«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»
Weber, 2009), we note these studies have investigated students’ reasoning within a single mathematical domain. Further, a general finding from the learning science literature on learning styles is that researchers and teachers alike are frequently too quick to assign students with learning styles based on limited evidence. In this talk, we present a case study of a student who displayed a strong semantic reasoning style when working on proving tasks in calculus, but only limited semantic reasoning when completing proving tasks in linear algebra. From his interview comments, we conjecture how the nature of the task, the way in which his courses were taught, and his familiarity with the material strongly influenced the proof processes he used. We then argue that there are factors researchers should consider before assigning proving styles to individual students.
Hence, our research question is: (1) Do students’ proving styles depend on the domain in which they are working? (2) What factors influence the proving processes that an individual student uses on a proof construction task?
Methods As a pilot study for our grant, we interviewed 12 undergraduate mathematics majors or recent graduates to participate in our study. These students met individually with the first or fourth author for two 90-minute interviews. During each interview, participants were instructed to “think aloud” while completing seven proof production tasks. They were permitted to write scratch work, but also told that they should write up their final solution as if it were to be graded as an exam question in a mathematics class. During their proof construction, participants were given access to definitions of every question involved in the proof and access to a graphing application on a computer. Participants were given ten minutes to write each proof. Afterwards, participants were asked general comments about their proving processes.
Seven proofs came from linear algebra and seven came from calculus. Proofs varied in terms of difficulty (one easy, three medium, three hard) and in terms of how accessible they were through semantic and syntactic reasoning (two semantic tasks, two syntactic tasks, and three neutral tasks). The labeling of tasks came from interviews with mathematicians and piloting the materials with roughly 20 students.
We coded participants tasks based on the nature of the semantic reasoning that was used. We coded every instance in which a participant created, referred to, or reasoned from an informal representation of a mathematical concept. We coded each as representing a concept, understanding a concept, recalling a definition, illustrating reasoning to an interviewer, verifying that a claim or the theorem was true, and seeing why the theorem was true. (We hope to receive feedback on this coding scheme from the audience).
498 15TH Annual Conference on Research in Undergraduate Mathematics Education This analysis focuses on the reasoning processes of one student, with the pseudonym Kevin, as he was unique in showing a strong propensity for semantic reasoning in calculus, but did not often use this reasoning in linear algebra.
Results In all seven of his calculus tasks, Kevin used semantic reasoning. They often played an important role in his reasoning. For instance, in five of his proof productions, he used graphs to see why the statement was true. Further, in Kevin’s comments on these tasks, he remarked that he viewed the task of proving as essentially “translating my intuition”, or to use the language of Raman (2003), he viewed the task of proving as generating key ideas.
In the linear algebra tasks, Kevin’s use of semantic reasoning was more sparse.
For only two of the tasks did Kevin generate a graph or example. For these tasks, the role of these informal representations was relatively minor. For instance, in one task, Kevin tested his recall that det(AB)=det(A)det(B) with two sample matrices (a fact that was not used in Kevin’s final proof of the statement), and sketched an example of a non-singular matrix before immediately crossing it out. When describing his proof processes, Kevin spoke of carefully reasoning from definitions to reach desired conclusions.
In his post-task comments, Kevin indicated that his differences in reasoning in calculus and linear algebra was not due to the conceptual differences in the two domains.
Rather, he cited three differences. First, he noted that he understood calculus better so he had more access to graphical interpretations of relevant concepts. Second, his real analysis professor illustrated every concept both graphically and with diagram, while his linear algebra professor focused more on procedures. Third, the time constraints of our study prevented Kevin from exploring concepts in linear algebra conceptually. He indicated, that given unlimited time, he preferred to explore all concepts conceptually, but given the time constraints of our study (and the time constraints of most classroom examinations!), he had to rely on syntactic reasoning to get a solution efficiently, even though he valued the understanding engendered by a semantic proof production more.
Discussion These results illustrate that students’ proving styles may be a function of the domain in which their proofs are situated. As most research assigning proving styles to individual students asks them to construct proofs within a singular domain, we suggest that researchers either qualify students’ proving styles to that domain or ask students to ask write proofs in multiple domains. Also, the results provide an existence proof that, at least in some cases, students’ proving processes are directly influenced by how they are taught, their familiarity with the content being studied, and how the external constraints (such as time) placed upon them.
Questions for the audience Methodologically, how can we determine if students are semantic or syntactic reasoners?
How can we improve our coding scheme for inferences based on semantic reasoning?
Are there any suggestions for how we can improve the direction of our project?
15TH Annual Conference on Research in Undergraduate Mathematics Education 499 References Alcock, L. & Inglis, M. (2008). Doctoral students' use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69, 111-129.
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), 1-32.
