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1) Do students’ efforts to formalize the concept of definite integral motivate a need (for them) to formalize the notion of convergence/limit?

2) Once students reinvent a formal definition of definite integral, can they use this formalization as a tool for formalizing other advanced calculus concepts? If so, which concepts?

Methods Because the intent of our research was to learn how students can leverage their informal notions of Riemann approximation and bounded area to formalize their understanding of definite integral, we adopted a developmental research design. Gravemeijer (1998) describes the goal of developmental research as follows: “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). Instead of presenting students with a formal definition of definite integral and asking them to interpret the definition based on their informal understanding, we utilized the guided reinvention heuristic, using the students’ informal knowledge as a starting point for constructing a formal definition. Thus, the students’ reinvention of definite integral more closely resembled the historical development of the idea, avoiding what Freudenthal (1973) critically referred to as the anti-didactical inversion in which the end results of mathematicians’ efforts (formal definitions) are taken as the starting points for students’ learning.

The teaching experiment consisted of fourteen 60-90 minute sessions, occurring roughly once a week. The study was conducted with two students (Betty and Kathy) at a large, public university in the Pacific Northwest. Both students had completed an introductory calculus sequence, earning high marks in each course, and demonstrating strong conceptual understanding on written assignments and exams. We purposely chose to work with students possessing robust concept images because we felt doing so increased the likelihood of us gaining insight into how students reason about definite integral (and other Calculus concepts) as they formalize their intuitive understandings.

Ongoing analysis between sessions consisted of analyzing video, creating content logs designed to describe the students’ mathematical activity, and making initial conjectures about 15TH Annual Conference on Research in Undergraduate Mathematics Education 561 students’ reasoning. Analysis also included weekly research team meetings, during which time an outline of the following week’s session was constructed based on analysis of the students’ reasoning to date. In the coming months, we will conduct a retrospective analysis of the data corpus, with the intent of better understanding the students’ reasoning related to all of the mathematical concepts that arose: area, definite integral, sigma notation, sequence convergence, derivative, limit at a point, limit at infinity (for a function), continuity, and the Fundamental Theorem of Calculus (FTC). Specifically, our goal will be to identify the challenges the students encountered in their reinvention of these concepts, as well as what supported the students in overcoming said challenges.

Initial Findings We found the concept of definite integral to be a promising starting point for exploring and formalizing advanced calculus concepts. For instance, as the students attempted to characterize precisely what it means for a definite integral to exist on a closed interval [a, b], they recognized a need to clarify their language. Specifically, they were struggling to operationalize what might be meant by Riemann approximations getting “closer and closer” to an actual sum. This spurred an exploration of what sequence convergence could mean. Unlike previous research (Oehrtman, Swinyard, Martin, Hart-Weber, and Roh, 2011; Swinyard, 2011), the reinvention of sequence convergence in this setting was purposeful from the students’ perspective – it served to help them better understand what is involved for a definite integral to exist for a function f on a stated interval [a, b]. Despite not having been introduced to a formal definition of sequence convergence prior to the teaching experiment, Betty and Kathy were able to construct the converges to X if for any distance ε from X there exists a following definition: A sequence K such that for all kK, ak satisfies |X-ak| ε. They then used that definition to help them formalize the notion of definite integral. After doing so, they wondered aloud why evaluating an antiderivative of a function f at the endpoints of an interval [a, b] results in the exact same area one would get by taking the limit of approximating rectangles under f on that same interval. This curiosity led Betty and Kathy to explore why the FTC works. Although their exploration was never fully resolved (the FTC was not “reinvented” from a mathematician’s perspective), it did lead the students to also wonder what it means for a function to have a derivative. A subsequent exploration included the students formalizing what it means for a derivative to exist at a specified point, using their experience of approximation in the integral context as a model. In our talk, we will provide an overview of what concepts Betty and Kathy reinvented, as well as some conjectures for why definite integral served as such a fruitful starting point.

–  –  –

With the intent of furthering our research, we provide the following questions for consideration from the audience

1) What advantages/disadvantages can you think of for beginning an advanced calculus inquiry based curriculum with definite integral?

2) Do the audience members have insights from their experiences teaching and learning advanced calculus that might suggest other points of contact/starting points for motivating students toward thinking deeply about definite integral?

–  –  –

Davis, R.B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of mathematical behavior, 5(3), 281-303.

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.

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

Kilpatrick (Eds.), Mathematics Education as a Research Domain: A Search for Identity (pp. 277Dordrecht, The Netherlands: Kluwer.

Monaghan, J. (1991). Problems with the language of limits. For the learning of mathematics, 11, 3, 20-24.

Oehrtman, M., Swinyard, C., Martin, J., Hart-Weber, C., and Roh, K. (2011). From Intuition to Rigor: Calculus Students’ Reinvention of the Definition of Sequence Convergence. Proceedings of the Fourteenth Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education Conference. Raleigh, NC. Retrieved from http://sigmaa.maa.org/rume/crume2010/Abstracts2010.html on October 6, 2011.

Swinyard, C. (2011). Reinventing the Formal Definition of Limit: The Case of Amy & Mike.

Journal of Mathematical Behavior, 30, 93-114.

