Today I tortured our department's printer by asking it to reproduce six weeks' worth of progressively more sophisticated reports crafted by this past summer's REU students. I hope to have some time in the next few weeks to sort through this mass of data and learn a bit about the development of the research students' writing skills over the course of the program. (Having spent most of today setting up next term's classes' websites and syllabi, I'm far enough ahead of the game that I can afford a few days of textual analysis.)

The data comprise end-of-week reports for weeks 2 through 7, a weekly report from each of the the eight students. The first week's reports are highly tentative, and only three of the eight concern what would later prove to be the respective student's summer project (the others concern sample topics the students gave some thought to investigating but soon abandoned).

What variables will I be paying attention to? Most of the following should be familiar to anyone who's taken a class with me during the past couple of years.

1. Mechanics. Is the raw .tex file, when available, compilable? Has the student been able to produce a more or less error-free LaTeXed document? Has she avoided common mistakes like "dumb quotes" and misuse of the math environment? Has she used the second person, plural voice common to mathematics papers? Does her document conform to the formatting and typographical conventions of a professionally-written mathematics article? (E.g., use of section headings, figure captions, and appropriate intertextual and post-textual citations.)

2. Composition. Has the student begun by laying out the context for the problem to be studied (history, prior work in the area, etc.)? Does the student then indicate what he will show in the course of his article? Does the exposition proceed in a clear and logical fashion from this point on, culminating in one or more primary results propped up on auxiliary ones? Does the student indicate applications of his work, or directions for future study? Do the aforementioned elements interact smoothly to create a single coherent text?

3. Clarity. Does the student make effective use of notation and teminology, including appropriate use of existing terms and apt introduction of new terms? Does the student eschew jargon for jargon's sake and avoid the passive voice? Are sentences manageably short? Has the student included diagrams where needed?

4. Completeness. Has the student answered, or at least addressed, all obvious questions? Has he ensured that his proofs consider all cases?

5. Correctness. Are the student's proofs correct? On a more pedestrian note, is the student's grammar correct? (Here we speak of both standard English grammar and the "mathematical" grammar that attends to sentences formed by "expanding" mathematical notation, replacing it with the corresponding verbatim text.)

6. Audience. Does the student's writing address the appropriate level of reader? (The students were asked to write as though their work would be submitted to a mainstream math research journal. In one case the finished product, a slight modification of the seventh week's report, was submitted, and very quickly accepted, to such a journal.)

In the hour-long course of printing the articles out I noticed unmistakable development along a few of these axes; greater perspicacity will be needed to discern development along others.

I've got my work cut out for me. Between this project, my two papers on math poetry (feedback on a draft of one of which I hope to receive from Lulabelle very soon), and three ongoing math research papers, it's going to be a busy break.

Before I go, in the interest of checking as many items as I can off of my recent "to-blog" list, I thought I'd give a brief update to my long-ago note about how much online homework systems suck.

One of my students in Precalc who's put up a mighty struggle this past semester came to me and complained about how she felt the homework system had kept her from really understanding the computations that she'd had to perform. "You just don't get the sense of it that you do when you write it. When you write math, it's like you really understand it, you're forced to think about it." I couldn't agree more.

Though I've encouraged my Precalc students over and over to do their computations by hand on scratch paper or better yet in notebooks (which they can refer back to later should the software not accept their answers), for many of them it's tempting to do as little of the work as possible by hand. Some students will scribble a few scratch marks on their paper before completing the computation in their heads and transcribing the solution directly onto the computer screen. The thought process that attends to this disjointed series of events is jumbled and chaotic; it's no wonder the students who do their work in this manner come away with an incomplete understanding of the material.

Writing is a tool for learning, and online homework systems diminish the efficacy of that tool.

Moreover, as the semester's gone on, this particular system's rigidity (already discussed in the post linked to above) has become more evident. It won't accept answers in forms that I feel are entirely valid. One egregious instance arose in a section on polynomial factoring and root-finding. Of course, certain polynomials have complex roots that appear in conjugate pairs. Let's say, for instance, that 1+i and 1-i are two roots to a given polynomial p.

Educo would require you to enter the roots individually, separated by commas: "1+i,1-i." Just so. If you enter the roots as "1±i" you're assured of receiving an "Incorrect" mark, despite the fact that this section's answer palette contains a "±" button, coquettishly inviting the user to press it.

It's asinine. The rigidity of thought that this sort of functionality encourages is sure to stifle any budding mathematician's creativity and curiosity.

These technological issues aside, the text is awful: it's full of errors, unnecessarily complicated algorithms the student is encouraged to memorize, dense and unintuitive exposition, and nonstandard notation. The text is unlikely to make converts of those who don't hope to go further than Precalc in their mathematical explorations. Moreover, its often addle-pated terminology and notation is likely to prove a minor stumbling block to the students who plan to continue on to Calc I as they try to build a bridge from the math Man M. Sharma's text has mapped out for them to the math Gilbert Strang will introduce.

Yes, folks, after much hemming and hawing, our department's sticking with Strang's free textbook next semester. While most of the people who used it to teach Calc I this past term (with one notable exception) felt at best ambivalent about it, we've decided to give it another chance. I've yet to form an opinion on it. It should prove a more challenging read for the students, but I've had success with challenging texts in the past (Rotman's algebra text is certainly a tougher row to hoe than Gallian's, but I think my Abstract class has come off quite well)...besides, a good teacher can teach effectively from any text.

I'll leave you with a final thought about teaching. I truly believe that the following statement is true:

If you take the trouble to learn your students' names quickly, show that you care about their learning, and do your best to figure out their individual learning styles as soon as you can, you've mastered 75% of what it takes to be an excellent teacher. (I don't know if 75% is the right number, but it seems like a good ballpark figure.) The rest is practice and perfecting of your various pedagogical skills.

## Friday, December 12, 2008

### A ream o' reading

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