{"id":8261,"date":"2019-01-15T16:59:45","date_gmt":"2019-01-15T16:59:45","guid":{"rendered":"https:\/\/citejournal.org\/\/\/"},"modified":"2019-06-05T11:07:51","modified_gmt":"2019-06-05T11:07:51","slug":"designing-to-provoke-disorienting-dilemmas-transforming-preservice-teachers-understanding-of-function-using-a-vending-machine-applet","status":"publish","type":"post","link":"https:\/\/citejournal.org\/volume-19\/issue-1-19\/mathematics\/designing-to-provoke-disorienting-dilemmas-transforming-preservice-teachers-understanding-of-function-using-a-vending-machine-applet","title":{"rendered":"Designing to Provoke Disorienting Dilemmas: Transforming Preservice Teachers\u2019 Understanding of Function Using a Vending Machine Applet"},"content":{"rendered":"
Functions could be considered one of the most important topics in high school mathematics due to their importance and relevance to a number of other mathematical topics and their foundational role in college-level mathematics and other related areas in the sciences (e.g., Cooney, Beckmann, & Lloyd, 2010; Dubinsky & Harel, 1992; Leinhardt, Zaslavsky, & Stein, 1990). Further, they are a critical base for mathematical understanding of science, technology, engineering, and mathematics (STEM) disciplines and are often regarded as the unifying element of much of secondary mathematics.<\/p>\n
Students begin studying informal function concepts as early as third grade, then they become a dominant feature throughout most high school courses. In the Common Core State Standards for Mathematics (CCSSM) the study of functions is given its own domain, separate from Algebra, in grades 9\u201312 (National Governors Association Center for Best Practice & Council of Chief State School Officers, 2010).<\/p>\n
Putting functions front and center in high school mathematics is accompanied by many conceptual obstacles that have been well documented in the literature (e.g., Even, 1990; Tall, McGowen, & DeMarois, 2000). Research has revealed common misconceptions with respect to the definition of function (Vinner & Dreyfus, 1989), use of function notation (Oehrtman, Carlson, & Thompson, 2008), and connections between function representations (e.g., Brenner, 1998; Clement, 2001; Dreher & Kuntze, 2015; Stylianou, 2011). Some of the same misconceptions have been identified in studies involving preservice and practicing secondary math teachers (Bannister, 2014; Chesler, 2012; Even, 1990, 1993; Kabael, 2011; Wilson, 1994). Due to the similarities among misconceptions held by K-16 students as well as teachers, preservice teachers need opportunities to examine their own knowledge of function and consider how to engage students in tasks that develop robust understandings of function.<\/p>\n
Foundational to developing deep and nuanced understandings of functions as they are positioned throughout secondary mathematics is understanding the concept of function itself. This need is highlighted in the recently published National Council of Teachers of Mathematics (NCTM) book, Developing Essential Understandings of Functions, Grades 9-12 <\/em>(Cooney et al., 2010), in which the authors identified five \u201cbig ideas\u201d in regard to function \u2013 the first of which is the function concept. Related to this big idea, the authors identified essential understandings that high school teachers must have and explicitly address when teaching. The following are the essential understandings presented for the function concept:<\/p>\n The authors suggested that attending to the development of the essential understandings is imperative: \u201cThe importance of understanding functions and the challenge of understanding them make them essential for teachers of mathematics in grades 9-12 to understand extremely well themselves\u201d (Cooney et al., 2010, p. 1).<\/p>\n In response to the call for opportunities for preservice teachers (PSTs) to deepen their knowledge of function, we designed and studied the implementation of an applet-based learning intervention focused on disrupting PSTs\u2019 current understanding of the function concept. The purpose of this study is to examine the ways in which an applet designed to challenge PSTs\u2019 understanding of function does so, and determine whether or not engaging with the applet results in shifts in their sense making related to the functions and nonfunctions represented in the applet.<\/p>\n Over the last 50 years an extensive body of research has focused on understanding of functions. The consensus in this body of work is that many students possess a weak understanding of the concept of function, and research has documented several misconceptions that appear to be quite common.<\/p>\n Much of the research on students\u2019 understandings of function has occurred in the context of college algebra, precalculus, or calculus classes. These studies have identified common understandings that students develop related to the concept of function. One common student understanding is that functions are defined by an algebraic formula (Breidenbach, Dubinsky, Hawks, & Nichols, 1992; Carlson, 1998; Clement, 2001; Sierpinska, 1992). This understanding is not surprising since functions are typically introduced as specific function types, such as linear and quadratic functions, in the middle school and high school curriculum (Cooney et al., 2010). Thompson (1994b) found that, not only do students view functions as algebraic formulas, some view functions as two expressions separated by an equal sign. While an equation view of function is not inherently wrong, it is narrow and can lead to difficulties for students as they work with functions in different contexts and with different representations (Cooney et al., 2010).