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Inspired by colleagues in English Composition, we wondered whether this drafting technique could be imported for teaching proof-writing. Specifically, we elected to explore whether having students submit the same proof multiple times (drafting) would help them better learn to write proofs. Our guiding research question was: “Does revising proofs lead to improved proof-writing skills for undergraduates in introductory proof-writing settings?” We hypothesized that proof-writing is analogous to essay writing. That is, when students are struggling with difficult mathematical content, their communication of their ideas often becomes unintelligible. Since drafting helps English students develop their ideas, and the writing mechanics automatically improve alongside, perhaps proof drafting would help math students develop their understanding of content, and the mechanics of writing a clear proof might naturally emerge.
Methodology During the fall and spring of 2009 we conducted a pilot study with Linear Algebra students: a control and drafting group both taught by the same professor under otherwise similar conditions. The control group was assigned approximately 15 proofs but none were formally revised. The spring section included 10 proofs (a subset of the control’s) and students were allowed to resubmit each proof up to 3 times total with instructor comments between submissions. Both sections’ (closed-book) finals included the same proof questions, none of which the students had encountered before. This allowed us to determine whether the drafting group could craft better proofs “spontaneously”. Had they learned to write better proofs overall or were any improvements limited within the drafts themselves? The setting was a small liberal arts college in central Texas where Linear Algebra serves as the introductory proof-writing course.
We required a coding method that could adapt to several different contexts: allowing us to compare the end of semester output of one class of students with a different class of students, to analyze individual student growth, and to determine if the types of errors differed between the two Linear Algebra sections. Such a coding scheme would provide a more fine-grained assessment of errors, rather than only measure the relative strength of an attempt.
556 15TH Annual Conference on Research in Undergraduate Mathematics Education We initially consulted coding schemes published by Selden & Selden (2003) and Andrew (2009), and began to classify our proofs. At times neither coding scheme seemed to fit our need.
For example, Selden & Selden’s classification was drawn from an Abstract Algebra course and at times the error categories were overly specific to that context. Andrew’s work was well-suited to a classroom grading context but not to a research comparison project. We attempted to create a coding scheme that would both fit our Linear Algebra context, but possibly apply to a broader range of courses as well.
Our basic process was to jointly categorize each error according to the codes from Selden & Selden’s or Andrew’s work, and to note errors that seemed to not fit anywhere. We then looked for clusters of outliers, or ways in which existing categories might be modified to include these. Then we reviewed whether any originally categorizations might fit better into the newer categories. This was a heavily iterative process.
Once we felt we had created a system that was neither too narrowly specific nor too broad we gathered proofs from other courses (a Modern Geometry and a Number Theory course) and individually coded them. We then came together and compared our individual assessments.
At this stage we did not find any errors requiring new categories, but we did refine our descriptions of the errors types to create better reliability between us. Our final coding scheme is attached but space restrictions prevent us from also including the more narrative guide to the codes themselves.
What emerged was a coding matrix. Rows are assigned to types of proof errors, such as “Misusing Theorem” and “False Implication”. Columns are assigned to a possible attribution of the source of the error: If evidence suggests that the student understands the key ideas, but is incorrectly communicating their ideas, the error would be coded as a Rhetorical error. If it appears that the student misunderstands the content of the statement(s), definitions, or related math, then the error would be coded as a Content error. If the student seems not to understand logical implications, or has grave misunderstandings of what makes a proof “prove”, then the error would be coded as a Fundamental Error.
Preliminary Results Unfortunately, the classes had very few students in any one semester (13 control and 8 drafting) making statistical comparisons difficult. Therefore we continued the research during the 2010-2011 academic year with two more Linear Algebra courses using the same protocols.
Results provided in this proposal are from 2009-2010, although by the conference, we will have results for both years.
The most basic result was that in the drafting course more proofs ultimately were fully correct than in the control group. This is in some ways obvious as the drafting group had three attempts for each proof. While the drafting group engaged with fewer overall proofs, they actually turned in their assignments far more often than in the control group. That is, the students in the control group often just skipped the assignment, while the students in the drafting group turned in each assignment at least once, and usually three times. Therefore engagement with proof writing was increased for the drafting group. In the final exams, we did not measure an overall difference in the nature of the errors, in part because of the small sample size. However, there is some suggestion that the drafting group were more likely to attempt the proofs on their final, mirroring what we saw throughout the semester. The control group skipped proofs at four times the rate of the drafting group on their final. Overall, the drafting group did better because they did more, but when the skipped proofs are controlled for, the results are more similar 15TH Annual Conference on Research in Undergraduate Mathematics Education 557 between the groups. Unrelated to the comparison, it was discovered that students made content errors that derailed proofs far more often than fundamental errors. On the final, for example the ratio of content errors to fundamental was 31:1 (control) and 24:1 (drafting).
Discussion At this stage in the work, it is unclear that either group performed better than the other in any of the categories with the exception of actually doing the work, where the drafting group outperformed the control. From a teaching perspective, the grading burden was roughly the same between the two groups. Drafting appears to lead to higher engagement, but higher engagement has not been shown here to produce better results. The drafting may then improve something else
- possibly perseverance, self-confidence, respect/understanding of the general process - but these qualities were not measured in the study and remain an important area to further investigate.
Course evaluations did suggest that students enjoyed the drafting approach; it would be a nice result if students found pleasure in proof-writing. Finally, in both groups, content errors were far more frequent than structural errors on the final exam. This suggests that students’ weak knowledge of definitions and theorems is an underlying cause of inadequate proofs.
