Mathematics Colloquium
Date:Lincoln NE 68508
Affiliation: New Mexico State University
Local Host: Yvonne Lai
Title: Helping Students with Proving: Two Teaching Experiments
Additional Public Info:
First, we will discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught at least eight times since Fall 2007 and each time we are learning something more about students’ proving capabilities and difficulties. For example, there are certain aspects of proving that mathematicians do automatically, but that students are often unaware of. We define the formal-rhetorical part of a proof to be those aspects of a proof that can be written by examining the logical structure of the statement of a theorem and by unpacking associated definitions. This is something we have called a proof framework. Examples include writing the first and last lines, “unpacking” the meaning of the last line, and considering what that means for the structure of a proof. Writing the formal-rhetorical part of a proof can expose “the real problem(s)” to be solved. We call the remainder of the proof the problem-centered part.
Second, we will discuss a voluntary proving supplement for an undergraduate real analysis class. This has been taught three times since Fall 2009. Each week, one proof problem was selected or created to “resemble in construction” an assigned homework proof problem that the undergraduate real analysis teacher intended to grade in detail, and that could be improved subsequently and resubmitted for additional credit. The supplement proof problem could be solved using actions similar to those useful in proving the corresponding assigned homework proof problem. However, the supplement proof problem was not a template problem, and “on the surface” would often not resemble the assigned proof problem.
The teaching for both the proofs course and the supplement has been informed by our theory of actions in the proving process, by our division of proofs into their formal-rhetorical and problem-centered parts, and by observations of students’ proving difficulties.
Refreshments will be served 3:30-4:00 in 348 Avery. The talk is free and open to the public.
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