Lesson study being situated in the context of teaching makes the integration very natural. Teachers often select their topic of study based on their classroom experiences, and even while studying the topic in its own right they naturally think about ways to make it more understandable and conceptual not only to themselves but also to their students. They incorporate study of common (mis)conceptions, and frequently interview students to gain a deeper understanding of their own student's conceptions of the topic. This all feeds into the lesson design, as teachers think not only of activities to highlight important concepts, but also of formative assessment in the classroom, and ways they might respond to students' questions and confusions.

Regarding measuring results, are you asking about assessing student learning in the research lesson? Designing that assessment is an integral part of the lesson planning process, often involving designing an "exit card" to provide evidence of student learning at the end of the lesson (in addition to observing, and sometimes collecting, student work during the lesson). The teachers then engage in post-lesson analysis, in which the exit cards (and possibly other student work) are examined and discussed to learn more about student thinking. This might also lead to additional interviews of students, to understand better the thinking behind their written work.

Yes, it would! 

I should note that in addition to the teacher learning communities supported and facilitated by project personnel, in the MSP project we built in to the design monthly self-facilitated teachers study/collaboration time. Thus the teachers met in small groups to focus on study and research of their own choosing. This either followed on the lesson study topics, or led to new lesson study topics, or involved topics in other areas. The goal was not only to support teacher learning during the project, but also to support development of independent teacher learning communities that would be sustained after the funding ended. 

In both projects we built in 10 full days of lesson study annually. Most of the teams went through two lesson study cycles each year in this time. 

As for impact on practice, it's difficult to single out the effect of the lesson study. In the MSP, for example, we provided a 2-week summer institute, monthly seminars, and self-facilitated collaboration time in addition to the lesson study. However, the Inland Counties Lesson Study Support Hub (mentioned by Jeremy Aikin in a previous response) has the potential to provide this kind of information, as it focusses primarily on lesson study.

I would agree with Davida that it would be difficult to discern the impact of just lesson study, as there were so many other professional development components of one of the grants showcased here. However, in terms of tracking measurable growth, in this grant the teachers were videotaped at least once every year that they participated in the the 5 year grant.  Those videos are now being coded, using the Mathematical Quality of Instruction (MQI) rubric where the data will be compared across the years to determine growth.

For the other grant however, lesson study was the main focus of the professional development.  These teachers were not videotaped annually (this grant had a very different funding structure).  However, we did periodic observations of these teachers (outside of lesson study), and are comparing notes from year to year.  I can say that what I noticed as a facilitator of lesson study in this grant, is that these teachers embraced the benefits of figuring out the underlying concept that students were struggling with (for instance, one group claimed that students struggled with solving equations because they did not understand inverse operations, but later came to the realization that it was really because students did not understand equivalence).  These types ah-has have led this group of teachers to think more about the big ideas in their planning, to be more purposeful in their questioning, and to approach instruction from a more conceptual perspective. 

Most of our teams include teachers of two or more grades, and we have found that to be extremely effective. Teachers are able to get a good sense of the progression of content through the grades, which helps them in understanding student needs, and then in lesson planning. They also develop a deeper sense of shared responsibility for students, with long-term learning in mind. 

We have also had facilitators with a variety of backgrounds and experiences, and that has enriched our facilitator community. And important aspect of our work has been to stay faithful to core aspects of lesson study (for example, the study component and the live observation of the research lesson) while allowing teams to develop their own characters and styles. Some of this is described in our paper "Teacher Learning in Lesson Study" (https://scholarworks.umt.edu/tme/vol10/iss3/5/).

One additional structural aspect: in planning the program, and considering the difficulty in making substantive changes in instruction, we asked that teachers apply as teams to participate in the project, and that they obtain administrator approval for their participation (there were additional reasons for the need for administrator approval). Those teams formed also the basis for determining lesson study teams, and the teams tended to prefer to work together in the summer institutes and academic year sessions (though sometimes we mixed them up during those sessions for various reasons). 

We found that the team approach worked well, and teachers formed strong communities of practice that provided both support for change and support for learning.