DescriptionThe objective of this analytic is to examine the contributions made specifically by Romina and observe her methods of reasoning and rationale in relationship to the group discussions and outcomes from the proposed task. This analytic focuses on a session occurring on June 25, 2003 that highlighted the development of the explanation of the Fundamental Theorem of Calculus through the eyes of a group of three female third-year college students, Romina, Magda, and Angela. The students (3 years out of AP Calculus and participants in a longitudinal study at Rutgers) were challenged with explaining The Fundamental Theorem of Calculus (FTC) to a “hypothetical” current student. Researcher Pantozzi (also their former high-school Calculus teacher) posed the task to the group. The participants were asked to explain to a student “what the theorem means, what it is used for, and why it is true” (B45). This prompt comes from the researcher’s own personal experience while taking college Calculus, whereby he reports that he was asked to answer a question using the FTC and was unable to do so successfully (Pantozzi, 2009).
The students were invited to use various resources and materials that were available to them, including their own work from their previous high-school Calculus course provided by the researcher. Accessible were the FTC task description, multiple textbooks, graphic organizers, calculators, and graph paper.
Throughout the events we see Romina engaged in the task and soliciting responses from her peers. She asks members of the group for clarification, rewording, and to trace back to previously mentioned ideas. She shares recollection of her past experiences and draws upon them to show how her learning in the high-school Calculus class had built upon ideas and concepts and had developed a foundation in the understanding of how to find the area under a curve using other techniques prior to being introduced to the FTC. She also relates that her previous experiences learning Calculus were memorable. She recollects when she decided to study Calculus and what concepts she remembered from her high-school course to solve the currently posed task.
Event 1 shows Romina giving a visual of a graph drawn in the air with her finger along with her description of the Riemann Sum, in order to make the description more understandable for the others. Event 2 features Romina’s use of her textbook resource as she revisits the Theorem in her former textbook and begins to uncover its meaning. Magda offers her perspective but is interrupted by Romina. It is evident that it is important for Romina, herself, to understand the meaning of the Theorem as she works to be able to express its meaning in her own words. Once she is able to articulate what the definition means using her own language, through description and a graph, she asks her group mates for further reinforcement by sharing what she thinks the theorem means by asserting and questioning, “The integral is like the area underneath the graph, right?”. Event 3 shows Romina’s recognition of her experiences in learning about the area under the “integral” (curve). She also recalls being taught the very topic in class. In Event 4 Romina asks the group questions in regards to an indefinite integral vs. a definite integral and how area can be calculated. She asks how an indefinite becomes a definite integral in order to compute area. She draws some graphs as possibilities are discussed, beginning with a bell curve. The group then creates other curve sketches and discusses whether or not they could find the area under that curve. Romina prods the group by asking if there are other ways to determine area without knowing the equation. She asks, “There’s not a way you could just figure it out, is there? You’d have to get the equation of the line”. We see more initiative come from Magda in Event 5, however it is Romina who orchestrates the conversation initially by referencing an old Calculus packet from her high-school work provided by the researcher and asks for clarification from Magda. Event 6 highlights Romina asking for the group to make connections. She poses her findings as a question to her peers attempting to confirm her own understanding of the graphs they are viewing. Romina makes an observation that the slope of the integral is equivalent to the derivative of the integral, which is the same as the original function. This is an extremely valuable concept, showing an understanding of precise vocabulary and its significance to the graphs represented and how they relate to the FTC. Event 7 shows Romina interacting with the researcher. She recollects the time he was trying to convince her as a high-school junior to study Calculus in her senior year. She recalls that while driving to a task session, he suggested what the study of Calculus might entail by drawing a curve in the dust of his car’s dashboard. The researcher, a high-school teacher at that time, had asked her questions about the curve in relationship to distance, speed and acceleration with respect to time. Romina recalls that this early interaction was how she would explain the FTC. The group members then share how often their teacher prompted them to write about their ideas and that this activity made use of language that was sometimes helpful; they report that graphical representations also helped uncover understanding of concepts. Note: this reference to distance, speed, acceleration and time was a topic of conversation several times in this session however, Romina insisted that the group put that aside until they had a better understanding of the FTC and could explain it well prior to applying the concept.
Conclusion
To recall past experiences in mathematics and then use this knowledge to help explain something familiar in a new problem have been previously observed with students working on prior tasks. We see this occur in The Dice Problem, where we hear Jeff say, “Didn’t we have this stuff in, like, second grade?” (retrieved from the analytic by Nasdha, 2017) and similarly, in Logothetis’ (2015) analytic in Event 2, when Ankur recalls doing something like this before, sharing a familiarity with the activity and making a connection to a previous experience. At the conclusion of that analytic the young lady presenting with Stephanie, along with Jeff, do the very same thing by referencing their old times table grids from second grade they found in their Trapper Keepers to that of Stephanie’s sum chart to show they understand the representation by the way it is organized and use this to help explain it to others along the way. Such connections spark a familiarity and resonate with the student, giving them a sense of confidence while approaching a new topic with a set of tools with which they are already familiar. This analytic traces Romina’s journey to call upon her past experiences from her Calculus class from 3 years earlier in order to make connections and build upon earlier knowledge to solve the task posed to the group. The recollection she made as a high-school student contemplating continuing math studies in her senior year when traveling in the car with her current teacher (a graduate-student researcher), Mr. Pantozzi (referred to as Researcher in this analytic), proved to be a defining moment in helping explain the FTC. Romina and her group also recalled their experiences of having to write about their mathematical ideas while in the classroom. Together, they began to “retrieve representations that had been built years before…in order to build and extend their mathematical knowledge and make connections to (other) mathematical ideas”, similar to the students working on Pizza, Towers and Binomials (Tarlow, 2011, p. 130). Revisiting earlier ideas and building upon them are important tools when problem solving. It shows the durability of a previous understanding and that an impression was made about that learning.
