Classrooms in the U.S., Germany, and Japan were compared, and the main insight that came out of the study was that the culture of the classroom was very different in these 3 countries. Although it's an oversimplification, what they saw was something like this: In Germany, the teacher directs the students in developing advanced procedures, in Japan, the class works individually and in groups on structured problem solving, and in the U.S., the teacher leads the class in learning terms and practicing procedures (pages 25-46). One researcher said he had trouble "finding the mathematics" (page 26) in the videos of U.S. classrooms. (Yikes!)
Classroom culture is hard to change, according to Stigler, because much of it is deeply imprinted in us, as what school is. "The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom." (page 87) Many teachers who've tried teaching with more of a problem-solving focus (including yours truly) can attest to how much resistance students put up: "That's not how math class is supposed to work! Just tell us how to do it!"
After viewing the Japanese lessons, a fourth-grade teacher decided to shift from his traditional approach to a more problem-solving approach such as we had seen on the videotapes. Instead of asking short-answer questions as he regularly did, he began his next lesson by presenting a problem and asking the students to spend ten minutes working on a solution. Although the teacher changed his behavior ... the students, not having seen the video or reflected upon their own participation, failed to respond as the students on the tape did. They played their traditional roles. They waited to be shown how to solve the problem. The lesson did not succeed. The students are part of the system. (page 99)
Which makes it clear that, however cool we think those Japanese classrooms are, we can't just bring their style over here as is. What we might be able to use here, however, is their lesson study process, modified to suit us. Teachers plan one lesson together in great depth, over a long period of time.
During lesson study, the teachers discussed what problem to start with, what materials to give students, what solutions and thoughts the students might come up with, what questions to ask, "how to use space on the chalkboard (Japanese teachers believe that organizing the chalkboard is a key ingredient to organizing students' thinking and understanding)", timing, working with different levels, and how to end the lesson. (from page 117, paraphrased)Then they all watch in the classroom while one teacher plays out their plan with the kids. Afterward they all discuss some more, modify, and try it again in another teacher's class.
"Virtually every elementary and middle school in Japan is engaged in kounaikenshuu [lesson study]." (page 110) What Dan, Kate, and others are doing online (here and here, for example) might come close. Wouldn't it be great if we could start our own kounaikenshuu movement here?!
Here are some quotes I liked:
page 49:
He [Japanese math teacher] concludes by posting the goal for mathematics: "To learn to think logically while searching for new properties and relationships." He asks students to repeat this goal several times and memorize it.
page 75: (paraphrased)
The chalkboard as used as a visual aid that helps focus students' attention in the U.S. versus as a cumulative record of the day's lesson in Japan.
page 93: (regarding chalkboard use)
[Japan] Apparently, it is not as important for students to attend at each moment of the lesson as it is for them to be able to go back and think again about earlier events, and to see connections between the different parts of the lesson.
page 89:
Teachers were asked what was the "main thing" they wanted students to learn from the lesson. 61% of U.S. teachers described skills they wanted their students to learn. ... 73% of Japanese teachers wanted their students to think about things in a new way... to see new relationships between mathematical ideas.
page 90:
[In the U.S. view,] practice should be relatively error-free, with high levels of success at each point. Confusion and frustration ... should be minimized; they are signs that earlier material was not mastered.
page 91:
[In Japan] frustration and confusion are taken to be a natural part of the process, because each person must struggle with a situation or problem first in order to make sense of the information he or she hears later. Constructing connections between methods and problems is thought to require time to explore and invent, to make mistakes, to reflect, and to receive the needed information at an appropriate time.
...
Students will learn to understand the process [of adding unlike fractions] more fully, says the [Japanese teachers'] manual, if they are allowed to make this mistake [of adding denominators] and then examine the consequences.
page 94:
Japanese teachers view individual differences as a natural characteristic of a group. They view differences in the mathematics class as a resource for both students and teachers. Individual differences are beneficial for the class because they produce a range of ideas and solution methods that provide the material for students' discussion and reflection. The variety of alternative methods allows students to compare them and construct connections among them. It is believed that all students benefit from the variety of ideas generated by their peers. In addition, tailoring instruction to specific students is seen as unfairly limiting and as prejudging what students are capable of learning...
In Japan, classroom lessons hold a privileged place in the activities of the school. It would be exaggerating only a little to say they are sacred. They are treated much as we treat lectures in university courses or religious services in church. A great deal of attention is given to their development. They are planned as complete experiences - as stories with a beginning, a middle, and an end. Their meaning is found in the connections between the parts. If you stay for only the beginning, or leave before the end, you miss the point.
page 119: [Japanese teacher speaking]
Conceptually it's easy to break 6 down into 5 and 1, and it's easy to break 7 into 2 and 5, but it's really hard for first-grade students to break 7 down into 3 and 4. [!]
[During a lesson study meeeting] the teachers consulted some of the teachers' manuals and found 5 common ways of solving simple subtraction problems with borrowing.
page 127:
[Japanese] culture genuinely values what teachers know, learn, and invent, and has developed a system to take advantage of teachers' ideas: evaluating them, adapting them, accumulating them into a professional knowledge base, and sharing them.
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*The letters originally stood for Third International Mathematics and Science Study, which was conducted in 1995. At the National Center for Education Statistics website, the letters now stand for Trends in International Mathematics and Science Study, which is conducted every 4 years.
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