A classroom teaching experiment was conducted to document prospective teachers' development of whole number concepts and operations. The purpose of this mixed-methods study was to examine the collective understanding of prospective teachers in an elementary mathematics content course. Design research methodology, specifically a classroom teaching experiment was the methodology selected for this study since it allows learning to be documented in a classroom environment and is iterative in nature. A revised hypothetical learning trajectory and instructional tasks from a previous classroom teaching experiment were used in this study (Andreasen, 2006). Research about children's development of whole number concepts and operations was used in developing instructional learning goals. In addition, research regarding prospective teachers' development supported the instructional modification that all tasks would be presented and expected to be reasoned about in base-8. Both qualitative data and quantitative data were collected. Qualitative data included whole class dialogue that was videotaped and transcribed, as well as student work samples. Quantitative data included items from the Content Knowledge for Teaching Mathematics database that were administered prior to and subsequent to the instructional sequence in base-8 (Hill, Schilling, & Ball, 2005). It should be noted that the items selected from the database were in base-10. The emergent perspective served as the interpretive framework of the collected qualitative data. This perspective reflexively coordinates the social or group perspective simultaneously with psychological or individual perspective. As stated, this study sought to describe the communal mathematics understanding of prospective teachers in an elementary mathematics content course. Toulmin's (1969) model of argumentation and Rasmussen and Stephan's three-phase methodology served to document normative ways of group reasoning called classroom mathematical practices. The following classroom mathematical practices were identified as taken-as-shared by prospective teachers: (a) developing small number relationships using Double 10-Frames, (b) developing two-digit thinking strategies using the open number line, (c) flexibly representing equivalent quantities using pictures or Inventory Forms, and (d) developing addition and subtraction strategies using pictures or an Inventory Form. Quantitative results indicated that prospective teachers were able to apply mathematical understandings grounded in base-8 to whole number concepts in base-10. In the end, counting and calculating in base-8 provides a meaningful context for prospective teachers to reconstruct their knowledge of whole number concepts and operations.

This research project aimed to extend the research literature by providing greater insight into the way individual prospective teachers develop their conceptual understanding of whole number concepts and operations in a social context. In this qualitative study, a case study analysis provided the opportunity for careful exploration of the manner in which prospective teachers' understanding changed and the ways two selected participants reorganized their mathematical thinking within a classroom teaching experiment. While previous research efforts insisted on creating a dichotomy of choosing the individual or the collective understanding, through the utilization of the emergent perspective both the individual and the social aspects were considered. Specifically, using the emergent perspective as a theoretical framework, this research endeavor has outlined the mathematical conceptions and activities of individual prospective teachers and thus has provided the psychological perspective correlate to the social perspective's classroom mathematical practices. As the research participants progressed through an instructional sequence taught entirely in base-8, a case study approach was used to select and analyze two individuals. In order to gain a more thorough understanding of the individual perspective, this research endeavor focused on whether teachers with varying initial content knowledge developed differently through this instructional sequence. The first participant initially demonstrated "Low-Content" knowledge according to the CKT-M instrument database questions which measure content knowledge for teaching mathematics. She developed a greater understanding of place value concepts and was able to apply this new knowledge to gain a deeper sense of the rationale behind counting strategies and addition and subtraction operations. She did not demonstrate the ability to consistently make sense of multiplication and division strategies. She participated in the classroom argumentation primarily by providing claims and data as she illustrated the way she would use different procedures to solve addition and subtraction problems. The second participant illustrated "High-Content" knowledge based on the CKT-M instrument. She already possessed a solid foundation in understanding place value concepts and throughout the instructional sequence developed various ways to connect and build on her initial understanding through the synthesis of multiple pedagogical content tools. She demonstrated conceptual understanding of counting strategies, and all four whole number operations. Furthermore, by exploring various ways that other prospective teachers solved the problems, she also presented a greater pedagogical perspective in how other prospective teachers think mathematically. This prospective teacher showed a shift in her participation in classroom argumentation as she began by providing claims and data at the outset of the instructional sequence. Later on, she predominantly provided the warrants and backings to integrate the mathematical concepts and pedagogical tools used to develop greater understanding of whole number operations. These results indicate the findings based on the individual case-study analysis of prospective elementary school teachers and the cross-case analysis that ensued. The researcher contends that through the synthesis of the findings of this project along with current relevant research efforts, teacher educators and educational policy makers can revisit and possibly revise instructional practices and sequences in order to develop teachers with greater conceptual understanding of concepts vital to elementary mathematics.