# Cartwheeling through CamMotion.

To many students, mathematics has only tenuous connections to their everyday lives and personal concerns. Therefore, current educational reform emphasizes teachers adopting curriculum activities that connect mathematics to students' lives. But what kinds of activities might there be? And how can students' concerns be integrated into mathematics instruction?The view project at the Technical Education Research Center (TERC) combines new video and computer tools to bring math and students together in previously impossible ways. With these tools, students explore and analyze their own experiences by making measurements on videos of real phenomena. Video allows them to slow down or speed up time; associated computer tools let them analyze events they have actually observed. By making measurements on single frames of video, students can explore the "fine structure" of actions that take place quickly, like bouncing balls or flying paper airplanes. They can examine patterns of motion through video analysis of their own bodies in such activities as sports and dance.

The innovation that makes this vision possible is called CamMotion. Using CamMotion to make measurements on the video, students construct meaningful connections between their own motions and conventional mathematical representations. We have coined a general term for such tools: video-based labs, or VBL.

A Data Question

The easiest way to see CamMotion's potential is to work through a simple example.

Two students, Kathy and Nadia, practice gymnastics. Kathy uses a camcorder to videotape Nadia doing cartwheels in the school auditorium, then digitizes the video into a machine-readable movie that CamMotion can read. Now they are ready to use CamMotion to explore how Nadia does a cartwheel.

To analyze Nadia's cartwheel, Kathy and Nadia start by tracking the positions of Nadia's left hand. While advancing through the frames of the movie, they click on Nadia's left hand in each frame to mark its position. Clicking causes CamMotion to store the position of Nadia's hand in (x, y) coordinates, along with the current time in the movie.

In addition to position data, CamMotion supports measurements of distances, angles, and areas, among others. Just as position data are used to track an object's motion over time, distance data track the distance between two objects over time, such as Nadia's two hands. Angle data can track the rotation of objects, such as a hamster's wheel, or examine the changing angle of parts of a moving object, such as Nadia's legs.

However, the process of mapping the real world to coordinates on the screen is not always straightforward. Like any robust measuring system, CamMotion has a set of tools that facilitate connecting the observed world to mathematical representations in a way that makes sense to the user. Two of these many tools allow the user to set the origin at any point on the movie and to change the scale of the coordinate system.

In all but a few cases, the size of an image on the screen will be significantly different from the size of the object in the world. CamMotion provides a scale tool that allows the user to specify the relationship between the two sizes. To set an appropriate scale, Kathy and Nadia indicate on the video a line that reflects a length they know in the world, then enter the real-world length as a scale factor. In examining the cartwheel, the girls use Nadia's arm, which they know is half a meter long, as their calibration length.

Viewing the Data

After Kathy and Nadia collect their samples, they can display their data as either a table or a graph. They choose to first use the table window to view their data in a spreadsheet, with each row representing measurements at a specific time. In looking at their data, Kathy and Nadia notice something strange: six rows with the very same data. They wonder if they might have made a mistake during data collection, or if a sequence of frames in the video really generated these data. What can this mean? Kathy and Nadia select these rows and request that the corresponding video frames be played in the replay window. They notice that the selected piece of video shows Nadia upside-down with both hands on the floor. Now they understand these numbers are not errors; Nadia's left hand did not move during this half second.

Kathy and Nadia are now eager to continue their exploration of Kathy's hypothesis: that at some point in her cartwheels Nadia has only one hand on the floor. They go through a similar process to collect data on Nadia's right hand, then build a graph combining the two datasets so they can compare the motion of her two hands.

The figure illustrates the graph that shows the motion of Nadia's right and left hands. As the girls examine it, Kathy says, "Look, Nadia, there's a small piece of the graph where you have only one hand on the floor!" As indicated by the arrow, a single data point shows where Nadia's left hand is on the floor and her right hand is off. But what part of the cartwheel corresponds to this part of the graph?

An Answer Appears

To find out, the girls select the graph segment that shows only one hand on the floor and use the replay button to see the corresponding video frames.

Having found the answer to their first question, the girls are off on another investigation. Observation, hypothesizing, and experimentation have led Kathy and Nadia to ponder other questions about the cartwheel. How fast are Nadia's hands and feet moving? Are her feet and hands always the same distance apart? Is there any part of her body that stays the same vertical distance from the floor during the entire cartwheel? With CamMotion, the fleeting seconds it takes Nadia to do a cartwheel can be stilled, and she and her friend can ask and answer questions they might have never thought of without such a tool.

Andee Rubin is a senior scientist and Scott Bresnahan is a programmer at Technical Education Research Centers in Cambridge, Mass. Ted Ducas teaches physics at Wellesley College in Wellesley, Mass.

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Title Annotation: | Modeling the Real World; video-based laboratory for teaching mathematics |
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Author: | Rubin, Andee; Bresnahan, Scott; Ducas, Ted |

Publication: | Communications of the ACM |

Date: | Aug 1, 1996 |

Words: | 1014 |

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