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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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This study seeks to address this gap in information. Students participating in this study have the ability to drop in to a study lounge to work together on their schedule where they are video recorded. Through these recordings, supplemented by the collection of journal entries at the end of each study session and interviews, this study seeks to answer the following research

questions:

1) What roles do students assume while working together in self-formed groups?

2) How are these roles impacted by the goals of the study group and the course content the group is working on?

3) What material resources, such as textbooks, class notes, or websites, are the groups utilizing? How are these resources being utilized?

2 Literature and framework The primary inspiration for this study is drawn from the work of Uri Treisman (1985).

During his dissertation work he observed that students working together in groups were more successful in learning calculus concepts. Although his work focuses on the creation of the workshop program at Berkeley and its subsequent impact on student learning, his references to how students created communities of study for themselves resonated with student behaviors I have observed as both a teaching assistant and an instructor.

Efforts to find further details of what transpired in student study groups or additional studies of student study behavior yielded mostly studies based on student self-reported data.

Some of this information was collected through anonymous surveys and tracked time each student spent working on the subject outside of the classroom or how confident the student felt about the material (Cerrito & Levi, 1999; Rohrer & Pashler, 2007). Other studies conducted interviews with students to uncover how they spend their time outside of the classroom preparing for tests (Danish Institute for Educational Research, 1970; Hong, Sas, & Sas, 2006). However, there is little in the way of observational data to back any of these findings up. Thus it is still unclear what it actually looks like when a student studies.

In order to observe students in their studying situations, an ethnographic approach is needed. Adopting the perspective that students in these situations socially construct their knowledge means the framework used to justify what data to collect and how to analyze it must be able to account for symbolic gestures in addition to dialogue. Using symbolic interactionism 436 15TH Annual Conference on Research in Undergraduate Mathematics Education as a lens for observing these interactions provides one way to combine a student's utterances, gestures, and other performed microtasks in order to interpret a student's intentions or the role the student has taken in the group (Blumer, 1969; Charon, 2010). The discourse analysis work of Goos, Galbraith, and Renshaw (2002) and Blanton, Stylianou, and David (2009) provide a compatible framework for coding student utterances in order to more carefully analyze a student’s contribution to the dialogue.

3 Data collection and methodology for analysis Participants for this study have been drawn from a second year undergraduate mathematics course that encourages students to work together in an inquiry-based tradition. This is achieved by using a lecture room equipped with round tables for the students to work at, assigning group projects, and allowing students to work together on homework assignments.

The participants in this study have been given access to a study lounge during hours scheduled to meet their studying needs. By restricting access to this lounge to only the participants there is no struggle for them to find seating or space to work and it provides a respite from the noisy dormitories if they dislike working in their rooms. The study lounge is equipped with two computers with internet access and several suites of mathematical software, round tables with chairs, and a white board. Students are video-recorded while working in this room, whether together, near each other, or alone. Students that have opted to do so are also invited to complete journal entries recording what materials they worked with during the session, which individuals they worked with during the session, what course content they worked on, and how successful they felt their study session was. Students will also be interviewed twice throughout the semester, once roughly halfway through the semester and once at the end of the semester.

These interviews will be designed to gather additional information about comments students leave on the journal entries and particular behaviors they exhibited during recorded study sessions.

All video recordings and audio recordings are being transcribed and coded. In particular, I am looking to create a catalog of microtasks that occur during the study sessions. Actions such as sharing a print out of the homework assignment, consulting class notes, and writing on the white board are considered microtasks. At this level each student utterance is also considered a microtask. The use of the coding scheme developed by Goos, Galbraith, and Renshaw (2002) and Blanton, Stylianou, and David (2009) will help identify types of student utterances and will serve as microtasks as well. Once this list of microtasks is compiled, this study intends to search for patterns in the microtasks performed by each participant in order to determine what sort of role that participant is playing in the group.





4 Results and significance Data collection is still underway at this time. Thus far however, some interesting phenomena have been observed. Discrepancies are arising between how I, as the researcher, would describe some of the events that have transpired and how the students appear to perceive these same events. For instance, in one event it happened that two individuals came to the study lounge around the same time to work on a homework assignment. Their arrival times were staggered and although they sat at the same table, they left an empty chair in between them so that they were not sitting immediately adjacent to each other. Although they engaged in some discussion over one problem the majority of their time was spent in silence as they worked on their individual tasks. From my position as observer, I would not have considered these two individuals to be working “as a group.” On the journal entry for that day however, one of the individuals reported that he “worked in a group” with the other individual that was present.

15TH Annual Conference on Research in Undergraduate Mathematics Education 437 Thus, “the reason why observation is so important is that it is not unusual for persons to say they are doing one thing but in reality they are doing something else” (Corbin & Strauss, 2008, p. 29). So one major implication of this study is its potential to confirm or contradict some of the earlier findings that have been published based on student self-reported data.

On another occasion, during one of the more lively study sessions, a group of 5 individuals came in to work together on a homework assignment. They proceeded to outline what homework problems still needed to be worked on and then split into 2 or 3 subgroups of individuals clarifying their understanding of a problem or checking their answers. They would suddenly converge as one group again, re-assess what problems individuals still needed to work on, then again split into 2 or 3 subgroups, comprised of different individuals than before. This process continued for the duration of their 2 hour study session. What is interesting about this is that there were very few times when the group focused on one homework question all at the same time. There was instead a very natural ebb and flow as students took turns being an authority on a question and aiding peers depending on which question a subgroup was working on. Yet their occasional convergence to assess everyone’s completion status indicates that they were organizing their efforts as an overall group.

