- Email: [email protected]
The catwalk task: Reflections and synthesis: Part 1
Journal of Mathematical Behavior 27 (2008) 250–254
Contents lists available at ScienceDirect
The Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb
The catwalk task: Reflections and synthesis: Part 1 Steven Case Mar Vista High School, 505 Elm Ave., Imperial Beach, CA 91932, United States
a r t i c l e
i n f o
Article history: Available online 17 March 2009
Keywords: Modeling Representation Teacher learning Task design
a b s t r a c t In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional. © 2009 Elsevier Inc. All rights reserved.
1. Reflections on my own work After getting the assignment of finding the cat’s speed late on a Thursday evening, I have to admit I was not as enthusiastic as I was once the assignment was completed. Given 20 min or so in class, it took me a bit of time to realize what I was actually looking at. Twenty-four pictures of a cat were taken at an almost instantaneous sequence. In fact, the microscopic separation between photos of only three one-hundredths of a second did not fully sink in right away. In actuality, I had listened to the directions, but when I looked at the photographs, it looked like twenty-four pictures of a cat at a bunch of different times, not necessarily in any sequential order. I was struggling to grasp that all of the pictures were connected in a way that represented continuous motion. After asking a few questions about the grid lines behind the cat, I took what little time I had left and my first intuition was to calculate average speeds in between the consecutive frames that surround the desired frame. Specifically, my plan was to calculate the average speeds between the ninth and tenth frames, tenth and eleventh frames and then take a simple average of the averages. My plan helped me understand the details of the frames as it was time to start making measurements. I knew the distance between each gridline behind the cat and the time that had elapsed between each photo. To start measuring the distance the cat had moved between each frame, I had to decide which part of the cat to measure in each frame. These photos had a striking similarity to photographs that are used to decide a narrowly contested horse race. As two horses lunge through the finish line a photograph is taken to see which nose crosses first. With that in mind, I chose to measure how far the nose of the cat had traveled between each frame. I measured 3 cm between the ninth and tenth frames for a speed of approximately 96.77 cm/s. Between the tenth and eleventh frame I estimated a distance traveled of 5 cm for a speed of approximately 161.29 cm/s. If I had abandoned the project that night, my answer would have averaged to 129.03 cm/s for the speed of the cat during the tenth frame. I remember feeling unsatisfied with the calculation. I knew that this kind of problem deserved more than just an average of averages. The cat was moving, accelerating, and its position had changed every frame. Certainly its movement could not be
E-mail address: [email protected]. 0732-3123/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jmathb.2009.01.003
S. Case / Journal of Mathematical Behavior 27 (2008) 250–254
251
summarized linearly as my average calculations would have assumed. There had to be more to this. I knew that the cat was moving, but there was just enough unknown to create uncertainty in any way I was going to redefine the cat’s speed. Why was this ‘average of averages’ the first thing I thought of and why did it leave me feeling so unsatisfied? Perhaps when faced with ‘how fast’ problems, students first think of relationships between the units. In our case it was how far the cat traveled in centimeters and how long it took the cat to travel such a distance measured in seconds. Naturally dividing the length traveled by the time it took to do so results in speeds of centimeters per second. This type of simple calculation is first introduced to students through situations and problems that involve uniform change; for instance, a car traveling 20 miles per hour, or a painter that can paint 4 walls in 3 h. In each situation the rate is constant. The situation assumes that the car (or the painter) will never speed up or slow down. An average calculation would model these situations perfectly. No matter how one divides the work done or distance traveled, the rate would remain the same. This same process applied to the cat problem would assume that the cat was traveling at the same speed throughout. Obviously, the varying average speeds in the ninth and tenth frames (96.77 cm/s and 161.29 cm/s) tell a very different story. The cat was clearly accelerating through the tenth frame. This could be the reason for my uneasiness over my ‘average of averages’ speed. I knew that my calculations assumed the cat was traveling at a constant rate. Even before I punched numbers into the calculator I knew that the two averages would be very different. Visually inspecting the pictures of those frames shows just why frame 10 becomes such a point of contention. The cat moves a considerable distance more at frame 10 than in the surrounding frames. Using an ‘average of averages’ was the only way I knew how to cope with the varying speeds. Since the first average described the cat’s average speed between the ninth and tenth frames and the second average represented the cat’s average speed between the tenth and eleventh frames, the average of those would be the closest I could get to the instantaneous speed of the cat at the tenth frame. I knew it would be close, but just not good enough. Averages could not possibly account for an instantaneous speed. Even though the time between frames was so tiny, there was still an aura of mystery as to what was going on in between the known frames. The average of the averages not only assumed a constant rate of change in between frames, but also assumed that even though the speeds were so different the instantaneous speed of frame 10 would occur exactly in the middle of the two speeds. There was just too much assumption in those calculations to make me feel comfortable in stating my final answer. I had to look at this in a different way to try to represent what was happening in another way. I awoke the next morning ready to redefine my calculations. To overcome the unknowns, or the position of the cat within the three one-hundredths of a second between each frame, I had to model the cat’s position in a way that was more comprehensible to me. After a night of thinking about it, the way the data was presented reminded me of a prior experience in my life. The data about the cat was given to us in chunks. For instance, we knew where the cat was at only specific times at the end of each frame. A few months earlier I had driven up the coast with my wife to see my mother. To pass the time as we went, I entered our distance traveled according to the time we had been on the road. I made a data entry point on a graphing calculator four times each hour. The data that I was entering looked very similar to the cat data that was given to us. Every 15 min, I looked over to the odometer and entered in how far we had traveled up until that point. It was fun to see the data points line up as we rolled up the coast. Even though the speedometer at each specific time was reading an explicit speed, it did not necessarily match our average speed for the trip. As we progressed, the graph almost told a story as you could see which stretches we were able to go faster and the plateaus that fit perfectly around where we had stopped to eat and stretch. Just as we could estimate the cat’s position at uniform times I had constructed data points of how far we had traveled according to uniformly timed entries. We knew where we were at each second of the trip, because we were the people traveling. If I had shown the data of our trip to a passerby not traveling with us, they would have only known where we were at specific times. Even though the data looked as if we were traveling at a constant speed, looking at four data points each hour does not confirm exactly what we did in between each 15-min increment. We could have stopped to change a tire in 5 min and then done 100 miles per hour to catch up to our original pace before the next timed entry was recorded. It was the in-between timed entries that people would have to guess and assume that we were maintaining a constant rate. The 15 min in-between each data entry was unknown to all but my wife and me. This experience inspired me to enter the cat’s position and time into a graphing calculator (see Fig. 1). My attempt was to try to fake the in-between unknown positions of the cat by looking at the pattern of the cat’s positions that were known. Certainly, there had to be a pattern, and after 11 data entry points, the story of the cat’s position started to take shape. I started to see a recognizable curve and was convinced that a well constructed regression could represent the unknown positions. If a curve were to fit nicely enough through the known positions, the unknown in-between time linking the frames together had a chance to be modeled very differently than the very linear assuming ‘average of averages.’ After fitting a curve to the points, all I had to do was take a quick derivative to find the cat’s instantaneous speed at any time desired. The first 11 points seemed extremely cubic to me and I could not help but to fit a curve to see how well it would have matched my rough average estimates. I quickly entered my cubic regression function into the calculator and watched intently as the curve passed so nicely through the second, third and fourth data points, as shown in Fig. 2. It scooted just under the next three points and passed nicely through the eighth and ninth points before passing just above the desired tenth point. Either way, the first derivative yielded a speed at the tenth frame that was only 7 cm per second off from my rough average calculations the night before. Even though the function did not pass through my desired point of interest, I was still more confident in the result. The curve that passed through the points so nicely seemed to connect the data points in a more natural way than rough averages would have.
252
S. Case / Journal of Mathematical Behavior 27 (2008) 250–254
Fig. 1. Cat’s position versus time.
