UC Berkeley

How do teachers and students create classroom environments in which mathematical errors are regarded as important opportunities for learning? What norms support students in learning from their errors, and how to these norms develop in a classroom community? This dissertation addresses these questions through a longitudinal case study investigating the emergence of classroom norms related to the treatment of errors. Classroom norms are here understood to be taken-as-shared expectations for behavior in a classroom. The fifth-grade case study classroom was selected because in prior research studies the experienced and highly-regarded teacher had engaged students in rich discussions of their mathematical errors and the opportunities presented by these errors for learning.

To answer the questions of (a) what norms related to mathematical errors were taken up, and (b) how these norms developed over the course of the school year, Saxe’s (2012) framework describing the relation of micro-, onto-, and sociogenetic processes was used as a guide for determining methods. Sociogenetic processes were the focus of this investigation, and Saxe’s framework points to microgenetic constructions, when studied collectively over time, as likely to illuminate otherwise difficult to observe sociogenetic processes, such as norm development. To provide information about microgenetic constructions related to errors, several types of evidence were collected throughout the school year during three data collection periods lasting two weeks each: at the beginning (September), middle (January), and end (April) of the 2014 to 2015 school year. During all three time periods, the teacher was interviewed, five “focal” students were interviewed, classroom mathematics lessons were video-recorded daily, and all students in the classroom answered a paper-and-pencil multiple choice survey about their expectations related to errors. Interviews were analyzed using grounded analysis methods, and video-recordings were analyzed using a focused coding procedure and StudioCode software. Sources of evidence were used in a triangulating fashion to identify norms in the classroom.

Through this analysis, seven norms were identified as having been taken up by the majority of members of the class by the end-of-year data collection period. Two of these norms were selected for in-depth description. The norm everyone has some mathematical understandings to which you should pay attention provides a good example of a norm that was closely tied to a specific collective practice, the “coaching” practice that was used frequently in the case-study classroom. The norm there are different types of errors, only some of which are acceptable provides an example of a norm that emerged part-way through the school year in response to a problem with the way errors were being treated. Classroom interactions and teacher and student interview statements exemplifying these norms are described. The process through which these two norms emerged in the case study classroom over the course of the school year is detailed, using evidence collected throughout the school year. In general, the teacher strongly promoted these norms by frequently and persistently modeling, describing, and praising behaviors consistent with these norms and by correcting inconsistent behaviors.

Implications for how Saxe’s framework may be productively applied in future investigations of classrooms norms are discussed. In particular, attention to ontogenetic processes – that is, individuals’ shifting expectations over time – was found to be useful as an access point for identifying the norms of a classroom community, and the teacher’s actions and expectations were found to be especially important indicators of classroom norms. Examination of collective practices related to errors was also useful for the identification of norms because some norms were strongly associated with collective practices, such as the “coaching” practice. The results of this study also have implications for teaching practice. The findings indicate that children are capable of taking up challenging practices related to the study of errors, and teachers who promote these practices in their classrooms may be successful if they are persistent in modeling, explaining, and praising the desired practices.