Affect and graphing calculator use

Journal of Mathematical Behavior 30 (2011) 166–179

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The Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb

Affect and graphing calculator use Allison W. McCulloch ∗ North Carolina State University, Department of Science, Technology, Engineering and Mathematics Education, 502L Poe Hall, 2310 Stinson Drive, Raleigh, NC 27695-7801, United States

a r t i c l e

i n f o

Keywords: Affect Graphing calculators Instrumentalization Problem solving Calculus

a b s t r a c t This article reports on a qualitative study of six high school calculus students designed to build an understanding about the affect associated with graphing calculator use in independent situations. DeBellis and Goldin’s (2006) framework for affect as a representational system was used as a lens through which to understand the ways in which graphing calculator use impacted students’ affective pathways. It was found that using the graphing calculator helped students maintain productive affective pathways for problem solving as long as they were using graphing calculator capabilities for which they had gone through a process of instrumental genesis (Artigue, 2002) with respect to the mathematical task they were working on. Furthermore, graphing calculator use and the affect that is associated with its use may be influenced by the perceived values of others, including parents and teachers (past, present and future). © 2011 Elsevier Inc. All rights reserved.

1. Introduction Imagine watching a high school calculus class getting ready to take an exam. Before even looking at the exam paper that has been handed to her you see one young lady pick up her graphing calculator to make sure it works. She sighs heavily, as if she is relieved, and then sets it down right next to her exam paper. Later as she is working on a problem you see her using her calculator and whisper a quiet “Yes!” when she sees something on her calculator screen that apparently makes her happy. Another young man towards the back of the room looks perplexed as he works on a problem, and then he picks up his calculator to do something, looks back and forth between his paper and his calculator and then actually reaches over his shoulder and gives himself a pat on the back. It is clear that for these students mathematics is not an emotionless activity. Furthermore, it is apparent that these students do not feel alone in their activity; each of them has a graphing calculator with which they are interacting and that they are responding to with outwardly articulated feelings. What was it about this tool that had these students expressing so much emotion? What are they actually doing with it and why? More than twenty years ago Kaput (1989) predicted that in the near future technology tools in mathematics education would have great impact on students’ affective experiences. He suggested that technology tools would be instrumental in helping students avoid the discrepancy between an individual’s expectations when beginning a problem solving activity and the ongoing activity required to complete the activity that often leads to negative emotions about the mathematics. This paper reports on a study that examines such a relationship between technology and problem solving, a qualitative study of six high school calculus students and the affect they associate with their graphing calculator use. It aims to answer questions like those motivated by the starting vignette. Specifically, in what ways does affect impact one’s graphing calculator use? And, in what ways does graphing calculator use impact one’s affect? I begin with a review of the relevant literature and theories

∗ Tel.: +1 919 513 2803; fax: +1 919 513 0505. E-mail address: allison [email protected] 0732-3123/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmathb.2011.02.002

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that provide the background for this study. Next, I describe the research methods, followed by a report of the findings. Finally, I conclude with a discussion of the implications the findings present for teacher education and future research. 2. Background Until recently most of the research on graphing calculator use in mathematics education has either been quantitative in nature focusing on student achievement and attitude or qualitative studies on the teaching and learning of a particular mathematical topic (Burrill et al., 2003; Ellington, 2003). However, there is a growing body of research on how students are adapting graphing calculator technology to their problem solving strategies (e.g. Ball & Stacey, 2004; Berry & Graham, 2005; Graham, Headlam, Honey, Sharp, & Smith, 2003; Harskamp, Suhre, & van Streun, 2000; Sheryn, 2006). Over the past decade researchers have been studying how students come to fully integrate graphing calculators as a tool for learning. Artigue (2002) and her colleagues have proposed the term instrumental genesis to describe this process of coming to understand the potentialities and constraints of an artifact such as the graphing calculator while at the same time developing mathematical knowledge. Most of the studies focused on understanding instrumental genesis have looked specifically at graphing calculators with Computer Algebra System (CAS) capabilities. They have found that using graphing calculators with CAS capabilities effectively is not easily learned and that students go through many phases before becoming proficient users (Artigue, 2002; Drijvers, 2000; Guin & Trouche, 1999). As we learn more about the effects that the adoption of graphing calculator technology seems to be having on the assessment and learning of particular topics it becomes apparent that we need to know more about what students are actually doing with their calculators when they work independently. Furthermore, it is apparent we need to learn more about the role of emotions, values and beliefs as they relate to graphing calculator use in problem solving. Few studies have looked specifically at how and why students use graphing calculators in particular ways (Burrill et al., 2003); two exceptions are studies by Doerr and Zangor (2000) and Goos, Galbraith, Renshaw, and Geiger (2003). These studies were both classroom based studies that aimed to understand the different roles that the graphing calculator takes on within a classroom community. Doerr and Zangor (2000) conducted an observational case study of two precalculus classes and their teacher. Within this case study they considered how the classroom as a community shaped the ways in which technology was used. It was determined that within the context of the class the graphing calculator was used by the students in five different modes: as a computational tool, transformational tool, data collection and analysis tool, visualizing tool, and checking tool. These five modes of tool use emerged from the interactions between the teacher and the students. While this study highlighted the ways that a teacher’s beliefs about the role of the graphing calculator can shape the ways that it is used in the context of the classroom, the methodology employed did not allow for examination of how the classroom modes of use translated to independent situations. Goos et al. (2003) conducted a longitudinal study of 5 secondary classrooms and how graphing calculators and their peripheral devices were used as a tool that was integral to the learning environment in the context of these classrooms. Unlike Doerr and Zangor who focused on the actual actions taken with the tool, they focused on the students’ and teachers’ relationship with the tool. They theorize that when technology like the graphing calculator is used in relation to teaching and learning interactions there are four roles it may take on: master, servant, partner, and extension of self. Each of the four categories is a metaphor for the interaction (or relationship) between the individual and the tool. These metaphors emphasize the social nature of technology use in a classroom setting. Each of these studies has added to the knowledge of the complex role that the graphing calculator plays in the mathematics that is produced and shared in the context of the classroom. However, there is scant research on students’ graphing calculator use that has attended to the role of affect (i.e. emotions, attitudes, values, beliefs) in students’ graphing calculator use. One aspect of affect that has been studied as it relates to graphing calculator use is attitude. Such studies most often focus on how graphing calculator use impacts one’s attitude toward mathematics. In her 2003 meta-analysis of calculator studies, Ellington identified 18 studies conducted between 1983 and 2002 that fell in this category. The studies all compared two groups of students taught by equivalent methods, one having access to calculator technology and one not. Each of the studies used Likert scale instruments for measuring attitude towards mathematics. Ellington grouped studies according to six attitudinal factors identified in the Minnesota Research and Evaluation Project (Sandman, 1980) including attitude toward mathematics, anxiety toward mathematics, self-concept in mathematics, motivation to increase mathematical knowledge, perception of mathematics teachers and value of mathematics in society with only the first (attitude toward mathematics) being measured by most studies. Ellington’s meta-analysis revealed that students who used calculators during instruction reported a better attitude toward mathematics than those who did not (weighted effect size, g = .32). This is an important finding, but it is just the tip of the iceberg. There are many affect related questions that remain unanswered. For example, why do students who have access to calculators have better attitudes towards mathematics? Some researchers have suggested that this phenomenon might be due to a feeling of confidence about engaging with mathematics when a tool is available (Dunham, 2000). However, if it is simply confidence, what is it about having the graphing calculator available that makes students more confident in mathematical situations? In the Educational Studies in Mathematics special edition on affect in mathematics education, Op’t Eynde and Hannula (2006) introduced the case of Frank which served as the unifying feature for all of the articles in the issue. Frank was a junior high school student who was participating in a project about the role of students’ beliefs and emotions during problem solving in the mathematics classroom. Frank was asked to solve a problem about a group of Kosovan refugees trying to go