Questions for the Audience
1) We are currently in the process of conducting more formal inter-rater reliability information and seek the RUME audience’s opinion and advice related to our tool.
2) We seek suggestions for the describing of results that would be most helpful to both other researchers and teachers.
3) We seek suggestions for other issues related to proof-writing where we might use our tool for research.
References Andrew, L. (2009). Creating a proof error evaluation tool for use in the grading of studentgenerated ‘‘proofs’’. Primus, 19, 447-462.
Bean, J. (2001). Engaging Ideas: The Professor’s Guide to Integrating Writing, Critical Thinking, and Active Learning in the Classroom. San Francisco, CA: Jossey-Bass.
Dubinsky, E., & Yiparaki, O. (2000). On student understanding of AE and EA quantification. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Issues in mathematics education: Vol.
8. Research in collegiate mathematics education. IV (pp. 239-289). Providence, RI:
American Mathematical Society Selden, A. & Selden, J. (2003). Errors and misconceptions in college level theorem proving.
Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics (Joseph D. Novak, Ed.), Vol. III, Cornell University, July 1987, 457-470. Available as Tennessee Technological University Tech Report No. 2003-3 at http://www.math.tntech.edu/techreports/techreports.html.
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge.
Mathematical Thinking and Learning, 12, 306-336.
15TH Annual Conference on Research in Undergraduate Mathematics Education Definite Integral: A Starting Point for Reinventing Multiple Advanced Calculus Concepts
While recent Realistic Mathematics Education (RME) studies have shed light on students’ abilities to formalize limit conceptions, it remains to be seen how students might make similar progress with other fundamental advanced calculus concepts like continuity, derivative, and integral at a depth required for success in upper-division courses. To address this gap in the literature, we conducted a fourteen session teaching experiment geared at students’ reinvention of the formal definition of definite integral. Our presentation will address the following research questions: 1) Do students’ efforts to formalize the concept of definite integral motivate a need (for them) to formalize the notion of convergence/limit?; and, 2) Once students reinvent a formal definition of definite integral, can they use this formalization as a tool for formalizing other advanced calculus concepts? If so, which concepts?
Keywords: Advanced Calculus, RME, Guided Reinvention, Definite Integral, Limit
Introduction and MotivationThe limit concept is one of the most fundamental ideas in advanced calculus, serving as a conceptual foundation for derivative, integral, and continuity, among other mathematical notions.
Until recently, the vast majority of research on students’ understanding of limit focused on informal misconceptions (e.g., Davis & Vinner, 1986; Monaghan, 1991; Tall & Vinner, 1981;
Williams, 1991) possessed by introductory calculus students, and little was known about what challenges students face in reasoning about limits more formally. In the past few years, however, Realistic Mathematics Education (RME) studies (Oehrtman, Swinyard, Martin, Hart-Weber, and Roh, 2011; Swinyard, 2011) have employed the heuristic of guided reinvention to provide insights into students’ reasoning about limit in the context of reinventing formal definitions.
Swinyard looked at limits of a function and found that students with robust concept images can construct formal concept definitions of limit at infinity and limit at a point. Similarly, students in a teaching experiment conducted by Oehrtman et al. constructed formal concept definitions of sequence convergence, series convergence, and Taylor series convergence. In both teaching experiments, the students’ success in constructing precise limit definitions appears to have been supported by their: 1) ability to shift their reasoning from an x-first perspective to a y-first perspective; and, 2) use of an arbitrary closeness perspective to operationalize what it means to be infinitely close to a point (Swinyard & Larsen, 2011).
In both of the aforementioned teaching experiments (Oehrtman, Swinyard, Martin, HartWeber, and Roh, 2011; Swinyard, 2011), reinvention was supported by first having the pairs of students generate examples and non-examples of limits. For instance, the two students in Swinyard’s study were given the following prompt: “Please generate as many distinct examples of how a function could have a limit of 2 at x=5.” Student-generated examples and non-examples of limits subsequently served as tools for motivating definition refinement. Additionally, in both studies the pairs of students used their formal concept definition of one idea as a template for generating a formal concept definition for a more sophisticated idea. For example, in Swinyard’s study, the students used the formal definition of limit at infinity as a template for constructing a formal definition of limit at a point, and similar findings emerged in Oehrtman et al.’s work.
560 15TH Annual Conference on Research in Undergraduate Mathematics Education The studies discussed above suggest that guided reinvention may be a productive means by which to: 1) gain insight into students’ reasoning about foundational Calculus concepts; and, 2) support students in transitioning to more advanced mathematical thinking. While these studies have shed light on students’ abilities to formalize limit conceptions, it remains to be seen how students might make similar progress with other fundamental concepts like continuity, derivative, and integral at a depth required for success in upper-division courses. Further, the students in the studies conducted by Swinyard and Oehrtman et al. reinvented formal definitions of limit/convergence in an attempt to precisely characterize the functional behavior of the examples and non-examples they had constructed. In other words, the students’ reinvention of these formal definitions was not motivated by a mathematical need in a separate context. Though the students’ reinvention of limit was ultimately successful in both studies, the following question arose: Is there a mathematical context conducive to motivating a need (for the students) to formalize the notion of convergence/limit? To answer this, we conducted a study geared at students’ reinvention of the formal definition of definite integral. Our work aims to address the
following research questions: