October 6, 2001

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

Homework. It was the bane of my mathematical existence in 11th and 12th grade. My calculus assignments took upward of 2-3 hours to almost-complete, and I had 2 years of the stuff. (Inevitably some stubborn problems would stump my classmates and me.)

In the 6th period Integrated 3 Math, assignments regularly come from the corresponding textbook by McDougal Littell. (For reasons familiar to students and teachers alike they tend to include even-numbered problems.) We frequently, though not always, go over the previous homework assignment toward the beginning of class. This observation occurred during one of those sessions.

First: What are the goals of homework in this class? One is to give students practice on concepts and procedures (mostly the latter, which may help with the former). Another, related to the first, seems to be to assess how well students are able to apply newly-learned concepts and procedures. The results, when students turn in homework at all, are generally discouraging to me. I can't tell you the number of notes I've written to show why, in general, (a+b)^2 is not equal to a^2 + b^2. (It IS generally true in the field with 2 elements, but that would overload them.) During the following class I did go over that fact (and how to derive it), though my master teacher observed that they're much more likely to notice and heed personal notes.

I guess I should look into why I feel disheartened by homework and, in general, evidence of students' (mis)understanding. Here's a list:

Coverage

I'm not entirely convinced that class "coverage" of material is as meaningful or helpful as I think it could be. (No doubt this is partly attributable to my own naive optimism.) This stems somewhat from the organization of the curriculum, its emphases, and presentation (in class or in the text).

The chapter topics, in order, are: Modeling Problem Situations; Exploring and Applying Functions; Logical Reasoning and Methods of Proof; Sequences and Series; Exponential and Logarithmic Functions; Modeling and Analyzing Data; Applying Probability Models; Angles, Trigonometry, and Vectors; Transformation of Graphs and Data; and Periodic Models. If there are particular compelling reasons for this organization I sure don't see it.

Let's look at topics in the 1st chapter: Algorithms; Using Systematic Lists; Using Statistics; Using Graphs and Equations; Using Systems of Equations. (We didn't cover the last 3 topics, which I've omitted.) Well, that looks more coherent except for the fact that as the class went through these topics in class, I got no sense of their common thread, i.e. using math to model problems. Now I realize what algorithms and that one random statistics activity had to do with anything, but I didn't see it at the time. I doubt the students did either--but math education seems to train students not to question how topics connect or have significance.

Am I expecting too much in terms of the problem-solving and exploratory processes I would like to see fostered in students? God only knows. I don't deny the importance of drill in learning how to use procedures and familiarize students with the oftentimes exotic language of math. But if I asked one of my students WHY they are learning about polynomials, I doubt s/he could answer. (I certainly had no clue when I learned about them.) I should try asking, but I can already see the blank "huh?" response, and this isn't something I fault the students for. Could they answer the "who cares?" criterion with anything but "I don't care" or "I don't know"? To put it another way, the math we have exists because someone, somewhen, thought it was interesting or useful enough to develop, but this sense gets lost in the classroom.

What I object to, then, is the prevalence of drill and little else. Even "scaffolded" word problems with parts a, b and c stump 90% of the class. Students ask me "how do I do this?" because, as far as I can tell, what they've learned of math is that they will be given nice, neatly-packaged problems for which they are handed a toolkit with which to find an answer, whether or not they understand what's going on or why method A works better or faster or more interestingly than method B. I want to start showing them how to choose their approaches and to try more than one when the first is fruitless; how to decide when an approach is going nowhere, and mine it for new approaches.

This is, of course, a tall order for homework alone. But a steady diet of generally unchallenging problems isn't showing great results, and isn't going to help develop these more "advanced" skills.

What's Interesting?

How do I make homework interesting or worthwhile? Some "boring" or routine problems are inevitable (and useful for the reasons mentioned above). The textbook tries with its flashy colors and real-world word problems to show the worthwhileness of math, but in practice the context could be so much Javanese to them. (I don't believe we have any Javanese speakers, incidentally.) Interesting problems offer slightly more incentive, too; the "eureka!" feeling when you figure a tricky problem out is something I'd like everyone to experience.

Perhaps "hand-tailored" homework problems, with a brief verbal explanation or spiel, would be a start in a slightly better direction. I know so little about problem-design, though, despite a couple attempts with Zap the Martian. My first "challenge" test question was a failure in terms of scaffolding. You can't just throw a more interesting problem at students who have seen few of them before and expect them to figure it out. It's something to think about, anyway. Thank goodness for C&I.

Completion

How do we get more students to do the homework? Especially since a number who persistently fail to get it done outside of class are active and at least quasi-interested participants during class. My master teacher told me, when I asked her about phoning home, that she felt she'd rather have these students (as mostly-juniors mostly-planning to attend college) figure out accountability for themselves. I've written notes on blatantly incomplete homework papers from the last assignment.

On Friday we tried assigning a number of students who either don't complete or don't turn in homework to the 2-3 PM tutoring block on Monday (with one agreeing to come in on Tuesday because he has work Mondays). I approached them privately, with post-it note reminders. Most of them took it matter-of-factly, though I saw 3 such students giving each other significant glances. One told me that "I just keep forgetting to do it," and I said that these things happen, but come in anyway and we could see about next week. It might be a wake-up call. We'll see how that goes.

Homework in this class is worth 25%, and the other 75% comes from exams. Rather steep. Personally I would rather give greater weight to homework, which is routine but encourages practice (I hope), and perhaps projects. Tests are valuable but I've known far too many people who work too slowly to demonstrate the things they know. I guess that's a problem with time restrictions in general.

Late Homework

The policy in this class is no late homework. Personally, I wouldn't mind taking homework up to 3 days late for half-credit, on the theory that it would encourage them to do it instead of just giving up, but penalize those who can't get it in on time.

I did ask my master teacher if the block schedule seemed to have affected the rate of doing homework, and her impression was that it hadn't in her class, though she'd heard from a foreign language teacher that it had. Hmm.

Review

How long to spend on in-class homework review? Solutions? Another big one, related to general time management. I really need a watch (my last few either self-destructed or I lost them). When I'm in front of the class I tend to go into fugue-state and lose even more track of time than usual.

Things to remember: Ask for questions and don't penalize for mistakes that arise in the course of discussion, but explore them. Find out what the stumbling blocks are. Spend more time on mathematically interesting questions.

Let students present their work, whether partial or correct or whatever. You DO learn from "characteristic" as well as idiosyncratic mistakes. We haven't been doing this, as the previous pattern was to dispense answers (to questions) or let students "check" their work against a posted answer sheet (in practice, this meant some hasty copying went on). Perhaps picking out problems of interest in case the class doesn't have questions (ha!) or anticipating trouble spots (conceptual, procedural, etc.) would also be wise. I should look a little more closely at the homework ahead of time, but it's sometimes hard to guess what'll stump students, especially since it feels like it's been ages since this material was new and unfamiliar to me.

Conclusions?

I don't really have any. I think even with a curriculum I'm not especially happy with I could find, or experiment with, ways to make homework a bit more meaningful. I'm not averse to the idea of rearranging topics or modifying problems. I guess I'll see what comes up next.

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