## Abstract

This chapter presents a methodology for studying classroom communities as microcultures, with a focus on processes of teaching and learning over significant spans of time. In Sect. 11.1, we present a conceptual framework that treats classroom activity at two levels of analysis, collective and individual. Both levels are geared for understanding the reproduction and alteration of a common ground of talk and action through time. Key concerns are the emergence of collective norms and individuals’ use of representational forms to serve varied functions in classroom communicative and problem solving activity. In Sect. 11.2, we show how the conceptual framework was used to organize two related programs of empirical research. First, we present design research that led to a 19-lesson sequence on integers and fractions , which uses the number line as a central representational form. Second, we use the framework to organize an empirical analysis of a single classroom community over the 19-lesson sequence. We illustrate empirical techniques for capturing the reproduction and alteration of a common ground with shifting lesson topics. The chapter concludes with an analysis of the way the analytic approach illuminates core processes of teaching and learning and the utility of the approach for future work.

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## Notes

- 1.
There are notable exceptions that do incorporate cross-lesson perspectives. One example is Paul Cobb, Kay McClain, and Koeno Gravemeijer’sdesign research on statistical reasoning about data in the middle school grades (Cobb 1999; McClain et al. 2000). The project is the focus of a special issue of

*The Journal of the Learning Sciences.*Anna Sfard and Kay McClain served as Editors of the special issue (Sfard and McClain 2002), and convergent analyses produced from a range of methodological perspectives (Cobb 2002; Forman and Ansell 2002; Macbeth 2002; McClain 2002; Saxe 2002; Schliemann 2002; Sfard 2002). - 2.
We take for granted that norms have no stable existence beyond the participation of individual actors. However, as a heuristic, we consider norms as a collective level construct.

- 3.
An analysis of a video record revealed that Carol appears to take the distance between −1,000 and her mark of −1,001 as a unit interval (see Fig. 11.3). She translates that interval to the unlabeled tick mark about 6 or 7 times, yielding the label “–1,006 or −1,007.”

- 4.
Cuisenaire™ rods are manipulative wooden rods used in Ms. R’s classroom.

- 5.
We note that conventions vary in the use of forms . For example, in the United States in elementary mathematics classrooms, number line s contain arrows on the left and right ends, indicating that lines are extended in both directions. In contrast in other countries, number lines are sometimes depicted as rays, with arrowheads on the right end only.

- 6.
Throughout the chapter we make use of the term “fractions ” where “rational number” would often be more appropriate. We take “fractions” to be forms of representing rational numbers—a number expressed with a numerator and denominator (common fraction, mixed numbers) or a number expressed as a decimal (decimal fraction), and rational numbers to be the numbers expressed by these representational forms. For simplicity, we use the expression “fractions” to refer sometimes to rational numbers and other times to the representational form of common fractions (including proper and improper common fractions).

- 7.
From this point we refer to “definitions and principles” simply as definitions, even though some of the ideas to which “definitions” refer are actually in the form of principles (e.g.,

*every number has a place**but doesn’t need to be shown*). - 8.
For example, for additive relations, the length of 1 red (2 units) plus 1 white (1 unit ) equals 1 light green (3 units); inversely, the length of 1 light green (3 units) minus the length of 1 red (2 units) equals the length of 1 white (1 unit). Similarly, the rods can be used to express multiplicative relations: 1 light green is the length of three whites, and 1 white is 1/3 the length of the light green.

- 9.
There were two classrooms that we treated as laboratories, Ms. R. ’s and another, both of whom were partner teachers who participated in the development of the LMR lessons (see classroom studies reviewed earlier). Our choice of Ms. R’s classroom over the other teacher’s classroom was arbitrary.

- 10.
Depending upon the focus of videotaping, we used between one and five cameras in the classroom on a given day. When multiple cameras were used, the focus was on teacher and the class (two cameras) and teacher, class, and partner work (five cameras). One of our team members led the development of a video manual to organize positioning of cameras to maximize coverage (Katherine Lewis).