I remember feeling unfulfilled after computing the speed by an ‘average of averages.’ It just assumed too much. Who knew what was going on inbetween each frame. Just as a person who saw my driving data would have assumed that we were traveling at a constant speed, the data as it was constructed had no way of proving that the car did not change speeds. Certainly each data entry did not account for slowing down for traffic, speeding up to pass slower trucks on the road and even going up and down hills that each 15 min might have included. Although the data may have seemed linear, the reality is that a car speeds up and slows down according to the environment that it is in at the time. Just as a car has to accelerate to the speed limit from a stop, the cat had to do the same as it was gaining speed before its gallop. Certainly a linear function would not take into account the fluid way a cat would gain and maintain speed the way a higher degreed function could. Even when my cubic function missed the tenth data point, I felt better about the speed that the first derivative revealed. I knew we were trying to find an instantaneous speed, and averages were just taking too much for granted in between each frame. Later I found time to model all of the data points and not only fit a cubic, but also a quadratic, quartric and exponential to the data as well. I was able to compare the accuracy of each speed I obtained using first derivatives by how well each function fit the data points. Little did I know how different my thinking of this problem was in respect to how my classmates and even the students I experimented with approached the mystery of the cat’s speed. 2. Reflections on students’ work After seeing all the different solutions to the problem from my classmates, I was excited to present the same problem to students attending the high school at which I teach. The high school is in southern San Diego and we have close to 2000 students. Many of the students come to our school underperforming in mathematics. However, we manage to always have at least one to two sections of Calculus. The classes are mixed with juniors and seniors. Many of the juniors studied an extra year of mathematics during the summer to be able to participate in more advanced mathematics courses before they graduate. The Mathematics Department at our high school is always willing to try new lessons with students. I was wondering how the Calculus students at my high school would perceive the catwalk problem. The Calculus classes had studied derivatives,
Fig. 2. Cubic regression graph.
S. Case / Journal of Mathematical Behavior 27 (2008) 250–254
253
but I was unsure if the students would see such a problem as an application of derivatives as I did. Would the students see this as an application of derivatives, or would they have to settle with an average of averages, or would they see it in a completely different way? As I presented the problem to the students I emphasized that I was more interested in their line of thinking rather than in the answer they may or may not come up with. I gave them the photos on paper and transparency as well as rulers and graph paper. I also offered string, tape, colored pens and pencils. I encouraged the students to think of anything else they might need in order to represent the cat in motion. All of the groups came up with a speed for the cat, some groups more confident than others. Every group started by discussing how to measure the pictures in order to determine distance. This discussion followed by all sorts of calculations of averages. Some groups took averages from the first frame to the last frame, some from the first frame through the tenth frame, and some from the ninth to the tenth, tenth to eleventh, while others felt the ninth through the eleventh would do. However groups agreed upon which averages to believe, only one group took it to a level past averages. It is this group’s story that really provided a different way of looking at things. Right away the group started to think of this project in light of derivatives. They spoke in terms of taking averages, but wanted to know how speeds (or averages) would relate from frame to frame. They concluded that what they needed to find was the acceleration of the cat. After a few averages were calculated, out came a value for the cat’s acceleration between frame 9 and 11. When prompted to explain, a student calmly said, “I just made it up. Well, 193 − 64 is the difference in velocities, then divided it again from the time.” Once the acceleration value was accepted by the group a lull came over their thinking. They had average velocities, but were not confident in them because they represented the speed “in between” the frames and not the speed of the cat at the exact moment they wanted. The desired acceleration value, once calculated, seemed of no consequence for the students. They projected the transparency of the pictures up on a board and labeled each inbetween portion of the frames with their respective average velocities along with their acceleration value. They stared at it, paced the room, sat in the chairs and couldn’t see how to come up with a value for the cat’s speed at the exact moment they wanted. They knew that the speed of the cat at the desired moment was somewhere within a range of velocities, but had a hard time finding just where to pinpoint the desired velocity. One student’s comment summarized their frustrations as well as their understandings, “Yeah, the change in speed per time, how he gets faster. If we can plug this into calculus somehow, and I could just use derivatives.” I could see that the group was struggling. The conversations seemed unstructured but focused. The students were definitely frustrated but kept challenging themselves to think of a way around their dilemma. They knew what they had, but did not really know how to handle it in such a way to get what they needed. Their conversations may have strayed a little from the cat per se, but they were still focused on speed. They talked about cars, speedometers and radar guns and asked if any one knew exactly how these devices worked. I let them talk and intently listened as their questions sparked my interest. I did not know how a speedometer or radar gun worked so I did what any curious academic might in a time of complete ignorance—I consulted the Internet. Slowly the students noticed what I was doing and one of the students quickly took over the Internet search and the group keyed in on a site that attempted to explain the difference between average and instantaneous velocities. A search for “instantaneous speed” revealed a page that provided their next breakthrough (see Fig. 3). A student quickly jumped up and shouted, “We can find the slope of the tangent line! Where did your graph go? Slope is the change in speed over time.” The students quickly crowded around the graph they had made of the cats position versus time and carefully sketched a curve through the desired points. A ruler was used to approximate a tangent line at the tenth frame data point. The following conversation convinced me that they had arrived at a value they were finally comfortable with:
Fig. 3. Web page on instantaneous speed.
