January 23, 2002

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

Wednesday, 23 January 2002, I planned to review Euler's number, e, and exponential functions as a lead in to logistic growth functions. I was a little worried about what students would remember as their last class had been a final, and then they (and I!) had enjoyed a 4-day weekend. (I don't know what genius came up with the school calendar. If there's a rationale, I would love to see it made evident.)

Fortunately, the hiatus didn't seem to have harmed the class' memory any more than the usual one to two days between sessions. After getting some housekeeping out of the way--returning last semester's finals and the like--I launched into a review of "last time." (I have to keep reminding myself not to say "yesterday," as a literalist will inevitably correct me.) With a little prompting, students were able to recall the most salient features of exponential functions. Students also gave some examples of situations where they had seen exponential functions used: bacterial population growth, compound interest, and radioactive half-life.

We then worked with y = 2^x as an example using the graphing calculators to remind students of what a "canonical" exponential function looked like. I brought up the topic of bacteria growth and asked the class to consider whether even immortal bacteria could continue population growth indefinitely (pathological case #1) and, after they came to the conclusion that it'd be a little silly, the question of a zero starting population (pathological case #2). My suspicion is that pathological cases help clarify the canonical ones; my other purpose was to highlight limitations in a purely exponential model of population growth with the goal of introducing logistic growth functions as a slightly better model.

The new scenario I brought it was an outbreak of flu, which I believed was modeled, with limitations, by y = 2^x (I was incorrect, but only caught it upon reflection after class). In a brief visual demonstration of a very simple flu outbreak, I came in "infected" two others students, whom I asked to stand up; each student infected two more students, who stood up, and so on. Very shortly the entire class was infected, and a couple students suggested going out to infect the rest of James Lick, but the point of the "speed" of exponential growth came through.

Not only did students see the limitations in an unlimited spread with a limited population, one student pointed out that having each person only infect two more was unrealistic. If I'd let the discussion go on a little longer, I'm sure that other points, such as redundancy of contact, incubation period, and mutation, might have emerged. I was glad to see students begin thinking about issues of accuracy and appropriateness in devising or critiquing mathematical models. In retrospect, I wish I had emphasized this more. As I am not deeply acquainted with their applications, logistic growth functions are of passing interest to me in themselves; their real value lies in (re)introducing the idea that one of the driving reasons for math to exist is the creation of successively more accurate models. (A dramatic example from physics is Newton's formulation of gravitational force, which was succeeded, but not entirely superseded, by Einstein's general relativity.) In other words, I feel that I allowed an essential question (to use Wiggins & McTighe's term) to slip by in favor of relatively mechanical explorations of a particular class of function.

The flu scenario was particularly handy as the book itself gave an equation modeling the situation, y = 1000 / (1 + 990e^(-.7x)). Students graphed the function and wrestled a bit with their calculators. I wish I'd thought to ask about a graphing calculator projector, which would have been a handy use of technology; none of my classes in high school ever used them, so I admit it didn't occur to me. (Too bad technology these days is designed so you have to think like a geek--I'm using the word as a classification--in order to troubleshoot it. I'm good at it, and my sister is excellent, but most people's reaction is: "I hate this computer." Or "This calculator isn't working." That's another topic for another time, though; I refer the interested reader to Jef Raskin's The Humane Interface for one example of alternate usability emphases in interface design, specifically for the personal computer.)

Another snafu on my part was that, when no one questioned the appearance of these magical numbers in this magical equation, I neglected to bring up the issue myself. Even the teacher's edition of the textbook produces the equation with a flourish and leaves you to flounder in figuring out where it came from (I presume some empirical data and curve-fitting was involved). Again, in retrospect this is symptomatic of what I'm starting to call the "high priest syndrome" in mathematics (with apologies/thanks to fellow steppie Manny Medina, from whom I first heard the term). After years of accepting whatever math falls into their paths, it is no wonder students have the questioning burned out of them. I feel remiss for perpetuating it, though, and hope I will learn to be more vigilant.

I manipulated the numbers into another off-the-cuff logistic growth function, and students graphed it for comparison. Features that emerged included the characteristic "S-shape" and the two asymptotes, which students intelligently dubbed "min" and "max." A couple were able to share with the class their understanding of what an asymptote signified, and we spent a few minutes discussing how to write equations for them.

The class then had a textbook assignment, doubling as homework if they didn't finish it in class, to try a few problems for themselves. After 10 minutes, I semi-reconvened the class to go over one of the problems. This is something that I need to remind myself to do more often when a question crops up throughout a large portion of the class.

I'm afraid the class ended in quite a rush, as my supervisor reminded me of the time. I had been relying on the clock at the back wall, which was broken, instead of my watch (having not worn one regularly for over a year it's hard to get back in the habit), and was therefore taken by surprise that the 100 minutes had gone by so quickly. In a panic I asked the students for a "quick write" on what they had learned and any suggestions or complaints. I also brought up the issue of homework and what would make it worthwhile for the class, which (as my supervisor pointed out) should have been saved for a separate occasion. As far as that last goes, I've set up an online discussion board for those who would like to discuss homework electronically and have that access, and my master teacher and I plan to institute a new policy this week that homework will be collected at the beginning of class, scanned, and those whose homework doesn't show some reasonable effort will be asked to stay in during class to work on it.

In any case, both my supervisor's comments and overwhelming response on the students' quickwrites, which also overwhelmingly frank but polite, suggested that I should walk students through the kinds of problems they would see on the homework. Well, not all students were that specific, but even the suggested "explain more" is highly suggestive.

This leads me to an interesting quandary. My knowledge of mathematics has been a consistent strength as well as Achilles heel. I'm rarely worried about having my own understanding badly shaken (as opposed to expanded; and this excludes statistics and probability, which I consider utterly counterintuitive), and this frees me to deal with the students' understandings rather than my own. On the other hand, it is necessary for me to deal with the students' understandings, because without actually speaking with them I constantly overestimate their ability to jump from fairly high-level theoretical discussions to the actual doing-of-math.

This suggests first of all that I need a lot more pedagogical content knowledge than I actually have, which is something that time and experience will mitigate. Second, I need to figure out how to scaffold the ability to go from theory to practice, which is something I wasn't even especially good at in high school (let alone in math). Third, is it a good idea to give the students what they want? I really dislike the Spartan "sink or swim" paradigm, but on the other hand, where should I draw the line between fostering teacher-dependence and easing their way toward an understanding of math? Is an understanding of a relatively tiny aspect of math (logistical growth functions) ever worth sacrificing a more general understanding (essential question: mathematical models and their strengths and limitations) or ability? Perhaps part of the issue is the apparent mismatch between what I'm teaching and what the homework asks students to do; surely I can balance procedural and conceptual reinforcement. Not only am I trying to figure out how to balance these concerns, I am faced with the real-life constraints of curriculum, time, and student frustration.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 21 May 2002