Natural Blogarithms

Thursday morning, three students presented homework problems at the board. I followed this with a lecture introducing the concepts of supremum and infimum of a set. Because I got a little sidetracked, I did not quite make it to the Completeness Axiom.

What does “Wayne’s World” have to do with Real Analysis?

Good question. I inadvertently structured my lecture in such a way that it reminded me of the old saturday-night-live-turned-box-office-money-maker. Both in the sketch and the movies, Wayne and Garth would broadcast their rock-n-roll show from Wayne’s basement. Anyways, whenever they would do a dream segment or a “back in time” segment, they would mimic the “fuzzy” screen effect and dreamy music my waving their hands frantically and saying something to the effect of “doodle-oodle-oodle-…” (DISCLAIMER: By referring to Wayne’s World, I in no way condone some of the associated raunchy humor, even though I probably laughed at some of it – shame on me.)[...]

We had no homework assignments due on Thursday, so I dove right into lecture. We proved a few more results following for our ordering of the reals. Upon completing these theorems, we were then able to do the basic sorts of solving of inequalities we teach in our lowest level of algebra. It is interesting that we get to one of final senior undergraduate courses and start doing things we do in the earliest mathematics courses, but this time, we’ve built our understanding of these rules from the ground up. Have no fear, analysis students, we will go much, much beyond those mathematical tools we used intermediate algebra.

During this morning’s class I surprised the students by letting them know that today was my turn to answer homework questions at the board. As I have mentioned in previous posts, the students are being randomly selected to present some of the homework problems at the board. For many of them this is a considerably stressful event and with today’s assignment being particularly challenging, I noticed that many were very relieved when I offered to answer some of the questions at the board. Plus, it saves me needing to grade those particular problems. [...]

On Thursday, we began the chapter on Real Numbers. We are taking the axiomatic approach to defining the set of real numbers. I demonstrated the difficulty of defining Real Numbers in the techniques that have been typically used in their earlier class, such as by decimal expansion or as points on the real number line. I alluded to the method of actually constructing Real Numbers through the method of Dedekind cuts but I have chosen to follow the text by describing the properties that uniquely define the set. [..]

During I.A. on Tuesday (Holy Cow 7:25 am is early to start a class) two students present problems at the board. I really enjoy having this part of the class. All of the students have had a colleague of mine for class and that has been a regular part of her upper level courses that she teaches. They are all used to it, but it is new to me. I enjoy hearing the though processes of the students. I even developed Rubric to grade these presentations. Don’t I sound like an education major? (By the way, I was never formally trained to teach.) [..]

In Intermediate Analysis this morning, three students each went to the board to present homework problems on basics of sets and mappings. This is first time that I have incorporated the oral presentation of homework as a part of an upper level mathematics course. It was one of my least favorite aspects of the graduate courses that I took and yet I realize just how formative it was in the development of my understanding for those courses. I have a stronger grasp of those courses in which such presentation was an essential portion of the course…