254 Jack: Manny: Jack:
Manny: Jack:
S. Case / Journal of Mathematical Behavior 27 (2008) 250–254 I just took the change in distance from our tangent line which was 10. Then I divided by the change in time which was .093 and got 107.52 cm/s. So, do you want to stick with that, because that sounds pretty cool. Because that’s that one spot, that one speed at the point. It’s the tangent at that dot, the slope. It’s in the middle [of the two average velocities], but not the exact middle. In this period of time it’s that one spot. [He marks approximately where 107 shows up in between the average velocities.] It means he would be going faster here [on the side closer to 193 cm/s]. First derivative is the speed, and it’s the tangent line Yeah, yeah. . . So if we just took the tangent of the cat at that point, I guess it would make sense that that is the speed. It wouldn’t make sense if our figure came out exactly in the middle, because then we would know he wasn’t accelerating, which he clearly is.
I was very impressed, not only with the students’ persistence to press on, but also with the creativeness they showed to arrive at an acceptable answer. I might have been so impressed because of their creativity to arrive at a solution. The students were not afraid to sketch in the curve of best fit. As I went through my progression, I never once thought to draw my own curve through the points. I was sure that a function could model the situation. The students on the other hand might not have had enough experience in curve regression. It is hard to say which curve would model the situation best. The students’ curve definitely fit the points better than any of my functions did. However, their curve was an approximation. As my regression functions might not have modeled the data points as well, there was no approximation aside from the initial measuring of the picture. One may never know the true speed of the cat at the tenth frame, but this is the beauty of the problem. So many different solutions can arrive at so many different but similar conclusions, each solution having strengths and weaknesses. Although the focus of each group may be to come as close as they can to the actual speed of the cat, in all actuality, the strength of the activity lies in its ability to have students collaborate and dialogue mathematically. 3. Looking back (and forward) on the entire experience After participating in this project from both sides of the desk, it has given me a chance to see many different perspectives. After doing the project myself, I could not fathom how to consider the cat’s motion other than from a graph of position versus time. When I got back to class the following week, I was presented with many other ways in which to consider the problem. Students not only graphed position versus time, they also graphed velocity versus time and position versus velocity. It opened my eyes to a deeper look into how to create motion from an otherwise motionless series of pictures. When the class saw my solutions someone asked why I never considered my group of data points as a piece wise linear function. It never came up in my thought process. I think I was so inclined to stay away from anything linear as I was trying to redefine my average velocities. The curve of a polynomial function seemed to take advantage of more of the data points given to us. A piece-wise linear function may just have modeled the motion just as well, if not better. However, reducing it to a linear function, even in to two pieces only requires the services of two points for each line, which is no different than taking an average between two frames. When I saw my students’ solution, it wowed me. Their line of thinking was very similar to my own: fit some sort of curve to the data points, then calculate the derivative at desired times. The difference between our two solutions is that the students probably did not have enough experience in regression equations, although their solution was still ingenious. I would have never thought to fit my ‘own’ curve to the data points. I was too focused on finding an equation that would fit the data points. Of course, fitting your own curve is subject to critique. There are an endless number of different curves that could be drawn through the points. Who is to say which one would fit it best? Even after a curve is chosen, how can we be sure how steep to draw a tangent line? The students’ method, although novel, did involve many educated guesses. While many solutions did involve graphs of some nature, the question still remained: Is that the best way to represent the cat’s motion? When you fit a curve to the cat’s data points you are making a huge assumption that the cat’s motion is fluid. To the naked eye, it may seem that way, but how can we know for sure? The nature of the cat’s gait may suggest otherwise. The way the cat accelerates is through a series of jumps with the back legs. The front two legs may hit the ground at different times and places, but all in preparation for placing the back two legs underneath a crouched up body ready to pounce forward. It all happens so fast in real time that it seems fluid. When you look at most graphs that depicted the cat’s position versus time, you can see the moment of pounce at the tenth frame. How can we be sure where the cat was in its pouncing motion? The pictures only tell us a part of the story. The cat is really the only one who knows.
