Log 1: April 16, 2002

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

Lesson Activities

What I Expected Students to Learn

As this was the first lesson of the unit, I wanted students to begin thinking about two essential questions: "How does mathematics represent and shape the world around us?" and "What is a number?" I didn't expect them to have any form of answer yet, as these are questions I continue to ponder, but I wanted them to know that there was a purpose beyond the things we were studying. I also wanted to give them a brief heads-up on what they would be doing so it wouldn't come as a shock.

I also wanted students to realize that the task of "giving directions" can be nontrivial, even in the confines of a classroom. This would motivate the further development of coordinates, which most of them had become accustomed to taking for granted, as something of significant mathematical value.

Lesson Activities & Objectives

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

The COORDINATE SYSTEMS PREVIEW was intended to give students a "sneak peek" at the things they would be doing through the rest of the unit, and to give me an idea of what they had seen before. Students worked on it individually but were encouraged to consult the textbook and their peers, as I believe there's value in being able to extract information from a text or from other people.

GIVING DIRECTIONS forced students to attempt to do exactly that within the classroom. They worked in groups of four that paired up with another group. The point was to show that giving accurate directions is not a trivial task, and to motivate further disscussion of coordinates.

Connections between Lessons

This was the opening of a new unit; the two previous ones were on data analysis and probability. I confess that I didn't make any overt attempt to connect coordinate systems and vectors with anything in the previous units. Frankly, this unit seemed to be most "expediently" taught at the end of the year, as the prior data analysis and probability units tied together nicely and I was not prepared to teach it before them.

I planned to use this as a segue into polar coordinates as a specific and new example of coordinates that described the familiar realm of Euclidean two-space. In addition, a later activity, GIVING DIRECTIONS: VECTORS, would likewise introduce a context for vectors.

Teacher Actions

Instructional Strategies

The day began, as always, with students reading announcements to the class, which I had prepared on slips of paper; I take volunteers for some and conscript the rest based on whimsy. (Those who are uncomfortable can pass the announcement to someone else, with the other person's consent.) This lets students know what the day's agenda is; the agenda is also written on the classroom's secondary whiteboard for easy reference.

A brief lecture introduced the unit's topics. I kept this within 5 minutes because I know that most students, at least in this class, don't learn by listening to me talk. This gave them some further indication of the questions we would be addressing. The PREVIEW allowed them to look at the kinds of mathematical problems they would see later on.

Finally, the majority of the class was spent in groupwork. I felt that GIVING DIRECTIONS was a rich enough task for students to engage in genuine discussion with each other on how to give directions, the effectiveness of directions, trying to figure out others' directions, and the mathematical connections involved.

Interactions & Promoting Participation

Whenever I am obliged to give a lecture, I attempt to walk around and make eye contact with students. In this particular instance, I needed to write on the whiteboard--I find that students are more likely to be able to follow what's going on in an introduction if they aren't overburdened with information without some explanation or context--so I ended up at the front of the class longer than I would have liked. As much as possible I try to solicit student response, even if it's a student's incredulous "You can what?" in response to my assertion that "you can add arrows [vectors] just like numbers." Also, by soliciting examples from students of things they already know ("What are examples of numbers?"), I can encourage them to make connections to prior knowledge.

During the GIVING DIRECTIONS group activity, I walked around the classroom to answer questions and take notes on what groups were actually doing (I am peripatetic almost to a fault). I make it a point to ask them to suggest ways that other group members can contribute if it seems a couple students are dominating the group. At one point, too, I had a talk with two groups about having a critical discussion without offending others; the incident was triggered by a student saying, "Your directions suck" to the other group, and I stay on the alert for remarks like this that can lead to quarreling.

Student Actions

Most students found most of the PREVIEW beyond their comprehension and gave up after 5 minutes, though a few attempted problems they had not seen before by looking through the textbook for explanations or examples. As I wanted students to learn what they'd be doing, not how to do these things, and did not plan on using the textbook much, this was fine, though I probably should have given less time.

In GIVING DIRECTIONS, students attempted to outguess the activity (memorizing the locations of the place-markers to give them an edge in decoding others' directions), discussed how to specify locations verbally, drew diagrams or maps (including an ingenious Rube Goldberg device for catapulting a paperclip toward the end location), walked around the room trying to find points, and prepared for and gave presentations (in pairs of reporters, one from each group). A few students were less ambulatory and seemed less engaged in the goings-on, but most of them became interested when they realized the problem of giving directions was "harder than I thought it was going to be."

Reflections

Assessment of Student Learning

The PREVIEW was a pre-assessment that confirmed my suspicions that much of the more "standard" material--polar coordinates and vectors--was entirely unfamiliar to students. A few neglected even to plot points using Cartesian coordinates, the one problem I was sure everyone would get, but their prior work made me suspect that they knew how to do it and elected not to. (Later classes proved this impression correct.)

My exchanges with students suggested that many of them were beginning to realize that specifying locations can be more problematic than they realizing. ("Ms. Lee, this is hard!" was a typical comment.) On the other hand, one pair of groups that drew maps of the classroom found the task "easy," and I tried to underscore the analogous importance of coordinates as a mathematical "map" in my own remark to the class after their presentation. The one student who opted for an alternate assignment had the chance to listen in on two groups' discussions, and additionally had reservations to share regarding the effectiveness of groupwork, which she personally found awkward.

Future Adjustments

In my future teaching, I would like to develop students' ability to extract information from a not-particularly-friendly (in the sense of being understandable and well-organized, as opposed to bright colors and strange fonts) textbook, as it is a skill that will serve them well in the more traditional math classrooms they are likely to encounter later.

Otherwise, this was a relatively "gentle" introductory lesson whose main value depended strongly on students' own insights and contributions. One thing that I would like to improve in general is scaffolding on giving presentations, which is something I addressed more in later lessons by requiring students to prepare outlines and give models for peer feedback.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 1 June 2002