Researcher Pantozzi Prompts Students in Open-Ended Exploration of the Fundamental Theorem of Calculus
Creators: name: Magno, Sarah
Publisher: name: No Publisher Supplied
Year: 2024

description: This analytic examines researcher Ralph Pantozzi’s questioning style when posing open-ended tasks to students. This analysis focuses on three female students: Romina, Magda, and Angela, who were former AP Calculus students of Researcher Pantozzi’s and at the time of the observed sessions were juniors in college, over the course of two sessions. The first session took place on June 25, 2003 and the second session occurred on July 24, 2003. Both sessions showcased evolving thoughts and explanations of the Fundamental Theorem of Calculus (FTC). Researcher Pantozzi poses the task, “Why does the Fundamental Theorem of Calculus work?” in the first session. After analyzing the participants’ contributions, he poses a “more directive” task in the second session: “Why is the function g the antiderivative of the function f in the symbolic statement of the Fundamental Theorem of Calculus?” (Pantozzi, 2009, p. 232). The students were encouraged to use any of a wide variety of resources that Researcher Pantozzi made available, including, but not limited to: calculus textbooks, calculators, paper, and the tests they had taken in Researcher Pantozzi’s AP Calculus class several years prior.Throughout the events, Researcher Pantozzi joins the group to probe the students’ thinking. He uses questioning to elicit summary statements, such as asking for a short phrase for a “bumper sticker” description of the FTC. He challenges their viewpoints by providing counterexamples to elicit higher-order thinking. He encourages the group to explain the FTC to him while he role-plays a hypothetical student. He uses technology, such as Geometer’s Sketchpad, to provide visual representations of the students’ notions, which prompts discussion. His apparent goal is to strengthen the participants’ arguments by quantifying their language and elaborating on questionable terms that they used to refer to concepts.Event 1- Romina, Magda, and Angela are deciding how to phrase their short “bumper sticker description” of the FTC, in response to Researcher Pantozzi’s task and their progress during the closing minutes of the first session. The students generally agreed on the description, after Romina gave a slight critique to Magda’s choice of language. Event 2- During the second session, the students are pondering the meaning of the antiderivative. Magda and Researcher Pantozzi engage in dialogue about the open-endedness of the task, leading Magda to discuss with her peers about why she thinks the Power Rule for integration involves increasing the power by one. Researcher Pantozzi interjects to provide counterexamples at this point, which serve as cognitive obstacles for the students as they consider other cases. Event 3- Researcher Pantozzi prompts discussion as the group expands upon that idea to visually explain why the power is increased by one. This is the first instance where a student, Romina, references the Accumulator Function: “You’re taking the area, so it goes to the next power because it’s like the cumulative function… that’s why it has to be bigger.” Event 4- Researcher Pantozzi uses Geometer’s Sketchpad to provide an estimate for the definite integral. As the animation is drawn on screen, Angela points out the case of negative area while Romina notes that the “integral increases and then decreases.” Event 5- Romina asks the group for clarification on language used by Researcher Pantozzi referring to antiderivatives. This leads to a discussion as to whether there is a difference between antiderivatives and integrals. Event 6- Researcher Pantozzi restates assumptions that can be made about the task, allowing Magda to expand upon her explanation of the “+C” in the antiderivative. Romina and Angela agree with this notion and discuss a hypothetical case where C equals -2. Event 7- Once the definition of antiderivative is clarified and accepted among the students, Researcher Pantozzi engages in a discussion with Romina about its meaning, in which she states that it is “the opposite of taking the derivative.” This prompts an investigation to uncover why these two seemingly unrelated operations are inverses of one another. Event 8- The students make the connection between derivatives and antiderivatives clearer by linking them in a discussion about slope and area. They discuss rates at which areas are accumulated on another Geometer’s Sketchpad illustration that Researcher Pantozzi shows them. Event 9- Researcher Pantozzi listens to Romina’s summarized statements about their progress, questioning her language when it is unclear. He elicits conversation from Magda and Angela who add their thoughts about the underlying construction process that dictates the relationship between derivatives and antiderivatives.ConclusionThe design of open-ended questions and eliciting students’ contributions can be seen in previous tasks and analytics. In Event 7 of Silber’s (2013) analytic, Robert B. Davis leads a discussion about “ethics and morality.” He elicits a conversation about building a mathematical community and doesn’t reveal his ideas about the problem despite repeated requests from the students. As he explained, “I’m waiting to see if anyone else knows [the secret], maybe I do, maybe I don’t!” When Romina, Angela, and Magda probe Researcher Pantozzi for further clarification on his questions, he directs them to the task at hand and facilitates the discussion in an open-ended manner. This analytic describes Researcher Pantozzi’s responses to the group’s behaviors as he supports a curious and motivated group of students who want to collaborate. Researcher Pantozzi removes himself from the dialogue early on, in order to discourage the notion that he is the holder of the content knowledge. Researcher Pantozzi wants his students to “gain ownership of a mathematical idea [which] involves a process by which the learner takes responsibility for knowing” (Maher et al., 2011, p. 57). Researcher Pantozzi elicits discussion from the three students in order to get them to state