Reflection on Spring 2012; Part III

Originally Posted 05-22-2012

Dynamical Systems

I had taught dynamical systems once before, in the spring of my first year at UNI. Then I used a textbook (Alligood, Sauer and Yorke) which turned out to be too advanced for my students, and I asked them to do too much. The class was lecture/homework with a big final project that involved reading a scientific paper from when Chaos was just being recognized everywhere. Mostly, the class was too hard, but the project bit went pretty well.

This time was going to be different! I planned an IBL approach and I wrote my own notes. Also, as an experiment, I went with a structure described to me at an IBL workshop at UT Austin run by Mike Starbird and Carol Schumacher: each meeting the class was responsible for a small snippet of tasks (usually 4 or 5 of them), and we would spend the first 25–30 minutes of class working in groups to solidify our ideas and refine arguments, and then discuss/present to the whole class.

I won’t try that particular structure again. It didn’t work for me. That is, students didn’t seem to take the personal responsibility to get all of the work done each meeting. Many used the group meetings as a way to avoid coming to grips with the material on their own. Etc, etc. All of this made it hard to evaluate some of my students, as they really did their best to hide and nothing in my written expectations said they couldn’t. In retrospect, one thing I changed from the structure as it was described to me likely made a big difference: I did not require them to turn in written work every class period for evaluation. I don’t have a graduate TA for this type of class, so the workload wasn’t feasible for me. But I see now that this might be a key part of making that particular IBL structure work.

Also, I was trying to get them to use Sage for the numerical work required. Basic dynamical systems has a nice experimental feel, where you can get the computer to do some basic things for you and then poke around the data looking for patterns. What I learned is that even fairly advanced mathematics students who otherwise seem comfortable with technology will balk at having to do something that requires a simple “for” loop. I’ll have to write code snippets for them next time. I had a few computer science double majors, and they got the class limping through that portion of the course. But really, that is when things hit a wall. The class never really recovered. I realize that I have to work on how to teach how to use the computer to do mathematical investigation. Computers are an incredibly powerful tool, but only if you have some basic way of using them. (Like Sage/Python confidence.)

In the end, we did not get through all that I think we could have, but we did “enough”. I certainly want to do more next time.

What will change?

First Item: More structure to the learning of how to use the computer.
Second Item: Pre-written scripts to hand out so that the computer is not the obstacle.
Third Item: Back to a structure I am more comfortable with–one that involves more personal responsibility on the part of the students.

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Oh, my. I botched things this time… OR How to frighten your students in seven weeks

Originally Posted 02-27-2012

Part One: An Issue

So, things had been going too well. That is, I had to have screwed up somewhere and not noticed, right? I was completely aware of the mediocre mess that is my linear algebra course, but my other courses seemed to be swimming along.

(By the way: check out the adjustments to the Liberal Arts Math Course Task sequence. I need to adjust the beginning day or two on surfaces, but it is going well!)

So, where was the lurking issue? Dynamics. The day-to-day operation of the class is going well enough. I mean, I think we are progressing a bit slowly, but this is my first IBL attempt at this material, so I likely have an unreasonable expectation of what can be done quickly. No big deal. We’ll get through it when we get through it. Students seem to be working hard, and we are making progress.

The precipitate cause of the issue was the midterm I gave last week. Here is what happened. They asked if we were going to have one. I said, “sure. How about insert day here. And by the way, here is a little ‘anti-midterm’ to make sure you really have the basic terminology down.” Then I gave them an exam. I thought it was reasonable. Most of the class stayed to the end of the hour and looked harried. uh-oh.

Over the weekend I received a message from a student who was very anxious about her/his grade, was sure that she/he had bombed the test, and faulted me for not being up front about what to expect on the exam. In particular, the ‘anti-midterm’ was straightforward and she/he completed it easily, but it was not a practice test for the real midterm and the student felt misled.

