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

The work that is currently available on student transfer of mathematics to physics has focused primarily on algebraic or calculus skills. While much work has been done in the field of mathematics education to uncover student understanding of trigonometry concepts, how these understandings impact student performance in physics has not yet been explored in depth. In particular, there seems to be some room for clarification on how student understanding of the multiple representations of trigonometric functions may impact their performance on graphing modified forms of the trigonometric functions such as those that model simple harmonic motion.

This study was inspired by a project performed with a physics educator that was interested in assisting students graph position versus time functions modeling simple harmonic motion. Using the framework provided by Moschkovich, Schoenfeld, and Arcavi (1993) to identify which graphical representation perspective a student possesses, this study sought to uncover which perspective was necessary or sufficient for students to be able to analyze the trigonometric equations that model simple harmonic motion. Students were also given instructional activities designed to help them understand the change from angular input of traditional cosine and sine functions to the expected input of time in the physical situation. Thus we sought answers to the

**following questions:**

1) Which perspective of the graphical representations of sine and cosine, if either, is !!

sufficient for the ability to correctly graph ෆ(ෆ) = 2ළ cos !! ෆ ?

!!

2) What connections does the student make between graphing ෆ(ෆ) = 2ළ cos !! ෆ and the motivation for the switch from angular measure as input to time as input?

Here we take “correctly graphing” to mean that the student is able to create the correct shape, show appropriate scaling, and accurately label intercepts and maxima and minima of the function.

2 Literature and Framework The work of Ozimek, et. al. (2004) suggests that students can successfully transfer their knowledge of trigonometry to applications in physics. However, these findings only confirmed transfer in the cases where the physics problems the students were given mirrored specific instances of right triangles.

Hence a review of the literature on student understanding of trigonometry was in order to address what aspects of student understanding of trigonometry impact student ability to work with equations modeling simple harmonic motion and oscillating behavior. Unfortunately, though much work has been done on student understanding of trigonometry none seems to focus specifically on students’ abilities to switch between function, tabular, and graphical representations. Luckily, looking at the available literature on students’ abilities to switch between representations at a more general level yields a useful framework for analyzing student work and understanding of functions.

15TH Annual Conference on Research in Undergraduate Mathematics Education 431 The framework developed by Moschkovich, Schoenfeld, and Arcavi (1993) has two dimensions. The first dimension encompasses the means available for representing the functions, namely algebraic, graphical, and tabular. The second dimension addresses the perspective, object or process, from which a function is viewed. In the object perspective a function or its graph are thought of as “entities” that can be picked up and rotated or translated whereas in the process perspective a function is thought of as linking x and y values (Moschkovich, et. al., 1993). These distinct perspectives are well applied in the case of sine and cosine where students are often encouraged to memorize a table of values, a process perspective, and a portion of the graph that can be replicated due to the periodic nature of the functions, an object perspective.

3 Data collection and methodology for analysis Data was collected in two phases. The first phase was conducted during the Spring 2011 semester with students enrolled in the second course of a two course algebra-based physics sequence. At the beginning of the semester the students were presented with a physics lab designed to review several trigonometric topics that would arise throughout physics that semester. One activity on the lab was a treatment for helping students connect angular measure with time. These labs were collected and scans were made so that the labs could be returned to the students. Based on answers to these trigonometry labs, a round of task-based interviews was conducted with 3 participants. The second phase was conducted during the Summer 2011 semester with students enrolled in the first course of the two course algebra-based physics sequence. During this phase only task-based interviews were conducted. The treatment for connecting angular measure to time and emphasizing their linear relationship was given in the form of a task during the interview.

**During both phases each interview contained the following four tasks:**

1) Sketch the graph of a basic function. This was to establish the participant’s ability to connect an algebraic representation of a function such as a parabola or line with its graphical representation. Here participants were allowed to proceed in whichever manner they chose, though it was anticipated that they demonstrate a process perspective. We wanted to be sure that they recognized the process perspective as a valid method for producing a graphical representation.

2) Sketch the graphs of ෆ = sin ෆ and ෆ = cos ෆ. This was to capture their natural perspective regarding the graphical representations of sine and cosine, to determine whether they first approached using an object perspective or process perspective.

Participants were then prompted to attempt to use the perspective not chosen in order to establish whether they were capable of both.

3) Sketch a position versus time graph to model a given physical situation, namely a glider on a track attached to a spring. This was to establish their inherent comfort with the situation being modeled and to determine the level to which they were comfortable with their intuition.

!!

4) Sketch the graphical representation of ෆ ෆ = 2ළ cos !! ෆ. It was during this task that we hoped to see how the student’s graphical representation perspective worked in combination with the angular measure treatment to enable the student to sketch this graph with greater facility.

All interviews were transcribed and open-coded using the framework developed by Moschkovich, Schoenfeld, and Arcavi (1993) in order to determine which perspective was used by the student during a task.

432 15TH Annual Conference on Research in Undergraduate Mathematics Education 4 Significance and directions for further research All of the students interviewed were able to sketch and correctly label a position versus time graph to model the oscillations of a glider attached to a spring. Thus it does not seem that !!

the difficulties they encountered in the final task, graphing ෆ ෆ = 2ළ cos ෆ, are due to a !!

lack of understanding the physical situation. However, the students may not have associated the equation as a representation of such a physical situation.

