Deep thoughts and more dilettantish dalliances

Hey, All! It's been a long time since I blogged here about math pedagogy, so I wanted put out a post that lets you to know that I do indeed still reflect on my teaching, and that I am keeping close watch on my classes as they develop this semester.

I thought long and hard about teaching math this morning as I walked into campus. Specifically, I thought about our use of software in Precalc, and I've come to the conclusion that I'm not very happy with it.

Let me start out by saying that I'm glad that I've had the chance to use the software this semester in teaching the course: it's been an opportunity to gain proficiency with a particular pedagogical technology I'd not used much before. Moreover, I recognize the usefulness of some aspects of the software for some students: the software's ability to generate problem after problem of samples fitting particular problem molds is useful for students who learn well by example and iteration. Nevertheless, while before I couldn't say with certainty that I would prefer to not use computer-graded homework in teaching introductory math courses, I feel that having made use of the software in my course I can credibly affirm that statement.

Without going into detail, let me lodge three objections (relatively briefly! I'll flesh these out later once I've had a chance to further reflect on them, most likely once this semester's behind me) to the software:

1. Its use foregrounds the medium at the expense of the message: in asking the students to master the software's often unnatural and indeed often byzantine commands for entering mathematical notation with a Flash-driven interface, the I feel that all too often the computations the students need to perform to generate a correct answer are overshadowed by the mechanical manipulations they must undertake to enter the answer into the computer.

2. The software places the students at a distance from the instructor, to an extent that effective bridging of that distance vitiates the need for the software in the first place. To wit, even with the ability to zoom in on a single student's solution to a single specific homework problem, the opacity of the software's interface does not allow the instructor to penetrate beyond the student's final response. Should this response be wrong, there's generally no way to deduce from it just what it is the student did incorrectly without asking the student to submit her or his handwritten notes. Of course, I am all for students' working out solutions by hand...I continually exhort them to do their work on paper and use the computer only to enter their solutions...yet if ultimately I have to dig up their handwritten notes in order to tell what it is they're doing right and wrong, what's the point in having the software in the first place? I might as well simply ask that they submit their homework directly to me, let me grade it, see it, be in contact with it, and reestablish the missing and much-missed bond between the students' understanding and my own, without the electronic intermediary.

3. Finally, and most fundamentally, I feel that the software system by its nature reinforces the common and erroneous perception of mathematics as a rigid, timeless, universal enterprise. As the computer is trained to expect only a very particular form of answer to each problem it provides, the student may come away with the mistaken notions that math is a field in which there is a single correct answer, in which there is no gray but only black and white, that process is unimportant if the product is ambiguous, that every instance of a certain calculation requires a single form of solution. All of these claims are preposterously wrong and further the view of math as far-removed from ordinary ways of thinking, as something undertaken only by pointy-heads who've mastered arcane rules of mathematical computation and communication. Human-graded homework, on the other hand, far more sensitive to idiosyncratic-but-correct responses, to slight variations in notation and style, to math's true nature as fluid, era-dependent, humanly-crafted enterprise, allows students to succeed by responding in various ways that reflect their own particular learning styles. The unmediated bond between student and teacher facilitates the latter's ability to convey the perception of mathematics as a ground in which critical thinking can be taught, and the former's ability to construct a personal mathematics all her own.

These are deeply-rooted philosophical objections about which I hope to say more later. I'd be interested in hearing others' take on this matter, I don't claim to have the final word!

On a wholly different note, I've finished my first "exotic" (G,φ)-gram, although I must admit that I'm unsatisfied with one of the steps I took in its construction. I'll admit up front that I wrote another Mathematica notebook to help crunch the noncommutative multiplications that go into the poem's analysis.

The poem is a (D₄,φ)-gram, D₄ the dihedral group of order 8. The homomorphism φ that governs the poem is one Mathematica chose at random; this is the part I'm not happy with, as I'd rather choose a φ that's "meaningful" in some way, that relates each letter to an element of D₄ in a "useful" way. But here's the rub: what choice of φ would work best? My thought was to assign to each letter the longest element of its stabilizer subgroup (relative to the presentation of D₄ in which a represents the reflection in the vertical and b the reflection in the SW-NE line)...but it turns out that practically every letter then goes to an element of the abelian subgroup {1,a,bab,abab}. (The only one that doesn't is "Q", with its funky NW-SE symmetry, which goes to aba...and how often is "Q" used?) Thus if one reflects (or in fact in any way permutes!) the letters of a given line, the value of that line under this particular φ is unchanged.

Boring!

Even worse, any choice of φ(α) from Stab(α) will lead to the same problem! Thus my opting for a randomly generated homomorphism.

My hope was to write a poem in which the value of each line is the group-theoretic inverse of the same line written backwards (letter-by-letter, not word-by-word). The poem below achieves this, although it's an admittedly simple poem. However, it was surprisingly easy to construct (owing probably to the smallness of the governing group), so expanding on this theme would likely not be hard.

So here's the poem, with the homomorphism following it:


Inverse

What kind of mirror symmetry
must a piece possess
for its value to invert itself
when we trade east for west?


Let me give the homomorphism by a listing of the preimages:

φ^-1(1) = {C,L,P,Q,X,Z}
φ^-1(a) = { }
φ^-1(b) = {A,F,O,R,S}
φ^-1(ab) = {I,U,V}
φ^-1(ba) = {G,Y}
φ^-1(aba) = {H,M,W}
φ^-1(bab) = {B,D,J,K,T}
φ^-1(abab) = {E}

Thus only 7 elements ended up in the center of the group, and only 4 of these ("C","P","L", and "E") appear in the poem.

Anyway, I'm having fun. My Mathematica code will enable me to work with dihedral groups of any order, so I may try out a more complicated example later if I have time. Now though, I've got to meet with Sylvester in a few; he and I are continuing research on caterpillar labelings this semester, and we're meeting to debrief after his presentation in the Senior Seminar yesterday afternoon.

Let me close with a note to my Abstract students: keep up the good work! Your committee presentations are already at a very high level. You're doing a great job of highlighting common difficulties and errors, and in giving credit to particularly insightful methods your peers apply. I like that you're all making note of the fact that there's generally more than one way to prove a given proposition. You're also demonstrating a good understanding of the "metamathematical" aspects of mathematical writing. In particular, I appreciate the attention you're all paying to the "Four Cs" criteria, and I hope those criteria are helping you to learn to discern good math writing from bad.

And now, adieu! Thank you for reading.