<\/p>\n Along with an algebraic view of functions, research has shown that students often rely on the vertical line test to identify a function from a nonfunction (Breidenbach et al., 1992; Fernandez, 2005). This view can lead to conceptual difficulties in determining functions from nonfunctions, including the tendency to apply rules to determining functions from nonfunctions (Breidenbach et al., 1992; Fernandez, 2005). Students whose view of function is algebraic and uses procedural techniques to identify functions and nonfunctions struggle to comprehend a general mapping of input values to a set of output values (Carlson, 1998; Thompson 1994a).<\/p>\n When students consider particular function families, studies have shown that many exhibit difficulties identifying constant functions as functions (Bakar & Tall, 1991; Carlson, 1998; Rasmussen, 2000). This difficulty appears to stem from the fact that constant functions do not vary (Oehrtman et al., 2008).<\/p>\n In a study of undergraduates enrolled in a college algebra course, Carlson (1998) found that only 7% of students who earned a grade of A in the course (A-students) could produce a correct example of a function whose output values are the same regardless of input value. Carlson also posed this question to second semester calculus undergraduates, and 25% of A-students produced the example y = x<\/em>.<\/p>\n Underlying the students\u2019 problematic understandings about function is their definition of function. Yet, particular attention to the definition itself has not been widely researched (exceptions include Breidenbach et al., 1992, and Vinner & Dreyfus, 1989). In many studies student descriptions of functions are not tied directly to their definition of function. The consistency of problematic understandings of function found across studies speaks to the need for pedagogical practices to specifically disrupt and correct these understandings.<\/p>\n When considering the development of student understanding of function, teachers\u2019 understandings are important as well. The majority of the research has focused on in-service teachers\u2019 or PSTs\u2019 descriptions of function, and similar themes to students\u2019 understandings of functions have emerged (Even, 1990, 1993; Kaebel, 2011; Wilson, 1994). For example, similar to students, practicing teachers and PSTs tend to view functions algebraically and discuss properties of graphs (e.g., vertical line test) in their descriptions (Even, 1990, 1993; Wilson, 1994). One exception is a study by Vinner and Dreyfus (1989) who examined junior high teachers\u2019 definitions of functions prior to a professional development. They found that 25 of 36 teachers provided a definition that pointed to a correspondence between two sets of elements.<\/p>\n More recent studies have reported similar findings, adding to the evidence that teachers hold some of the same misconceptions as K-16 students with respect to the definition of function and connecting representations (Bannister, 2014; Chesler, 2012). In particular, Chesler (2012) noted that many secondary mathathematics PSTs lack \u201cflexibility and expertise in interpreting and using mathematical definitions\u201d (p. 38) as they examined the equivalence of various definitions of function.<\/p>\n Teachers\u2019 understanding of function has been shown to impact the pedagogical choices they make during instruction of the concept. For example, in her study of PSTs, Even (1993) found that PSTs could not justify the need for univalence (one-to-one) and did not know why it was important to distinguish between functions and nonfunctions. Because of this lack of content knowledge, the PSTs were limited in their pedagogical approaches, resulting in teaching students to procedurally identify functions using the vertical line test. Moreover, Bannister (2014) suggested that PSTs who are adept at translating between algebraic and graphical representations of functions may be better prepared to understand diverse student conceptions when they encounter them during instruction. Studies such as these point to the need to develop opportunities for teachers to not only examine their own understanding of function, but also analyze student thinking about functions in more productive ways.<\/p>\n In considering PSTs\u2019 learning related to function, we adopted a theoretical lens of transformation theory (Mezirow, 2009). Transformation theory is consistent with constructivist assumptions, specifically that meaning resides within each person and is constructed through experiences (Confrey, 1990). Mezirow (2009) described four forms of learning that lie at the heart of the theory: elaborating existing meaning schemes, learning new meaning schemes, transforming meaning schemes, and transforming meaning perspectives (p. 22). According to Peters (2014) meaning perspectives are the broad predispositions a person has toward a concept based on their prior experiences and culture. In contrast, meaning schemes, which are situated within meaning perspectives, are the specific expectations, knowledge, beliefs, attitudes or feelings that are used to interpret experiences (Cranton, 2006; Peters, 2014).<\/p>\n In the context of this study we were interested in a PSTs\u2019 meaning schemes related to the function concept. Furthermore, given that they have previous knowledge of function, we are very specifically interested in provoking an elaboration or transformation of their existing meaning schemes. For example, a PST might elaborate on her meaning scheme for function by reconsidering her prior conception of function as a graph that passes the vertical line test and adopt a broader view of function that includes representations beyond merely graphs. Another might transform his incorrect understanding of function by rejecting his prior conception of a function being any equation and replacing it with a conception of a function being a mapping between two sets in which that mapping has specific properties.