Log 5: May 16, 2002

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

Lesson Activities

What I Expected Students to Learn

I wanted to give exposure to the idea of "angular distance" (relevant in, say, aiming water guns from a fixed position) and review finding distances between two points when one is not at the origin as two uses of coordinate systems when it's not practical to go out and measure something directly. In a brief paper-airplane demonstration I hoped to expose them to 3-dimensional Cartesian coordinates. I also wanted them to think about the issues involved in assessment, in which they'd have an opportunity to participate.

Lesson Activities & Objectives

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

The warm-up had one problem on finding the distance between two Cartesian points, which was a distance metric students had seen before; and one problem on "angular" distance, which was meant to show an advantage of polar coordinates in finding one type of "distance." The last problem was similar to the previous lesson's chess example.

I demonstrated three-dimensional Cartesian coordinates not so much because I expected students to learn them, but so they could be exposed to something neat. The "gosh-wow" factor in math is sometimes downplayed, which I think it's a shame. If I can whet their interest in future math by taking a few minutes to show students something they wouldn't otherwise see in a math classroom, I'm all for it. If it hadn't been for tantalizing glimpses of hyperspheres, fractals, and Klein bottles, I probably would never have persisted with math in college, after all.

In GRADING LETTER TO ZAP #1, students used the given criteria to decide how to allocate points and assign a grade to another pair's letter. Then students had to then write feedback on the grade report they received. I hoped this would get them to think about what grades actually mean and the process a teacher goes through in giving them. At the end, I invited students to put up a comment on the question "What did I learn about grading?" on a "poster" (a large sheet of butcher paper).

Connections between Lessons

The question of finding distances tied in directly to some questions on HOMEWORK #5 and to the previous day's warm-up problems. Grading the letters, which I returned to students after having glanced over them, also made direct use of the previous lesson's LETTER TO ZAP #1 activity. As for grading in itself, it was something that I figured would be directly relevant to students no matter when it came.

Teacher Actions

Instructional Strategies

The warm-up gave students a chance to settle down and work out some math with others around whom they could ask for help. When I realized students were confused by the idea of "angular distance," I did an on-the-spot demonstration involving aiming an imaginary water gun from a fixed position, using students as targets. It's all about the low-tech demonstrations.

Similarly, 3-dimensional Cartesian coordinates involved a low-tech demonstration involving masking tape x- and y-axes on the floor, a meterstick for the z-axis, and two paper airplanes with their noses colored in to provide demonstration "points." I solicited two student volunteers to hold the paper airplanes, both for practical reasons (I have only so many hands) and so there would be as much student "ownership" as possible even within a fairly teacher-centered lesson fragment. After giving a couple examples I asked students to call out the coordinates of various points in space. This sort of anonymized mass-participation can be useful in allowing students to get involved under the cover of other voices.

Grading the letters was something the students did on their own once I gave them the guidelines. I used the grading "poster" as a way for students to summarize their impressions and thoughts, and share them with their classmates.

Interactions & Promoting Participation

I responded to students' confusion by giving the water-gun scenario above. In general, of course, I prefer that they consult each other, but there has to be a starting-point in terms of knowledge. (It would be rather silly for me to ask them to explain something to each other that they had no reasonable way of knowing a priori.) By "aiming" at particular students and making sure to make eye contact with everyone at some point, I hoped to capture their interest.

During letter-grading I circulated around the room to listen in on students' conversations. I was especially alert to any hard feelings that might result from less-than-stellar grades; students are inevitably harsher to each other in their assessments, whether for a presentation or a homework problem, than I would be, and there have been occasional incidents of less-than-tactful phrasing.

Student Actions

Students were more comfortable with the warm-up problem that also involved "no solution," a case of movement on a grid where a piece that could only move two points at a time yet could not at all reach a point that a piece that could only move one point at a time could plod toward. Students also seemed amused by the paper airplane demonstration; at least, their eyes were tracking.

Students generally took the task of assessing each other seriously. One pair of students acknowledged that their grade was fair since they hadn't put that much effort into the work. Also, while not everyone elected to put a thought up on the "poster," the comments showed an interesting variety, from the several who remarked that grading was harder than they thought, to the humorous "Show no mercy," to the pragmatic "Don't grade friends." I was honored that they took it seriously, in the main.

Reflections

Assessment of Student Learning

Most of the learning on this day consisted of reconsolidation of things the students had encountered before. No matter how much I despise "drill and kill," I imagine that all those sets of math worksheets came from the recognition that most people, most of the time, need practice to be able to get something "right." I even remember a certain serene pleasure in going through a page of integrals, but it seems counterproductive to to foist that page of drill onto students before they have some glimpse of that pleasure. I will never forget the time two students came in for tutoring, worked through some problems with me, then looked up at me with astonishment and said, "Ms. Lee, it's fun when you understand how to do it!" Next stop: showing the fun in finding out how to do it....

It seemed from the poster and various comments that students also learned more about assessment and how it happens. They've assessed each other's work before, but much more informally. They also once assessed a couple of my decidedly-incomplete test responses for a grade, but never had to formally grade each other before. All things considered, it turned out well.

Future Adjustments

The lesson went fairly well overall, though in the future I'd like to spend more time on 3-dimensional coordinates, assuming I end up teaching a class that isn't rushed for coverage. I also should have given a brief coordinate-conversion quiz on which students could use the letters for reference, then grade the letters. That way they would have a more immediate way of being able to tell how helpful/comprehensible they found the letters.

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