Well, yeah. My bad. I misled my class. Now, I still gave the exam I wanted to give, and I’m not sure I’d change much about the daily run of class. Here is where I failed: I did not communicate clearly how I assess student progress in this course.

Total, gobsmacking failure on my part. I don’t have a good excuse. I do know how it happened, but it is terrible all the same. You see, I usually avoid discussion of grading policy until the students bring it up. In a class full of new students, someone usually asks about my cryptic grading statement in the first month, and then we take ten minutes to discuss it as a class, and that settles everybody’s nerves. But in dynamics this term, about half of the class has taken a course with me before. The rest of them are just trying to fit in… and no one has asked. I should have noticed this and addressed it. But it was never part of my daily plan, and I just didn’t do it.

There is nothing to be done for it except to try and alleviate anxiety that already exists. I will have to do a lot of talking, I suspect. I have started this process.

In fact, the student who wrote me to complain phrased the point of view of a student new to IBL very well and I will eventually work up some courage to ask to quote the email to make a fuller discussion of how IBL works differently and how every little facet of change must be explained to students. (Wouldn’t this make a fetching title for MathFest, Dana? “A catalog of how to fail as an IBL instructor, Part One: communication issues” No. that is terrible. I bet I can do better. See even this title is a failure.)

Part Two: A Little Discussion

How do those of you who run IBL-ish courses handle examinations? What kind of test do you use? What do you aim for?

I usually write a short test with new stuff on it. Not out-of-left-field new, but still not “please do this problem which is exactly like something I already told you was important.” I try to pick a variation of the kind of thing we’ve been doing in class and ask the student to demonstrate their mastery of both the basic ideas and simple exploratory skills. Am I out of line on that? I don’t really think so, but I guess I want to know if I am off base.

Part Three: A Broader Discussion

Now, the students message has brought up a point I have been worring about for a while. I might as well let it out.

How the hell am I supposed to grade an Inquiry-Based-Learning (IBL) course?

Here is what I have been doing: I run a careful course, with lots of interaction with the students. Usually I have an exam or two, just to keep them on their toes, but that just refines my opinion, it doesn’t form it. I invite them to discuss their progress with me whenever they would like. At the end of the term, I just know the appropriate grade. I am a professional, and I take seriously my responsibilities to fairly evaluate the students and to safeguard some reasonable standards. I can’t believe it, but no one has ever complained about a grade after the term is over. I hope it is because they can see that I am being as fair as I can be.

Here is what I want to do: use a standards-based-grading scheme. I should set up a formal list of learning objectives, provide cues to which ones have been assessed at what point, and indicate progress on individual objectives when giving feedback.

I just don’t have that set-up, yet. I learned about this sometime last semester thanks to the magic of the internet, and I haven’t yet implemented it.

So… what do you think? Is a good SBG scheme the magic bullet? It seems like a way to formalize what I have been doing implicitly. I really do have a decent list of the kinds of skills and habits I expect students to form during the semester. It also seems like a lot of work, and full of surprizes waiting to bite me… like the dreaded “reassessment process.”

But if there is a time to try SBG, it is next term! I will be teaching only classes I have taught before, including one that I have taught so often that I could do it in my sleep. [by the way, I think it is the first time since I have arrived at UNI that I don’t have a new prep. Ten straight semesters. At least one new prep every term. Seriously.]

I’d be happy to hear your thoughts. Please use the comment boxes.

Oh, No! The first unit of class ended. Now What?

Originally Posted 02-14-2012

The first part of class went smoothly enough. But, suddenly, the first unit of each class was done, and roughly at about the same time: one to two weeks ago.

So I have been scrambling to make it through each day and meet my scheduled obligations. I said something to my wife about how the problem was that the measured time between me and the next expected due date had become negative. She made fun of me. You know, I might have used the phrase “event horizon” in there, too, so I bet she was justified.

Anyway, what has been going on?

