Central High School
|
Newark, NJ
Dr. Roberta Schorr
|
|
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 couldn’t
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 student’s 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.
Picture 14
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."
Picture 16
Picture 18
Picture 19
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 character’s 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 clown’s 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.
Picture 18n
Picture 10
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
Dude’s 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 Clown’s 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).
Picture 12
Picture 22
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. I’ll 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 Clown’s motion).
This one is with negative velocity, but that means that he is walking
back.
The teachers and supervisor were amazed by this student’s
work. His clear facility with the technology and mathematics was
truly exciting to all.
|
|
|
