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«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

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During the follow-up interviews, each of the four students were given the opportunity to clarify their responses to the item. Though they used a variety of verbs in describing the convergence, e.g. “comes to”, “resolves”, “approaches”, etc., none of the four referenced partial sums in any sense. This is of particular concern in that the notion of partial sums provides the foundational link between series and sequences upon which all other definitions and theorems reside. For A and B level students to have so completely ignored this connection between sequences and series is noteworthy. For the most part, all of the participants use similar language in describing the convergence of sequences and the convergence of series.

For one participant, she was able to recognize that the convergence was different for sequences and series, yet like the other three participants, describing that difference proved impossible for her. One student, AJ, did mention that the “sum begins to approach a point” for a series, which could be interpreted as a referent to partial sums. However, he failed to make any more mention of this type of thinking when probed.

Misconception 2: Sequences and functions are related but series are not related to either.

In order to tease out each student’s conception of the “big picture” in second-semester calculus, each was asked to describe the relationships among sequences, series, and functions. In true design research fashion, this question was drafted after the interview with the first student, = () in one case, they struggled to connect series with sequences or with functions.

Teri, so her response is absent. The predominant finding from responses to this question was that while students acknowledged a link between sequences and functions, even indicating Though one student did take more time, he did recognize that functions can be represented as power series, especially for the purposes of integration and differentiation.

Preliminary findings from videotape analysis:

There is evidence that misconception 1 comes from instruction.

On the first day in which series were discussed, the instructor drew a clear distinction between the mechanics of sequences and the mechanics of series, labeling sequences as “Easy”, and series as “Hard” on the board to emphasize the distinction. He then went on to define series partial sums as 1 = 1, 2 = 1 + 2, etc., and gave the example ∑∞ . In writing the partial as converging “when the partial sums converged”. No mention was made of the partial sums as

–  –  –

=1 find out if it converges or diverges we’re going to take the limit as n goes to infinity”. Again, this language distinctly references the process for determining the limit of a sequence, but the connection to sequences is not made explicit for students.

15TH Annual Conference on Research in Undergraduate Mathematics Education 481 Is there evidence that misconception 2 comes from instruction?

This will be investigated by conducting additional analysis of the videotapes.

Conclusions While students from a single section of second-semester calculus did hold a variety of conceptions about series in general, student conceptions regarding the convergence of a series appeared to agree in key areas, and evidence was found that this understanding was fostered during classroom discourse. Further study will reveal the extent to which the instruction influenced other aspects of students’ conceptions of infinite series.

Questions to be addressed during the session:

1. What are the methods by which large volumes of videotape can be analyzed qualitatively in a manner which reduces researcher bias?

2. Would assessing the instructor’s understanding via a survey be relevant? Or could concept maps be used in a way to address the research questions?

482 15TH Annual Conference on Research in Undergraduate Mathematics Education

References:

Alcock, L., & Simpson, A. (2004). Convergence of sequences and series: Interactions between visual reasoning and the learner’s beliefs about their own role.

Educational Studies in Mathematics, 57, 1-32.

Carlson, M.P. (1998). A cross-sectional investigation of the development of the function concept. In A. Schoenfeld, J. Kaput, & E. Dubinsky, (Eds.), Research

in collegiate mathematics education III (pp. 71-80). Providence, RI:

American Mathematical Society.

Creswell, T. (1998). Qualitative inquire and research design. Thousand Oaks, CA: Sage Publications.

Hiebert, J., & Grouws, D. (2007).The effects of classroom mathematics teaching on students’ learning. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 843-908). Charlotte, NC: Information Age Publishing.

Keynes, H., Lindaman, B., & Schmitz, R. (2009). Student understanding and misconceptions about foundations of sequences and series. In D. Wessels (Ed.), The Seventh Southern Right Delta Conference of the Teaching and Learning of Undergraduate Mathematics and Statistics (pp. 148-158). Stellenbosch: International Delta Steering Committee.

Lindaman, B. (2007). Making sense of the infinite: A study comparing the effectiveness of two approaches to teaching infinite series in calculus. Unpublished doctoral dissertation.





Lindaman, B., & Gay, S. (In press). Improving the instruction of infinite series. Problems, Resources, and Issues in Mathematics Undergraduate Studies.

Lithner, J. (2003). Students’ mathematical reasoning in university textbook exercises.

Education Studies in Mathematics, 52, 29-55.

Mamona, J. (1990). Sequences and series-sequences and functions: Students’ confusions. International Journal of Mathematical Education in Science and Technology, 21, 333-337.

Mamona-Downs, J. (2001). Letting the intuitive bear on the formal: A didactical approach for the understanding of the limit of a sequence. Educational Studies in Mathematics, 48, 259-288.

McDonald, M., Mathews, D., & Strobel, K. (2000). Understanding sequences:

A tale of two objects. In E. Dubinsky, A. Schoenfeld, & J. Kaput, (Eds.), Research in collegiate mathematics education IV (pp. 77-102). Providence, RI: American Mathematical Society.

Monk, G.S., & Nemirovsky, R. (1994). The case of Dan: Student construction of 15TH Annual Conference on Research in Undergraduate Mathematics Education 483 a functional situation through visual attributes. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education I (pp.