Studying the way the group breaks out into subgroups and then reconvenes in addition to understanding the roles that arose in those subgroups and in the overall group, provide a way to describe different study groups of students based on their dynamics and the roles they are composed of. Hence another implication of this study is that it lays the groundwork for comparing groups to assess efficacy by providing a means of describing the group based on dynamics and role composition.

Finally, this study also contributes information regarding what resources students are using to find answers to their questions when the instructor is not around. In addition to simply generating a list of textbooks referenced and websites visited, this study provides a means for assessing how these materials are being used. For instance, from video-recordings and interviews, it can be determined whether a website was used to generate a correct answer to a homework question or whether it was used to gather further information about the concept in order to develop an improved solution strategy. With such knowledge instructors can create assignments that take these material utilizations into account.

5 Questions for discussion There are many questions that could be raised for discussion regarding the methodology, the chosen framework, or even the implications of the findings. I am choosing to focus on the

following questions for discussion:

1) What other perspectives or frameworks may provide an insightful analysis of the data being collected?

2) What information is there to be learned from observing students working alone or silently near each other?

References

Blanton, M., Stylianou, D., & David, M. (2009). Understanding instructional scaffolding in classroom discourse on proof. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective, (pp. 290-306). New York, NY: Routledge.

Blumer, H. (1969). Symbolic interactionism: Perspective and method. Englewood Cliffs, NJ:

438 15TH Annual Conference on Research in Undergraduate Mathematics Education Prentice-Hall, Inc.

Cerrito, P., & Levi, I. (1999). An investigation of student habits in mathematics courses. College Student Journal, 33(4), 584-588.

Charon, J. (2010). Symbolic interactionism: An introduction, an interpretation, an integration.

(10th ed.). Boston, MA: Prentice Hall.

Corbin, J. & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory. (3rd ed.). Thousand Oaks, CA: Sage Publications, Inc.

Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics, 49, 193-223.

Greenman, J. (1993). Math power: Next generation strategies and information, for “success” in the study of mathematics. Retrieved from: http://www.eric.ed.gov/ERICWebPortal/detail?

accno=ED398047 Accessed: 5-14-11.

Hong, E., Sas, M., & Sas, J. (2006). Test-taking strategies of high and low mathematics achievers. The Journal of Educational Research, 99(3), 144-155.

Rohrer, D., & Pashler, H. (2007). Increasing retention without increasing study time. Current Directions in Psychological Science, 132, 354-380.

Swain, H. (1970). How to study mathematics. Washington, D.C.: National Council of Teachers of Mathematics.

The Danish Institute for Educational Research. (1970). Study skills and mathematics achievement. Copenhagen, N. DK: Poulsen, S.

Treisman, P. (1985). A study of the mathematics performance of black students at the University of California, Berkeley (Doctoral dissertation, University of California, Berkeley, 1986).

Dissertation abstracts International, 47, 1641-A.

Wu, H. (1999). The joy of lecturing – with a critique of the romantic tradition in education writing. In S. G. Krantz (Ed.), How to teach mathematics (2nd ed.), (pp. 261-271). Providence, RI: American Mathematical Society.

15TH Annual Conference on Research in Undergraduate Mathematics Education 439 Undergraduate Proof in the Context of Inquiry-Based Learning Preliminary Research Report Todd A. Grundmeier, Alyssa Eubank, Shawn Garrity, Alyssa N. Hamlin and Dylan Retsek California Polytechnic State University, San Luis Obispo This research project explores students’ proof abilities in the context of an inquiry-based learning (IBL) approach to teaching an introductory proofs course. IBL is a teaching method that focuses on student discussion and exploration in contrast to lecture based instruction. Data was collected from three sections of an introductory proofs course, which included 70 students total. Data collection included a portfolio from each student, consisting of their work on every proof assigned throughout the course, as well as each student’s final exam. Contrary to previously published research relating to courses taught in a more traditional lecture based setting, this data analysis suggests that students developed a strong grasp on how to correctly use definitions and assumptions within the context of their proofs. Results also suggest that within the IBL setting, students generally organized their proofs in an efficient, thoughtful, and logical manner.

Key-Words: Proof, Inquiry-Based Learning, Undergraduates, Definitions, Assumptions Current methods for teaching mathematics often consist of lecture-based lessons followed by students completing homework on their own. This classroom structure does little to encourage the development of deep problem solving techniques that will stay with students after they have moved on to higher-level classes. An emerging method to combat these potential problems is Inquiry Based Learning (IBL). Stemming from the Modified Moore Method, IBL focuses on student discussion and exploration in contrast to lecture-based instruction. Instructors typically place a high responsibility on students for their own learning and use leading questions to prompt students’ problem solving. “As mathematics education researchers turn their attention to IBL, evidence mounts that this approach to the teaching of mathematics is ideal for the teaching of proof” (Schinck 2011). Studies conducted by Boaler (1998) and Rasmussen and Kwon (2007), summarized in Schinck’s (2011) article, deduce that IBL students experience mathematics in a way that deepens their comprehension of abstract ideas essential to proofs.



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