Dynamics

This course is going along well enough. We are now just a bit behind where I hoped to be, and I have a little knot of students who are too quiet for their own good. I recenly gave the class a “not-a-midterm” list of questions that I think they should be able to handle at this point. Perhaps that will motivate some quiet students to come and discuss things with me. If not, I’ll plan another intervention.

I don’t mind about the schedule–it probably means I was too optimistic in my planning. We have definitely gotten through the idea of a dynamical system as a modeling tool, how to use basic terminology correctly (like dynamics, orbit, phase space), basic orbit types like fixed points and periodic points, and we’ve learned to draw good pictures like phase diagrams, cobweb plots and something more like a “folding diagram”. (Does anyone know if those have a more standard name? I don’t recall ever learning one, but I know I am not the only one to draw this kind of picture.) I think we have at least understood how the intermediate value theorem can guarantee a fixed point, and we are very close to nailing down how the mean value theorem can tell us when a fixed point is attracting and repelling.

The next phase of study will be to look at some classical families of systems and study bifurcations. We can then draw some different diagrams and ask deeper questions. I have three different families I can use for the rest of the course, each of which has chaotic behavior in it:

  • the family of tent maps
  • the family of quadratic maps x \mapsto x^2 + c
  • the logistic family x \mapsto a\cdot x\cdot (1-x).

It seems to me that we need the following things out of our next unit:

  • types of bifurcations, including the period doubling route to chaos
  • an actual discussion of what “chaos” means as a mathematical term
  • some symbolic dynamics to provide proofs (hiding here is the notion of conjugacy of two systems)
  • an appearance of the Cantor Middle Thirds Set.

More advanced things that I’ll want my graduate students to do include:

  • presenting Sharkovski’s Theorem
  • Discussing some numerical algorithms for finding periodic orbits.

The last phase of the course should be about Newton’s method, Julia sets for the complex dynamical systems z \mapsto z^2 +c and possibly the Mandlebrot set.

Math in Decision Making

This has been going really well. Basically my class came around on what it means to say two sets have the same size, and then we explored how weird that is for infintie sets. They managed to see that the natural numbers, the evens, the odds, the positive rationals, the integers, the rationals, the set of all `mathematical words’ on the standard English alphabet, and the rationals all have the same size. Then we saw that the real numbers are different by way of Cantor’s diagonal argument. For a kicker, I gave a lecture day about the Middle Thirds set and we saw that a set could be “huge” but “hard to see” at the same time. A few of them were suitably impressed.

I am still grading exams. They did not take me seriously when I talked about writing to explain clearly…so they will be rewriting their exam papers as soon as I get them returned. I think it is an appropriate time to talk about the process of writing as something that includes revisions.

This week we are starting our unit on surfaces. We just started, so yesterday’s class was a bit of a mind-bender. We successfully noted that a torus was different from a sphere because there are simple closed curves on a torus that are non-separating. And we managed to see that a donut can be deformed into a coffee cup. I had my regular coffee mug and a tire innertube which is just too big to be worn as a hat for visual aids, and I managed to hit and stick the chalk tray with a thrown piece of chalk three times running. That was clearly the best performance of my chalk-throwing carreer.

I plan on hitting many questions about what curves can live on a surface and what shapes are created by cutting along those curves until we feel comfortable enough to go in the opposite direction and make surfaces out of polygons with “sewing patterns.” I still want to talk a bit more about the idea of “stretch equivalence” and later differentiate it from “cut and paste” equivalence.“ Also, I have a week to do about Mobius bands to introduce the ideas of boundary components and orientability. When the ground is suitably prepared, we will talk about the Euler characteristic, and then the classification will have to happen. I am not completely decided what proof I will use. There is the ”standard scissors and glue“ proof I learned in graduate school which involves putting a cut-up surface into normal form, and then there is the Conway ”ZIP proof." These are equivalent, of course, and I’ll have to think about which one I find conceptually easier.

I still hope to have enough time to do a third unit on classifying wallpaper patterns.

