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For this paper, we detail the ninth day in which the students participated in an activity using The Geometer’s Sketchpad where they had to interpret their understandings of sequence convergence on premade manipulate-able dynamic visualizations of sequences. We hypothesize that by using these dynamic visualizations, the definition and their resolutions to problems were reinforced by strengthening their connections between their definition and these visual representations.

Keywords: Limit, Definition, Guided Reinvention, Sequences, Dynamic Visualizations Introduction and Research Questions A consensus of research on student understanding of limits has revealed great difficulty in reasoning coherently about formal definitions (Artigue, 2000; Bezuidenhout, 2001; Cornu, 1991;

Tall, 1992; Williams, 1991). Recently, some studies have begun to outline how students come to understand formal limit definitions (Cory & Garofalo, 2011; Cottrill et al., 1996; Roh, 2010;

Oehrtman et al., 2011; Swinyard, 2011). But even after seemingly successful teaching experiments where students articulate understandings consistent with formal theory, Martin et al.

(2012) point out that students can still struggle in recalling their formal limit definitions after short periods of time. Fortunately, dynamic visualizations used by Cory & Garofalo (2011) seemed to effectively increase retention of formal limit definitions by allowing students to manipulate key elements of the definitions within the constraints of relevant relationships to strengthen their connections and their various representations. We recruited a pair of students who had just completed a Calculus II course covering sequences for a teaching experiment to reinvent the formal definition for the limit of a sequence. Following their construction of a formal limit definition, the students participated in activities using a computer-generated dynamic visualization designed to reinforce relationships in the students’ definition. Six months after the teaching experiment, the participants will be asked to reconstruct their formal definition.

This study attempts to address the following research questions: (a) How did the dynamic visualizations reinforce the students’ prior reinvention activities? (b) How did the dynamic visualizations play a part in their reconstruction of their formal definition six months later?

15TH Annual Conference on Research in Undergraduate Mathematics Education 385 Theoretical Perspective and Methods To investigate our research questions, we adopted a developmental research design, described by Gravemeijer (1998) “to design instructional activities that (a) link up with the informal situated knowledge of the students, and (b) enable them to develop more sophisticated, abstract, formal knowledge, while (c) complying with the basic principle of intellectual autonomy” (p. 279). Guided reinvention, “a process by which students formalize their informal understandings and intuitions,” supported the task design (Gravemeijer et al., 2000, p. 237).

Over a month’s time, the authors conducted a teaching experiment at a small, southwest university with a pair of students, selected based on their experience with sequences but lack of experience with formal limit definitions. The central objective of the teaching experiment (comprised of nine, 120-minute sessions) was for the students to generate a rigorous definition of sequence convergence. The instructional activities, adapted from Oehrtman et al. (2011), engaged the students in an iterative refinement process involving definition creation, definition evaluation against examples and non-examples, conflict acknowledgement of identified problems with the current definition, discussion of potential solutions, and the creation of a modified definition, thus restarting a new iteration. Oehrtman et al. (2011) noted that during the refinement process, the problems identified by the students were the most meaningful and supported the formation of ideas that remained stable through multiple iterations.

During Day 1, Joann and David (pseudonyms) produced and subsequently unpacked details of convergent sequences graphically. By Day 2, they had produced nine graphs of what they viewed as qualitatively different examples of sequences converging to 5 and nine graphs of sequences not converging to 5. During Day 2, the facilitator prompted the students to create a definition for sequences convergence by completing the statement, “A sequence converges to 5 provided….” This was continually qualified by the facilitator as “construct a statement that will keep all of your examples in and keep all of your non-examples out.” Days 2 through 8 consisted of the students engaging in the iterative refinement process and unpacking their intended meanings for individual elements within their evolving definition. By the beginning of the 9th session, Joann and David had produced a definition that they felt correctly captured the meaning of sequence convergence (see Figure 1). Probes by the facilitator revealed that their understanding of this definition was consistent with formal theory.

Figure 1. The participants’ final definition.

During Session 9, the students used The Geometer’s Sketchpad (Jackiw, 2002) to manipulate dynamic graphs of sequences in hopes of improving upon the lack of retention observed by Martin et al. (2012). This instructional activity’s design was adapted from Cory and Garofalo (2011) who found that students strengthened their understanding of sequence convergence by engaging dynamically with a consistent visual representation of the formal definition and by 386 15TH Annual Conference on Research in Undergraduate Mathematics Education reflecting on their evolving conceptions as they compared their interactions with the visual representation to the written formal definition. Cory and Garofalo (2011) put forth the possibility that as result, their participants demonstrated a more coherent, enduring understanding of limit ideas eight months later. Their findings are consistent with the principle of manipulation (Plass et al., 2009) which suggests that learning from visualizations is improved when learners manipulate the content of the visualization and with Mayer’s (2009) Theory of Multimedia Learning which holds that a crucial step in learning involves integrating one’s pictorial model of a concept with one’s verbal model. For the present study, we adapted Cory and Garofalo’s (2011) dynamic sketches to coincide with the language and symbols the students used to create their definition and to include several graphs the students generated during Day 1. Six months later, the students will repeat the reinvention process so that the dynamic visualization’s impact, if any, on their re-development of the definition can be investigated.

Emerging Results Leading up to producing their final definition, Joann and David engaged in many challenges that provided opportunities for learning through the thoughtful resolution of identified problems during the creation of their sequence convergence definition. On the last day of the teaching experiment, the students used Sketchpad’s dynamic capabilities to continue to explore their definition and how it applied to many of the sequences they generated earlier. In many ways, the sketches appeared to reinforce the students’ ideas about sequence convergence by giving them opportunities to manipulate a coherent visual representation of their definition. We describe three challenges encountered by Joann and David, how they resolved these problems, and how Sketchpad’s dynamic capabilities seemed to reinforce the resolutions they had made.

