Tuesday, November 10, 2009

This and that

This week's gotten off to a good start, though Tuesday already feels like Thursday, and Friday will feel long overdue once it's come.

Today I played host to one of my colleagues from Samford University. Having driven seven hours from Birmingham, Alabama, Colin spent last night and today with me and my colleagues here, giving a great talk, chatting with me about REUs and the Sectional MAA, and meeting with various faculty and students from the department.

His talk was fantastic, offering the audience a unique blend of real analysis, linear algebra, and introductory proof techniques. There were about a dozen students present, and many of them are currently enrolled in...well...Real Analysis, Linear Algebra, and Foundations. For the analysts there were metrics, and orthogonal families of functions, and convergence; for the linear algebraists there were opportunities to apply eigenvalues to compute the closed forms for the terms of the Fibonacci sequence. For my MATH 280 students there were both implicit and explicit references to a number of the core concepts from the course: bijections, the pigeonhole principle, induction, proofs by contradiction, and equivalence classes and partitions. The talk was challenging but, I hope, accessible, and there were knowing smiles on a number of the students' faces as Colin reached his deftly delivered denouement.

In the afternoon, after his talk, Colin spent a few hours with me in my office talking about the design and execution of REUs, as he's hoping to submit a proposal to start one up at his own institution. I think I was able to give him some pointers and step through the process I followed as I put my own program together, but I couldn't answer every question. I honestly don't know what in particular about our program, aside from hard work and dedication on the part of the participating faculty and students, has made it so successful.

Colin will be heading home tomorrow; I've already been invited to join him at Samford in April, where he'll return the favor of hospitality he granted him during his stay here.

What else is new?

I realized yesterday that I was so busy bitching about grading over the weekend that I neglected to mention even once that on this past Thursday Algebra al Fresco sponsored the building of our second full Level-2 Menger sponge. (Pictures soon, I promise!) This one came together on the quad, on the steps leading up to the library. Working from 10:45 in the morning until nearly 7:00 that night, last Thursday several different students joined me in making the monster which now rests on a card table in my office, right where this past summer's sponge sat for a few weeks before moving on to the Engineering Department to get shellacked for display (so I'm told...it's yet to reappear).

A single student, Nighthawk, was singlehandedly responsible for about half of the cube's construction. The guy's a born folder. By 5:00, when I had to head home, Nighthawk and my current Calc I student Lambert, having overseen the splicing of 16 of the 20 Level-1s needed to complete the Level-2, decided they'd not rest that night unless they'd finished the sponge, and so they worked away in the Math Lab for a few more hours, wrapping up over eight hours after construction had begun.

Nighthawk swears that he'll be able to set the unofficial world record for solo construction of a Level-2 sponge (current record: 15 hours). I believe he'll be able to do so, maybe after a few practice runs. Speedy construction poses an interesting operations research problem, actually: imagine a team of four builders working together to complete a Level-2 sponge. How best to use their time? All four should start out building Level-0s, and at a certain point one or two should switch to sewing together the Level-1s, and at a later point still one of these should switch over to the making of the Level-2, all while their two friends keep plugging away at the basic building blocks.

But when should the switches occur in order to minimize construction time?

And is there a more efficient means of splicing the lower-level cubes to form the higher levels? (There surely is...the question is more "what is the most efficient method?")

As I said above, I'll soon post some pictures of the construction. Most of it took place on an unseasonably warm and sunny day on the library steps. It was a pleasant Thursday.

What else is new?

Perhaps an update on the Fall 2009 Calc I Homework Debacle is in order.

After a good deal of thought, I decided to make all homework for my Calc I students optional for the remainder of the semester. It's simply not worth my time to grade half-hearted attempts at homework completed (or, more to the point, incompleted) by undermotivated students who are more often than not cribbing their answers from the solutions manual. To those (who I suspect will make up the majority of the class) who still wish to complete the homework, I promised to continue providing the same robust feedback and the same careful attention I've always given. (Not once have I begrudged granting such feedback and attention to deserving students; I'm frustrated only when a dozen hours of my time spent grading sloppy work remains unreciprocated and undervalued.) To these students I also promised to "lock in" their current homework grades, ensuring them that their grades will not fall but can only see improvement between now and the semester's end.

I can't stay mad at these students: for the most part they're hard-working, well-intentioned, bright, and fun to work with. As I said to them in class, I'm not frustrated with them so much as I am frustrated with the process. And as I said to one or two of them in the cozy confines of my office, I'm not disappointed that they come to me seeking ways to maximize their grades, I'm just disappointed that they and I have been caged in a system in which they feel it's necessary that they maximize their grades in the first place.

The students' relatively strong performance on the applications handouts from two weeks back has convinced me that such assignments may be able to form the backbone of a yet more student-centered Calc II course. Next semester's homework schedule might look something like this (assuming a four-day class meeting on MTWF):

Week 1, Tuesday: suggested textbook problems from Section x

Week 1, Wednesday: suggested textbook problems from Section x+1

Week 1, Friday: suggested textbook problems from Section x+2; due for feedback only: textbook problems from previous week; due for a grade, or for inclusion in a student's portfolio: applications handout regarding Sections x-3 through x-1

Week 2, Monday: applications handout regarding Sections x through x+2

And so on.

There's that "p" word again: "portfolio." I've thought a bit more about portfolios, and about what might go in them. Whereas, as I've said before recently, students might be able to demonstrate their achievement of very skills-oriented learning goals (like mastery of derivatives or integrals, for example) through including in their portfolios more traditional exams or quizzes, suitably suggestive applications handouts could provide students with relatively uncomplicated low-stakes writing assignments through which they might demonstrate achievement of some of the harder-to-get-at goals, such as maintenance of skepticism and application of problem-solving methodologies.

Speaking of skepticism, it delighted me to no end to hear Uriah, one of my Foundations students, talk about the ways in which our class has begun to change his perspective on mathematics. "You just can't take anything for granted," he said as we sat at the dinner table with our guest speaker. "I want to question everything, and prove everything to make sure it's true."

His comments reminded me of the Calc I learning goal I recently discussed on this blog: "Demonstrate (through informed question-asking) a healthy skepticism regarding mathematical and scientific arguments." His comments assured me that he, like a number of his peers, is getting a lot from our class.

And speaking of getting a lot from our class, I'm getting more and more excited about the textbook as it begins to come together, and as several of the students are expressing increasing interest in ensuring that it's executed as cleanly, completely, and correctly as possible. "I intend to share it with future 'generations' of students who come through this course, so please keep in mind as you write it that you ought to be writing to help them." It's got tremendous potential, and I hope to share it was as wide an audience as I can. You can bet I'll bragging on it at the Southeast Sectional Meeting of the MAA in March.

Okay, I'm clocking out for the night. I'll leave with a notice of publication: I found out a week or two ago that my article on using poetry in the mathematics classroom, complete with poems by several wonderful students whose work first appeared here and here, has now appeared in The WAC Journal. Let the celebration commence.

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