Contents lists available at ScienceDirect
The Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb
The catwalk task: Reflections and synthesis: Part 1 Steven Case Mar Vista High School, 505 Elm Ave., Imperial Beach, CA 91932, United States
a r t i c l e
i n f o
Article history: Available online 17 March 2009
Keywords: Modeling Representation Teacher learning Task design
a b s t r a c t In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional. © 2009 Elsevier Inc. All rights reserved.
1. Reflections on my own work After getting the assignment of finding the cat’s speed late on a Thursday evening, I have to admit I was not as enthusiastic as I was once the assignment was completed. Given 20 min or so in class, it took me a bit of time to realize what I was actually looking at. Twenty-four pictures of a cat were taken at an almost instantaneous sequence. In fact, the microscopic separation between photos of only three one-hundredths of a second did not fully sink in right away. In actuality, I had listened to the directions, but when I looked at the photographs, it looked like twenty-four pictures of a cat at a bunch of different times, not necessarily in any sequential order. I was struggling to grasp that all of the pictures were connected in a way that represented continuous motion. After asking a few questions about the grid lines behind the cat, I took what little time I had left and my first intuition was to calculate average speeds in between the consecutive frames that surround the desired frame. Specifically, my plan was to calculate the average speeds between the ninth and tenth frames, tenth and eleventh frames and then take a simple average of the averages. My plan helped me understand the details of the frames as it was time to start making measurements. I knew the distance between each gridline behind the cat and the time that had elapsed between each photo. To start measuring the distance the cat had moved between each frame, I had to decide which part of the cat to measure in each frame. These photos had a striking similarity to photographs that are used to decide a narrowly contested horse race. As two horses lunge through the finish line a photograph is taken to see which nose crosses first. With that in mind, I chose to measure how far the nose of the cat had traveled between each frame. I measured 3 cm between the ninth and tenth frames for a speed of approximately 96.77 cm/s. Between the tenth and eleventh frame I estimated a distance traveled of 5 cm for a speed of approximately 161.29 cm/s. If I had abandoned the project that night, my answer would have averaged to 129.03 cm/s for the speed of the cat during the tenth frame. I remember feeling unsatisfied with the calculation. I knew that this kind of problem deserved more than just an average of averages. The cat was moving, accelerating, and its position had changed every frame. Certainly its movement could not be
E-mail address: [email protected]. 0732-3123/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jmathb.2009.01.003
S. Case / Journal of Mathematical Behavior 27 (2008) 250–254
251
summarized linearly as my average calculations would have assumed. There had to be more to this. I knew that the cat was moving, but there was just enough unknown to create uncertainty in any way I was going to redefine the cat’s speed. Why was this ‘average of averages’ the first thing I thought of and why did it leave me feeling so unsatisfied? Perhaps when faced with ‘how fast’ problems, students first think of relationships between the units. In our case it was how far the cat traveled in centimeters and how long it took the cat to travel such a distance measured in seconds. Naturally dividing the length traveled by the time it took to do so results in speeds of centimeters per second. This type of simple calculation is first introduced to students through situations and problems that involve uniform change; for instance, a car traveling 20 miles per hour, or a painter that can paint 4 walls in 3 h. In each situation the rate is constant. The situation assumes that the car (or the painter) will never speed up or slow down. An average calculation would model these situations perfectly. No matter how one divides the work done or distance traveled, the rate would remain the same. This same process applied to the cat problem would assume that the cat was traveling at the same speed throughout. Obviously, the varying average speeds in the ninth and tenth frames (96.77 cm/s and 161.29 cm/s) tell a very different story. The cat was clearly accelerating through the tenth frame. This could be the reason for my uneasiness over my ‘average of averages’ speed. I knew that my calculations assumed the cat was traveling at a constant rate. Even before I punched numbers into the calculator I knew that the two averages would be very different. Visually inspecting the pictures of those frames shows just why frame 10 becomes such a point of contention. The cat moves a considerable distance more at frame 10 than in the surrounding frames. Using an ‘average of averages’ was the only way I knew how to cope with the varying speeds. Since the first average described the cat’s average speed between the ninth and tenth frames and the second average represented the cat’s average speed between the tenth and eleventh frames, the average of those would be the closest I could get to the instantaneous speed of the cat at the tenth frame. I knew it would be close, but just not good enough. Averages could not possibly account for an instantaneous speed. Even though the time between frames was so tiny, there was still an aura of mystery as to what was going on in between the known frames. The average of the averages not only assumed a constant rate of change in between frames, but also assumed that even though the speeds were so different the instantaneous speed of frame 10 would occur exactly in the middle of the two speeds. There was just too much assumption in those calculations to make me feel comfortable in stating my final answer. I had to look at this in a different way to try to represent what was happening in another way. I awoke the next morning ready to redefine my calculations. To overcome the unknowns, or the position of the cat within the three one-hundredths of a second between each frame, I had to model the cat’s position in a way that was more comprehensible to me. After a night of thinking about it, the way the data was presented reminded me of a prior experience in my life. The data about the cat was given to us in chunks. For instance, we knew where the cat was at only specific times at the end of each frame. A few months earlier I had driven up the coast with my wife to see my mother. To pass the time as we went, I entered our distance traveled according to the time we had been on the road. I made a data entry point on a graphing calculator four times each hour. The data that I was entering looked very similar to the cat data that was given to us. Every 15 min, I looked over to the odometer and entered in how far we had traveled up until that point. It was fun to see the data points line up as we rolled up the coast. Even though the speedometer at each specific time was reading an explicit speed, it did not necessarily match our average speed for the trip. As we progressed, the graph almost told a story as you could see which stretches we were able to go faster and the plateaus that fit perfectly around where we had stopped to eat and stretch. Just as we could estimate the cat’s position at uniform times I had constructed data points of how far we had traveled according to uniformly timed entries. We knew where we were at each second of the trip, because we were the people traveling. If I had shown the data of our trip to a passerby not traveling with us, they would have only known where we were at specific times. Even though the data looked as if we were traveling at a constant speed, looking at four data points each hour does not confirm exactly what we did in between each 15-min increment. We could have stopped to change a tire in 5 min and then done 100 miles per hour to catch up to our original pace before the next timed entry was recorded. It was the in-between timed entries that people would have to guess and assume that we were maintaining a constant rate. The 15 min in-between each data entry was unknown to all but my wife and me. This experience inspired me to enter the cat’s position and time into a graphing calculator (see Fig. 1). My attempt was to try to fake the in-between unknown positions of the cat by looking at the pattern of the cat’s positions that were known. Certainly, there had to be a pattern, and after 11 data entry points, the story of the cat’s position started to take shape. I started to see a recognizable curve and was convinced that a well constructed regression could represent the unknown positions. If a curve were to fit nicely enough through the known positions, the unknown in-between time linking the frames together had a chance to be modeled very differently than the very linear assuming ‘average of averages.’ After fitting a curve to the points, all I had to do was take a quick derivative to find the cat’s instantaneous speed at any time desired. The first 11 points seemed extremely cubic to me and I could not help but to fit a curve to see how well it would have matched my rough average estimates. I quickly entered my cubic regression function into the calculator and watched intently as the curve passed so nicely through the second, third and fourth data points, as shown in Fig. 2. It scooted just under the next three points and passed nicely through the eighth and ninth points before passing just above the desired tenth point. Either way, the first derivative yielded a speed at the tenth frame that was only 7 cm per second off from my rough average calculations the night before. Even though the function did not pass through my desired point of interest, I was still more confident in the result. The curve that passed through the points so nicely seemed to connect the data points in a more natural way than rough averages would have.
252
S. Case / Journal of Mathematical Behavior 27 (2008) 250–254
Fig. 1. Cat’s position versus time.
