The purpose of this study was to advance knowledge about the relationship between graphing calculator use and classroom norm development. An interpretive case study design incorporating qualitative and quantitative research methods was used to explore the question: What happens when an exemplary teacher uses graphing calculators in a college algebra class? The purposively selected participants were the teacher and eleven students of a Texas community college algebra course. All 29 classes of the 14-week spring 2006 semester were observed in their entirety by the researcher. The theoretical frameworks guiding the study were the affective representation system and the Multiple Representations Model of Learning and Teaching with the use of the Mathematics and Science Classroom Observation System for data collection, analysis, and profiling of classroom lessons. Originally developed for grades K-12, the use of the instrument was extended to college algebra. Triangulation of data sources using constant comparative and content analysis methods were used to support the three major findings: (1) The instructor's proactive orchestration of specialized instruction, support materials, and designed activities contributed to the establishment of graphing calculator use as an essential part of classroom norms and promoted students' independent use of the tool; (2) The dynamic and interactive features of the TI-84 Plus graphing calculator facilitated the delivery of instruction at high cognitive levels during student interactive activities providing access to, exploration of, and use of multiple representations for some mathematical concepts and solutions not easily attainable using traditional methods; and (3) Although the majority of students had never used a graphing calculator before the course, all students used the tool at appropriate times during instructional activities, self-reporting that their use of the calculator was generally beneficial for enhancing their understanding of lessons and supporting class interactions. Additionally, all students independently chose to use the calculator during major assessments and reported knowledgeable use of the tool to facilitate improved test performance. Replication of the study is limited because the norms developed in this case are unique to the teacher and students who negotiated their establishment. Suggestions are given regarding educational policies, reform practices, and research extensions.

Recent scholarly work in mathematics education has included a focus on learning mathematics with understanding (Hiebert & Carpenter, 1992; Hiebert et al., 1997; Fennema & Romberg, 1999; National Council of Teachers of Mathematics, 2000). Hiebert et al. (1997) discussed two processes that they suggested increase understanding and that are central to this study: reflection and communication. Learning mathematics with understanding requires that the students create a deeper knowledge of mathematics through reflection and communication. The environment in which such learning can take place must include patterns of behavior, known as social norms that promote deeper thinking. When the social norms encourage reflection and communication among the members of the classroom community, or supports learning with understanding, it becomes what I term a productive discourse community. The purpose of this study is to find out what a teacher does to create and maintain a productive discourse community where students can reason and learn with understanding. To accomplish this purpose, this research asks the following question: In what ways does the teacher in the study direct mathematical discourse in order to facilitate understanding? To answer this research question, data was gathered from eight class periods. The classroom discourse was analyzed and six discourse generating tools were found to be used by the teacher: (1) using lower-order questions to engage students, (2) persisting in eliciting students' reasoning, (3) encouraging as many student participations as possible, (4) encouraging students to analyze and evaluate each other's comments, (5) encouraging students to share as many strategies as possible and (6) using a focusing discourse pattern. There were also three social norms found to be established in the classroom at the time of the data collection. These norms are: all students are expected to (a) participate (b) share their reasoning when called upon, and (c) listen to, analyze, and evaluate each other's comments. Through further analysis, it was found that the six discourse generating tools reinforced the social norms, while the social norms supported the six discourse generating tools. Thus creating an environment where reflection and communication occurred in a way that promoted learning mathematics with understanding.