Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 281-294). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 281-294). Also available on the Internet.

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This qualitative study investigated how confidence chats affect how students with disabilities (SWD) position themselves during mathematics discourse and the impact the confidence chats had on a student’s mathematical identity. A critical discourse analysis was conducted over a three-month period using transcripts from classroom observations, student-teacher confidence chats, and teacher interview debriefs. Findings revealed that following the confidence chats, SWD participated in discourse, and students were able to adjust their positioning in relation to others. Over the course of the study, student collaboration increased and their reliance on teacher support decreased. In addition, the findings indicated that confidence chats provided a window into students’ macro-identities which the teacher was not seeing during classroom interactions. Students were positioned in relation to tests and grades and were not seeing the relevance of how mathematics is preparing them for college or career readiness. The implications of this study suggest that students may not know what collaboration looks and sounds like. Taking time to set norms and expectations is a critical element when providing opportunities for discourse in the classroom. Additionally, student strengths need to be affirmed, so students build confidence in their mathematical abilities and connect their mathematics identities to positive experiences and the real-world.

The vision of K-12 standards-based mathematics reform embraces a greater emphasis on students’ ability to communicate their understandings of mathematics by utilizing adaptive reasoning (i.e., reflection, explanation, and justification of thinking) through mathematics discourse. However, recent studies suggest that many students lack the socio-cognitive capacity needed to succeed in learner-centered, discussion-intensive mathematics classrooms. A multiple case study design was used to examine the nature of participation in mathematics discourse among two low- and two high-performing sixth grade female students while solving rational number tasks in a standards-based classroom. Data collected through classroom observations, student interviews, and student work samples were analyzed via a multiple-cycle coding process that yielded several important within-case and cross-case findings. Within-case analyses revealed that (a) students’ access to participation was mediated by the degree of space they were afforded and how they attempted to utilize that space, as well as the meaning they were able to construct through providing and listening to explanations; and (b) participation was greatly influenced by peer interactional tendencies that either promoted or impeded productive contributions, as well as teacher interactions that helped to offset some of the problems related to unequal access to participation. Cross-case findings suggested that (a) students’ willingness to contribute to task discussions was related to their goal orientations as well as the degree of social risk perceived with providing incorrect solutions before their peers; and (b) differences between the kinds of peer and teacher interactions that low- and high-performers engaged in were directly related to the types of challenges they faced during discussion of these tasks. An important implication of this study’s findings is that the provision of space and meaning for students to participate equitably in rich mathematics discourse depends greatly on teacher interaction, especially in small-group instructional settings where unequal peer status often leads to unequal peer interactions. Research and practice should continue to focus on addressing ways in which students can learn how to help provide adequate space and meaning in small-group mathematics discussion contexts so that all students involved are allowed access to an optimally rich learning experience.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 143-150). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 143-150). Also available on the Internet.

Different types of ambiguous student contributions occur in mathematics classrooms. In this study I identified (1) different types of ambiguous student contributions and (2) the different ways teachers respond to one particular kind of ambiguous contribution, clarifiable ambiguity. Note that clarifiable ambiguity is ambiguity that stems from a student who uses an unclear referent in their contribution and can be clarified in the moment by the particular student. Literature has focused only on ambiguity that has the potential to further the development of mathematical concepts and has only theorized about teacher responses to this specific type of ambiguity. This study identified an additional three types of ambiguous student contributions: Student Appropriation of Teacher Ambiguity, Irrelevant Ambiguity, and General Ambiguity. It was important to identify all the different types of ambiguous student contributions because teacher responses should likely be different to the different types of ambiguity. In addition, through analyzing the teacher responses to the clarifiably ambiguous student contributions, this study found that teachers addressed the clarifiably ambiguous student contributions about half the time. When the teachers did address the clarifiable ambiguity, the majority of the time the teacher clarified the ambiguity themselves with the most common response being the teacher honed in on the clarifiably ambiguity and asked for confirmation from the student on the accuracy of the clarification.

Teaching mathematics is a difficult and complicated task. For student teachers, who are extremely new to the mathematics classroom, this difficulty is magnified. One of the biggest challenges for student teachers is learning how to effectively use the student thinking that emerges during mathematics lessons. I report the results of a case study of two mathematics education student teachers. I focus on how they make decisions while teaching in order to use their students' mathematical thinking. I also present analysis of the student teachers' discourse patterns, the reasons they gave to justify these patterns, and how their reasons affected how they used their students' thinking. I found that generally the student teachers used student thinking in ineffective ways. However, the reasons the student teachers gave for using student thinking always showed the best of intentions. Though given with the best of intentions, most of the reasons for using student thinking given by the student teachers were correlated with the student teachers ineffectively using their student's thinking. However, some of the reasons given by the STs for using student thinking seemed to help the student teachers more effectively use their students' thinking. I conclude with implications for preparing future student teachers to better use student thinking.

While the use of student thinking to help build mathematical understandings in a classroom has been emphasized in best teaching practices, teachers still struggle with this practice and research still lacks a full understanding of how such learning can and should occur. To help understand this complex practice, I analyzed every instance of student thinking and every teacher response to that thinking during a high school geometry teacher's whole class discussion and used these codes as evidence of alignment or misalignment with principles of effective use of student mathematical thinking. I explored the teacher's practice both in small and large grains by considering each of her responses to student thinking, and then considered the larger practice through multiple teacher responses unified under a single topic or theme in the class discussion. From these codes, I moved to an even larger grain to consider how the teacher's practice in general aligned with the principles. These combined coding schemes proved effective in providing a lens to both view and make sense of the complex practice of teachers responding to student thinking. I found that when responding to student thinking the teacher tended to not allow student thinking to be at the forefront of classroom discussion because of misinterpretation of the student thinking or only using the student thinking in a local sense to help advance the discussion as framed by the teacher's thinking. The results showed that allowing student thinking to be at the forefront of classroom discussion is one way to position students as legitimate mathematical thinkers, though this position can be weakened if the teacher makes a move to correct inaccurate or incorrect student thinking. Furthermore, when teachers respond to student thinking students are only able to be involved in sense making if the teacher turns the ideas back to the students in such a way that positions them to make sense of the mathematics. Finally, in order to allow students to collaborate a teacher must turn the mathematics to the students with time and space for them to meaningfully discuss the mathematics. I conclude that the teacher's practice that I analyzed is somewhat aligned with honoring student mathematical thinking and allowing student thinking to be at the forefront of class discussion. On the other hand, the teacher's practice was strongly misaligned with collaboration and sense making. In this teacher's class, then, students were rarely engaged in sense making or collaborating in their mathematical work.