I remember feeling unfulfilled after computing the speed by an ‘average of averages.’ It just assumed too much. Who knew what was going on inbetween each frame. Just as a person who saw my driving data would have assumed that we were traveling at a constant speed, the data as it was constructed had no way of proving that the car did not change speeds. Certainly each data entry did not account for slowing down for traffic, speeding up to pass slower trucks on the road and even going up and down hills that each 15 min might have included. Although the data may have seemed linear, the reality is that a car speeds up and slows down according to the environment that it is in at the time. Just as a car has to accelerate to the speed limit from a stop, the cat had to do the same as it was gaining speed before its gallop. Certainly a linear function would not take into account the fluid way a cat would gain and maintain speed the way a higher degreed function could. Even when my cubic function missed the tenth data point, I felt better about the speed that the first derivative revealed. I knew we were trying to find an instantaneous speed, and averages were just taking too much for granted in between each frame. Later I found time to model all of the data points and not only fit a cubic, but also a quadratic, quartric and exponential to the data as well. I was able to compare the accuracy of each speed I obtained using first derivatives by how well each function fit the data points. Little did I know how different my thinking of this problem was in respect to how my classmates and even the students I experimented with approached the mystery of the cat’s speed. 2. Reflections on students’ work After seeing all the different solutions to the problem from my classmates, I was excited to present the same problem to students attending the high school at which I teach. The high school is in southern San Diego and we have close to 2000 students. Many of the students come to our school underperforming in mathematics. However, we manage to always have at least one to two sections of Calculus. The classes are mixed with juniors and seniors. Many of the juniors studied an extra year of mathematics during the summer to be able to participate in more advanced mathematics courses before they graduate. The Mathematics Department at our high school is always willing to try new lessons with students. I was wondering how the Calculus students at my high school would perceive the catwalk problem. The Calculus classes had studied derivatives,
Fig. 2. Cubic regression graph.
S. Case / Journal of Mathematical Behavior 27 (2008) 250–254
253
but I was unsure if the students would see such a problem as an application of derivatives as I did. Would the students see this as an application of derivatives, or would they have to settle with an average of averages, or would they see it in a completely different way? As I presented the problem to the students I emphasized that I was more interested in their line of thinking rather than in the answer they may or may not come up with. I gave them the photos on paper and transparency as well as rulers and graph paper. I also offered string, tape, colored pens and pencils. I encouraged the students to think of anything else they might need in order to represent the cat in motion. All of the groups came up with a speed for the cat, some groups more confident than others. Every group started by discussing how to measure the pictures in order to determine distance. This discussion followed by all sorts of calculations of averages. Some groups took averages from the first frame to the last frame, some from the first frame through the tenth frame, and some from the ninth to the tenth, tenth to eleventh, while others felt the ninth through the eleventh would do. However groups agreed upon which averages to believe, only one group took it to a level past averages. It is this group’s story that really provided a different way of looking at things. Right away the group started to think of this project in light of derivatives. They spoke in terms of taking averages, but wanted to know how speeds (or averages) would relate from frame to frame. They concluded that what they needed to find was the acceleration of the cat. After a few averages were calculated, out came a value for the cat’s acceleration between frame 9 and 11. When prompted to explain, a student calmly said, “I just made it up. Well, 193 − 64 is the difference in velocities, then divided it again from the time.” Once the acceleration value was accepted by the group a lull came over their thinking. They had average velocities, but were not confident in them because they represented the speed “in between” the frames and not the speed of the cat at the exact moment they wanted. The desired acceleration value, once calculated, seemed of no consequence for the students. They projected the transparency of the pictures up on a board and labeled each inbetween portion of the frames with their respective average velocities along with their acceleration value. They stared at it, paced the room, sat in the chairs and couldn’t see how to come up with a value for the cat’s speed at the exact moment they wanted. They knew that the speed of the cat at the desired moment was somewhere within a range of velocities, but had a hard time finding just where to pinpoint the desired velocity. One student’s comment summarized their frustrations as well as their understandings, “Yeah, the change in speed per time, how he gets faster. If we can plug this into calculus somehow, and I could just use derivatives.” I could see that the group was struggling. The conversations seemed unstructured but focused. The students were definitely frustrated but kept challenging themselves to think of a way around their dilemma. They knew what they had, but did not really know how to handle it in such a way to get what they needed. Their conversations may have strayed a little from the cat per se, but they were still focused on speed. They talked about cars, speedometers and radar guns and asked if any one knew exactly how these devices worked. I let them talk and intently listened as their questions sparked my interest. I did not know how a speedometer or radar gun worked so I did what any curious academic might in a time of complete ignorance—I consulted the Internet. Slowly the students noticed what I was doing and one of the students quickly took over the Internet search and the group keyed in on a site that attempted to explain the difference between average and instantaneous velocities. A search for “instantaneous speed” revealed a page that provided their next breakthrough (see Fig. 3). A student quickly jumped up and shouted, “We can find the slope of the tangent line! Where did your graph go? Slope is the change in speed over time.” The students quickly crowded around the graph they had made of the cats position versus time and carefully sketched a curve through the desired points. A ruler was used to approximate a tangent line at the tenth frame data point. The following conversation convinced me that they had arrived at a value they were finally comfortable with:
Fig. 3. Web page on instantaneous speed.