Oh, at the conclusion of yesterday’s class, a student told my teaching assistant that this is the most interesting and fun math course he has ever had. This was unsolicited, and wasn’t directed at me. It made me feel awesome for a bit.

Linear Algebra

Here I am just surviving. Though I do think we finished an interesting week of work on Monday. We got to a spot where we had made a model of the internet as a weighted directed graph, and a model for the behavior of a “random web surfer” and then set up a Markov Chain to describe the evolution of the probability that the random surfer is currently at page X. Then we showed how the long term behavior should be goverened by an eigenvector with eigenvalue 1, and discussed the basics of the “power method” for finding that vector. In short, we took a week to explore the basic structure of a naive Google PageRank algorithm.

The next application will be least squares and other polynomial approximation problems. I hope to use that as a springboard for more abstract material, since the notion of an abstract vector space made of polynomials will just happen, and we will see things like column and row spaces.

Not quite Research

My student research projects are coming along at vastly different rates. The undergraduate I have worked with since last May has hit a bit of a roadblock. We are now casting about again for something else interesting to say.

The graduate student I am supervising for just this term has made serious progress on some programming and graphics functions, and I still have hope that he will draw the limit sets of some Kleinian groups by the end of the term. (My secret hope is that he will get far enough to make an animation of how the limit sets change as we change one of the generators of a group inside PSL(2,C).)

The graduate student who will be doing a thesis option with me has come up with a neat sounding question on his own. I have no idea how to do it, nor any idea what is known, so I have sent him to the library as a feasibility check. If he can find some relevant literature, I can help him carve out a project.

Research

I haven’t done enough of this lately, but a colleague and I are having conversations about a problem of mutual interest. We are taking baby steps.

One thing I would like to explore is the Birman-Williams result about the kinds of knots that can appear in the classical Lorenz system, and Ghys’ stuff about how that is the same as the knots that can appear in the geodesic flow of the modular surface. It just looks so cool. Maybe I’ll need to write some expository blog posts to make myself really work the details.

I am going to not make a list of all the other projects in various stages of “incomplete”. But it is long.

Other Stuff

I have a problem solving contest to take students to in about ten days. Math club is starting to get rolling properly again. And now that hiring season is over, it will be time to start convening meetings about writing an REU grant.

Unexpected Classroom Dynamics, and a plain linear algebra class

Originally Posted 01-25-2012

when teaching today, one each for dynamics and linear algebra.

First, dynamical systems. I wanted to give the students their first real taste of the interesting behavior that makes a chaotic dynamical system, so I gave them some stuff to do about the doubling map x \mapsto 2x \mod 1. I was trying to set up a conversation about the dangers of using a computer and just trusting the output without thinking about what the computer is really doing, so I asked the students to compute a bunch of orbits both “exactly” and by starting with numerical approximations to their inputs. Specifically, I asked the students to find the nature of the orbits of 1/k for k from 2 to 30. I didn’t expect anything deep to come out of this particular conversation, except maybe the students would notice something about the orbits of rational numbers. But, being well-trained in the mystic arts, some students noticed that there were almost patterns to the periods involved. This blew up my plan for the day in the most wonderful way. The best way to see what happens is to try it yourself and see the apparently simple pattern with just a few oddballs. And the oddballs aren’t some simple to describe phenomenon. So, we have a big task staring us in the face: “What is going on here?” The students asked it themselves, they made a couple of conjectures in class, and they seem pretty motivated. I expect we will be productively sidetracked for a couple of meetings. It is just wonderful.

Second, to follow up my good linear algebra activity, I had a very blah class today. I just couldn’t figure out how it wasn’t meeting my expectations or how to adjust it. I was preoccupied with this during my unattended office hour, and I finally came to a realization: I never set out my expectations. That is, in the flurry of last minute planning, I never took the time to set some basic learning goals for my linear algebra class. This is why I feel aimless. So, tomorrow I am going to spend some time fixing this problem. I think I will share my list directly with the students and ask them which ones they think they have met. We’ll set straight to work on ones they don’t feel are comfortable. I hope it resets and refocuses our study.