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to a hospital in Albania through the mountains. Part of the problem asked the time travel by car from the village to the city where the hospital was located. Frank was video taped while solving this problem and shortly thereafter participated in a video-stimulated recall interview in which he watched himself working on the problem and was asked about how he felt and what he did during the problem solving session. Eynde and Hannula report that at one point in the session Frank took out his calculator. When he was asked about this Frank replied, “Actually, I did not really need the calculator there. I wasn’t thinking properly, and then I panic, and then I immediately want to go to my calculator, and then if I stop and think for a moment, I probably know again what I have to do” (p. 126). When he is further probed about his feelings regarding this action Frank says, “. . .I don’t know how to put it, you don’t feel well because you need to go to the calculator” (p. 127). Though this is a short episode, Frank provides further evidence that the decision to use technology tools like the graphing calculator is laden with emotion and deserves to be studied.

3. Theoretical framework This exploration of students’ graphing calculator use was guided by two theoretical constructs, affect as a representational system (DeBellis & Goldin, 2006) and instrumental genesis (Artigue, 2002). Each is described in the sections that follow.

3.1. Affect as a representational system The study of affect in mathematics education has its roots in the investigation of emotions and problem solving. Most early work in this field focused on ‘mathematics anxiety’ and ‘attitude toward mathematics’; there was little consensus of what made up the affect domain or how it was best studied (McLeod, 1992; Zan, Brown, Evans, & Hannula, 2006). More recently there has been a concerted effort to both define affect and develop frameworks to help coordinate discussion and research within the mathematics education community (Zan, Brown, Evans, & Hannula, 2006). Mandler’s (1989) cognitiveconstructivist model of the process of emotional experience has been the ground on which theoretical frameworks for the study of affect in mathematics education have been built. Based on Mandler’s theory McLeod (1992) offered up a framework in which the affective domain was defined as the internal system that includes emotions, attitudes and beliefs. McLeod suggested that these three constructs fall in line ranging in stability and intensity with emotions being the most intense and the least stable, beliefs being the least intense and most stable, and with attitudes falling in between. Later DeBellis and Goldin (1997) added a fourth dimension to McLeod’s framework for the affective system by including values as a part of the affective domain. They represented these four dimensions using a tetrahedral model rather than by a continuum of stability and intensity to show the interaction between them. They further operationalized this addition with the introduction of their framework for affect as a representational system (DeBellis & Goldin, 2006). The methodology in this study is grounded in DeBellis and Goldin’s research-based theoretical framework on affect as a representational system. By stating that this system is representational, they mean that it exchanges information with cognitive systems. DeBellis and Goldin describe two categories of affect, local affect and global affect. Local affect refers to changing states of feeling during problem solving. Global affect refers to more stable, longer-term affective constructs such as attitudes, values, and beliefs. Implicit in the framework is the conjecture that over time local affective experiences that are similar and powerful can come to influence the more stable constructs of global affect. For example, if a student has repeated experiences of frustration when trying to create and use a table on a graphing calculator to solve a problem, that student may start to have a negative attitude toward the table tool and even possibly develop the belief that the table is not a useful tool. In addition, an individual’s affect (both local and global) is also influenced by the affect of others, social and cultural conditions and external contextual factors. This is referred to as external affect. For example, a student who repeatedly hears from a teacher that the graphing calculator should only be used as a last resort when problem solving may possibly feel guilty when the decision is made to use one. Goldin (2000a) has pointed out that students use emotions to provide useful information, to facilitate monitoring and to evoke heuristic processes. He suggests that affect is not inessential, but critical to the structure of competencies that account for success or failure in problem solving. To better understand the role of local affect in problem solving Goldin (2000a) introduced the notion of affective pathways. Affective pathways are sequences of states of feeling that interact with cognitive representations in problem solving. An example of an affective pathway follows: In an (idealized) model, the initial feelings are of curiosity. If the problem has significant depth for the solver, a sense of puzzlement will follow, as it proves impossible to satisfy the curiosity quickly. Puzzlement does not in itself have unpleasant overtones – but bewilderment, the next state in the sequence, may. The latter can include disorientation, a sense of having “lost the thread of the argument” of being “at sea” in the problem. . .If independent problem solving continues, a lack of perceived progress may result in frustration, where the negative affect becomes more powerful and more intrusive. This is associated with the occurrence of an impasse. However, there is still the possibility that a new approach will move the solver back to the sequence of predominately positive affect. Encouragement can be followed by pleasure as the problem begins to yield, by elation as major insights occur, and by satisfaction with the sense of a problem well solved and with learning that has occurred. (p. 211)

