Log 4: May 14, 2002

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

Lesson Activities

What I Expected Students to Learn

I wanted students to review Cartesian distances and develop a fledgling sense of "unconventional" distance metrics (though not, alas, in any real depth). Also, the activity that took up the bulk of the class, LETTER TO ZAP #1, aimed to develop their skills at communicating math conceptually and procedurally, specifically in explaining polar and Cartesian coordinates and conversions.

Lesson Activities & Objectives

The lesson activities are described in detail in the Curriculum Unit.

The first part of the warmup was a problem on finding the distance between a Cartesian point and the origin. Students could either use the distance formula, which they had seen before, or the Pythagorean theorem, which they had also seen before. (The distance formula is derived from the Pythagorean formula in any case.)

The purpose of the second part of the warm-up, based on chess, was to give an accessible introduction to the notion of an "infinite" distance within a finite space: a bishop can't move to an adjacent square no matter how close it looks, due to the distance metric that its method of moving defines on the chessboard.

Connections between Lessons

The chess-distance brainteaser tied in directly to an optional problem on HOMEWORK #5 (which everyone who did the homework ended up trying, actually) asking whether a chess knight or rook moves "faster" if you count the number of moves. Also, LETTER TO ZAP #1 drew directly on students' prior experiences with coordinates and conversions, and they had been asked to write such letters sporadically in the past, albeit with far less guidance. During the next class, GRADING LETTER TO ZAP #1 would give them an opportunity to read each others' letters and assess them for clarity, correctness, and completion.

Teacher Actions

Instructional Strategies

Students found the chess-based warm-up, which I'd written up on the whiteboard, extremely frustrating--moreso than I would have expected. I did have to make some clarifications in phrasing, particularly for students less familiar with chess, as to what constituted a "move." However, other than those clarifications, students were able to resolve the question among themselves after some heated discussion, so the hands-off strategy worked out.

Students also wrote the LETTER TO ZAP #1 in pairs. Physical materials they might have desired to use, such as polar graph paper, markers, or construction paper, were in the Materials Corner at the back of the room; everything the students needed were either on the handout they received or in their own heads.Other than walking around to peek at what students were doing and checking explanations when requested to do so, I did very little directly after giving them the assignment.

Interactions & Promoting Participation

Other than walking around to peek at what students were doing and checking explanations when requested to do so, I did very little directly after giving them the assignment. By this time of year, students seemed to need less external prompting to get involved in discussions, listen in on the ideas floating around, or work out division of labor in paired activities.

Student Actions

When they're frustrated, students tend to skip any intervening steps and ask me directly. This isn't bad in itself, as they often have legitimate needs for clarification. In the case of the brainteaser, though, I listened to the first few students to raise their hands, reassured them that their thinking processes made sense to me at least, and turned them loose on the rest, which resulted in a spirited debate. Part of the controversy was whether the problem had a "solution" in a conventional numerical sense; "it's impossible," while a perfectly legitimate "answer" in mathematics, is often ostracized in the math classroom, and thus I imagine they had some psychological difficulty in accepting it.

The letter-writing went fairly smoothly. Students worked out division of labor, or took turns writing, computing examples, and discussing what needed to be done. A few groups seemed slower-moving, and there were occasional off-task conversations, but everyone was able to hand in something by the end.

Reflections

Assessment of Student Learning

I found it enlightening both to listen to students discussing the brainteaser among themselves, and to read the writeups that were turned in, many of which revealed that they understood the basic features of the problem. I don't expect them to have, say, a deep understanding of metric spaces based on such a brief exposure. Also, some of them came away with a glimmer that "legitimate" math problems may have no solution, which is worthwhile in itself.

The letters, when I glanced over them, were pretty impressive. I did notice a few errors of formula, which I planned to address; but the whole point of estimations was to build in a way to check for errors. I came away with the impression that students would be able to work with coordinates in the future if they were given a little prompting and the necessary formulas, which is fine by me; memorization will come naturally, with practice, as needed.

Future Adjustments

I'm torn on whether I should have spent more time on distance metrics in general. There's a great book in the STEP library--whose title and author I unfortunately don't recall--on "taxicab" distance, which is another example of a "nonstandard" distance metric that people encounter in daily life, but may not see addressed in a classroom. I realize, though, that vectors--which I wanted to make time for, as the second part of the unit--are something that students who continue in math will very likely encounter, while distance metrics are not. I think I could happily spend a lot of time on either topic, or better yet, both.

The brainteaser worked out well, though in the future I hope to incorporate chess more consistently throughout the unit, as not all students will be familiar with it, and games are a quick way to gain and sustain students' attention (or mine, come to think of it). I wonder if it would have been more effective to have students work individually on the letters, though, if they could consult with each other or the textbook as desired. Since they do a lot of pair- or groupwork, it's nice to give them some opportunities to work alone once in a while, too.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 2 June 2002