254 Jack: Manny: Jack:
Manny: Jack:
S. Case / Journal of Mathematical Behavior 27 (2008) 250–254 I just took the change in distance from our tangent line which was 10. Then I divided by the change in time which was .093 and got 107.52 cm/s. So, do you want to stick with that, because that sounds pretty cool. Because that’s that one spot, that one speed at the point. It’s the tangent at that dot, the slope. It’s in the middle [of the two average velocities], but not the exact middle. In this period of time it’s that one spot. [He marks approximately where 107 shows up in between the average velocities.] It means he would be going faster here [on the side closer to 193 cm/s]. First derivative is the speed, and it’s the tangent line Yeah, yeah. . . So if we just took the tangent of the cat at that point, I guess it would make sense that that is the speed. It wouldn’t make sense if our figure came out exactly in the middle, because then we would know he wasn’t accelerating, which he clearly is.
I was very impressed, not only with the students’ persistence to press on, but also with the creativeness they showed to arrive at an acceptable answer. I might have been so impressed because of their creativity to arrive at a solution. The students were not afraid to sketch in the curve of best fit. As I went through my progression, I never once thought to draw my own curve through the points. I was sure that a function could model the situation. The students on the other hand might not have had enough experience in curve regression. It is hard to say which curve would model the situation best. The students’ curve definitely fit the points better than any of my functions did. However, their curve was an approximation. As my regression functions might not have modeled the data points as well, there was no approximation aside from the initial measuring of the picture. One may never know the true speed of the cat at the tenth frame, but this is the beauty of the problem. So many different solutions can arrive at so many different but similar conclusions, each solution having strengths and weaknesses. Although the focus of each group may be to come as close as they can to the actual speed of the cat, in all actuality, the strength of the activity lies in its ability to have students collaborate and dialogue mathematically. 3. Looking back (and forward) on the entire experience After participating in this project from both sides of the desk, it has given me a chance to see many different perspectives. After doing the project myself, I could not fathom how to consider the cat’s motion other than from a graph of position versus time. When I got back to class the following week, I was presented with many other ways in which to consider the problem. Students not only graphed position versus time, they also graphed velocity versus time and position versus velocity. It opened my eyes to a deeper look into how to create motion from an otherwise motionless series of pictures. When the class saw my solutions someone asked why I never considered my group of data points as a piece wise linear function. It never came up in my thought process. I think I was so inclined to stay away from anything linear as I was trying to redefine my average velocities. The curve of a polynomial function seemed to take advantage of more of the data points given to us. A piece-wise linear function may just have modeled the motion just as well, if not better. However, reducing it to a linear function, even in to two pieces only requires the services of two points for each line, which is no different than taking an average between two frames. When I saw my students’ solution, it wowed me. Their line of thinking was very similar to my own: fit some sort of curve to the data points, then calculate the derivative at desired times. The difference between our two solutions is that the students probably did not have enough experience in regression equations, although their solution was still ingenious. I would have never thought to fit my ‘own’ curve to the data points. I was too focused on finding an equation that would fit the data points. Of course, fitting your own curve is subject to critique. There are an endless number of different curves that could be drawn through the points. Who is to say which one would fit it best? Even after a curve is chosen, how can we be sure how steep to draw a tangent line? The students’ method, although novel, did involve many educated guesses. While many solutions did involve graphs of some nature, the question still remained: Is that the best way to represent the cat’s motion? When you fit a curve to the cat’s data points you are making a huge assumption that the cat’s motion is fluid. To the naked eye, it may seem that way, but how can we know for sure? The nature of the cat’s gait may suggest otherwise. The way the cat accelerates is through a series of jumps with the back legs. The front two legs may hit the ground at different times and places, but all in preparation for placing the back two legs underneath a crouched up body ready to pounce forward. It all happens so fast in real time that it seems fluid. When you look at most graphs that depicted the cat’s position versus time, you can see the moment of pounce at the tenth frame. How can we be sure where the cat was in its pouncing motion? The pictures only tell us a part of the story. The cat is really the only one who knows.