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This idealized model illustrates how local affect might influence the heuristics employed by a problem solver. In the context of this paper the focus is on how the availability of a tool like the graphing calculator might further influence an affective pathway like the one described above. For example, if a student is facing feelings of bewilderment or disorientation it is possible that the introduction of a useful tool might invoke feelings that are of a more positive sequence. Meta-affect, or affect and/or cognition about affect, is the monitoring system that allows for people to recognize affective representations with cognitive representations and produce other affective representations (Goldin, 2002). For example, a student who has hit a road block in a problem is frustrated. However, if that student reflects on the feeling of frustration and remembers that frustration is temporary when one tries another approach and is successful, then the student might become curious about the problem and move forward. DeBellis and Goldin believe that meta-affect is one of the most important aspects of affect. It is through this monitoring system that problem solvers learn to create productive affective pathways for themselves. At this point, graphing calculator studies have addressed the more stable affective construct of attitude, but few, if any, have addressed graphing calculator use as it relates to local affect. It is through the study of local affect that we might move beyond classifying students’ attitudes and come to understand the influence that graphing calculator use has on problem solving experiences. The DeBellis and Goldin view of affect and problem solving is consistent with many frameworks that have been offered by the research community in that it seriously considers the interactions between affective and cognitive systems (e.g. Evans, Morgan, & Tsatsoroni, 2006; Malmivuori, 2006; Op’t Eynde & Hannula, 2006). While the view is consistent, it studies affect from a different perspective in that it was designed in the context of individual problem solving, not classroom interactions. As a result, though social and cultural conditions are included, it does not regard the interactions between these conditions and students’ emotions in the way that others do, by focusing on interactions in the classroom (e.g. Evans et al., 2006; Op’t Eynde, De Corte, & Verschaffel, 2006). Nevertheless, DeBellis and Goldin’s separation of local and global affect along with the importance placed on meta-affect and recognition of external affect make it an appropriate and powerful framework for this particular study. 3.2. The process of instrumental genesis The premise that students’ understandings are shaped by the tools that they use and by their relationship with those tools is consistent with socio-cultural theories of learning (e.g. Vygotsky, 1978). When studying how CAS tools are used in mathematics, specifically how they mediate student learning, Artigue (2002) and her colleagues found it useful to turn to Chevallard’s work in anthropology (e.g. Chevallard, 1992) to better understand the ways in which students develop a relationship with a tool that takes into account the context of classroom learning. When considering the graphing calculator it is necessary to point out that I consider the graphing calculator to be an artifact that is actually made up of many tools. For example, there are both tools for visualization and tools for computation. However, I believe that these tools are useless until they become “instruments” through a process that Artigue calls “instrumental genesis”. In her explanation of this theory and its relationship to graphing calculator use, Artigue explains that an instrument is “a mixed entity, part artifact, part cognitive schemes which make it an instrument. For a given individual, the artifact at the outset does not have an instrumental value” (p. 250). The process of instrumental genesis involves one coming to understand the potentialities and constraints of the tool while at the same time developing mathematical knowledge. In other words, “users shape the artifacts they use and the artifacts shape the users, and that yields instruments” (Artigue & Kilpatrick, 2008, p. 6). So, while a student may own a graphing calculator, that alone does not make it an instrument. It is possible that particular modes (or tools) on the graphing calculator become instruments to a student before others. For example, in the context of linear functions one student might have instrumented visualization tools of a graphing calculator and meaningfully use graphical representations to understand and engage in solving a linear function problem. That same student may not have developed the CAS capabilities as instruments with respect to linear functions and thus will not find them helpful in this context. It is important to note that the mathematical context matters, as in the process of instrumental genesis the tool and the mathematical knowledge are shaping each other. The same student just described above might have developed the CAS capabilities in a different mathematical context, say function limits. Kilpatrick (2009) noted that the theory of instrumental genesis appears to have considerable promise in research regarding the ways in which technology is and is not being used. In the context of this study, whether or not students have developed particular capabilities of the graphing calculator as instruments could have considerable impact an affective pathway when attempts are made to incorporate its use. 4. Methodology To build an understanding of how the availability of a graphing calculator impacts students’ local affect, or affective pathway, during independent problem solving qualitative methods were used. The choice of design methods was influenced by a series of pilot studies. Three lessons were learned from those experiences that influenced the design of this study. First, students with different math backgrounds appear to have different perceptions of the usefulness of the graphing calculator. Therefore the settings for this study were chosen carefully to increase the possibility of identifying unique individual graphing calculator stories. Second, students have many different purposes for actually using the graphing calculator. Third, task selection is very important to the development of meaningful conversations about graphing calculator use. The tasks selected

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Table 1 Study participants.