- 11.
Though we administered four assessments, we made use of three different forms , with the post-assessment and final assessment being the same form. Each of the three measures consisted of about 30 items. 18 problems were shared across measures (to enable scaling using an IRT model). On each of the assessments, we included number linenumber line and non-number line items for both fractions and integersintegers . Of particular interest was whether learning gains for students in Ms. R’s class and all LMR classrooms might be limited to number line items or whether student learning might be reflected on both number line and non-number line items.

- 12.
In our coding procedure, we noted that sometimes references to definitions varied from their canonical forms inscribed on the poster in front of the class. For example, the following variants were also coded as “unit interval ”: “unit,” “interval of one,” “distance of one.” We used this scheme to code all references to definitions or principles across all 19 lessons for both partner teachers. We established the reliability of the references to the definitions in one partner teacher’s classroom by having two coders independently re-code reference to definitions/principles for 20 % of the video data evenly distributed across all of the lessons. We computed percent agreement to be 86 %.

- 13.
The graph also reflects students’ performances on the final assessment , about 16 weeks after LMR instruction. We find that most students reproduced their post-assessment solutions to the problem, 8/7, while three shifted to 1/7 as the solution. No students reverted to a solution for which 8 was the denominator .

- 14.
To determine proportions, we counted the number of total references to each of the five definitions for each lesson. In this computation, we divided the number of references by the total number of references for each lesson. Note that total number of references to the selected definitions (n) differs markedly by lesson. There are two reasons for these differences. First, other definitions were referenced in the lessons, and so the number of instances does not reflect the total number of instances referenced in a given lesson. Second, Ms. R decided to devote two math periods to F-3; as a result, more references were made to definitions for that lesson than the other two lessons.

- 15.
We note an issue that is important but that we have not taken up in our analysis of sociogenesis:sociogenesis In Ms. R’s classroom during any particular lesson, students may well have been using the same form, like the unit interval , to serve different functions in classroom displays during the same period of time (like a whole class discussion). Our coding schemes did not allow us to document these form-function distributions. Variant functions for the same form are to be expected in sociogenetic processes, and we expect that these phenomena were very much a part of the nuance of classroom discourse in Ms. R’s community.

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## Acknowledgments

The research reported here was supported in part by the Institute of Education Sciences grants R305B070299 (research grant) and R305B090026 (predoctoral training grant) to the University of California, Berkeley. The opinions expressed are those of the authors and do not represent views of the Institute of Education Sciences or the U.S. Department of Education. We are grateful to the teachers and students who participated in our research, to the many people who contributed to various phases of the project. Geoffrey Saxe and Maryl Gearhart were principal collaborators in crafting the LMR curriculum (and Maryl Gearhart leads a team that is working on additional revisions and additional lessons). In addition to the those who contributed to this chapter as authors, those doctoral students who contributed to various phases of the research, and the development of the curriculum include: Ronli Diakow, Darrell Earnest, Lina Chopra Haldar, Nicole Leveille Buchanan, Jen Collett, Jennifer Langer-Osuna, Katherine Lewis, Anna McGee, Meghan Shaughnessy, Alison Miller Singley, Josh Sussman, David Torres Irribarra, and Ying Zheng. Rick Kleine and Jenn Pfotenhauer worked on the project as collaborating teachers in the development of the LMR lesson sequence. We are grateful to Professors Deborah Loewenberg Ball and Hyman Bass for feedback on the design of the curriculum and feedback on the mathematics content of our student assessment . We are also grateful to Vinci Daro, Sten Ludvigsen, and Fady El Chidiac who have made valuable comments as we crafted the manuscript. Email correspondence: Geoffrey Saxe (saxe@berkeley.edu)

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Saxe, G.B., de Kirby, K., Le, M., Sitabkhan, Y., Kang, B. (2015). Understanding Learning Across Lessons in Classroom Communities: A Multi-leveled Analytic Approach. In: Bikner-Ahsbahs, A., Knipping, C., Presmeg, N. (eds) Approaches to Qualitative Research in Mathematics Education. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9181-6_11

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