Posts Tagged ‘Analysis’


August 22, 2011

“One idea people had was to check out Calibrated Peer Review. I have only scratched the surface at that site but I’m grateful for being pointed to it.”

That was a sentence from Andy Rundquist’s blog. As much as Andy ever has a throwaway line in a blog entry, this was it—this was his only mention of Calibrated Peer Review (CPR). I imagine that Andy simply put it in his weblog so he could find it later, on the off-chance that he ever thought about it again. But it changed my semester.

I have decided to use CPR in my real analysis courses this semester. Here is what CPR is in a nutshell:

  1. Students log on to CPR to get a writing assignment.
  2. Students complete the assignment and upload it to the CPR website.
  3. Students view three copies of the same assignment, all written by the instructor. These three copies are examples of differing quality.
  4. Students need to make judgements about the quality of each of the three instructor-written examples. The students answer specific questions about each article. If a student’s assessment of each of the three pieces agrees with instructor’s, the student moves to the next step. Otherwise, she must start the evaluation process again. This repeats until the student agrees with the instructor’s assessment.

    The purpose of this step is to “train” students to critically example these assignments; this is the “calibrated” part of “Calibrated Peer Review.”

  5. The student reads an anonymous article from a peer and rates it on the same criteria as the previous step. This happens a total of three times.
  6. The student evaluates his/her own article.
  7. The student sees the results from other people’s evaluation of his/her article.

By the end of this process, the student will have evaluated a total of seven different versions of the writing assignment, and will have thought about what makes a good piece of writing seven times.

I was planning on doing peer review, and I was planning on having students evaluate three different versions of the same proof. This combines the two in a nice way.

[Edit: A member of the CPR team emailed me to tell me that there is a pay version of CPR that supports a direct upload of PDF files (among other things). I don’t think that I can make it work this school year, but that would render the rest of the post irrelevant.]

[Edit: Also, here is a link to a screencast on the perhaps-unnecessary process below.]

The one catch: the CPR website only accepts text and html, which does not work well with mathematics. My workaround is this:

  1. The student writes up the solution offline in \LaTeX.
  2. The student uploads the resulting PDF to our Moodle site.
  3. The student copies the URL from the Moodle site, and simply creates a link to the Moodle site within the CPR website.

This is not the most elegant workaround, but it should work. If you have a better idea, I would love to hear it.

Analysis Standards Draft

June 29, 2011

I spent most of the day on Monday designing the course standards for my real analysis class in the fall. I ended up with a rough draft and went home.

Later that night, I got this mysteriously clairvoyant tweet from Joss Ives: “writing learning goals is a task well-suited to collaboration.” He is right, of course, but I was astounded that—seemingly out of nowhere—he decided to tell me exactly what I needed to hear at exactly the time when I needed to hear it. He was some sort of JIT Jedi mentor.

(What actually happened was this: Joss was tweeted last week that he was writing learning goals, I replied, and he was just replying back. I like the Jedi version better, though).

Anyway, here is the first draft of the standards. I welcome feedback in the comments (what big ideas are missing, which of these aren’t so important, etc):

Core Topics: repeatedly demonstrate

Apply the Completeness Axiom
Determine convergence/divergence of sequences
Determine if a function is differentiable at a point
Apply the Mean Value Theorem to solve problems.
Use sequences when working with functions to show divergence
Determine if functions are continuous
Determine if functions are uniformly continuous
Determine if a function is integrable

Supporting Topics: demonstrate at least once

Determine limits of sequences
Determine limits of functions
Determine convergence/divergence of series
Apply Bolzano-Weierstrass
Know three equivalences for compactness on R (compact, closed bounded, finite subcover)
Determine if a sequence of functions converges pointwise; if so, determine the limit
Determine if a sequence of functions converges uniformly; if so, determine the limit
Apply Abel’s Theorem
Determine radii of convergence for power series
Apply the Fundamental Theorem of Calculus

Cold Problem Solving

April 5, 2011

I am teaching real analysis in the fall, and I am beginning to plan it out. Here is one more idea that I would like to record before I forget.

First, some background. I was never very good at real analysis. I like it a lot, but it was over my head as an undergrad (I got B’s in the course, largely because my relative difficulty with the material outpaced my study skills) and I only took one course as a graduate student (similar). Part of the reason why I am teaching it is that I want to learn more about this beautiful subject.

So I am not a great analyst. To use this to my advantage, I am planning on—perhaps weekly—asking my students to give me an analysis problem to solve “cold.” I won’t prepare for it at all; I will solve it on the spot.

I was inspired by a lost blog posting (please comment if it was you—I will happily edit this to credit you) about the difference between the way we discover mathematics and the way we communicate mathematics. Sadly, we normally teach by showing the “communication” rather than the “discovering.” By

  1. Doing problems “cold” in front of the students, and
  2. Not being very good at analysis, but
  3. Having general problem solving skills,

I can hopefully give the students a glimpse into the “discovery” world of mathematics. If I am really on the ball, I will (at least occasionally) take the time to re-write my “discovery” into a “communication” form.

Abbott wins

March 30, 2011

Recall that I am teaching real analysis next semester, and I asked for help in deciding which textbook to use. I started by considering texts by Abbott, Beardon, Ross, and Trench, but I started considering Strichartz due to Adam Glesser’s comment.

First, I have decided to dive head-first into the inverted/flipped classroom and screencasting pool in the fall. This just seems like it makes sense: the professor should be around the students when they are doing the harder work, not the easier (and more passive) work.

With this in mind, I wanted a textbook that would complement this system well. Here are my thoughts on each:

  1. Strichartz is too expensive. I did not spend much time considering it (sorry, Adam!) because of the price.
  2. Beardon is too technical and too expensive, although I like his attempt at integrating all of the ideas.
  3. Trench is the right price (it would be about $25 to print and bind a copy for a student), but there is too little exposition—t seemed like he basically hopped from theorem to theorem. Since I am going to have my students read the text (in addition to screencasts), I wanted a readable book.
  4. Ross is a reasonable price and reasonably readable, but I do not like his treatment of continuity (he defines it in terms of sequences of points).
  5. Abbott is left standing. It is a reasonable price (though not the cheapest), it is the most readable, and I like his motivating questions. I have heard complaints about the amount of typos, but I can fix this by making this part of my first homework assignment.

So Abbott wins. Thanks for your input.

Undergraduate Real Analysis

February 7, 2011

I am teaching undergraduate real analysis for the first time in the fall. This means that I will need to choose a textbook soon: I am looking for advice in the comments. Here are the textbooks I am considering so far:

These texts seem to be at the correct level for my students—we have used Abbott and Ross here before. I think that Beardon is intriguing—he hammers on limits at the beginning of the text, and then shows that everything else (derivatives, integrals, etc.) is a mere consequence of the notion of a limit—but I cannot find any reviews online (save for one on Amazon).

Here are some textbooks that I am not considering:

  • Rudin—too advanced
  • Pugh—too advanced
  • Spivak—too calculus-y, not analysis-y enough

Convince me if I am wrong to disregard these (keeping in mind that Ross/Abbott is definitely the right level). In particular, I would really like to use the fabled Calculus by Spivak, but it seems more like advanced calculus than real analysis.

While I am at it: I am teaching complex analysis in the spring of next year. So far, I am considering Churchill. I would love ideas for this text, too.