Classroom mathematical practices in a preservice elementary mathematics education course using an instructional sequence related to place value and operations

This qualitative study documents a classroom teaching experiment in a semester-long undergraduate mathematics education course for 16 prospective elementary school teachers. The purpose of this study was to investigate how social aspects of the classroom environment facilitated the collective mathematical learning of place value and whole number operations by preservice elementary school teachers. Design-based research methodology was used for formulating the study. A hypothetical learning trajectory and instructional sequence related to place value and operations were created and refined in the two semesters prior to this study. The instructional sequence was in its third iteration for this study.; The developmental levels that children progress through in learning place value and operations were used in identifying the learning trajectory and supporting tasks in which the preservice teachers were asked to engage. A large portion of the instructional sequence involved a setting of base eight instead of base ten. The sequence returned to base ten in order to discuss whole number operations and alternative strategies for operations in an effort to further develop the preservice teachers' conceptual understandings of place value and operations and to examine children's thinking strategies.; Data were collected through video-taped recordings of class sessions, audio-taped recordings of table discussions and research team meetings, field notes, and journals written by the research team. Sixteen preservice teachers participated in the study which lasted over 5 class sessions of 3 hours and 10 minutes each.; The emergent perspective which attempts to coordinate the individual learning and the social aspects of the classroom that support collective learning was used as an interpretive lens for data collection and analysis. The social aspects along with some aspects of individual student understandings together give an indication of collective mathematical understandings of the students as a whole group.; Social norms established were: (a) the expectation of providing explanations and justifications for solutions and solution methods, (b) making sense of each other's solutions and (c) asking questions of classmates or the instructor. Sociomathematical norms that were valued but not fully established were: (a) criteria for different solutions and solution methods and (b) criteria for what constituted a good explanation. Data analysis for the establishment of classroom mathematical practices was conducted using Toulmin's argumentation model (Toulmin, 1969). A three phase approach described by Rasmussen and Stephan (in press) was used in determining what constituted a classroom mathematical practice. The classroom mathematical practices that facilitated student learning in this study were: (a) unitizing, (b) flexibly representing numbers, and (c) reasoning about operations. This study led to the refinement of the hypothetical learning trajectory and further progress in defining an instructional theory of how preservice teachers may come to understand place value and whole number operations.