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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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The issue of using an elementary school context appears to be useful in some ways but not in others. Students are engaged early on with the ideas and willing to participate with little encouragement. However, there is some drawback, as the students seem to expect the task to focus on elementary mathematics and may not stay involved throughout. We continue to struggle with this idea that making the context elementary does not accomplish our goal of deep understanding of calculus concepts.

474 15TH Annual Conference on Research in Undergraduate Mathematics Education Student learning of advanced mathematical ideas. The limit task was not successful in helping students avoid some common misconceptions of limit of a sequence. Specifically, students were apt to describe limits using imprecise language (e.g. “getting closer and closer”) and a common conception held by students in ensuing lessons/activities was that a sequence could never reach its limit. Students were successful in attending to the difference between the physical act of walking across a bridge and the specific mathematical task this activity presents.

This activity successfully helped transition students from an informal understanding of the context and the mathematics to a more abstract, formal setting.

Conclusion

Research continues all over the United States about teacher knowledge and its relationship to good teaching. The research reported here contributes to that research base in that we are developing instruction that will allow future teachers to develop deep understanding of complex mathematical ideas and connect them to the mathematics that they will teach in elementary school. The work will be disseminated, as the idea of STEM-focused elementary school teachers is growing, and the use of calculus as a base course has great potential. Further work is necessary in this project to follow these teachers who are learning calculus and evaluate how it affects their teaching and students.

Questions

1. How could we implement more dynamic ways of looking at limit?

2. How can we move students into more formal thinking about function, limit, and derivative- or we do need to?

3. What ways does calculus tie to earlier mathematics?

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Ball, D. L. (1990). The Mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90, 449-466.

Ball, D.L., Hill, H., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American

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Ball, D.L., Thames, M.H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 383-388.

Collins, A., Joseph, D., & Bielaczyc, K. (2004). Design Research: Theoretical and Methodological Issues, The Journal Of The Learning Sciences, 13(1), 15-42.

Davis, R. B. and Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconceptional stages. Journal of Mathematical Behavior, 5(3), 281–303.

Enochs, L. G., Smith, P., & Huinker, D. (2000). Establishing factorial validity of the Mathematics Teaching Efficacy Beliefs Instrument. School Science and Mathematics, 4, 194-202.

Fennema & Franke (1992), Teachers knowledge and its impact. In Grouws, D. Ed. Handbook of Research on Mathematics Teaching and Learning. Reston: NCTM.

Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence

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476 15TH Annual Conference on Research in Undergraduate Mathematics Education Mamona-Downs, J. (2002). Letting the intuitive bear on the formal: A didactical approach for the understanding of the limit of a sequence. Educational Studies in Mathematics, 48(2Mathematical Association of America (2004). Committee on the Undergraduate Program

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Mewborn, D.S. (2001). Teaching, teachers’ knowledge, and their professional development. In J.

Kilpatrick, W.G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 45-52). Reston, VA: NCTM.

National Mathematics Advisory Panel (2008). The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.

Oehrtman, M. (2008). Layers of abstraction: Theory and design for the instruction of limit concepts. In M. Carlson & C. Rasmussen (Eds.) Making the Connection: Research and Teaching in Undergraduate Mathematics Education. Washington, D.C.: American Mathematical Society.

Papert, S. (1971). “Teaching children to be mathematicians vs. teaching about mathematics.” (Al Memo No. 249 and Logo Memo No. 4). Cambridge MA: MIT.

Phillip, R. A. (2007). Mathematics teachers’ beliefs and affects. In F. K. Lester (Ed.) Second handbook of research on mathematics teaching and learning.

Roh, K. (2008). Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics, 69(3), 217-233.

15TH Annual Conference on Research in Undergraduate Mathematics Education 477 Seaman, C. E. & Szydlik, J. E. (2007). Mathematical sophistication among preservice elementary teachers. Journal of Mathematics Teacher Education, 10, 167-182.





Thompson A. G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D. A.

Grouws (Ed.) Handbook of research on mathematics teaching and learning. Reston:

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Utley, J., Bryant, R., & Moseley, C. (2005). Relationship between science and mathematics teaching efficacy of preservice elementary teachers. School Science and Mathematics, 105 (2) 82-88.

Utley, J. & Moseley, C. (2006). The effect of an integrated science and mathematics contentbased course on science and mathematics teaching efficacy of preservice elementary teachers. Journal of Elementary Science Education, 18(2), 1-12.

Wu, H. (2006). How mathematicians can contribute to K–12 mathematics education. In Proceedings of the International Congress of Mathematicians, Madrid 2006, Volume III, European Mathematical Society, Zurich, 1676-1688.

478 15TH Annual Conference on Research in Undergraduate Mathematics Education Title: Instructional Influence on Student Understanding of Infinite Series Author: Brian J. Lindaman, Montana State University

Abstract:

Many studies have documented the nature of student conceptions for various topics in college calculus. Of interest in this study are the sources of these understandings. In particular, the instructor’s discourse seems to significantly impact students’ views and conceptualizations of a topic. The nature and scope of this influence, on a particularly troublesome topic, infinite series,

is the object of study in this research. Research questions:

1. What are the sources of students’ misconceptions about infinite series?

2. Are there student misconceptions about infinite series which arise from the classroom discourse in college calculus?

The data collected consisted of survey responses, transcripts from interviews, and videotapes of instruction. Student conceptions regarding the convergence of a series agreed in key areas, and evidence indicated 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.

