There appears to be a clear distinction between the script that Japanese teachers use to generate lessons, and the ones used by German and US teachers. These different scripts follow from different instructional goals (see previous page) and are probably based on different assumptions about the role of problem solving in the lesson, about the way students learn from instruction, and about what the proper role of the teacher should be.

US and German lessons tend to have two phases: an initial acquisition phase and a subsequent application phase. In the acquisition phase, the teacher demonstrates and/or explains how to solve an example problem. The explanation might be purely procedural (as most often happens in the US) or may include development of concepts (more often the case in Germany). Yet still, the goal in both countries is to teach students a method for solving the example problem(s). In the application phase, students practice solving examples on their own while the teacher helps individual students who are experiencing difficulty.

Japanese lessons generally follow a different script. Problem solving comes first, followed by a time in which students share the solution methods they have generated, and jointly work to develop explicit understandings of the underlying mathematical concepts. Whereas students in the US and German classrooms must follow the teacher as she leads them through the solution of example problems, the Japanese students have a different job: to invent their own solutions, then reflect on those solutions in an attempt to increase understanding.