see also Rost et al., 2003).
In order to optimally foster the learning of both girls and boys, e.g., with regard to functions, and hence to counteract possible gender differences, the extent of such gender differences needs to be empirically investigated. In addition, it is of interest to identify whether there are specific types of representational changes that are particularly difficult for girls or boys. This is one of the research focuses of the present study. We will outline the corresponding research questions in the next section.