Aaron Enoch Maryanne Melissa Rudy Shemika

Gender

Teacher rated ability

School

G.C. type

M M F F M F

Average Strong Strong Weak Weak Strong

C A B C A A

TI-89 TI-83+ TI-83+ TI-89 TI-83+ TI-83+

for this study were chosen based on the responses students provided in pilot studies. These tasks along with the details of the methods are described in the following sections. 4.1. Setting and sample Advanced Placement (AP) Calculus classes were chosen as the focus of this study because the curriculum and expectation of calculator use is relatively consistent nationwide since it is set by The College Board. To ensure that the population of students was as diverse as possible, students from four high schools in the northeastern United States were surveyed and six students were chosen based on the results of the survey. These high schools were purposefully selected based on access, presence of an AP Calculus program that uses graphing calculator technology, and their ethnic and socio-economic status (SES) make-up. High school A is located in a low-income urban community. It serves approximately 2000 students in grades 9 through 12. High school B serves approximately 2800 students in grades 9 through 12 from both suburban and rural communities. High school C serves approximately 1100 students in grades 9 through 12 in an affluent suburban community. High school D serves approximately 1700 students in grades 9 through 12 in a middle class suburban community. All four of these schools have provided the AP Calculus students a graphing calculator to use at home and at school. High schools A, B, and D provided their students with a TI-83+, while high school C provided the TI-89 (which has Computer Algebra System, or CAS, capabilities). Students may have used the calculator provided by the school or possibly their own personal calculator. For the purposes of this study, the term graphing calculator refers to both calculators with and without CAS capabilities. Every student in every AP Calculus class at high schools A, B, C and D was asked to participate in the survey phase of this study (n = 111). The survey provided data on student demographics, mathematical achievement, frequency of graphing calculator use, modes of graphing calculator use, comfort with the graphing calculator, and reasons for graphing calculator use. In addition, the students’ teachers provided a rating of weak, average or strong relative math ability for each student. From this rich pool of data six students that were representative of the extremes in the survey responses while also being representative of the types of schools participating in the study were selected to participate in the main study. The students are introduced below as well as in Table 1. Aaron (all names have been changed) attends high school C. His teacher considers him to be an average calculus student compared to his peers. His mathematics grades throughout high school indicate that he has been very successful in his course work. Aaron says that he is confident in his calculus abilities, but does not like when he is expected to use a graphing calculator. Graphing calculators have been integrated into the curriculum of all of his high school math classes, but he admits that he has not taken the time to learn how to use them. Enoch attends high school A and is considered to be one of the strongest math students at his school. However, Enoch is not very confident in his calculus abilities and says that he relies heavily on the graphing calculator. He explained that his previous coursework did not prepare him for calculus and the graphing calculator helps him with skills that he should have perfected in those courses. Maryanne attends high school B. Her calculus teacher identified her as one of his strongest calculus students. She is the only junior in this high school of over 2500 students who is taking calculus. Maryanne’s parents are both scientists and have influenced her views on graphing calculator use significantly. She feels strongly that students should not rely on the graphing calculator. Maryanne explained that she rarely uses her graphing calculator because it is very important for her to be able to do everything without one if she is going to be successful in college. Melissa attends high school C and was identified by her teacher as one of the weaker calculus students at her school. Melissa is not confident in her calculus abilities and says that she relies heavily on her graphing calculator. It is important to point out that though she is weak compared to the calculus students at her school; she is not a weak student. This was evident by her AP exam score, which she shared with me through an informal conversation, and in her ability to conceptualize each of the four tasks in the study. Melissa uses her calculator to graph and check quite often, but does not use the CAS capabilities because she likes to do the manipulations on her own. Though she is not very confident in her abilities, she has shown that she is a capable calculus student. Rudy attends high school A and is one of only six students enrolled in calculus at his school. However, he is not a very strong math student. Rudy is very aware that his less than proficient algebra skills are making learning calculus difficult. He points out that his high school did not offer traditional algebra and geometry courses and he feels that the courses that were

