The TTP Classroom

A Research Lesson I Observed

I observed a lesson, as a part of a district-wide Lesson Study in 1979, when I was a novice teacher at a public school in Tokyo. It was hosted at another public school in the same district and taught by a second-year teacher who worked there. This teacher and her team developed a lesson that introduced a new mathematics topic to her third-grade class by creating a problematic classroom situation that the students then had to resolve.

The teacher began by telling her students that they were going to play a game:

Teacher (T): Let’s play a game of ring toss!

Students (S): Yay, sounds fun!

T: Move all your desks and chairs to the back of the classroom so

we can have enough space to play.

The students moved their desks and chairs and then stood in a group in the front half of the classroom. In the front half of the room there was a line of tape on the floor stretching straight across which the teacher had prepared prior to the lesson (Figure 1.4.01).

How the classroom was arranged when teacher posed the problem during the ring toss research lesson

Figure 1.4.01 How the classroom was arranged when teacher posed the problem during the ring toss research lesson.

T: Each of you will get a ring to throw. Everyone will line up on this line and then throw your rings all at the same time. Those whose rings land on the pole will be the winners!

As the students lined up, one student at the far end of the line started to

complain:

S: Ms. [Teacher], I don’t think it’s fair that I have to throw my ring from all the way over here, everyone in the middle of the line is way closer to the pole.

Their classmates agreed:

S: Yeah, I think this line might be unfair.

T: Well, if this line isn’t fair, what kind of line or shape do you think would make the game fair?

S: Maybe a “V” shape?

S: We could make a square around the pole?

T: Okay, let’s come up with a better idea.

The teacher then wrote the problem on the board, “Think about how we can all stand side by side in order to make this ring toss game fair.” Students had a short time to think on their own and then they discussed it in small groups. The teacher then initiated a classroom discussion of everyone’s ideas. She had students draw their ideas on the blackboard under the problem she had written. They came up with several different ideas (Figure 1.4.02).

l .4.02 Some of the ideas the students came up with during the ring toss research lesson

Figure l .4.02 Some of the ideas the students came up with during the ring toss research lesson.

T: How did you come up with your ideas?

S: 1 tried to come up with a way that each of us would be the same distance from the pole.

T: Do you think if we line up using your ideas that all of you will be the same distance from the pole?

S: We can try and see.

S: We can use a piece of string to measure to make sure we are all the same distance from the pole.

The teacher asked a few volunteers to come stand around the pole to help her

make the figures the students had drawn on the blackboard. Students began

to realize that maybe they needed to make a circle so everyone could stand an

equal distance from the pole.

S: It should be a circle!

T: Okay, let’s use a piece of string to help us make sure everyone can stand an equal distance from the pole. Then we can play our ring toss game.

The solution students came up with at the end of the ring toss research lesson

Figure 1.4.03 The solution students came up with at the end of the ring toss research lesson.

The teacher let the students make the shape.

T: Are you happy with this shape?

S: Yes, now we’re all the same distance from the pole.

T: Do you know what we call this shape?

All the students said, “A circle.”

T: Did you know that we could form a circle by making all the positions equally distant from the pole?

S: No, 1 thought that “circles” is just what we call round shapes.

T: Let’s study more about circles. We can play our ring toss game during recess.

This concluded the lesson. I was impressed that the students came up with the definition of a circle on their own in order to make the ring toss game fair.

TTP as an Integral Part of the Mathematics Curriculum

I observed this lesson more than forty years ago. It was an innovative way to invite students to discover what a circle is. The concept behind this TTP lesson has since become part of contemporary government authorized textbooks in Japan.

As I discussed in section 1.3, Japanese teachers initially developed TTP lessons on their own, but many of the most successful lessons were later incorporated into official textbooks. The ring toss lesson I observed in 1979 is no exception. It has since been used in several textbooks over the years (e.g., Fujii & Majirna, 2020a). One such textbook has been translated into English, titled New Mathematics for Elementary School (Fujii & Majirna, 2020b). This third-grade textbook uses a version of the ring toss lesson as the opening lesson for the unit on circles (Figure 1.4.04).

Opening page of the unit on circles

Figure 1.4.04 Opening page of the unit on circles.

Reprinted with permission from New inathematics far Elementary School ЗА (Fujii &. Majima, 2020b, p. 121)

An outline of the four lessons in the unit on circles in New Mathematics for Elementary School (Fujii & Majima, 2020b). These lessons address Japans national curriculum standards for third grade

Figure 1.4-05 An outline of the four lessons in the unit on circles in New Mathematics for Elementary School (Fujii & Majima, 2020b). These lessons address Japans national curriculum standards for third grade: to understand circles as a geometric figure and to understand the concepts of center, radius, and diameter.

The circle unit in New Mathematics for Elementary School is a series of TTP lessons (Figure 1.4.05). The second lesson asks students to construct several paths made up of points that are all an equal distance from a center point (Fujii &. Majima, 2020b). After performing this task, the students will realize that all the paths they made look like circles. The third lesson asks students to find the radius of the circle without being given the location of the center (Figure 1.4.06). This hands-on investigation deepens students’ understanding of a circle and helps them explore the relationships among the center, radius, and diameter. In this investigation, students trace a circle on a sheet of paper using an everyday object, such as a coffee mug. Students then cut out the circle and fold it in half to find the center, the diameter, and the radius. In the fourth lesson, they draw circles using a compass (Fujii & Majima, 2020b).

The four lessons in this unit use TTP lessons to address third grade mathematics standards. Each lesson invites students to engage in mathematics by solving everyday problems and performing hands-on activities. So while the development of Japanese TTP lessons historically began as isolated lessons, they are now an integral part of the curriculum. TTP lessons are recognized as the most effective way to help students acquire both mathematical knowledge and thinking skills.

 
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