Dr. Schorr began working with two groups of students from Central High School in March of 1998, and continued working with them until their school year ended (June, 1998). Since they could not use the computer lab at the high school, the students came over to the lab at Rutgers-Newark. This may not seem like a major issue, but it was. Each and every time that the students came to our lab, they had to have their parents sign a new permission slip. (Anyone who knows anything about high school students can appreciate how difficult it is to have students actually get these permission slips signed.) Despite this, most students got their slips signed regularly (some even ran home to get a slip signed when they realized that they couldnt go without it). The students repeatedly shared their enjoyment in doing math this way. One student wrote this note to me: "I hope we can go back and learn some more because once again it was exciting. Little simple things like that make me want to no more and more about math." Another student wrote this note to me after realizing that there were many different ways to solve a problem: "It seemed I was the most outstanding student in the class, because of the way I used a certain technique, it was different from the way everybody else did it and I was still correct. I think that we should have many trips to go on to educate our minds and see more of what the world has to offer us as the people of today."
Dr. Schorr and her sturdents began by working with the elevators. The students had to figure out how to get the elevator to stop at the same floor in many different ways. One student wrote this: "On March 18th, 1998, our math class went to the Rutgers Computer Lab. Me and my classmates worked on the computers and learned about velocity. We used worlds. The world we was working on dealt with Elevators. We had to make the elevators give them names and send them to the floor that we wanted on by using velocity. There were three elevators. All of the elevators had to be sent to the 8th floor first.... Then we had to send the elevator to the 12th fl. The first went 6 fl. per 2 seconds the second was 3 fl. per 4 seconds and the last was 2 fl. down and 8 fl. per 3 second." Notice the students language to describe the motion of the elevators. While she did indeed get the elevators to the 12th floor, she had great difficulty in describing what she had done in words. In fact, her description seemed to indicate a basic misunderstanding of the graph, the units involved, and the notion of rate. This was not at all uncommon among the students. They could get the elevators to go where they wanted them to go, they could make sensible predictions about the motion (slow, medium and fast) of the elevators and the final position of the elevators, but they had great difficulty describing, in words-verbally or written, what they had actually done. Most of the students were not accustomed to talking or writing about their thought processes since they had not had many opportunities to do so in the past. Therefore, over the course of the project, Dr. Schorr repeatedly encouraged them to talk about their thinking. By the end of the project, most students would readily share their solutions and mathematical thinking with others.
After Dr. Schorr and her students completed the elevator activity, they were challenged to think about how changing the order of velocity segments could produce different motions. They actually walked slowly and quickly, in different combinations and they were asked to describe how their motions were the same and different. Next, they did a version of the permutations activity. While doing this, they were challenged to think about the number of velocity pieces used, the time interval for each piece, the total time for the motion, and the total distance traveled, and the order of the velocity segments.
Students could predict where the character would end up by "counting the boxes", and could tell if the motion was fast medium or slow by looking at "how high the line is." The students had to consider how the motions could be so different even though the number of velocity segments, the time interval for each segment, and the total distance traveled were the same. Ultimately, they were challenged to come up with a single segment that would get the character to the final position in the given amount of time--and they were able to do it. The language that they used slowly began to change as well. For example, when referring to a simulation that she had generated, the girl in picture 14 said that she could predict that Clown would get to the 18 meter mark because he moved "3 meters every second for 6 seconds."
After this, they worked on an activity in which the students had to build graphs to reproduce a motion (Follow that Clown and Final Positions). They had to make velocity graphs with slow, medium, and fast velocity segments based on viewing a characters motion. While most students began by using trial and error, they were consistently encouraged to consider how the velocities were related to time and observable in the clowns motion (see also boy in Pictures 16, 18, 19). The same girl referred to above (Picture 14) described a fast and a slow motion in which Clown and Dude end up in the same position: "Here he went 4 meters in one second and here he went 2 meters for 1 second, but for 2 seconds."
Perhaps the most challenging task involved getting Clown and Dude to exchange positions. The goals here involved developing an understanding of how to relate negative values on a velocity graph to the motions of characters, and to see how different velocities and time durations can affect total distance traveled. Before using the computers, the students enacted several scenarios by actually walking and meeting in the middle of the room, to the left of middle, and to the right of middle.
Initially, some of the students thought that there was only one way to get the two characters to exchange positions and meet in the middle.
One boy (see picture 18n and picture 10) thought that this could only be accomplished by using one velocity segment for Dude (15m/s for 1 sec) and one for Clown (-15m/s for 1 sec). Dr. Schorr had to challenge him to come up with another way. He decided to reduce Dudes velocity to 5m/s for 2 seconds. She asked him where that would make Dude land. Without answering, he extended the time to 3 seconds. He went on to adjust Clowns motion so that he moved -5m/s for 3 seconds. Without saying anything else, he proceeded to generate 2 other solutions (3 m/s for 5 seconds for Dude and -3m/s for 5 seconds for Clown; 1 m/s for 15 seconds for Dude and -1 m/s for 15 seconds for Clown).
He was convinced that there were no other solutions until a girl (see Pictures 12 and 22) found another way. She shared the following with me and the other students: "Well, I took the velocity graph and put two for both and then I was playin around with it and I was gettin them to meet at 5 but I had to come closer so I got the, um, Dude to go 8 sec, meters for a second, and then 5, ugh, 7 meters for another second, then I did the same but I switched it around so that they could meet at the same time. See em meetin ya all?" Several students began to experiment with other ways, while others went on to get the characters to meet to the left of middle, right of middle, meet and go back to their original places, etc.
At the end of the project, one student made a presentation before the mathematics faculty and supervisor of the high school. He chose to demonstrate how Clown and Dude could exchange positions and meet in different places using multiple velocity pieces. While Dr. Schorr did not have an exact transcript, she could paraphrase some of his comments, based upon some field notes that she took about that afternoon:
You see, you can get them to exchange positions by bringing in these velocity pieces. I can do it lots of ways. Like this, you have Dude going 5 meters in one second but for 3 seconds, and Clown doing the same thing, only he is at -5. See you can tell its gonna work by lookin at the area under these (points to velocity pieces). I coulda done it where they go slower or faster. Ill show you slower. (He generates a graph in which Dude walks at 1 m/s for 15 seconds, and Clown walks at -1 m/s for 15 seconds). See, you can tell its gonna work because of the area under the velocity segment (he points to the graph). You also coulda done it using more than one velocity segment for each. (He proceeds to use 2 pieces for each, again explaining how he could use area under the curve to justify his solution.) You add up the two areas, and see they equal the areas of this one (points to the segments representing Clowns motion). This one is with negative velocity, but that means that he is walking back.
The teachers and supervisor were amazed by this students work. His clear facility with the technology and mathematics was truly exciting to all.