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offered did not prepare him well for upper level mathematics. Rudy uses his graphing calculator often and in very creative ways. He says that he would not have been able to do any calculus without it. Shemika attends high school A. Her calculus teacher says that she is by far his strongest student. Shemika is very busy both in school and after school and feels that the graphing calculator is a necessary tool given her other commitments. Shemika is very knowledgeable about how the graphing calculator actually works, both its strengths and limitations. She uses it very precisely, which is possibly because of her deep understanding of the mathematics. 4.2. Data collection The opening vignette was constructed based on a classroom observation conducted during a visit to a calculus class at high school C. In this vignette there were many instances in which students appeared to be experiencing affect related to the graphing calculator that was influencing their mathematical engagement. However, simply observing students in the classroom does not provide sufficient information about the nature of their affect or its impact (if any) on their mathematical work. As a result, this study was designed to unveil the range of emotions the students related to graphing calculator use through a series of interviews. Each student participated in three interviews. The first was a semi-structured interview (Rubin & Rubin, 2005) followed immediately by a task-based interview (Davis, 1996; Goldin, 2000b). The third interview was a video stimulated response (video-SR) interview (Lyle, 2003). A semi-structured interview was chosen for the initial meeting to assure similar structure and content of the interviews with each participant, while still allowing for the additional follow-up questions to provide elaboration when needed. A task-based interview is one in which participants are posed a carefully selected task to solve and asked to explain their solution methods. Such interviews have been shown to be valuable in studies of students’ mathematical thinking (Davis, 1996). Often task-based interviews are supplemented with a think aloud protocol in which participants are asked to explain their thinking while they work. Video-SR is a procedure in which videotaped behavior is replayed to an individual to stimulate recall of their activity (Lyle, 2003). Lyle (2003) suggests that a video-SR methodology is appropriate for studies that “benefit from minimal intervention in the activity” (p. 862), such as those in which asking students to think aloud might be considered disruptive. In this study the semi-structured interview was utilized to collect background information. This interview provided initial insight into the students’ perceptions of graphing calculator use in mathematics in addition to the ways they actually use it when problem solving. In the subsequent task-based interview students were posed five tasks (described below) and provided the graphing calculator they regularly used (i.e. TI-83+ or TI-89). This interview was both audio and video taped. In addition, video of the students’ calculator screen was also collected. In order to address the need to ask questions about strategy and emotion without influencing the problem solving process a video-SR design was used in the third interview. In this case the purpose of the video-SR interview was to stimulate the recall of both cognitive and affective activity. Real-time side-by-side video clips, of both written and calculator work, from each task were used to prompt the students to reflect on the tasks, the role of the graphing calculator in their solution strategies, and the emotions they experienced during the session. The students were asked to attend to not only their problem solving actions, but also to their facial expressions and body language. The video-SR interviews took place the day after the semi-structured and task-based interviews. All interview data were transcribed verbatim. The task-based interview data were also transcribed; meaning student work was recreated using the video of the students’ working on paper along with video captured from their calculator screens. Participants were invited to review and edit the transcriptions to ensure that the information was accurate. This process, called member checking, added to the validity of the study (Creswell, 1998). No changes in the transcripts were requested by the participants. The tasks used in the task-based interviews were all designed so that they could be solved using mathematics that is typically taught before the second semester of AP Calculus AB, when the students were taking part in this study. However, they were neither necessarily familiar problem types, nor were they all “graphing calculator friendly.” The hope was that these tasks would challenge the students and require them to use problem solving methods that were not necessarily regularly used in their calculus courses. This work would then serve as a stimulus for a conversation about how the graphing calculator impacted (or not) their mathematical experience. The four tasks are described in detail and possible non-calculator solutions are presented in Table 2. 4.3. Analysis The interview data were analyzed as it relates to DeBellis and Goldin’s framework for affect as a representational system. The data were first coded for talk and actions related to modes of graphing calculator use (computation, use of graphs, use of tables, and use of statistical tools). Next the data were coded for referents to emotions and values, those of self and those perceived of others, with respect to graphing calculator use as they emerged from the data. For each student reported affective pathways were constructed for each task. By reported affective pathway, I mean the path of emotions that the student reports having felt with respect to the activity that has been recalled for a particular task. These pathways are referred to as reported because the researcher does not have access to what students were actually feeling in the moment, only what they reported that they remembered feeling. However, it must be noted that since the students were watching video of themselves working, their reports were not simply a recall of memories, but were actually stimulated by the video thus increasing the possibility that their responses were consistent with what they actually felt at the time. Once the coding

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Table 2 Description of the tasks. The task

Characteristics of the task

Possible solution

Task #1: Find a rational function that satisfies the given conditions: a. It has a vertical asymptote at x = 3 b. It has a horizontal asymptote at y = −2 Task #2: Find the maximum rate of change of the graph of y = − x3 + 3x2 + 9x − 27

The student must have a concept of what horizontal and vertical asymptotes are

Given that the solution is a rational function such that f(3) ∈ /  and lim f (x) = −2, one

Task #3: For what values of x is −2 < |k − x| < 5 Task #4: Give an example of a function for which |f(x)| = f(|x|)

x→±∞

possible solution is f(x) = − 2x/(x − 3) The student must first recognize that they are being asked to find maximum slope of the first derivative and then have a method for finding that solution

The student must make sense of the compound inequality and consider the influence of different values of k on the solution set for x The student must decode the equality and consider functions that would make the statement true

The task asks for the maximum rate of change of the derivative, y ’ = − 3x2 + 6x + 9. Using the first and second derivative test, one could determine that the maximum rate of change of y = − x3 + 3x2 + 9x − 27 is 12 and occurs at x = 1 The absolute value will always be greater than −2, so that leaves −5 < k − x < 5 to solve. The solution is k − 5 < x < k + 5 Possible solutions include f(x) = x2 and f(x) = cos x + 1