An Interesting Juxtaposition

Originally Posted 01-23-2012

Today’s time in the classroom showed an interesting juxtaposition. The “low level’’ class that is focusing on big, deep ideas is on track, but the more advanced class working on “straightforward material” spent a lot of time not going very far.

Math in Decision Making Success?

My liberal arts course students have made good progress on understanding the idea of a bijection. We haven’t completely formalized it, but I think most of the class basically “gets it.” In fact, today’s four exercises proved to be not enough. They dispatched them quickly and we were done with ten minutes to spare! I’m afraid that means I might need to come up with something meatier to add to Wednesday’s activities. This shouldn’t be a problem—coming up with hard problems is easier than coming up with approachable ones.

Linear Algebra: The End of the Activity

It took a surprisingly long time to get through the “solve and sort” phase of the activity I started last time. But I am happy with the results. I asked the class to distill some lessons from the experience, and I got this list:
1. The size of the solution set depends on the number of pivots in the matrix.
2. The origin is a solution exactly when the system is homogeneous.
3. If you are working by hand, it can be helpful to look for clever tricks instead of just blindly following the algorithm.
4. If you had to do a lot of these, or even just one of any appreciable size, you want to use a computer. It is not difficult, just tedious.

So, it took a lot longer than I planned, but the main points came across. I used the end of our discussion to introduce the term “rank” and we talked a little about writing the solution set in the standard parametrized form using vector operations.

Dynamics

If you have bothered to look at these notes, and especially if you are doing something as odd as working through them, know that problem 29 was a total failure.

That is, I am sure the problem is fine, but it is totally misplaced. The students had no idea what to say about it. I’ll have to find the right time to bring that point back up.

Also, one of the more advanced students wrote up a nice presentation of what it means for a sequence to converge along with his proof for task 15. I hope it helps out some for those students who haven’t taken a real analysis course, yet.

Some Complicated Dynamical Systems

Originally Posted 01-22-2012

Dynamics Notes Updated

I spent some time today designing new tasks in my activity sequence for dynamical systems. The new tasks confront the students with several problems at once:

  • First, the systems have much more complicated behavior than the simple systems we have dealt with so far.
  • Second, they need to use the computer to do some numerical investigation.

The second item is a bit subtle. Of course, using a computer to model precise computations often leads to difficulties. But I have the students using Sage. Sage can often handle exact symbolic computations! I say often, because even when I figured out how to set up my orbit computations to be exact, at some point things got too nasty and it threw back an error message. So, the big message of “the computer might lie” is a bit muddled. If you know how to use it carefully, you can make Sage do exact computations, except when you can’t.

To get through these points, I have chosen the doubling map D: x \mapsto 2x \mod{1} and the tent map where T(x) = 2x if x \in [0,1/2] and T(x) = 2 - 2x if x \in [1/2, 1]. These are much more exciting than our previous systems, and they will help motivate most of the rest of the semester.

Also, I have included some rather open-ended and hopeful tasks asking the students to make up “graphical methods” for plotting orbits. I hope that we get out cobweb plots and phase portraits. I think we will, but the students could honestly not see where I am aiming and have no idea what to try. Fingers Crossed!

Other Achievements

I managed to do some other small bits of business today.

  • I did a bunch of organizational work for our Undergraduate Research Committee. (I am the chair.)

  • I read another chapter of Indra’s Pearls to prepare for a meeting with a student tomorrow.

  • I finished the update to my web page for the new semester. Since the pages only needed the slightest tweaks, this really came down to figuring out how to get the Sphinx Python library working on my computer and figuring out how to make ssh and sftp work on my new OS. (Turns out it is easy. It helps that I am old enough to have used a Unix machine on campus for my first email checker.)

  • I played Contre Jour on the ol’ iPad and completed six or seven levels. (Two of these took me a few days to get right. So, yes, this is an achievement.)