terms and concepts in their own voices, making the FTC more meaningful for all of them.When working in the collaborative mathematical space that Researcher Pantozzi nurtured, there were times when language usage was unclear to certain members of the group. As a natural consequence of group discussion and the journey of mathematical learners, occasionally Researcher Pantozzi and/or one of the students needed to step in to clarify another’s language. Just as Romina worked with Brian, Jeff, Ankur, and the other students on Ankur’s Challenge: “As Romina tried to communicate her ideas so that Jeff would follow, she revised her representations” (Maher &amp; Muter, 2011, p. 94).The way Researcher Pantozzi facilitated the second session enabled him to prompt the students’ discovery regarding nuances of the FTC. As he noted in his dissertation (Pantozzi, 2009, p. 232), he altered the task design in the second session to be “more directive,” in order to elicit contributions about the FTC that were not discussed in the first session due to its broader design. As discussed in a personal interview with my cooperating teacher, Dr. Jennifer Corboy, who has been teaching calculus for more than twenty years, “Students can really lose important details of the concept if they are not fully exploring that. So for instance, with the FTC, sometimes you get positive and negative areas, things that can cancel out, or if you accumulate backwards it changes the sign, and things like that” (Corboy, 2024). Researcher Pantozzi directed the group towards carefully crafted Geometer’s Sketchpad illustrations to spark a conversation about positive and negative areas, for example, in Event 4. It was important that this discussion was elicited before moving on to other nuances of the FTC in following events. It appears that the sense of “mathematical ownership” and opportunities for dialogue with others helped Romina gain a deeper, conceptual understanding of the FTC. When she was interviewed six years later, after earning her MBA, as part of the longitudinal study, Romina confidently stated, “But I probably could explain to them The Fundamental Theorem of Calculus and kind of explain to them how all these things happened and worked and visually how everything was represented” (Steffero, 2009). This instance of problem solving, prompted by Researcher Pantozzi’s open-ended tasks, gave Romina “ownership” over the underlying ideas of the FTC. Her experiences working with Magda and Angela made her feel confident in expressing this idea to others.Overall, Researcher Pantozzi’s design of his tasks led to “durability of understanding” among the students (Uptegrove, 2011, p. 140). After Romina and some other students from the study, such as Mike and Ankur, worked on the Pascal’s Identity exploration in terms of the Pizza Problems, it was determined that “all… were able when prompted to write the formula, and all… were able to provide a cogent explanation of the addition rule” (Uptegrove, 2011, p. 140). As Romina explained in her 2009 interview, one of the only theorems she recalled by name was the FTC. The task design of the FTC group sessions that Researcher Pantozzi facilitated solidified her knowledge of the subject. Reflecting on her experiences in Researcher Pantozzi’s AP Calculus class, along with the two sessions conducted nearly three years later, served to strengthen her conceptual understanding of the topic.Definitions and TermsThe Fundamental Theorem of Calculus (FTC)If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b], then the integral of f(x) from interval a to b is F(b) – F(a). “Informally, the theorem states that differentiation and (definite) integration are inverse operations, in the same sense that division and multiplication are inverse operations… The Fundamental Theorem of Calculus states that the limit processes (used to define the derivative and definite integral) preserve this inverse relationship” (Larson et al., 1998, p. 274).Problem TasksTask in Session 1 (and Event 1): “Why does the Fundamental Theorem of Calculus work?”Task in Session 2 (and Events 2 – 9): “Why is the function g the antiderivative of the function f in the symbolic statement of the Fundamental Theorem of Calculus?”Video Clip ReferencesB45, The fundamental theorem of calculus, Session 1 of 2: Discussing the meaning of the theorem (student view), College, June 25, 2003, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3HQ3X26B47, The fundamental theorem of calculus, Session 2 of 2: Discussing why the antiderivative can be used to calculate the area under a graph (student view), College, July 24, 2003, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3HM56MDReferencesCorboy, J. (2024, March 16). Personal Communication [Zoom Interview].Larson, R., Hostetler, R., &amp; Edwards, B. (1998). Calculus of a Single Variable (6th ed.). Boston, MA: Houghton Mifflin.Maher, C. A. &amp; Muter, E. M. (2011). Responding to Ankur’s Challenge: Co-Construction of Argument Leading to Proof. In Maher, C. A., Powell, A. B., &amp; Uptegrove, E. B. (Eds.). Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (Vol. 47). (pp.89-96). Springer, Dordrecht.Maher, C. A., Sran, M. K., &amp; Yankelewitz, D. (2011). Building an Inductive Argument. In Maher, C. A., Powell, A. B., &amp; Uptegrove, E. B. (Eds.). Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (Vol. 47). (pp.45-58). Springer, Dordrecht.Pantozzi, R. (2009). Making Sense of the Fundamental Theorem of Calculus (Doctoral dissertation, Graduate School – New Brunswick, Rutgers, The State University of New Jersey). Available from ProQuest Dissertations. UMI Number: 3373679Silber, C. (2013, August 21). An Experiential Investigation of Algebraic Ideas Created and Implemented by Robert B. Davis. Retrieved from https://doi.org/doi:10.7282/T3NV9G6HSteffero, M. (2009, July 15). Romina Interview Reflections (M.B.A. Graduate): Understanding the Ideas [Video]. Retrieved from https://doi.org/doi:10.7282/T3TX3CSVUptegrove, E. B. (2011). Representations and Standard Notation. In Maher, C. A., Powell, A. B., &amp; Uptegrove, E. B. (Eds.). Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms (Vol. 47). (pp.133-144). Springer, Dordrecht.