was complete, the data were put back together and reread looking for emerging themes regarding affect and student decision making as it relates to using their graphing calculators when solving problems. Throughout the analysis the interview data were constantly compared between and among cases to determine whether or not the emerging themes were consistent throughout. 5. Findings Though the tasks designed for this study were not chosen because they were easily solved using a graphing calculator, all six of the students used their graphing calculator at least once. A brief summary of each student’s graphing calculator use across the four tasks is presented in Table 3. The construction of reported affective pathways for each of the graphing calculator based solution methods revealed two consistencies: (1) the graphing calculator has the potential of helping students maintain productive affective pathways and (2) students’ perceptions of external affect sometimes influences decisions as to whether to employ graphing calculators. Since this was a qualitative study, my aim here is to richly describe these findings; as such I have chosen to share instances that best exemplify the results that were consistent across all six students. To that end, three of the six students will be highlighted in the following sections as exemplars of the types of affective pathways that were identified, two to illustrate number 1 above and one to illustrate number 2. First Rudy’s work and reported affective pathway on Task 3 and Aaron’s on Task 4 are presented to illustrate two very different ways in which the availability of a graphing calculator can influence one’s affective pathway. Next, Maryanne’s reported affective pathway for Task 2 is presented as an illustration of the potential influence that perceived external affect could have on one’s mathematical experience. 5.1. Maintaining productive affective pathways A comparison of the reported affective pathways constructed from the students’ work revealed that the availability of the graphing calculator, and subsequent decision to use it, was often instrumental for maintaining a productive affective pathway. As described above, a productive affective pathway is one that is not necessarily positive, but keeps the student engaged in the mathematics. All six of the students noted anecdotally that having the graphing calculator available during problem solving was important to them. They often shared examples of problems that they would not have tried and places they would have stopped if they did not have a graphing calculator available. In this section two examples will be presented, one in which a reported affective pathway remained productive and one in which it did not. During the task-based interview Rudy was presented with Task 3: For what values of x is −2 < |k − x| < 5. A description of his work on this problem follows: Rudy read this task for quite a while. He asked, “Can k be anything?” To which the researcher replied “it could be any real number.” Rudy then wrote down “7 − x = ” and picked up his calculator. He entered “y1 = 7 − x ”, graphed it, and quickly went to the table. He scrolled up and down the table of values between x = 2 and x = 9 and paused for a moment. Next he returned to the y = screen and changed the function from y1 = 7 − x to “y1 = 6 − x ” and went back to the table. On the table he scrolled between x = 1 and x = 8. Then he wrote on his paper, “Depending on k there are 6 numbers that make x greater than −2 and less than 5.” Rudy’s reported affective pathway for this task appears in Fig. 1. When asked how he felt when he first read the problem Rudy replied, “I was frustrated. At first I didn’t understand it. I was trying to figure out what they meant.” Once he picked up his graphing calculator, he actually sat up straighter in his seat. Rudy noted that he was much more comfortable at that point.

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Table 3 Summary of student task solutions. Task #1

Task #2

Task #3

Task #4

Aaron

No GC No solution

No GC Correct solution

No GC Incorrect solution

Enoch

No GC Incorrect solution

No GC Incorrect solution

Maryanne

GC Correct solution Solved all other tasks first, returned to this task and used trial and error to determine a numerator that resulted in a correct graph GC Correct solution Turned on GC before even reading the problem, sketched coordinate plane with asymptotes on paper then tested functions on GC until identifying one that met criteria No GC Incorrect solution

GC Correct solution Started working by hand, got stuck and turned to GC to view the graph of the function and use the ‘calc’ tools GC Correct solution Solved by hand, noted an error and turned to GC to graph the function and its derivative then used ‘calc’ tools to solve

GC Correct solution Used trial and error to test particular functions at single points on the home screen GC Correct solution

No GC Incorrect solution

No GC Incorrect solution

GC Correct solution Tried to solve by hand and did not remember how, then used CG to graph the derivative and used ‘calc’ tools to solve

GC Incorrect solution Rewrote the inequality without the absolute value, chose values for k and then used the table to determine how many integral solutions there were for this new inequality No GC Incorrect solution

Melissa

Rudy

Shemika

No GC No solution

GC Correct solution Immediately graphed derivative on GC and used ‘calc’ tools to solve

No GC Correct solution

Activity

SR-Associated Affect

Read task and thought about it for a while

Frustration

Substituted 7 for k and put into GC, scrolled up and down table

Curiosity and comfort

Substituted 6 for k, scrolled up and down table

Curiosity

Noticed a pattern and recorded solution

Contentment

Fig. 1. Rudy’s reported affective pathway.

GC Correct solution Substituted functions into each absolute value expression and compared graphs to identify functions for which the equality was true No GC Correct solution

GC Incorrect solution Chose functions to evaluate at both 2 and −2 on the home screen to determine whether or not made the given statement true No GC Correct solution

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Fig. 2. Aaron’s written work.

He explained that he often uses his graphing calculator to test ideas. As he started noticing a pattern, he became curious and tried a few more numbers to test the pattern further. When he wrote down his answer he said he “felt good” because he was able to write something down, especially since it was a problem that he did not know how to do when he first read it. (Note: Rudy was unaware that he had not solved the problem as it was stated.) When Rudy was asked how he would have done this problem if he had not had his graphing calculator he said “I wouldn’t have tried it.” For Rudy, his graphing calculator use on this task was central to his maintaining a productive affective pathway, even if it was productive toward a goal different than that stated in the given task. Due to the context of this study (i.e. a graphing calculator was available to students at all times) there are not any examples for which the absence of a graphing calculator resulted in an unproductive affective pathway. However, in the examples there were situations in which students had not developed the graphing calculator capabilities they wished to use as instruments (i.e. the process of instrumental genesis was incomplete), which ultimately resulted in an unproductive affective pathway. Aaron’s work on Task 4 (Give an example of a function for which |f(x)| = f(|x|)), presented below, is an excellent example. His work follows: After reading through the task Aaron immediately wrote down “x2 ” as a solution. The researcher then asked him if he could come up with a few more examples, which prompted him √ to write down “sin x”. Next he picked up his calculator, a TI-89, √ and entered “sin(/3) ” on the home screen and got “ − 3/2” as a solution. Then he entered “sin( − /3) ” and got “ − 3/2”. He stared at this for a long while. After a few moments he drew a 30–60–90 triangle on his paper and labeled the lengths of the sides (see Fig. 2 below). Once again, he paused. He then picked up his calculator and calculated both “cos(/3) ” and “cos( − /3) ” and got “1/2 ” as an answer both times. He then wrote down “sin x” and “cos x” as solutions. ˆ and “(−3)3” ˆ on his calculator and wrote “x3 ” as a solution as well. Finally, he calculated both “(3)3” Aaron’s reported affective pathway for this task appears in Fig. 3. When asked about how he felt when he first read the problem Aaron replied that he was “fine” because he immediately knew a function that would satisfy the conditions stated. However, once he was asked if he could find any other solutions he reported feeling both “curious” and “a little nervous”. He was curious to see if he could find other solutions, but also nervous because he was being put on the spot to do so. Aaron said that he figured either the sine or cosine function should work, so he decided to use his graphing calculator to test them out. However, when he picked up his graphing calculator he could not figure out how to input an absolute value sign to which he said he was “kind of annoyed”. He went on to explain, “It’s frustrating when I can’t figure out how to use my calculator to help me.” When he couldn’t figure out how to use his graphing calculator to support his work he instead drew a 30–60–90 triangle because “I knew it has something to do with trigonometry and it might be helpful. . .but it wasn’t.” Finally, Aaron gave up on the sine and cosine functions and tested two values for the function x3 (without using the absolute value function) and wrote that down as a solution. He put his pencil down and said, “I give up.” As he watched himself working during the video-SR interview he kept repeating that “I was so frustrated that I couldn’t make my calculator do what I needed” and finally that he was “embarrassed to have to give up.” Though Aaron had a graphing calculator available to him, he did not know how to use it in such a way that it could be of help to him. In the context of this problem, he clearly had not developed the instruments he needed to make the graphing calculator useful. Though there is no evidence to support such a claim, it is plausible that if Aaron had developed the absolute value capability as an instrument with respect to absolute value functions this episode might have had a very different outcome. Though these are only two examples they serve to illustrate the possible impact that technology tools, like the graphing calculator, can have on a student’s affective pathway and ultimate engagement in problem solving. In fact, all six of the students in this study reported multiple situations in which having a graphing calculator available provided them with a means to maintain a productive affective pathway, including Aaron who reported having had many instances of contexts in which graphing calculator use did result in productive affective pathways. The only exceptions were those in which graphing calculator capabilities being used were not developed as instruments with respect to the task that was posed, like Aaron’s work on the absolute value task above.