Random Thoughts

  1. I still have linear algebra to prepare for tomorrow. This is becoming a pattern.
  2. Thanks to the magic of Google+, I got alerted to Treesaver. This looks like a possible way to make your own ebook using HTML5. I wonder if it would be worth it to edit and write up my old linear algebra course now that I’m thinking about it again…

A Long Day, with Fun Bookends

Originally Posted 01-19-2012

Today was a very long day at work. Long, but mostly enjoyable. There is still more to do, so I’ll keep this post short.

I began by taking a job candidate out for breakfast. This was very enjoyable. The candidate is a specialist in mathematics education, and we had a good discussion. I am not on the hiring committee, so it was easy (for me) to just talk and not have every word carry weight. It was interesting that most of the breakfast was pretty genial and smiley, but the candidate’s face really fell at one point when I said I disagreed with something. I had forgotten that this was an interview for the other half of the conversation… Upon reflection, I wish I had instead asked, “really? What makes you say so?” Somehow I have Mike Wolf’s voice in my head saying that. He was always good at disagreeing with me while being direct but friendly. Anyway, I enjoyed breakfast. Oh, and today was a terrible day to be wearing interview clothes. It is absolutely frigid.

After breakfast I took a little time to poke around with Apple’s new iBooks Author package. It doesn’t look very friendly for making technical documents at this point: the editor is WYSIWYG and there is insufficient mathematics support. And they are apparently just replacing the publishing industry instead of reinventing it. Somehow, you are allowed to make a book (for free), but the EULA demands that you only sell it through the iBooks Store. So that is three strikes already. Nothing revolutionary, just a shiny anodized aluminum case for the old publishing paradigm.

I had meetings with two students about research today. The first is making slow progress. I suspect that he is not spending that much time on the problem right now, and I’m trying to make up my mind on when to say something about it. The second is only getting started, so we had a brief but productive conversation and he was off to work again.

In between I negotiated a letter of recommendation request. Rather, I presented my demands. If they are met, I’ll write the letter. I think that brings the current pile to four waiting to get done–so I’ll have to save some time in the afternoon next week to get those written and sent.

After my meetings, I had a blissful few hours to myself. So I read a bit in the notices I got and the January issue of the Monthly over a mug of tea. Then I worked out some new problems for my Math in Decision Making class. If you’d like to see them, they are here. Basically, I think the class has come around to the fact that two sets have the same size if there is a ‘matching’ between them, so the next collection of problems is meant to sharpen that idea and push on the weirdness of this notion for infinite sets. I have enough material to last at least four more classes, likely six. I think somewhere in here they will have to slow down. I am just not sure which problem is going to be the stickiest. By the way, it bears repeating that this sequence is adapted from one written by Doug Shaw, though responsibility for the awful condition of the current set of tasks is all mine.

I haven’t updated my dynamics sequence, yet, because I haven’t needed to. I suspect that the students will force me to change that this weekend. They are doing a pretty good job at keeping up with the pace I have set. I think the next example they need is the doubling map. It will serve as a good gateway to all sorts of things for the rest of the term.

The last event for the day was a Math Club meeting, which was very enjoyable. There was no real agenda for today, and the students just played games. One of my colleagues brought her ten-year-old daughter who demolished me (and three students) in a game of Set. A small group of students tried out a game called Sleuth, which is a bit like Clue and looks like a lot of fun. One of my students said something about my cheating at the game so I could home and blog about it. If he is reading this one, he knows I have to fail him now.

I’m off to prepare materials for linear algebra tomorrow. I think our activity will involve reverse engineering some systems of equations to satisfy certain conditions. If the students use what we discussed about reduced row echelon form (or about the geometry of the dot product), they will go far. Perhaps I’ll turn it into a game. Pairs of students will create (and then disguise) systems which have a certain set of conditions, and then we will trade and see if we can remove the disguises. I also intend to quickly introduce the terms rank and determinant for matrices. It is about time I got off the stage in that class.

Good night internet. Sleep tight.