1. Quotes from The Teaching Gap Stigler & Hiebert, 1999
“In Japanese lessons, there is the mathematics on one hand, and the students on the other. The students engage with the mathematics, and the teacher mediates the relationship between the two. In Germany, there is the mathematics as well, but the teacher owns the mathematics and parcels it out to students as he sees fit, giving facts and explanations at just the right time. In U.S. lessons, there are the students and there is the teacher. I have trouble finding the mathematics; I just see interactions between students and teachers.”
(p – a summary from an observer)

2. Mottoes: Germany: “developing advanced procedures”
Japan: “structured problem solving”
U.S.: “learning terms and practicing procedures” (p. 27)

3. “[U.S.] Students are seen working in small groups or engaging in a discussion about solution methods, but the mathematics is simple compared with that encountered by their German and Japanese peers, and the work and discussion are mostly about memorizing definitions for terms and following rules and procedures.” (p. 52)

6. “It has now been documented in several studies that teachers asked to change features of their teaching often modify the features to fit within their pre-existing system instead of changing the system itself.” (p. 98)

7. Major ideas of the lesson listed in a way that summarizes the lesson
Meanings students must have at the outset if they are to participate productively in the lesson. (This is not the same as things they must be able to do)