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Activity

SR-Associated Affect

Read task and immediately wrote down a solution

Comfort

Prompted to find more solutions

Curious and nervous

Tried to use abs function on GC but did not know how

Discouraged

Drew 30-60-90 triangle

Helpless and annoyed

Tested function for single value on GC

Uncomfortable

Said, “I give up.”

Embarrassed

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Fig. 3. Aaron’s reported affective pathway.

5.2. Perceptions of external affect It is evident in the examples presented above that both Rudy and Aaron’s local affect (i.e. affective pathway) while solving their respective tasks was impacted by the availability (or lack there of) of a technology instrument. However, the availability and use of the graphing calculator was not the only thing that impacted students’ local affect during problem solving, their local affect was also affected by their perceptions of what others might think of their problem solving activities. For example, as the students took part in the video-SR interviews they reported emotions related to their decisions regarding the graphing calculator that they tied to what they perceived their parents, teachers, and even their future professors might think of those decisions. Maryanne’s work on Task 2 (Find the maximum rate of change of the graph of y = −x3 + 3x2 + 9x − 2) follows as an example of the impact of external affect, in the sense of DeBellis and Goldin, on a student’s reported affective pathway. Maryanne began this task by working on paper. She first wrote down “ y ’ = 27 +” and then crossed it out (see Fig. 4). Next she wrote the first derivative, set it equal to zero, and factored it (forgetting to square the x). Directly below she sketched a number line on which she denoted “x = −1 and x =3” and the sign of all values on either side of these points. She looked at this for a moment and then crossed out the number line. Next she wrote down the second derivative and a second number line denoting “x = −1”. She then evaluated the first derivative at −1, looked at it for awhile, and left it and went to work on another task. When Maryanne returned to the task she immediately crossed out her previous work. She re-read the problem, and above the phrase ‘rate of change’ she wrote “slope.” Next, she wrote the first and second derivative and sketched another number line with x = −1 denoted on it. Then she picked up her graphing calculator, a TI-83+, and graphed the original function, y1 = − x3 + 3x2 + 9x − 27. After changing the window using the zoom standard command she looked at the graph for a moment. Next she returned to the y = screen and inserted “nderiv” in front of the original function she had entered, changed her mind and instead entered “nderiv(y1, x, x)” in y2 which commanded the calculator to sketch the graph of the original function, y1, and the first derivative of y1 on the same screen. After looking at the graph of both functions together, she zoomed out so that she could see the functions in a larger window. She followed this inspection of the graph with the use of the max tool on the calculator to find the maximum of y2, the derivative. The calculator determined the maximum to be 12 at x = 1. Maryanne considered this for awhile, returned to her written work where she crossed out the number line she had drawn with x = −1. She wrote down her solution, “maximum slope at x = 1 is 12.” Finally, she finished by sketching a number line one more time with x = 1 denoted.

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Fig. 4. Maryanne’s written work.

Maryanne’s reported affective pathway for this task appears in Fig. 5 below. As Maryanne watched herself look at the graph she created on her graphing calculator she recalled being very comfortable with the problem until she evaluated the first derivative at −1 and found it equal to 0. She recalled that at that point she knew she had made a mistake, but was unsure what it was. “I was really confused.” As she watched herself pick up the graphing calculator and construct the graphs of both

Activity

SR-Associated Affect

Read task, immediately did 1st derivative and zeros on paper

Comfort

2nd derivative and zero (incorrect) on paper

Evaluated 1st derivative at -1 and found equal to 0

Confused

Graphed functions on GC

Comfort and guilt

Recognized error

Relief

Found and recorded solution

Happiness

Fig. 5. Maryanne’s reported affective pathway.

