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

**1. Research Goals We will investigate the following questions:**

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1) What is the relationship between the written, spoken and gestural text of a lecture and the students’ written text of their notes? What do students include in their notes and how do they decide what to write?

2) Viewing the lecture as a text, who is the implied reader of this text, the actual readers and what is the relationship between them?

2. Theoretical Perspective We will adapt the theoretical framework described by Weinberg and Wiesner (2011), who applied ideas of literary criticism to describe factors that impact the ways students read and understand mathematics textbooks. The two concepts we will use are those of the implied reader and reading models.

The implied reader of a mathematics text is the “embodiment of the behaviors, codes, and competencies that are required for an empirical reader to respond to the text in a way that is both meaningful and accurate” (Weinberg & Wiesner, 2011 p. 52). The behaviors of the implied reader are “sequences of actions (physical or mental) enacted by the implied reader” (Weinberg & Wiesner, 2011 p. 52). For example, the implied reader of the lecture might actively think about previous examples and theorems and trying to make connections with what they are currently observing. The codes of the implied reader are the ways that the implied reader interprets the language, symbols, words, gestures (etc.) that are part of the lecture. For example, a lecturer might say (and write): “Let G be a group….” ; the implied reader might interpret the word “group” as an algebraic object and recognize G as the standard symbol to represent it.

Finally, the competencies are the “mathematical knowledge, skills, and understandings” to understand the text (Weinberg & Wiesner, 2011 p. 55). For example, the implied reader might know what a group is and be familiar with the axioms that are related to the mathematical context in which the group is being discussed.

The empirical reader of a mathematics text is the person who is attempting to interpret the text—in this case, the students in the class. The students’ reading models—their strategies for reading and and beliefs about their role in a classroom shape the transaction between the students and the lecture. Weinberg and Wiesner (2011) describe two key types of beliefs that affect students’ strategies. Students who have a text-centered model “believe that they are receivers of meaning” (Weinberg & Wiesner, 2011 p. 56); they may be likely to try to transcribe aspects of the lecture as literally and “accurately” as possible for later memorization and replication. In contrast, students who have a reader-centered model if they think of their participation in the lecture—even if it is passive—as a meaning-making process; these students may be likely to be selective about the aspects of the lecture that they record and construct their own interpretations of important aspects of the lecture.

3. Methods Data is being collected on an on-going basis in an introductory abstract algebra class during the Fall 2011 semester. The instructor self-identifies as a traditional teacher who uses lecture as his principal in-class pedagogical technique and maintains nearly complete control over the content.

Classroom observations. Approximately seven class meetings will be observed and video recorded throughout the Fall 2011 semester. Field notes will focus on the relationship between gestures, speech, and the text written on the board. Thus far two class meetings have been observed and recorded. The observations will be selected to capture a variety of typical 426 15TH Annual Conference on Research in Undergraduate Mathematics Education lecture content (including the introduction of a new concept, a proof writing episode, and working of a homework type problem) and a variety of presentation formats (including episodes where the instructor discusses an idea without writing on the board and an episode where the instructor writes ideas on the board non linearly. All of the instructor’s talk and board work during the relevant portion of the lesson will be transcribed—including the order in which the board work was developed—and all gestures will be described. We will construct tables with 3 columns to describe the written text, spoken text and described gestures.

Analysis of classroom data. Data will be coded in two distinct manners that will then be synthesized during analysis. First, each piece of text will be analyzed to explicitly describe the mathematical meaning that it conveys to an expert reader, including a description of any explicit links to mathematical ideas from outside the lecture. Each of these pieces will be marked for the expert observers’ perception of the instructor’s emphasis of importance. In order to describe the implied reader, we will create a set of notes that capture a possible “expert” observer’s explanation of content, mental habits, and required competencies for learning advanced mathematics that incorporates all aspects of the text. Some specific aspects of analysis include describing the requirements in terms of symbols, proof-skills, knowledge of examples and properties, and the various verbal, symbolic, and gestural codes. Finally, we will compare and contrast the aspects of the implied reader across the different aspects of the text (written, spoken and gestural) to describe the barriers and supports to understanding the mathematics that these different aspects may provide.

Data from students. Seven students will participate in this study. We will collect their classroom notes from the observed course meetings and assess their understanding of the relevant content with a written instrument. We will conduct semi-structured interviews with each student them about their note-taking habits and beliefs about their role in a lecture-based classroom, and ask them to give a short summary of how they use their notes as part of doing homework and preparing for exams.

Prior to conducting each interview, we will identify excerpts where the student’s notes differed from what the instructor wrote on the board, what the instructor said, or the gestures that the instructor used; these excerpts will be used during the interview to prompt discussion. The

**interview will include the following questions:**

1. How do you take notes in this class? For you personally, what is the purpose of taking notes in this class?

2. How do you plan to use your class notes?

3. [Using a video clip where the instructor attempted to convey a difficult idea or example:] Was there something in [this video clip] that you felt was difficult for you to take notes on? 4. [Using a video clip where there was (or wasn’t) something in the lecture—verbal, written, or gestural—that wasn't recorded in the notes:] How come you didn't (or did) record this aspect of the lecture in your notes?