Revisiting, rewording, and rewriting are all important forms of editing that are crucial when it comes to showing understanding of tasks. Romina was constantly revising her thoughts and representations so that they could be convincing not only for her audience and her group mates, but for her own benefit as well. According to Speiser (2017), “Children can build fundamental mathematical understanding, over time, through extended task-based explorations. They create models, invent notation, and justify, reorganize, and extend previous ideas and understandings to address new challenges” (p. 74).
These creations lead to ownership and were “…central in promoting students’ successful problem solving in the longitudinal study”, as noted by Francisco and Maher (2005). From her experiences throughout the study, Romina shared that she may not be able to remember a specific name to a theorem or the mathematician connected to it. However, she, demonstrated “the importance of explaining and knowing things in her own way so as to develop personally meaningful knowledge (Francisco & Maher, 2005). Romina’s “sense of ownership of the problem solving was accompanied by an increase in confidence and engagement” (p. 367-8).
Learning the foundation of a topic or idea enables one to build upon it. Many proofs and theorems in mathematics build upon the understanding of a concept that has been previously learned. Having a foundation to build on a newly introduced topic was insightful for Romina in uncovering the meaning of the FTC. It is important to know where earlier ideas come from and to appreciate the meaning of the idea (theorem) because of the tedium required to build skill in equivalent methods taught earlier, such as finding the sums of the areas of geometric shapes represented under the curve and finding the Riemann Sum. It is also important to understand The Fundamental Theorem of Calculus means. Too often students are successful in carrying out computations that are appropriate as a result of the claims of a theorem and are often are unable to articulate why those calculations are valid. Romina’s experiences in building an understanding of Calculus concepts, in conjunction with her growth in mathematical learning as a participant in a longitudinal study, provide a rich environment for her to express and convey understanding of a very complex theorem by making use of her own representations. Overall, we can learn that “Teachers should aim to help students to develop a powerful organization, one that lends itself to a mapping onto formal notation. In that way, the formal notation can be seen as the solution to a problem that arises during the students’ own investigations…” (Uptegrove, 2015, p. 144).
Collaboration was also an integral part in helping solidify Romina’s understanding of the FTC. By having others to test ideas and receive feedback, Romina was able to gain confidence in her ideas and had opportunity to reorganize her thinking. Similar to Stephanie’s journey in solving the Towers problem, “She would validate or reject her own ideas and the ideas of others, based on whether they made sense to her or not” (Speiser, 2011, p. 74). Tarlow (2011) observed, “Given the opportunity to think and reason together the students constructed deep and powerful mathematical ideas” (p. 131).
The Task:
A current student of calculus has asked you to help them understand the Fundamental Theorem of Calculus. The student, knowing that you have taken calculus in the past, is interested in what the theorem means, what the theorem is "for," and why the theorem is true. You have agreed to the student’s request. In preparing to respond to the student, you can use any materials that you feel would be helpful to you, including textbooks, other calculus resource materials, specific calculus questions, calculators, and computers. You may also discuss your ideas with the other students here with you today.
Definitions and Terms:
The 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 ∫_a^b▒〖f(x) dx=F(a)- 〗 F(b).
“Roughly, 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 the definite integral) preserve this inverse relationship.” (Larson et al., 1994, p. 280)
Definition of a Riemann Sum
Let f be defined on the closed interval [a, b], and let Δ be a partition of [a, b] given by a=x_(0 )< x_1
“In the previous section, the limit of a sum was used to define the area of a region in the plane. Finding the area by this means is only one of many applications involving the limit of a sum. A similar approach can be used to determine quantities as diverse as arc length, average value, centroids, volume, work, and surface areas. The following development is named after Georg Friedrich Bernhard Riemann (1826 – 1866). Although the definite integral had been defined and used long before the time of Riemann, he generalized the concept to cover a broader category of functions.
In the following definition of Riemann sum, note that the function f has no restrictions other than being defined on the interval [a,b]. (In the previous section, the function f was assumed to be continuous and nonnegative because we were dealing with the area under a curve.)” (Larson et al., 1994, p. 271)
Theorem 4.5 The Definite Integral as the Area of a Region
“If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is given by
Area = ∫_a^b▒〖f(x) dx 〗.” (Larson et al., 1994, p. 273)
References:
Francisco, J. M., & Maher, C. A. (2005). Conditions for promoting reasoning in problem solving: Insights from a longitudinal study. The Journal of Mathematical Behavior, 24(3-4), 361-372.
Larson, R., Hostetler, R., & Edwards, B., (1994). Calculus of a Single Variable. D.C. Heath and Company.
Logothetis, 2015. Discovering Probability with Dice Games and the Evolution of a Convincing Argument. Retrieved from https://doi.org/doi:10.7282/T3PN97F0
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: 3373679
Speiser (2011). Block Towers: From Concrete Objects to Conceptual Imagination. In Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). Combinatorics and reasoning: Representing, justifying and building isomorphisms (Vol. 47). (pp. 73-86). Springer, Dordrecht.
Tarlow, L. D. (2011). Pizzas, towers, and binomials. In Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). Combinatorics and reasoning: Representing, justifying and building isomorphisms (Vol. 47). (pp. 121-131). Springer, Dordrecht.
Uptegrove, E. B. (2011). Representations and standard notation. In Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). Combinatorics and reasoning: Representing, justifying and building isomorphisms (Vol. 47). (pp. 133-144). Springer, Dordrecht.