Swinyard, C., & Larsen, S. (2011). What Does it Mean to Understand the Formal Definition of Limit?: Insights Gained from Engaging Students in Reinvention. To appear in Journal for Research in Mathematics Education.

Tall, D.O., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational studies in mathematics, 12, 151-169.

Williams, S. (1991). Models of limit held by college calculus students. Journal for research in mathematics education, 22, 3, 219-236.

15TH Annual Conference on Research in Undergraduate Mathematics Education 563



Gail Tang University of Illinois at Chicago Abstract: While the limit in the definitions of the derivative is troubling to many students, a difculty that preceded this confusion was observed: students were not able to correctly set up the difference quotients as required in the definitions. The purpose of this study is to investigate the cognitive processes involved in setting up the difference quotients and the associated errors. This explanatory case study seeks to explain why these particular errors occur from the perspective of student thinking of function composition. At the end of the study, a framework that aggregates criteria used (by past studies and this study) to assign student membership into a function conception category will be produced in an attempt to move towards a systematic classification of students’ cognitive processes. Implications from this study can inform teaching practices by exposing students to expected errors. As observed from the data, this can lead to rich discussions on the concept of function itself.

Keywords: case study, function composition, difference quotient, APOS Theory, Pre-Calculus

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564 15TH Annual Conference on Research in Undergraduate Mathematics Education rational function, and 58 students – 54% – for the trigonometric function. The most frequent error made was failure to compute or correctly compute f (a). Astonishingly, students also had quite a bit of difficulty recognizing what to place in the f (x) portion of the difference quotient. These are alarming observations that warrant further investigation.

Even though setting up of the two difference quotients seems to be at different levels of mathematical difficulty, the errors students made in the respective first terms of the numerators, f (x + h) and f (x), were very similar in nature. The aim of this study is to investigate the errors made in the difference quotients and to connect them with the cognitive processes of students when setting up the difference quotients.

Literature Review Functions are foundational building blocks of mathematics. The importance of functions is well known within the mathematics community and documented and acknowledged among mathematics education researchers (Dubinsky, 1991; Dubinsky & Harel, 1992; Tall, 1991; Thompson, 1994). The concept has been widely explored both theoretically (Eisenberg, 1991; Harel & Kaput,

1991) and empirically (Ayers, Davis, Dubinsky, & Lewin, 1988; Carlson, Oehrtman, & Engelke, 2010; Cottrill, 1999; Sfard, 1992). Although more recent research on undergraduate mathematics education has moved away from functions and towards upper level topics such as Linear Algebra and Mathematical Proof, student difficulty with function has not disappeared; as indicated in the title of the Gooya & Javadi (2011) paper, “Unversity Students’ Understanding of Function is Still a Problem!” We can see from the results of the exploratory study that indeed, functions still pose a problem for students. Similar to computation of f (x + h), Carlson (1998) briefly documented difficulty with computing f (x + a) for a quadratic expression as part of a larger cross-sectional study. While the most common error that surfaced in Carlson’s study was adding a to the expression, other more common errors were found in the exploratory part of my study. Research from my study seeks to further Carlson’s study by describing the additional errors and explaining why these errors occur with respect to student cognitive processes while setting up the difference quotients.

Unlike Carlson (1998), this study looks at computing f (x + h) from a function composition perspective; it explores the possibility of using the concept of function composition as a learning tool to evaluate f (x + h). The few studies on composition focus on the topic itself, its relation to chain rule, or its place in secondary and post-secondary curricula (Ayers et al., 1988; Cottrill, 1999;

Horvath, 2011). At the time of this study, no studies view composition in relation to evaluating f (x + h), even though it is recognized as a composition (Horvath, 2011).

Comparative Framework In Carlson (1998) and in other studies (Ayers et al., 1988; Breidenbach, Dubinsky, Hawks, & Nichols, 1992; Zandieh, 2000) that investigate student cognitive processes, researchers categorize students as having action, process, or object conceptions of function by using the operational definitions of the conceptions. During data analysis, raw student interview data are fit to these definitions at the discretion of the researchers. As a result, student function conception classification can vary from study to study. Recent studies (Carlson et al., 2010; Oehrtman, Carlson, & Thompson, 2008) attempt to mitigate this problem by breaking the conceptions into specific function topics such as domain, inverses, etc., and providing operational definitions for them. Some 15TH Annual Conference on Research in Undergraduate Mathematics Education 565 definitions are so specific that they detail raw student interview observations.

In an attempt to move towards a systematic classification of students’ cognitive processes, this study seeks to produce a framework to aid researchers in assigning student membership into a function conception category. This data analysis tool will be created from criteria used by this study and by past studies.

Research Methodology This explanatory case study (Yin, 2009) seeks to explain why these particular errors occur.

There are many trajectories leading to student error: students’ interactions with past texts, teachers, curricula and other learning resources, and their thinking as a result of the complexities of their mathematical background. This study looks at reasons for error from the perspective of student thinking.

Five cases from the previously identified errors were chosen to be studied in-depth. To learn about the complexities of these errors, ten student sources were interviewed. These ten student volunteers came from the pool of participants from the two large lectures of the exploratory study.

For each error, the students were split into two groups: those whose answers were in the error category and those whose answers were not. Triangulating data from these two groups is necessary to create a robust study.

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