Using a Theoretical Perspective to Teach a Proving Supplement for an Undergraduate Real Analysis Course John Selden Annie Selden New Mexico State University Department of Mathematical Sciences
[email protected] [email protected]
ICME-13 TSG 2 July 2016
Undergraduate Real Analysis Undergraduate real analysis in the U.S. is typically a 3-credit (3rd year) math course serving: • Math majors • Pre-service secondary math teachers • Occasional math grad students needing remediation. 2
• Typically, students are required to construct original (new to them) proofs on homework, tests, and a final exam. • The ability to do this is often treated as the major evidence of student understanding.
3
The Invitation • Dr. R, who was teaching real analysis, invited us to teach a proving supplement for her analysis course. • Dr. R had heard from some graduate students that our “proofs course” for beginning graduate students was helpful. • We agreed and developed and taught the supplement for three semesters. 4
• In an interview, Dr. R said that the course tries to be all things to all students, which is virtually impossible in three hours per week. • Dr. R went on to say, “In my opinion students learn to do proofs by doing proofs and not [by] reading them or doing exercises” and this cannot always be done in the normal class setting.
5
• Dr. R gave us no suggestions as to the supplement teaching and we made no suggestions about the course teaching. • We used the theoretical perspective we had developed during the teaching of our “proofs course” and modified its application to the new circumstances.
6
• We implemented, videoed, and analyzed the supplement for 3 semesters. • We worked closely with Dr. R, who often viewed the videos with us.
7
Our “Proofs” Course • A 3-credit, one-semester graduate course/design experiment in proof construction at our university. • The students were beginning mathematics graduate students who wanted a little help with writing proofs.
8
• The course was taught from notes of our own design and topics included sets, functions, real analysis, abstract algebra, and a little topology when time permitted. • In class there were no lectures, rather students presented their proofs and we critiqued them, sometimes extensively, including occasional explanatory comments such as on logic.
9
• Documentation of the proofs course was first by retrospective notes, but later changed to field notes taken by a mathematics education graduate student. • Videos were made of both the proofs class and our planning sessions during which we reviewed the class videos.
10
Our Theoretical Perspective • We view the proving process as a sequence of mental and physical actions. • Some actions, such as looking up a definition, drawing a sketch, or focusing on a particular part of the proof, are not easily noticed or visible in the final written proof.
11
• Making such actions, and the reasons for them, explicit and visible facilitates student reflection and the autonomous enactment of future similar actions. • Some repeated actions in the proving process, when paired with triggering situations, can become automated. We call such (small) lasting mental structures, behavioral schemas. (Selden, McKee, & Selden, 2010)
12
• Enacting behavioral schemas does not require conscious processing and reduces the burden on working memory. This allows working memory to be better applied to the more difficult problem-solving parts of proof construction. (Selden & Selden, 2009)
13
• Changing a detrimental behavioral schema requires more than just understanding the need for the change. (Selden, McKee, Selden, 2010)
• This perspective is consistent with that of psychologists who discuss the automated nature of much of everyday life. (Bargh,1997) 14
Proof Frameworks • We call the beginning and end of a proof, with blank space in between, the first-level proof framework. • It can be determined just from the statement of the theorem given in customary “if…, then…” format.
15
• For example, if the theorem has the form “For all x∊X, if P(x) then Q(x)”, then the first-level framework looks like: Let x∊X. Suppose P(x).
… Therefore Q(x).
QED 16
• By unpacking the meaning of Q(x), one often gets something to be proved in a way that produces a second-level proof framework, which can be written between the beginning and end of the first-level framework. • This often requires remembering (or looking up) and unpacking a definition (e.g., continuous) 17
• Thus, a proof “grows” from both ends, instead of being written from the top down. • A proof need not show evidence of a proof framework to be correct, but proof frameworks help beginning undergraduate students write correct, well-organized, and easy-to-read proofs. (McKee, Savic, Selden, & Selden, 2010) 18
Preparing for the Supplement • Customarily, each week Dr. R selected one homework “proof problem” on which she furnished students extensive written feedback. She allowed them to resubmit this problem to improve their grades. • Well before a supplement session, Dr. R would email us the weekly homework assignment.
19
• A theorem was found (or invented) whose proof could not be obtained by using one of the homework theorems as a template, but whose proof calls on similar actions, perhaps in a different order. [See upcoming example.]
20
The proof of the new theorem was written out (to be given to students at the end of a supplement session). It: • Mentioned all actions (including mental actions) • Carried them out, and • Entered the actions into the partly constructed proof, to produce a sequence of “snapshots” of the partly finished proof construction. 21
• This process was repeated until the proof was completed. • One might term this a hypothetical proof construction trajectory. (Simon, 1995) N.B. Done in this way, the handout for a halfpage proof could take up to 4 (handwritten) pages to record all the actions.
22
Facilitating the Supplement • The weekly supplement was about 75 minutes long. • We began a typical supplement class by writing our selected, or invented, theorem statement on the blackboard. • The supplement students were encouraged to first co-construct the firstand second-level proof frameworks. 23
• This consisted of a student first supposing the hypotheses at the beginning of their proof. • Then, after leaving a space for the body of the proof, another student 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.”
24
• Students volunteered and came up to the blackboard to add something. • We did not write for the students. • Students often need to build a sense of self-efficacy. That is built by either contributing or observing a comparable person contribute. (Bandura, 1995) • Class discussion and questions were encouraged (provided they do not prevent finishing the proof by the end of the supplement session). 25
Sample Paired Supplement Theorem • Theorem from Supplement: Let {an} and {bn} be sequences, both converging to P. If {cn} is the sequence given by cn=an when n is even and cn=bn when n is odd, then {cn} converges to P.
• Theorem from Class Homework: Show that {an} converges to A if and only if {an- A} converges to 0.
26
• Proof of Supplement Theorem: [1] Let {an} and {bn} be sequences and P be a number so that {an} and {bn} converge to P. Suppose {cn} is the sequence given by cn=an when n is even and cn=bn when n is odd. [3] Let e>0. [5] As {an} converges, there exists an Na such that for all i> Na,|ai-P| Nb,|bj-P|N. [8] Case 1: Suppose n is even. Then |cn-P|=|an-P|