139-168). Providence, RI: American Mathematical Society.

Rasslan, S., & Tall, D. O. (2002). Definitions and images for the definite integral concept. In A. Cockburn & E. Nardi (Eds), Proceedings of the 26th conference of the international group for the psychology of mathematics education, UK, 4 89-96.

Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum.

In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education I (pp. 21-44). Providence, RI: American Mathematical Society.

Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371-397.

Tall, D., & Razali, M. (1993). Diagnosing students difficulties in learning mathematics.

International Journal of Mathematics Education, Science & Technology, 24, 209202.

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

Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. Schoenfeld, & J. Kaput, (Eds.), Research in collegiate mathematics education IV (pp. 103-127).

Providence, RI: American Mathematical Society.

484 15TH Annual Conference on Research in Undergraduate Mathematics Education Reading Comprehension of Series Convergence Proofs in Calculus II

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Abstract This study examines the effect of activities and assessments concerning reading comprehension of series convergence proofs in Calculus II on students’ exam performance. Two sections of Calculus II taught during a summer semester were compared. Both sections primarily used traditional lecture methods, and one section was also given reading assignments with openended questions and in-class quizzes evaluating reading comprehension. We compare test scores and interview data from the two sections.

Keywords: Calculus, series convergence proofs, reading comprehension, teaching experiment Literature Review Standard evaluation methods in lower division mathematics courses measure students’ ability to work problems, with the result that many students focus on mimicking algorithms rather than understanding the underlying mathematics. In a Calculus II course, the level of difficulty increases significantly because students are expected to determine convergence of an infinite series, a very abstract task, and to write an argument justifying their conclusion.

Traditional instruction in Calculus II does not emphasize reading in class, and so any reading of the textbook or other related materials that students might do in order to learn these techniques must be done on their own. Students often have a hard time understanding the dense and symbolheavy style of most mathematical writing (Watkins, 1979). It has been found that in an inquiryoriented classroom, reading can serve multiple roles, such as focusing the inquiry, carrying out the inquiry, and communicating results (Siegel, Borasi & Fonzi, 1998). The importance of writing mathematics in Calculus has also been documented (Brandau, 1990; Porter, 1996). We believe that requiring students to critically read mathematical arguments and reflect upon their reading is a promising pedagogical technique that should contribute both to better facility with determining convergence and greater fluency in writing convergence arguments.

Stickles & Stickles (2008) found that giving students assignments that directly address their assigned reading can help motivate students to read their textbooks, and can have a positive effect on their success in Calculus. Reader-oriented theory suggests that a reader's understanding of a text is shaped in part by their goals and motivation as they read (Weinberg & Wiesner, 2011). It may be that students will read mathematical texts differently when they know that they will be evaluated based on their comprehension.

In order to design assessment instruments for reading comprehension, we need a model of reading comprehension. Mejia-Ramos et al (2010) have developed a framework of proof comprehension that can be used to create assessment tools. To illustrate their model, they presented a Calculus-level proof, and several multiple-choice items to assess the different dimensions of proof comprehension. The proof that they chose intentionally highlighted all of their dimensions, but the dimensions are not always easy to assess for every proof. We have 15TH Annual Conference on Research in Undergraduate Mathematics Education 485 adapted their dimensions to the types of arguments that appear during the discussion of infinite series in Calculus II.

Several prior studies show that sequences and series arguments are problematic for students at the university level. The literature shows that students think about series in a wide variety of ways, including visual, verbal and algebraic, shaped by their own view of their role as a learner (Alcock & Simpson, 2004; Alcock & Simpson, 2005). A number of different methods for presenting the idea of convergence have been proposed (Burn, 2005; Roh, 2008; Roh, 2010).

We are proposing to evaluate the effectiveness of traditional instructional methods augmented by our reading comprehension tasks on exam performance and on reading comprehension tasks as evaluated in interviews of selected students.

Research Questions  Do students read mathematical arguments differently after activities that emphasize and assess reading comprehension?

 Do students comprehend more of what they read after activities that emphasize and assess reading comprehension?

 Will students’ facility in determining series convergence or divergence improve after activities that emphasize and assess reading comprehension of series arguments?

 Will students have more fluency in writing justifications of series convergence or divergence after exercises assessing reading comprehension of convergence arguments?

Methods Two sections of Calculus II were taught during a summer semester by instructors with similar styles and similar teaching experience. Both instructors were advanced doctoral students in mathematics who had not previously taught Calculus II. Both sections were taught in a traditional way, with the majority of each class period devoted to lecture and additional time spent on class discussion and problem solving by students. Students self-selected between the two sections, which met at the same time, with 19 students enrolling in the first section (control) and 29 students enrolling in the second (test). The two sections used identical examinations given four times during the semester and identical assignments in an online homework system.

The first in-class examination, covering techniques of integration and applications, was used as the study’s pre-test. All three researchers will score students’ test papers both to provide numerical scores and a catalogue of student errors on each problem. These data are used to provide a cross-section comparison of students’ knowledge base and frequency of various types of errors.



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