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the first and second derivative she groaned. When asked about this reaction she replied that she “shouldn’t have to rely on the graphing calculator. . .ever.” However, at the same time she admitted that she felt relief when she recognized her error, “Whew, I knew that wasn’t right (smiles), but I knew I knew what I was doing.” As a matter of fact she pointed out exactly when she recognized what her error was. She said, “Well, I knew that the slope is clearly the steepest in this area so I could tell that to begin with. . . I found the maximum of the slope just using the maximum function. Then I looked back and it said it was at one and I was like that’s where I made my simple algebra mistake!” Though the graphing calculator obviously impacted Maryanne’s ability to maintain a productive affective pathway, her response that she should never have to rely on the graphing calculator suggests that she associated a certain amount of guilt for choosing to use it. When asked to provide further explanation Maryanne clarified that these feelings of guilt were prompted by what she perceived her mother would think. She said, “My mom doesn’t, really does not like teaching to the calculator. She knows my teachers haven’t done that, but she talks about how she doesn’t like seeing kids using calculators at all.” Thus Maryanne’s affective pathway was not only impacted by her graphing calculator use, but also by what she perceives others, like her mother, might think of that use. Other students in the study also reported their graphing calculator use was sometimes influenced by their perceptions of external affect. For example, as Rudy reflected on his work on the absolute value inequality task presented above, in addition to being content with his solution, he mentioned that although he could not have solved the problem without the graphing calculator he did not feel good about having to use it. This was because he did not think his teacher would have approved of the way that he used it. Enoch mentioned that he felt good when he used his graphing calculator because he was under the impression that it was expected that he would be able to use it when he got to college, and “I’ll look better to my professors next year if I am good at using it.” It appears that for these students when technology tools are involved, perceived external affect impacts their affective pathways during problem solving and sometimes even their decisions regarding whether or not to use the tool at all. 6. Conclusion, limitations, and continuing research Goldin (2000a) has argued that it is important to understand students’ affective pathways when problem solving so that we can figure out how to help students on unproductive affective pathways deal with frustration and impasse and turn those into more productive emotions like curiosity, bewilderment, motivation and maybe even elation. The results of this study suggest that one way to help students maintain productive affective pathways is by providing an instrument with which they can act in response to feelings of frustration or impasse. It is notable that Rudy indicated that he most likely would have given up on the absolute value inequality task if he had not had his graphing calculator. Using the graphing calculator changed the situation from one of defeat and frustration to one in which he was curious because he had a tool that he could use to easily explore. It seems as if one of the most important roles of graphing calculator use, no matter what actions are being taken, is to help maintain a productive affective pathway that supports mathematical success. While students look to their graphing calculators to help maintain a productive affective pathway, Aaron’s experience with the absolute value task is an important one. We see in this example, that having a tool available is not enough. If the particular graphing calculator capability that the student wishes to employ has not been developed as an instrument with respect to the concepts at hand, it is useless and can even have adverse affects on one’s affective pathway, possibly resulting in feelings of defeat as we saw with Aaron. Researchers (e.g. Artigue, 2002; Guin & Trouche, 1999) have documented the importance and complexity of mathematics instruction that aims toward instrumentation of graphing calculator and CAS tools. Aaron’s experience here is further evidence of the importance of such work. We cannot expect that simply providing tools will necessarily result in the more frequent occurrence of productive affective pathways; instead we need to think carefully about how those tools are incorporated into the classroom so that students are supported in the instrumentation process. Through the use of reported affective pathways it was possible to examine the role of meta-affect with respect to graphing calculator use. The data show that these students did employ meta-affect and that their meta-affect was greatly influenced by perceived external affect of parents and teachers. The influence of external affect on graphing calculator use is not really surprising given the literature on the promotion of technology in the classroom. Previous work has considered how promotion by teachers might impact student technology use, but has not considered other external influences on its use (e.g. parents, future teachers) or even its impact on the emotional experience of problem solving. The students in this study were influenced not only by their current and past teachers, but also by parents and future teachers. This suggests that this is an area in need of further study. As Goldin (2002) points out, “Powerful affective representation that fosters mathematical success inheres not so much in the surface-level affect as it does in the meta-affect (p. 63).” As we think about meta-affect and its potential role in problem solving, it is necessary to understand the constructs that might possibly be impacting it. Admittedly this study has major limitations. It was a very small exploratory study. With only six students, I can only speculate that the consistencies found in their experiences might be true for a larger group. This can only be determined with further research. However, the consistency seen here does suggest that further research should be undertaken. Additionally, it is important to recognize the possible limitations of a video-SR methodology for unveiling in-the-moment emotions. Some might argue that these emotions may not actually represent what the student felt at the origin of the mathematical experience. However, they do represent what the student recalled feeling at the time. Therefore, at the very least this recall reveals the emotional memories that the student is taking with them from the mathematical experience. Finally, given the

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role that perceived external affect ended up playing for these students in their graphing calculator use it is important to note that this study was not done in the context of the classroom. As such, the only information regarding the values and beliefs related to graphing calculator use that was available was from the students’ perspective. The purpose of this study was to look at independent graphing calculator use through the lens of affect as a representational system. It was found that using the graphing calculator helped students maintain productive affective pathways for problem solving as long as they were using graphing calculator capabilities that they had developed as instruments with respect to the mathematical task they were working on. Furthermore, the results of this study indicate that perceived external affective representations of others (e.g. teachers past, present and future; parents) does influence meta-affect and has the potential to either help maintain or interrupt a productive affective pathway. In light of this, future studies should be designed to understand the complexity of how graphing calculators are promoted by external forces (i.e. teachers and parents), how students interpret such promotion and finally how this promotion influences their independent use. Additionally, further studies like the one described are needed to determine if the nature of the reported affective pathways here are common, or if there are other ways in which the availability of graphing calculators might influence one. 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