The students seemed to be primarily relying on their understanding of sine and cosine as !!

functions in order to produce the graph of ෆ ෆ = 2ළ cos ෆ. Based on their responses to the !!

first task, sketching the graph of a linear or quadratic function, all students were capable of using a process perspective in order to produce the graph. When it came to graphing sine and cosine as functions, the participants clearly split on their perspectives. Three of the participants were able to use both a process perspective and object perspective in discussing the graphs. Two participants only possessed an object perspective and were unable to identify intercepts, maxima, and minima. The remaining two participants had no object perspective of the graphs of sine and cosine and were only able to demonstrate a process perspective using integer inputs.

Based on student responses and preliminary analysis of the final task, it seems neither the object nor process perspective is sufficient on its own for students to be successful. Those individuals that showed only a process perspective, continued to use a process perspective using integer inputs rather than more informed inputs. The participants that showed a preference for an object perspective easily recognized what shape the graph should have and identified the new amplitude, but froze in identifying the new period and often wouldn’t even sketch the shape.

Even the ability to switch between object and process perspectives wasn’t enough to guarantee success. Those individuals started by identifying shape and amplitude, but didn’t initially sketch the cosine shape and resorted to a process perspective of inputting integers in order to try to determine how the period of the function changed.

The main implication of these findings is that there seems to be some element lacking.

The students rarely referred to the instructional motivations meant to help them identify the new period. Either a new motivation technique, some sort of “informed” process perspective, or an improved object perspective where the student feels more confident in his or her ability to correctly scale the base function appears to be needed.

One way this could be addressed is that mathematics instructors could spend more time emphasizing the validity of multiple representations of functions and how to translate between them. Students that froze on an object perspective were often reluctant to use a process perspective. This result is confirmed by Leinhardt, Zaslavsky, and Stein (1990) who found that students seem to steer clear of process perspective as the focus in their classroom instruction is primarily on using an object perspective. Another issue that has arisen as a result of these findings is that another method is needed for guiding students to understand the change in input from angular measure to time. The two methods investigated during his study seemed to have no lasting impact.

5 Questions for discussion

1) As an alternate motivation, what about introducing the translation from angular measure in radians as input to time in seconds or minutes as input as the conversion of units?

2) What other aspects of student knowledge, besides graphical representation perspective, should be taken into account when observing students translating to a graphical representation of a function?

15TH Annual Conference on Research in Undergraduate Mathematics Education 433 References Cui, L., Rebello, S., Fletcher, P., & Bennett, A. (2006). Transfer of learning from college calculus to physics courses. Proceedings of the National Association for Research in Science Teaching 2006 Annual Meeting, San Francisco, CA.

Leinhardt, G., Zaslavsky, O., & Stein, M. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60, 1-64.

Moschkovich, J., Schoenfeld, A., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relationships and connections among them. In T.

Romberg, E. Fennema, & T. Carpenter (Eds.). Integrating research on the graphical representation of functions (pp. 69-100). Hillsdale, NJ: Erlbaum.

Ozimek, D., Engelhardt, P., Bennett, A., & Rebello, N. (2004). Retention and transfer from trigonometry to physics. In J. Marx, P. Heron, & S. Franklin (Eds.), 2004 Physics Education Research Conference (pp. 173 - 176). Sacramento, CA: American Institute of Physics.

434 15TH Annual Conference on Research in Undergraduate Mathematics Education Title: What do Students do in Self-formed Mathematics Study Groups?

Category: Preliminary Research Report Abstract: While it is widely taken as understood that students should be spending additional time outside of the typical undergraduate mathematics classroom studying, little is known about how students spend that study time. Currently available research has investigated how much time students spend outside of the classroom studying, whether they work alone or with others, and what materials students keep on hand while studying. However all of these studies rely on self-reported data in the form of interviews or anonymous surveys. This ethnographic study undertakes to expand our understanding of what activities students are engaged in when they say that they are “studying” through direct observation, journal entries, and interviews. Particular attention is given to how students study together in groups and how students make use of the materials they bring with them for studying purposes.

Keywords: group work, study habits, discourse analysis

**Author information:**

Gillian Galle Mathematics and Statistics, University of New Hampshire get7@wildcats.unh.edu 15TH Annual Conference on Research in Undergraduate Mathematics Education 435 1 Introduction and research questions The majority of the university system is set up with the didactic contract that students are expected to spend up to 3 hours per hour spent in class studying outside of the classroom context (Wu, 1999). In fact, the predominant advice both from instructors and study guides (Greenman, 1993; Swain, 1970) to a student that is struggling in a mathematics course is to spend more time outside of the classroom working through the material to improve his or her understanding. This advice is often supplemented with suggestions to work with peers from the class. However, when this advice is given little instruction or guidance is given regarding how to effectively work on the material with peers or what resources to have on hand while studying.

In order to provide more accurate and valuable advice to improve student study habits, we need a better idea of what goes on when students are working together outside of the classroom beyond the reach of the instructor. Though there is some literature that addresses students working in groups, these groups are often situated within classrooms where a goal has been established by the instructor and the instructor is available as a resource to answer questions. There is nothing to inform us on how students set their own goals as a study group or how they proceed without an instructor nearby to keep them on task.