One challenge Joann and David faced was to understand the importance of the universal quantifier on the “barrier b” (corresponding to ε in standard formulations). This concept appeared on Day 2 when Joann first mentioned the idea of “breaking a barrier” while investigating the graph of a monotonically increasing sequence converging to 4.9 rather than 5.

She explained, “If [the sequence] was going to 5, then it would cross the 4.9 and it would cross the 4.99 and…the 4.999….You have to…break that barrier of 4.9.” After some discussion, the participants wrote the definition in Figure 2. As the students were invited to compare various definitions they had developed and to make their ideas more concise and precise, they created the phrase, “for all decreasing decimal barriers”, and ultimately settled on the words, “for any barrier, b.” Later, as they manipulated the b-value on the Sketchpad sketches to show any value they desired, the concept of the universal quantifier was reinforced (see Figure 3).

Figure 2. Participants’ definition involving decimal “barriers” 15TH Annual Conference on Research in Undergraduate Mathematics Education 387 Figure 3.

An interactive sketch of the formal definition of the limit of a sequence.

During Days 3 to 5, one of the participants’ challenges was recognizing that using a “peak” on a graph was not an effective way to establish an error-bound. For them, a peak became a local extremum as seen in an oscillating sequence or could be any particular point in a monotonically decreasing sequence since all subsequent points are “below” that particular point. After much discussion, they developed the definition in Figure 4. Immediately the students had difficulty clearly defining a peak and struggled in applying their “peaks” definition graphically, eventually realizing that their definition did not exclude some of their non-examples. In addition, they interpreted this “peak” idea as setting the error bound for subsequent points, a conception in which an error bound is dependent upon a peak’s height. Following this, the facilitator guided them to think of the error-bound as an independent variable which could be placed anywhere on the y-axis and to compare their “peaks” definition to their earlier definition. The removal of their “peaks” idea and eventual acceptance of error bounds were reinforced on the last day of the teaching experiment as both participants took turns manipulating values for the b-band on the sketches (see Figure 3) which no longer needed to correspond with points on the graph.

Figure 4. Participants’ definition involving peaks.

388 15TH Annual Conference on Research in Undergraduate Mathematics Education As the teaching experiment progressed, the participants also worked to resolve problems leading to their acceptance that all terms past some point must fall within a barrier and that this point depends on the barrier. The “past some point” idea had already appeared in their “peaks” definition (see Figure 4), and after returning to their “decreasing decimal barriers” definition and attending to their convergent graph with early random behavior, they immediately reincorporated the “past some point” idea in a new “decreasing decimal barriers” definition. They eventually chose “s” to identify the point after which all terms must fall within a specified barrier. Finally, after exploring the relationship between s and b using various graphs, they revised their definition so that s depended on the barrier (see Figure 5). These resolutions were repeatedly reinforced on the sketches, as one participant chose a b-value for which the other participant chose a “good” s by sliding the s-line along the graph until all dots beyond s fell inside the horizontal lines set by b (see Figure 3). Each time they carried out these manipulations, the participants were guided to explain why their s “worked,” thus giving them opportunity to connect the visual representation to their verbal model and the written definition.

Figure 5. Participants’ definition involving the dependence of s on b.

Discussion and Questions As in Oehrtman et al. (2011), Joann and David wrestled with the problem of rigorously articulating their ideas as they focused on relevant quantities and their relationships. The universal quantification of the barriers, the move away from terms determining the value of barriers, and the cognitive shift to focus on s as a function of b were all seen as viable solutions to problems. On the teaching experiment’s last day, we gave the students an occasion to strengthen connections between their definition and their visual representations by using interactive dynamic visualizations. Our remaining questions include: How might we better isolate the dynamic visualizations’ effects? How might the dynamic visualizations be incorporated into the reinvention itself? How could the dynamic visualizations be modified to support students in using their definitions to address genuine mathematical problems?

References Artigue, M. (2000). Teaching and learning calculus: What can be learned from education research and curricular change in France? In E. Dubinsky, A. Schoenfeld, & J. J. Kaput (Eds.), Research in Collegiate Mathematics Education IV (pp. 1-15).

Providence, RI:

American Mathematical Society.

Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students.

International Journal of Mathematical Education in Science & Technology, 32(4), 487-500.

Cornu, B. (1991). Limits. In D. O. Tall (Ed.), Advanced Mathematical Thinking (pp. 153-166).

Dordrecht, The Netherlands: Kluwer Academic Publishers.

15TH Annual Conference on Research in Undergraduate Mathematics Education 389 Cory, B., & Garofalo, J. (2011). Using dynamic sketches to enhance preservice secondary mathematics teachers’ understanding of limits of sequences. Journal for Research in Mathematics Education, 42(1), 68-100.Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinate process schema. Journal of Mathematical Behavior, 15, 167-192.

Gravemeijer, K. (1998). Developmental research as a research method. In J. Kilpatrick & A.

Sierpinska (Eds.), Mathematics Education as a Research Domain: A Search for Identity (ICMI Study Publication) (Book 2, pp. 277-297). Dordrecht, The Netherlands: Kluwer.

Gravemeijer, K., Cobb, P., Bowers, J., and Whitenack, J. (2000). Symbolizing, modeling and instructional design. In P. Cobb, E. Yackel and K. McClain (Eds.), Symbolizing and Communicating in Mathematics Classrooms (pp. 225–273). Mahwah, NJ: Erlbaum.

Jackiw, N. (2002). The Geometer’s Sketchpad (Version 4.03). [Computer software]. Berkeley, CA: Key Curriculum Press.

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