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] JMM 2016
Undergraduate Real Analysis Undergraduate real analysis 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 an exam. • The ability to do this is often treated as the major evidence of student understanding.
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The Need for a Proving Supplement • It appears that in a typical undergraduate real analysis course, at least a few (often many), students need help with proving.
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• Also, the NCTM Standards emphasis on “proofs and reasoning” throughout the school math curriculum suggests that preservice secondary math teachers may eventually be called on to teach aspects of proof construction. • Often real analysis teachers feel class time should be devoted to content, rather than the teaching of the proof construction process.
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A Practical Supplement In order for the kind of supplement described here (involving what we call co-construction) to be widely usable it should : • Not require any change in the analysis course • Not require supplement homework (that might compete with normal course homework) 6
• Not require additional faculty teacher time (except to occasionally monitor/supervise student facilitators) • Not carry credit or grades (except possibly as a lab).
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The Invitation • Dr. R, who was teaching real analysis, invited us to teach a supplement for her analysis course. • Dr. R had heard from some graduate students that our “proofs course” for beginning graduate students was helpful. 8
• 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. 9
Our Theoretical Framework • This is useful in understanding how we guide the students. • 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. 10
• Making such actions, and the reasons for them, explicit and visible facilitates 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)
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• 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 difficuolt problem-solving parts of proof construction. (Selden & Selden, 2009)
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• 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) 13
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.
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• 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 15
• 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.
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Preparing to Facilitate a Supplement Session • Well before a supplement session, the facilitator(s) should obtain the cooperating analysis teacher’s weekly homework assignment.
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• A theorem should be found (or invented) whose proof cannot 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.]
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The proof of the new theorem should be written out in a special way that: • Mentions all actions (including mental actions) • Carries them out, and • Enters the actions into the partly constructed proof, to produce a sequence of “snapshots” of the partly finished proof. 19
• This process is repeated until the proof is completed. N.B. Done in this way, a half-page proof might take up to 4 (handwritten) pages to record the actions.
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There are two uses of the above record of the proof construction: • A copy should be given to each student when he/she leaves the facilitator’s session. • The facilitator(s) should use it as a guide for suggestions for the students’ proof co-construction.
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Facilitating a Supplement Session • Students should bring their books and paper for personal scratch work. • They need not prepare for the session and should not take notes. • Pre-prepared notes on the (hypothetical) proving actions should be provided as they leave. [See sample attached to our handout.] 22
• The supplement should be about 75 minutes long. • The facilitator(s), guided by the preprepared notes, should ask the students to contribute small parts of the proof. • Each student should come up to the blackboard and add something after volunteering. 23
• It is important for the facilitator(s) not to write for the student. • Students often need not build a sense of self-efficacy. That is built by either contributing or observing a comparable person contribute. • Facilitator(s) are not normally as comparable as other students in the class. 24
• Class discussion and questions should be encouraged (provided they do not prevent finishing the proof by the end of the supplement session).
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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.
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• 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|
