Video Commentary

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

Unit Goals & Lesson Objectives

The unit's essential questions, "How does math describe the world?" and "What is a number?" were written on the whiteboard at the beginning of class, as the video's opening frames show. Students were given a chance to hear the day's agenda by reading announcements to the class; the agenda was also written on the secondary whiteboard for easy reference. Thus, students had some idea of what the unit, as well as this particular lesson, was going to be about before embarking on anything.

After some introductory comments on the essential questions and their relationship to coordinate systems and vectors, and a PREVIEW that would help me find out what students already knew, students chose groups of four to work on GIVING DIRECTIONS. This activity, which took the majority of the 100-minute period, was student-centered in that, while I gave them the guidelines, they chose their modes of participation. My role during this time consisted of walking around to keep an eye and ear on what students were doing and to address, if not answer, questions that came up. I believe that there can sometimes be a significant difference between giving students an answer, and giving them the tools to determine an answer or answers for themselves.

Due to the animated discussions going on in the classroom, I regret that it's difficult for a listener to pick out the various and fascinating strands of conversation captured in the video. These included logistical concerns about usage of materials and division of labor, a consideration of whether "[foot]steps" made an adequate measure of distance; how detailed directions should be; where the beginning and ending locations had been set up at the beginning (a legitimate strategy since they had had access to that information!); and others that generally indicated that students were thinking about the issues involving in describing places and routes.

Also evident is one particular pair of groups: group A was attempting to locate their starting point, which a student (in the blue-and-red jacket) indicated by pointing to a clip atop the whiteboard, then following the directions to the ending point. In the meantime, their partner-group B, consisting of the three girls who appear to the left, watched to see if the group A was able to locate the points using group B's directions.

This experience, which all groups participated in at their own pace, formed the foundation of students' learning for this lesson. Students' initial reaction when they saw the assignment tended to be along the lines of, "Why are we doing this? This is easy." Once they started actually writing the directions, the reactions shifted to, "Ms. Lee, this looked easy but it's kind of hard!" To some extent, it seems that the lesson succeeded in addressing the preconception that coordinates and specifying location is something that is a priori "obvious" or &easy."

Finally, student presentations, of which a typical example is given in the video, forced them to consolidate and communicate some of their discoveries. The ability to speak math, even in a relatively friendly and nontechnical activity, is something I've attempted to develop this semester. I regret in retrospect that I didn't ask more students to participate in the presentations. As it stood, pairs of groups chose a reporter from each group.

Student Interaction/Learning

I promoted student interaction and learning in this lesson, perhaps paradoxically, by taking a back seat. Though the day's announcements and the guidelines for GIVING DIRECTIONS came from my hand, I attempted as much as possible to encourage them to interact with each other rather than with me, whether this meant keeping an eye on groups to make sure they stayed on task, directing them toward each other when they asked a question of me, or facilitating presentations; at the end of the two students' presentation shown you'll hear my (disembodied) voice asking if students had any questions at the end; I was sitting in the audience at a desk with other students. (In later lessons I wised up and gave more formal guidelines for presentations and feedback thereon.)

As the (unfortunately blurry) text at the video's end states:

Not once did I appear
as the teacher
in this collage
of one day's lesson,
though I was present
(and even lectured
briefly, alas).

That is as it should be.

My goal in teaching is to
make myself obsolete,
so when the time comes,
students will unfetter
their thoughts and soar
with no more need of me.

Teaching Strategies

My conviction is that student engagement depends strongly on what they're asked to do. If it's too easy, many students will get bored and tune out. If it's too difficult, many students will get frustrated and tune out. If it's too rigid or too unstructured, many students will--well, you get the point.

My first line of defense, then, was to attempt to create an activity that students would feel was easy enough to tackle, yet rich enough that, in tackling it, they would discover unanticipated complexities to whet their curiosity. In addition, I wanted it to have connections to real-life and, as ever, to the mathematics so they would have a mental "pigeonhole" to use as a starting-point. As one of the major uses of coordinates is in describing the physical world--as any cartographer can tell you--and just about everyone has had to give directions to a place at some point in his/her life, challenging students to give directions in the classroom seemed ideal. From the nonstop conversation that took place, most of it directly task-related, it seems that this enjoyed some measure of success.

The second line of defense, since there is only one of me, was to structure the class so students could realistically refer to each other more than they referred to me. Groups by themselves are not enough: students have to have the knowledge-base to answer their peers' questions (it is not reasonable to expect a student with no prior exposure to explain Gödel's incompleteness theorems!), and the comfort-level to desire to ask and answer. The common-denominator nature of the activity ensured the former; a semester spent building groupwork norms helped toward the latter. Nevertheless, I made a point of walking around and paying active attention to students; I also made a point of trying not to "hover" too much and inhibit the opportunity for students to learn self-regulation.

There is a certain bias in the footage shown in that, to the students, I would have been visible walking around the room. During the groupwork segment, however, I was operating the video camera for some of the time and was therefore not in the camera's eye. On the other hand, instances in which I was in front of the class and the focus of all (or nearly all) students' attention were limited to no more than 10 minutes out of the entire period.

Evidence of Student Learning

Once students came to the realization that "this is harder than I thought," and some notion of why, my main objective was accomplished. The sample presentation shows that students gave the problem of giving directions some thought, and made tentative connections to coordinates and real-world contexts. Those particular groups gained some understanding of the need for precision that coordinates can satisfy. Individual reports, not shown in the video, also ensured some level of individual accountability.

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