In 1995, the Third International Mathematics and Science Study (TIMSS) included a Videotape Classroom Study. This video study was an international videotape survey of eighth-grade mathematics lessons in Germany, Japan, and the United States. Funded by the National Center for Education Statistics (NCES) and the National Science Foundation, the 1995 video study was the first attempt to collect videotaped records of classroom instruction from nationally representative samples of teachers. The study was conducted in a total of 231 classrooms in Germany, Japan, and the United States and used multimedia database technology to manage and analyze the videos.
The Videotape Classroom Study had four goals:
For the report on the methods and findings of the Videotape Classroom Study, click here.
Example lessons from the TIMSS 1995 Video Study were made available in the form of video vignettes of six eighth-grade lessons, two each from Germany, Japan, and the United States. These example lessons were taught by teachers who volunteered to be videotaped for the project. The video vignettes were originally made available on a CD-ROM: Video Examples from the TIMSS Videotape Classroom Study: Eighth Grade Mathematics in Germany, Japan, and the United States (NCES 98092). Now they are all available for viewing through the links below.
In the previous lesson students explored the areas of triangles constructed between two parallel lines. In this lesson they apply this concept to solve a problem involving the areas of irregularly shaped quadrilaterals. After working the problem on their own, students share their solution methods with the class.
Linking Yesterday's Lesson to Today's Topic
|The class begins with a ritual greeting, as in almost all Japanese classrooms, where the students stand up and bow to the teacher. The teacher begins the lesson by asking, "Do you remember what we did last period?" Then he walks to the TV monitor in front of the classroom, which is connected to a computer, and shows a triangle between two parallel lines. A student replies that they studied how to obtain the area of a triangle constructed between parallel lines. As the teacher shows various triangles that can be formed by moving the top vertex along the top line, he reminds the students that the areas of these triangles are the same because the base and the height are always the same. The teachers says they will use this result as "the foundation today."|
Posing the Problem
|The teacher draws a figure on the board representing two pieces of land, each piece owned by a student in the class. The boundary is a line bent in the middle. The owners would like to make the boundary straight without changing the areas of the two pieces of land. The teacher asks where he should draw the boundary. After a brief question and answer session to clarify the problem, and several predictions by the students, the teacher asks the students to work on the problem, "First of all, please think about it individually for three minutes."|
Working on the Problem
|The students work individually on the problem while the teacher circulates around the room, observing and assisting students. Because the task for the students is to develop a method to solve the problem, the teacher mainly gives hints to the students instead of showing them what to do. For example, the teacher asks one student, "Is there a method that uses the area of triangles?" and says to another student, "The question is... that there are parallel lines somewhere." After three minutes, the teacher suggests that students may want to work together. He adds: "And for now I have placed some hint cards up here so people who want to refer to this can refer to it." He tells students they can think about the problem themselves, with a friend, or discuss it with the assistant teacher.|
Students Presenting Solutions
|For about ten minutes, the students discuss the problem with each other, the teacher, or the assistant teacher. The teacher asks two students to draw their solution methods on the chalkboard while the other students finish their discussions. The teacher then asks everyone to return to their seats and attend to the students’ presentations. While a series of students explain their solutions, the rest of the students and the teacher ask questions and request clarifications. The solutions involve drawing parallel line segments, one connecting the two endpoints of the boundary and the other passing through the point of the bend in the boundary. By moving the vertex along the line segment, a new straight boundary can now be formed that retains the same areas.|
Reviewing Students' Methods and Posing Another Problem
|The teacher reviews and clarifies the students’ methods and asks how many students used each method. Then the teacher presents a follow-up problem, which is to change a quadrilateral into a triangle without changing the area. The teacher puts a figure on the board and says, "This shape of quadrilateral, without changing the area, please try to make it into a triangle. Please think for three minutes and try doing it in your own way." The students work on the problem at their desks, and the teacher walks around the room and assists individual students. After about three minutes, the teacher again tells them to discuss their solution methods with one another.|
Summarizing the Results
|During the students’ worktime, the teacher encourages them to find as many solutions as possible. He draws ten quadrilaterals on the chalkboard and asks individual students to show their solutions on the figures. After about 23 minutes, he briefly reviews the solutions and asks which students found each solution. All of the methods involve drawing a line that divides the quadrilateral into two triangles, then drawing a line parallel to the first through the opposite vertex of one of the triangles, and then changing the shape of that triangle by moving the vertex along the parallel line until the entire figure is a triangle. The teacher ends the lesson by suggesting that, for homework, the students try to change other polygons, such as pentagons, into triangles with equal areas. After the customary bow, students are dismissed.|