«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»
To address this difficulty, several mathematics educators have suggested alternative formats for presenting proofs, such as using generic proofs (e.g., Rowland, 2001; Malek & MovshovitzHadar, 2011), e-proofs (Alcock, 2009), explanatory proofs emphasizing informal argumentation (e.g., Hanna, 1990; Hersh, 1993), and structured proofs (Leron, 1983). These suggestions have an obvious appeal; if changing the format of a proof can increase students’ understanding of its content, then these alternative proof formats provide a practical way to improve the effectiveness of lectures and textbooks in advanced mathematics courses. When Roy, Alcock, and Inglis (2010) attempted to see if Alcock’s (2009) e-proofs improve students’ comprehension of proofs in a pilot study, they found that students who studied an e-proof performed significantly worse on a post-test than students who studied the same proof from a lecture or textbook. Also, (year) investigated whether Leron’s structured proofs improve students’ comprehension. They found that students who read a structured proof were better than students reading a linear proof at identifying a good summary of the proof, but performed slightly (though not statistically reliably) worse on questions pertaining to justifications within the proof, transferring the ideas from the proof to another context, and illustrating the ideas of the proof using examples.
Moreover, many students complained that structured proofs “jumped around,” requiring them to scan different parts of the proof to coordinate information.
The goal of this study is to examine the extent to which generic proofs will improve student understanding of written proofs they read. A generic proof, also known as a proof by generic example, illustrates general steps of reasoning in terms of a particular mathematical object 594 15TH Annual Conference on Research in Undergraduate Mathematics Education without relying on specific properties of that object. Generic proofs are claimed to aid student comprehension, particularly in number theory, where proofs can be illustrated using appropriately chosen numbers (Rowland, 2001). Malek and Movshovitz-Hadar (2011) used the term “transparent pseudo-proof” (TPP) for the same idea, highlighting that it is not a formal proof but allows one to “see through” the particular case that is illustrated. They examined the impact of exposing linear algebra students to TPPs as compared with exposing them to formal proofs of the same theorems. They found that exposure to TPPs made no difference for “algorithmic” proofs, but for non-algorithmic proofs it improved students ability to (a) reconstruct a proof, (b) explain the main idea of the proof, and (c) construct a similar proof of a new statement. Malek and Movshovitz-Hadar posited that TPPs help students construct meaning for a proof by providing a concrete model of its flow of ideas. They acknowledge that their small sample size—ten students, only three or four of whom read each TPP—and particular domain limit the strength of their interpretations. Our study builds on these results by further investigating the performance of mathematics majors who see either a generic or a traditional proof of the same statement.
2. Theoretical perspective Our model of assessing proof comprehension is based on Mejia-Ramos et al (2012). This model posits that students’ proof comprehension can be measured along seven dimensions: (1) understanding terms and statements in the proof, (2) logical status of statements and proof framework, (3) justification of claims, (4) summarizing via high-level ideas, (5) identifying the modular structure, (6) transferring the general ideas or methods to another context, and (7) illustrating the ideas of the proof with examples.
3. Methods Ten students were interviewed for this study—all were in their fourth or fifth (final) year of a joint B.A. and Ed.M. mathematics education program. Each student met individually with a coauthor of this paper and was presented with two generic proofs. The first was a generic proof of the claim “There are 2n−1 ways to express n as an ordered sum of natural numbers” and the second was a generic proof of the claim “There are infinitely many triadic primes [primes of the form 4k+3].” We will refer to these two proofs as the Partition proof and the Triadic Primes proof, respectively.
First, participants were given instructions on the format of generic proofs to reduce potential confusion due to lack of familiarity. Participants then read the Partition proof until they had studied it to their satisfaction. At this point, participants reported how well they felt they understood the proof (on a scale of 1 to 5), how convincing they found the argument (on a scale of 1 to 5), and whether they were confident that the proof would work in general. The participants then returned the proof to the interviewer and answered open-ended questions about the proof’s content—these questions were based on the model of Mejia-Ramos et al (2012).
Finally, students were asked to comment on the format of the proof, whether they preferred a proof in a traditional format, and if there was anything about the generic proof that aided or hindered their understanding of the content. This procedure was then repeated for the Triadic Primes proof.
15TH Annual Conference on Research in Undergraduate Mathematics Education 595 Our analysis concentrates on (1) the participants’ comments on the format of generic proofs, (2) the participants’ self-reported levels of understanding, conviction, and confidence that the proofs work in general, and (3) the participants’ performance on the assessment questions.
4. Results Overall, participants appeared to have positive opinions of generic proofs. Of the ten participants, nine commented that they could see how generic proofs could improve student comprehension (their own and others’). Particular positive features of generic proofs that were mentioned include reducing abstraction and eliminating confusing notation and jargon. Five of the ten participants expressed some reservations about generic proofs, focusing in particular on whether these were true proofs, sufficiently general, or sufficiently rigorous.
For the Partition proof, the participants on average reported an understanding of 4.1, that they were convinced with a score of 3.89, and 8 of 10 participants were confident of the generality of the proof. Participants answered an average 6.1 out of 10 questions correctly (61%).
For the Triadic Primes proof, the participants on average reported an understanding of 3.1, that they were convinced with a score of 3.49, and 8 of 10 participants were confident of the generality of the proof. Participants answered an average of 3.2 out of 7 questions correctly (45.7%). In a similar study by Fuller et al (2011), a group of six mathematics majors answered assessment questions based on the same model after reading a traditional linear version of the Triadic Primes proof and answered an average of 2 of 7 questions correctly (29%).
5. Discussion The above results provide preliminary evidence that generic proofs can increase comprehension, since students seeing a generic version of the Triadic Primes proof were able to answer more comprehension questions correctly than those seeing a linear version of the Triadic Primes proof. Moreover, students mentioned several ways in which they felt generic proofs were helpful for their understanding. However, more evidence is needed before formulating any conclusions.
We are currently conducting a larger-scale internet study in which math majors from various universities will be shown either a linear or generic version (chosen at random) of either the Partition or Triadic Primes proof. Following this, they will answer comprehension questions (a subset of those used in the interview study). By comparing the performance of students seeing the linear versus generic proofs, we can begin to answer the question of whether generic proofs improve comprehension.
6. Questions for the audience Under what conditions might we see the benefits of generic proofs? Are there any series of studies or interventions that might convince you that generic proofs (or any alternative proof format) are not effective at improving student understanding? What setting and methodology might be appropriate for investigating a longer-term intervention involving generic proofs?
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ReferencesAlcock, L. (2009). e-Proofs: Students experience of online resources to aid understanding of mathematical proofs. In Proceedings of the 12th Conference for Research in Undergraduate
Mathematics Education. Available for download at:
http://sigmaa.maa.org/rume/crume2009/proceedings.html. Last downloaded April 10, 2010.
Fuller, E., Mejia-Ramos, J.P., Weber, K., Samkoff, A. Rhoads, K., Doongaji, D., & Lew, K.
(2011). Comprehending Leron’s structured proofs. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehertman (Eds) Proceedings of the 14th Conference for Research in Mathematics Education (Vol 1), 84-102. Portland, OR.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13.
Harel, G. (1998). Two dual assertions: The first on learning and the second on teaching (or vice versa). American Mathematical Monthly, 105, 497-507.
Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389-399.
Leron, U. (1983). Structuring mathematical proofs. American Mathematical Monthly, 90(3), 174-184.
Malek, A. & Movshovitz-Hadar, N. (2011). The effect of using Transparent Pseudo-Proofs in linear algebra. Research in Mathematics Education, 13, 33-58.
Mejia-Ramos, J.P., Fuller E.,Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79, 3-18.
Porteous, K. (1986). Children’s appreciation of the significance of proof. Proceedings of the Tenth International Conference of the Psychology of Mathematics Education (pp. 392-397).
Rowland, T. (2001). Generic proofs in number theory. In S. Campbell and R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction. (pp. 157Westport, CT: Ablex Publishing.
Roy, S., Alcock, L., & Inglis, M. (2010). Supporting proof comprehension: A comparison study of three forms of comprehension. In Proceedings of the 13th Conference for Research in
Undergraduate Mathematics Education. Available for download from:
http://sigmaa.maa.org/rume/crume2010/Abstracts2010.htm Selden A. & Selden J. (2003) Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for research in mathematics education, 34(1) 4-36.
Thurston, W.P. (1994). On proof and progress in mathematics, Bulletin of the American Mathematical Society, 30, 161-177.
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AbstractWe have conducted interviews with children using integer-related tasks, and we have identified various ways of reasoning that children bring to bear on these tasks.
One product of this work is a collection of compelling video clips. We will share examples of children's reasoning, and the audience will be engaged in discussions of children's reasoning and use of video in instruction. Attendees will receive a free DVD with video clips that can be used with preservice teachers.
Keywords: Children’s thinking, integers, preservice teachers, video
We have conducted interviews with children using integer-related tasks, and we have identified various ways of reasoning that children bring to bear on these tasks. One product of this work is a collection of compelling video clips. We will share examples of children's reasoning, and the audience will be engaged in discussions of children's reasoning and use of video in instruction. Attendees will receive a free DVD with video clips that can be used with preservice teachers.
Theoretical Perspective and Prior Research We approach this research from a children’s thinking perspective. That is, we seek to understand the mathematics through the lens of children’s conceptions (Carpenter, Fennema, Franke, Levi, & Empson, 1999). Research has yielded valuable information regarding children’s mathematical thinking in the whole-number domain, including a framework describing developmental trajectories of students’ strategies and conceptions related to multi-digit arithmetic (Carpenter et al., 1999). This work has benefited elementary teachers and their students. Teachers who participated in professional development focused on understanding children’s mathematical thinking changed their beliefs about teaching and their teaching practices, and these changes were related to improvements in student achievement (Fennema, Carpenter, Franke, Levi, Jacobson, & Empson, 1996).
One compelling way of engaging preservice or practicing teachers with children’s mathematical thinking is through the use of video. Often, those who choose to pursue careers in elementary education are not strong mathematically (e.g., Ball, 1990; Ma, 1999). However, they care about children, and their interest in helping children can be leveraged to get them interested in mathematics via children’s mathematical thinking (Philipp, 2008). Studying children’s mathematical thinking has been found to positively influence prospective teachers’ beliefs about mathematics, teaching, and learning, as well as their mathematics content knowledge (Philipp et al., 2007). Prolonged involvement in professional development with a focus on children’s 598 15TH Annual Conference on Research in Undergraduate Mathematics Education mathematical thinking can help teachers to develop expertise in professional noticing of children’s mathematical thinking (Jacobs, Lamb, & Philipp, 2010).