Using a Theoretical Perspective to Teach

  Hamburg, 24-31 July 2016

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 from the hypothesis to the conclusion or trying to recall a relevant theorem). The sequence of actions that eventually produces a proof is usually considerably longer than the final proof text itself. We have found the concepts of situation-action links, automaticity, and behavioral schemas useful (Selden, McKee, & Selden, 2010; Selden & Selden, in press-a, in press-b). In addition, the

  “clarity 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 converges to A, then they { }

  � �=1

  would write into their proof   “Let ” 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. Towards the end of each supplemental class period, students were given the handout that went through a proof of the supplement theorem and described the actions of a hypothetical proof co-construction trajectory (Simon, 1995). This handout was not identical to the proof the students had produced, but was close enough and sufficiently detailed so that the actions were exposed.

  Sample from the supplement

  �

  �

  ⇐ [2] Suppose {

  [1]

  converges to 0.

  �=1 ∞

  − �}

  �

  − �| < �. [3] Therefore {

  �

  − � − 0| = [11] |

  �

  − �| < �. [8] Let � � be a positive integer. [4,5,6,9] Then |

  � > 0. [10] By definition, there exists a positive integer � such that for all � �, |

  �=1 ∞

  [7] Let

  converges to �.

  �=1 ∞

  ⇒ [2] Suppose { � }

  Proof. [1]

  converges to P.

  �=1 ∞

  − �| < �. [2] Therefore, { � }

  �

  |

  − �| < �, because � � . [3,4,5] In either case,

  �

  |

  − �}

  converges to 0.

  − �| =

  [9] Use definition of {

  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 more obvious than others but most definitely it was [there]

  [11] Algebra [12] Repeat for opposite direction Effect of the supplement on the students

  � } converging to �

  [10] Use definition of {

  | � − �| . . . < �.

  � > 0 [8] Let �=? Let � � be a positive integer. [9]

  [2] Write the first line [3] Write last line [4] Unpack conclusion [5] Write definition of convergence [6] Change notation in definition of convergence [7] Let

  [12] Algebra Actions in the Proving Process: [1] Break into two parts for the “if and only if“

  � is defined.

  [10] Find � using � and � . [11] Break into cases because of how

  � } converging to �

  � } and {

  | � − �| . . . < �.

  [7] Let

  � > 0 [7] Let �=? Let � � be a positive integer. [8]

  Actions in the Proving Process: [1] Write first line [2] Write last line [3] Unpack conclusion [4] Write definition of convergence [5] Change notation in definition of convergence [6] Let

  converges to �.

  �=1 ∞

  { � }

  − � − 0| < �. [3] Therefore

  �

  − �| = [11] |

  �

  � � be a positive integer. [4,5,6,9] Then |

  [8] Let

  � − � − 0| < �.

  � > 0. [10] By definition, there exists a positive integer � such that for all � �, |

  [12]

  �

  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]).

  �

  − �} �=1 ∞ converges to 0.

  { �

  } �=1 ∞ converges to � ⇔

  { �

  Theorem.

  converges to �.

  �=1 ∞

  }

  � , � odd , then { �

  , � even

  �

  = {

  is the sequence given by

  { � }

  �=1 ∞

  }

  �

  be sequences, both converging to �. If {

  �=1 ∞

  }

  �

  and {

  �=1 ∞

  }

  �

  {

  Problem from the Supplement: Paired Homework Problem from the Textbook: Theorem. Let

  Proof. [1] Let

  �=1 ∞

  � is odd. Then [8]|

  is the sequence given by

  [11] Case 2: Suppose

  � − �| < �, because � � .

  |

  [12]

  − �| =

  �

  � is even. Then [8]|

  [11] Case 1: Suppose

  � , then | − �| < �. [7,10] Let � = max{� , � }. Let � � be a positive integer.

  , � odd . [6] Let � > 0. [9] There is a positive integer � so that for every positive integer , if � , then | − �| < �. [9] Also There is a positive integer � so that for every positive integer , if

  � , � even �

  = {

  �

  �=1 ∞

  and { � }

  }

  �

  {

  both converge to P. Suppose

  �=1 ∞

  }

  �

  and {

  �=1 ∞

  }

  �

  be sequences, and P be a number so that {

  �=1 ∞

  ” Three supplement students were interviewed in the subsequent semester. When asked how the supplement had impacted how they construct proofs in their current courses, they replied that they now know where to start a proof, know 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 ”].

  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

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b). A perspective for university students’ proof construction. Proc. 18 Annual Conference on Research in Undergraduate Mathematics Education.

  

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