5. [Using an example where the board work isn’t developed linearly:] What aspects of this part of the class/lecture do you think are significant? What aspects did you decide to capture in your notes? Analysis of student data. The analysis of the students’ notes will focus on differences and similarities between the text of the lecture (as described by the “expert” observers) and the students’ notes. We will identify the implied reader of the lecture and use this construct to try to 15TH Annual Conference on Research in Undergraduate Mathematics Education 427 understand some of these differences. For example, the course instructor may rely on various proof heuristics—such as an “onto proof”—and we will describe whether these heuristics are part of the implied reader, whether the students’ notes are guided by this heuristic, and whether the students’ notes and their interview responses indicate that the underlying codes are meaningful to them. In addition, we will compare the students’ notes across types of episodes (definitions, examples, etc.) and during instances where the board work is not developed linearly to understand and characterize the implied reader.

We will also characterize the students’ reading models and use these to interpret patterns in their note-taking habits and the extent to which their notes match the written part of the lecture-text.

4. Proposed discussion questions:

1) What would you most want to know about how students take notes?

2) What are the benefits and drawbacks of framing this study using the ideas of the implied reader and reading models? What critiques would you offer?

3) Is this a fruitful line of inquiry for mathematics education? Given our interest in nonlecture-based classes, is the RUME community interested in this focus of research?

4) The implied reader of a mathematics lecture may very well be different from the empirical readers. As a result, using this framework for analysis will very likely portray lecturers as “out of touch” with their students. What kinds of things should we be thinking about and doing in order to help ensure the continued engagement of our colleagues with research that they might see as adversarial to their teaching practices?

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**References:**

Bligh, D. (2000). What’s the use of lectures? San Francisco: Jossey Bass. Davidson, N., & Gulick, F. (1976). Abstract algebra, an active learning approach. Boston: Houghton Mifflin. Dubinsky, E., & Leron, U. (1994). Learning abstract algebra with ISTEL. New York: Springer Verlag. Johnstone, A. H., & Su, W. Y. (1994). Lectures a learning experience? Education in Chemistry, 31(1), 75 76, 79. Kiewra, K. A., DuBois, N., Christian, D., McShane, A., Meyerhoffer, M., & Roskelley, D. (1991). Note taking functions and techniques. Journal of Educational Psychology, 83 (2), 240 245. Larsen, S. (2004). Supporting the guided reinvention of the concepts of group and isomorphism: A developmental research project. Unpublished Doctoral Dissertation, Arizona State University. Mejia Ramos, J., Weber, K., Fuller, E., Samkoff, A., Search, R., Rhoads, K. (2010). Modeling the comprehension of proofs in undergraduate mathematics. Proceedings of the Conference on Research in Undergraduate Mathematics Education, Raleigh, NC. http://sigmaa.maa.org/rume/crume2010/Archive/Mejia Ramos%20et%20al..pdf Weinberg, A. & Wiesner, E. (2011) Understanding mathematics textbooks through reader oriented theory. Educational Studies in Mathematics 76 (1), 49 63. Weinberg, A., Wiesner, E., Benesh, B., & Boester, T. (2011) Undergraduate students' self reported use of mathematics textbooks. PRIMUS. Retrieved from http://www.tandfonline.com/doi/abs/10.1080/10511970.2010.509336.

Category: Preliminary Research Report Abstract: Many studies exist on student difficulty transferring mathematical knowledge to physics, on student understanding of trigonometry, and student ability to create graphical representations of functions. However, there are no studies that exist in the intersection of these issues. This study sought to explore student understanding of simple harmonic motion by examining how their approach to graphing the sine and cosine functions impacted their ability to graph sine and cosine based models of simple harmonic motion. The findings of this study conclude that neither an object perspective or process perspective of the graphical representations of sine and cosine is sufficient for the ability to graph simple harmonic motion modeled based on cosine. There seems to be an element missing, a connection students must make between the changes in input type, that needs to be addressed in order for students to create a graphical representation of a cosine-based simple harmonic motion model.

Keywords: student understanding, trigonometric functions, simple harmonic motion, graphical representations

**Author information:**

Gillian Galle Mathematics and Statistics, University of New Hampshire get7@wildcats.unh.edu 430 15TH Annual Conference on Research in Undergraduate Mathematics Education 1 Introduction and research questions Many students find physics, both algebra and calculus-based, to be a challenging subject.

This may be in part due to some of the difficulties that students experience when they are expected transfer their mathematical knowledge to models of various physical concepts, such as simple harmonic motion. Several studies have sought to describe these difficulties by studying students’ abilities to transfer knowledge of algebra and calculus concepts to applications within physics (Cui, et. al., 2006; Ozimek, et. al., 2004). Students that are enrolled in algebra-based physics may face additional difficulties with this transfer the equivalent of a high school algebra course is often the only prerequisite to taking the course.