<\/p>\n Learning by changing meaning schemes (through elaboration or transformation) often begins with a stimulus, a disorienting dilemma<\/em>, which requires students to question their current understandings that have been formed from experiences (Mezirow, 2009). This type of learning experience was of particular interest \u2013 both in our design of stimuli for it and the ways that meaning schemes are changed as a result. Given the evidence that PSTs often have a view of function that is limited to algebraic expressions<\/strong> and their associated graphs (e.g., Carlson 1998; Even, 1990)<\/strong> and that such understandings typically result in a vertical-line-test-related definition of function (e.g., Carlson, 1998)<\/strong>, we aimed to design experiences that would problematize these understandings, thereby creating a stimulus for change.<\/p>\n One strategy that has been suggested for mitigating common misunderstandings related to function is the use of a function machine as a cognitive root. The idea of a cognitive root<\/em> was introduced by Tall et al. (2000) as an \u201canchoring concept which the learner finds easy to comprehend, yet forms a basis on which a theory may be built\u201d as he was developing a cognitive approach to calculus (p. 497). As an example of a cognitive root for function concepts, Tall et al. suggested the use of a function machine<\/em> (sometimes referred to as a function box). The machine metaphor Tall and colleagues described is typically a \u201cguess my rule\u201d activity. In such activities, the inputs and associated outputs are provided, and students are challenged to determine what happened in the function machine (i.e., determine the function rule). While students are presented with a machine to embody the function concept, the rules used by the machine are algebraic in nature. In their studies using such machines proved quite promising, yet some students still struggled with connecting representations and determining what is and is not a function (McGowen, DeMarois, & Tall, 2000).<\/p>\n Given the promise of a machine metaphor as a cognitive root for function coupled with our desire to present a dilemma with which PSTs would need to grapple with given their current understandings, we set out to design a machine-based experience using representations that were unfamiliar for PSTs as a stimulus for examining their meaning schemes of function.<\/p>\n Unlike typical function machines (i.e., a number is put in, inside the machine the number is transformed by a function rule, and then a number is spit out), the applet we designed to trigger a disorienting dilemma in PSTs\u2019 understanding of function contained no numerical or algebraic expressions. Instead it was built on the metaphor of a vending machine. Our intention was to avoid misconceptions that occur when students rely on an algebraic, and often procedural, view of functions (i.e., inconsistent use of the vertical line test).<\/p>\n Additionally, we hoped to emphasize the essential understandings identified by Cooney et al. (2010), in particular that functions apply to a wide range of situations and their domain and range do not have to be numbers. The Vending Machine applet (https:\/\/ggbm.at\/rCtUxApF<\/a><\/u>) is a GeoGebra file that contains five soda vending machines each with buttons for Red Cola, Diet Blue, Silver Mist, and Green Dew. The instructions direct users to explore the five machines and determine which are functions (Figure 1; McCulloch, Lee, & Hollebrands, 2015). When the user presses a button (input), one or more cans appear in the bottom of the machine (output). To remove the can(s) from the machine, the user clicks a reset button.<\/p>\n The functionality of each machine was designed to address misconceptions from the literature on distinguishing functions and nonfunctions. Machine A is the identity function; each button produces a can of the corresponding color. Machine B is the same as A, except when Silver Mist is selected, it produces two silver cans. This machine requires students to wrestle with the notion of what represents an element in the range. For Machine C, every button results in a single green can, the purpose of which is to present PSTs with a constant function to consider (i.e., the same number of cans of the same color for each button). For each button on Machine D, a single can is produced, but the color is different from the color of the button pressed. This machine was designed to problematize their occasional use of the term unique<\/em> when thinking about outputs.<\/p>\n Finally, Machine E is similar to D, except the Silver Mist button randomly produces a can of a different color each time it is pressed. The purpose of Machine E is to provide a context in which testing the buttons on the machine once is not sufficient for determining whether or not the object is a function. The idea of testing the buttons more than once is foundational, and something that should be expected on all machines, as functions are predictable \u2013 meaning that if users know the function rule they can predict the output for any input.<\/p>\n\n
Background<\/h2>\n
Students\u2019 Understandings of the Function Concept<\/h3>\n
Teachers\u2019 Understandings of the Function Concept<\/h3>\n
Theoretical Framework<\/h3>\n
Applet Design<\/h3>\n