Alcock, L., & Simpson, A. (2005). Convergence of sequences and series 2: Interactions between nonvisual reasoning and the learner’s beliefs about their own role.
Educational Studies in Mathematics, 58 (1), 77-100.
Alcock, L. and Weber, K. (2010a). Undergraduates’ example use in proof production:
Purposes and effectiveness. Investigations in Mathematical Learning, 3(1), 1-22.
Alcock, L. and Weber, K. (2010b). Referential and syntactic approaches to proving: Case studies from a transition-to-proof course. Research in Collegiate Mathematics Education, 7, 101-123.
Burton, L. (2004). Mathematicians an enquirers. Dodrecht: Kluwer Academic Publishers.
Moutsios-Rentzos, A. (2009). Styles and strategies in exam-type questions. In Tzekaki, M., Kaldrimidou, M. & Sakonidis, C. (Eds.). Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 1.
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Pinto, M., & Tall, D. (1999). Student constructions of formal theory: Giving and extracting meaning. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 65– 73). Haifa, Israel: PME.
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Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52, 319-325.
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.) Advanced Mathematical Thinking (pp. 65-81). Dodrecht: Kluwer Academic Publishers.
Weber, K. (2009). How syntactic reasoners can develop understanding, evaluate conjectures, and construct counterexamples in advanced mathematics. Journal of Mathematical Behavior, 28, 200-208.
Weber, K. and Alcock, L. (2004) Semantic and syntactic proof productions. Educational Studies in Mathematics, 56(3), 209-234.
500 15TH Annual Conference on Research in Undergraduate Mathematics Education Weber, K., Alcock, L. & Radu, I. (2007). Proving styles in advanced mathematics. In Proceedings of the 10th Conference for Research in Undergraduate Mathematics Education. Available for download: http://cresmet.asu.edu/crume2007/eproc.html.
15TH Annual Conference on Research in Undergraduate Mathematics Education 501 Using Community College Students’ Understanding of a Trigonometric Statement to Study Their Instructors’ Practical Rationality in Teaching Preliminary Research Report Vilma Mesa, Elaine Lande, Tim Whittemore University of Michigan Abstract This preliminary research report documents work in progress from a study that seeks to understand community college trigonometry instructors’ practical rationality regarding instructional decisions, using students’ understanding of trigonometry notions as a trigger for the conversations about those decisions. Students’ answers to one set of tasks were used to prompt discussions between two full-time instructors. We describe the task, the students’ responses, teachers’ anticipations of students’ difficulties, and their reactions and interpretations of students’ understanding of the task. The process provides insights into the nature of the obligations that instructors respond to, and instructors’ impressions of the role of curriculum and the demands that it imposes on teachers and students when pressures for increasing transfer rates are high. As a preliminary research report, we seek guidance from the audience on furthering the analyses of teachers’ data.
Keywords: practical rationality, teaching, trigonometry, community colleges, students’ conceptions 502 15TH Annual Conference on Research in Undergraduate Mathematics Education The purpose of this study is to investigate the practical rationality for decisions that teachers make in teaching (Herbst & Chazan, 2011) in the context of community college mathematics (Mesa & Herbst, 2011). We sought to create a dissonance between what instructors thought their students understood about trigonometry and what the students revealed through questionnaires and in-depth interviews and use the dissonance to generated discussions between teachers that
would allow us to answer the following questions:
1. What are the obligations thatStudents’ experience as theyateach trigonometry?
Using Community College teachers Understanding of Trigonometric Statement to Study Their Instructors’ Practical Rationality in Teaching
2. How do teachers manage those obligations in real time?
The study was not designed to alterPreliminary Research Report as an opportunity for us to teachers’ practices but rather understand how teachers make sense of the the practical rationality for decisions that teachers The purpose of this study is to investigate decisions they make when they are confronted with information about what their students understand about topicscommunity in their courses. We make in teaching (Herbst & Chazan, 2011) in the context of they teach college mathematics describe& Herbst, the students’ responses, teachers’ anticipations of students’ difficulties, andtheir (Mesa the task, 2011). We sought to create a dissonance between what instructors thought teachers’ reactions and interpretations of students’ understanding of the task. Data collection is students understood about trigonometry and what the students revealed through questionnaires described within each section. study was not designed to generate a change in teachers’ practices and in-depth interviews. The The Task as an opportunity for us to understand the obligations that teachers experience as they but rather teach and how they manage those obligations. We gathered students’ interpretations and We gatheredof various interpretations ideas, in particular various trigonometric ideas, in particular knowledge students’ trigonometric and knowledge of those about the statement below, which those about thetrigonometry textbook that was beingaused by several of our participating used appeared in a statement below, which appeared in trigonometry textbook that was being
by several of our participating instructors: