13th International Congress on Mathematical Education
Hamburg, 24-31 July 2016
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USING A THEORETICAL PERSPECTIVETO TEACH A PROVING
SUPPLEMENT FOR AN UNDERGRADUATE REAL ANALYSIS COURSE
Annie Selden John Selden
New Mexico State University
We will describe a voluntary 75-minute per week proving supplement for an undergraduate real analysis course, which we studied and facilitated for three semesters. Both the research and the facilitation were guided by our theoretical perspective (Selden & Selden, in press-a, in press-b). We briefly mention our theoretical perspective, where it came from, and how we came to teach the supplement/intervention. After that, we describe the actual teaching of the supplement. Finally, we discuss the effectiveness of the supplement and provide some evidence that it “worked”. Since no major reorganization of the real analysis course itself was undertaken, we feel such a supplement could be implemented practically using an advanced mathematics graduate student by many mathematics departments.
BACKGROUND
For more than ten years we have taught a 3-credit, one-semester graduate course/design experiment
in proof construction. The students were beginning mathematics graduate students who wanted “a little help” with writing proofs. The course was taught from notes of our own design and topics
included sets, functions, real analysis, abstract algebra, and a little topology. In class there were no lectures, rather students presented their proofs and we critiqued them, sometimes extensively. Documentation was first by retrospective notes, but soon changed to field notes taken by a mathematics education graduate student, and eventually videos were made of both the class and our planning sessions during which we reviewed the class videos.
The supplement to the undergraduate real analysis course, described below, was developed when a mathematician colleague, Dr. R, was assigned to teach undergraduate real analysis. She had heard from some mathematics graduate students that our above mentioned “proofs course” was helpful. So, Dr. R, feeling that she had a great deal of content to cover and that her students needed additional help with proving, invited us to teach a one-hour per week voluntary “proving supplement” to her real analysis course. However, she gave us no suggestions as to its teaching. We agreed and developed and taught the supplement for three semesters. We used the theoretical perspective we had developed during the teaching of our “proofs” course and modified it as appropriate to the new circumstances.
THEORETICAL PERSPECTIVE
Our theoretical perspective emerged from the above design experiments combined with ideas from psychology (e.g., Bargh & Chartrand, 2000). We view proving as a sequence of actions, which can
be physical (e.g., writing a line of the proof or drawing a sketch) or mental (e.g., changing one’s focus
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of a proof’s structure” results from using proof frameworks (Selden, Benkhalti, & Selden, 2014; Selden & Selden, 1995) in its construction.
We also divide a proof text into its formal-rhetorical and problem-centered parts. The formal-rhetorical part of a proof is the part that depends only on unpacking and using the logical structure of the statement of the theorem, associated definitions, and earlier results. In general, this part does
not depend on a deep understanding of, or intuition about, the concepts involved or on genuine problem solving in the sense of Schoenfeld (1985, p. 74). Here proof frameworks are especially helpful. The remaining part of a proof is called the problem-centered part. It is the part that does
depend on genuine problem solving, intuition, and a deeper understanding of the concepts involved.
THE TEACHING OF THE SUPPLEMENT
Each week Dr. R selected one homework “proof problem” on which she furnished students extensive written feedback and allowed them to resubmit to improve their grades. She provided us that problem several days in advance of the supplement class period. We then selected, or invented, a theorem whose proof construction sequence was similar to that of the “proof problem” but could not easily
function as a template. To do this, we first proved Dr. R’s assigned proof problem, noting such things
as the first- and second-level proof frameworks, as well as an entire sequence of actions used to produce a proof (McKee, Savic, Selden, & Selden, 2010). After selecting this non-template theorem, we also wrote a very detailed handout similar to one that students would probably construct – a
hypothetical proof construction trajectory. This was given to students at the end of the supplement
class so they could focus on the proof’s co-construction and not have to take notes. We did not lecture or mention most of our theoretical perspective to the students.
We began a typical supplement class by writing our selected, or invented, theorem on the blackboard. (See sample below.) The supplement students were encouraged to first co-construct the formal-rhetorical part of the proof. This consisted of first supposing the hypotheses at the beginning of their proof. Then, after leaving a space for the body of the proof, they would write the conclusion at the end of their proof. Next students would unpack the conclusion and write the relevant definitions, such as that of sequence convergence, on the side board, which had been set aside for “scratch work.”
Then the students would change the notation in the definition to “match” that of the theorem to be
proved. They would then examine this definition to see where to start and end the body of the proof. For example, if the proof problem called for showing a sequence { �}�=1∞ converges to A, then they would write into their proof “Let 0” immediately after supposing the hypotheses, leave a space for the determination of N, write “Let � �”, leave some space, and finally write “Then | �− �| <
ɛ” prior to the conclusion at the bottom of their proof. This would complete the framework and brought them to the problem-centered part of the proof, where some “exploration” or “brainstorming” on the side board would ensue. All writing, finding definitions, etc. was done by the students. The co-construction process, and accompanying discussions, were so slow, even with guidance, that only one theorem was proved and discussed in detail in each supplement class period.
During the entire co-construction process, student discussion and questions were actively encouraged.
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Sample from the supplement
Below is a supplement problem that was designed to be similar in actions to the assigned real analysis
homework “proof problem”. The corresponding actions are numbered in bold (e.g., [1]).
Problem from the Supplement: Paired Homework Problem from the Textbook:
Theorem. Let { �}�=1∞ and { �}�=1∞ be sequences, both
[5] Change notation in definition of convergence
[6] Let � > 0
Effect of the supplement on the students
Both Dr. R and the supplement students believed that the supplement was helpful. In describing the attempts of the supplement students to produce a proof on her tests, Dr. R said “I would see the first
line [the hypotheses], I would see the last line [the conclusion]… I can see the technique… some
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how to unpack the conclusion, know how to use definitions, and know how to use “fixed, but
arbitrary” [i.e., in proofs of convergence and continuity where one begins “Let 0”].
We compared some homework that the supplement students submitted to Dr. R with that of students who did not attend the supplement. The former wrote their proofs in a more concise, clear manner.
Dr. R volunteered that even when the supplement students’ homework was not entirely correct, it was
more clearly structured, so she could provide more useful feedback.
Finally, we feel confident that this supplement/intervention could be used widely. The idea originated with a mathematician whose course was unchanged; an advanced mathematics graduate student could handle its facilitation and did so a few times; the students and Dr. R liked it; and it worked. We also believe the students developed a greater sense of self-efficacy (Bandura, 1995; Selden & Selden, 2013). This is important because otherwise some students may stop trying hard.
References
Bandura, A. (1995). Self-efficacy in changing societies. Cambridge: Cambridge University Press.
Bargh, J. A., & Chartrand, T. L. (2000). Studying the mind in the middle: A practical guide to priming and automaticity research. In H. T. Reid & C. M. Judd (Eds.), Handbook of research methods in social psychology (pp. 253-285). New York: Cambridge University Press.
McKee, K., Savic, M., Selden, J., & Selden, A. (2010). Making actions in the proving process explicit, visible,
and ‘reflectable,’ In Proc. 13th Annual Conference on Research in Undergraduate Mathematics Education. Downloaded June 18, 2014 from http://sigma.maa.org/rume/crume2010/Archive/McKee.pdf.
Schoenfeld, A. H. (1985). Mathematical problem solving.Orlando, FL: Academic Press.
Selden, A., McKee, K., & Selden, J. (2010) Affect, behavioural schemas, and the proving process. International Journal of Mathematical Education in Science and Technology, 41(2), 199-215.
Selden, A., & Selden, J. (2013). Persistence and self-efficacy in proving. In M. V. Martinez & A. Castro Superfine (Eds.), Proc.35th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 304-307), Chicago, IL: University of Chicago at Illinois
Selden, J., Benkhalti, A., & Selden, A. (2014). An analysis of transition-to-proof course students’ proof constructions with a view towards course redesign. In T. Fukawa-Connolly, G. Karakok, K. Keene, & M. Zandieh (Eds.), Proc. 17th Annual Conference on Research in Undergraduate Mathematics Education (pp. 246-259). Denver, Colorado.
Selden, J., & Selden, A. (2014). The roles of behavioral schemas, persistence, and self-efficacy in proof construction. In B. Ubuz, C. Hasar, & M. A. Mariotti (Eds.), Proc. 8th Congress of the European Society for Research in Mathematics Education [CERME-8] (pp. 246-255). Ankara, Turkey: Middle East Technical University.
Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151.
Selden, J., & Selden A. (in press-a). A theoretical perspective for proof construction. Proc. 9th Congress of the European Society for Research in Mathematics Education [CERME-9], Working Group on Argumentation and Proof.
Selden, J., & Selden, A. (in press-b). A perspective for university students’ proof construction. Proc. 18th Annual Conference on Research in Undergraduate Mathematics Education.