Key words: calculus instruction, infinite series, sequences, conceptual understanding Introduction “When I say/write/do ________, do my students follow me?” This common and eminently practical question that instructors often ask presents a challenge to researchers. On the one hand, it is to be expected that student understanding of a topic is related to the specific content, such as examples, definitions, and diagrams, presented by the instructor. On the other hand, students also develop conceptions of topics that can deviate wildly from the material presented in class, leaving the instructor to wonder “Where are they getting this?” Certainly, there is a variety of sources for students’ conceptions of a topic, but instruction is a key component (Hiebert and Grouws, 2007). This study focuses on exploring the influence of instruction on student conceptions of a duly nefarious calculus topic: series and sequences.

Relevant Literature and Research Questions Many of the studies on calculus learning have found a lack of conceptual understanding among students regarding specific topics in calculus, including functions (Carlson, 1998;

Thompson, 1994); limits (Sierpinska, 1987; Tall & Vinner, 1981), derivatives (Monk & Nemirovsky, 1994; Zandieh, 2000), integrals (Rasslan & Tall, 2002), sequences (Mamona, 1990;

McDonald, Mathews, & Strobel, 2000), and infinite series (Alcock & Simpson, 2004; Lithner, 2003). While many studies have documented the nature of student conceptions, few have traced these conceptions back to instruction or curriculum. Infinite series is known to be a particularly difficult topic for calculus students to learn, rife with misconceptions and faulty understandings (Tall&Razali, 1993; McDonald, Matthews, and Strobel, 2000; Keynes, Lindaman, and Schmitz, 2009). Certainly, students’ conceptions of prior topics plays a role, as it does in the learning of other calculus topics. In the case of limits, and limit processes, the sources of these conceptions have been traced back to students’ prior conceptions of functions (Carlson, 1998), or knowledge of the real line (Mamona-Downs, 2001; Sierpenska, 1987).

However, other sources for their conceptions could exist, extrinsic to the students’ body of knowledge. Certainly, several relationships come to mind, but the three relationships which are most likely to influence students’ conceptions are: instructor-student, student-student, and 15TH Annual Conference on Research in Undergraduate Mathematics Education 479 curriculum-student. In particular, the instructor’s role in creating classroom discourse seems to have a significant impact on students’ views and conceptualizations of a topic. The nature and scope of this influence, on a particularly troublesome topic, is the object of study in the research.

That is, this study investigates the link between student conceptions about infinite series, and the

instructor’s presentation of the curriculum. The research questions are:

1. What are the sources of students’ misconceptions about infinite series?

2. Are there student misconceptions about infinite series which arise from the classroom discourse in college calculus?

Methodology Data collection focused on a single section of second-semester calculus, taught by an adjunct faculty with multiple years of teaching experience. The teaching style was predominantly lecture-based, with an emphasis on the use of examples to illustrate key concepts.

The classroom interactions were mostly teacher-centered with occasional group work, although students asked several (5-10) questions per class period. The instructor selected three or four students, by drawing out notecards with their names, to provide the solution to a problem, or answer a factual question. Technology played a limited role in the course; the instructor made little use of technology during instruction, and calculators were not permitted on course exams.

The data collected consisted of responses on a short in-class survey, transcripts from four student interviews, and from videos of the classroom instruction during the unit on infinite series. During the final week of class, over 27 students in a second-semester calculus class completed a written survey in class. The survey instrument contained one item asking students to describe convergence for a series, and the other asking students whether a repeating decimal equaled an integer. Other items collected demographic information as well as preference for various topics seen in the course. Students were also asked to participate in a voluntary followup interview. An email was sent to each of the six students who indicated their willingness to participate in an interview. Four students responded and an interview schedule was arranged according to the students’ preferences.

An interview protocol was created, based on an instrument used in prior doctoral research (Lindaman, 2007). The protocol consisted of 11 questions, focused primarily on gathering information about student understanding of series. The four participants were all undergraduate students, with three females and one male. They ranged from 19 to 23 years of age. For all four, this was the first time they had taken Calculus II. For two of the students their anticipated grade was a B, and the other two anticipated earning an A.

Grounded theory, as described in Creswell (1998), was used to analyze the transcript data from the interviews. Phrases and words were coded according to frequency and similarity. Then, codes were condensed into several categories. The videotapes are being coded by time and topic.

Transcripts will then be generated specific to various subtopics of series, such as convergence, various convergence tests, convergence criteria, etc., in order to match moments in the instruction with discussion in the interviews.

Preliminary Findings from the Student Interviews Misconception 1: The concept image for series convergence includes addition and sequence terminology, but neglects partial sums.

480 15TH Annual Conference on Research in Undergraduate Mathematics Education On all 27 surveys, students were asked to “Explain what it means for a series to converge.” The most frequent responses were “it adds to a number/sum”, followed by “it reaches/approaches a number”. The first response indicates an understanding grounded in the language of addition, while the second response indicates a limit process at work. Other responses referenced graphs, oscillations, and infinity. In all cases, however, no mention was made of the sequence of partial sums, a finding which is consistent to prior work (Lindaman and Gay, in press).



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