This paper describes a theoretical model for systemic change as it concerns the learning and teaching of mathematics in K – 12 schools, with particular attention being paid to the rural context. Systemic change is the active process of establishing change in the community through lasting, long-term relationships, practices, and procedures (Adelman & Taylor, 2003). Our purpose is to describe the mechanics of such change provided by the strategic, continuous, and monitored support of all three of the constituents: Teachers, administrators and community, and externally supported by a temporary catalyst. Systemic change is achieved when the removal of the external catalyst does not affect the rest of the model. Evidence to support this claim has been derived from our case studies.

Not only are the problems teachers pose throughout their teaching of great importance but also the ways in which they use those problems make this a critical component of teaching. A problem-posing episode includes the problem setup, the statement of the problem, and the follow-up questions. Analysis of problem-posing episodes of precalculus instructors suggests that their mindset influences how they pose problems. In this paper, we describe the reflexive relationship between an instructor’s mindset and problem-posing episodes.

This reflective paper develops a repertoire of questions for teachers to use in their classrooms during episodes of mathematics discussions with and among students. These questions are motivated by an examination of questions posed by Wittgenstein in Zettel, and are connected to underlying tacit assumptions about mathematics, most of which lie subtly below the generally accepted milieu of math-talk. Once classrooms norms have been established to encourage participation by all students in a democratic and just classroom environment, these questions can be used effectively to stimulate meaningful discourse. These questions provide important examples of problem posing designed to encourage student reflection.

Many students drop out or do not pass their first. college mathematics class. This sometimes causes them to change their major to one that does not require a strong mathematics background. To increase student success rates, we used K-12 research (Cobb, 2000) to investigate the role of class structure on the mathematical learning of undergraduate students. Each K-12 mathematics classroom is a unique community created by the instructor and students (Davis & Simmt, 2003) and is influenced by the class structure. Research on discourse in K-12 mathematics classrooms suggests that specific characteristics of mathematical discourse influence student learning. (Davis & Simmt, 2003; Kazemi & Steipek, 2001; Ladson-Billings, 1995; Sfard, 2008).This research project sought to adapt some aspects of classroom structure and discourse and investigate the results on students in a college pre-calculus class. Small class size provided opportunities for increased interaction between students and their instructor, and the ensuing discourse had the potential for teachers to assess student needs and acquisition of mathematical knowledge. In comparing student learning in problem-based learning (PBL) environments with traditional content-based instruction, Boaler (1998) found that. students who learn through a problem-based approach exhibit higher achievement on standardized tests and on problem solving tests dealing with realistic situations than students from classrooms using a traditional approach. We describe the teacher and student discourse in four pre-calculus classes, and focus on two particular classes, one employing PBL and the other a traditional lecture-based class, in order to compare and contrast the two classes. We attempt to explain the reasons for the differences in student achievement by examining three key components of teaching that influenced student learning in these two classrooms: teacher beliefs about their roles and that of their students; the classroom discourse; and the classroom environment.

The purpose of this article is to report on the importance of providing pre-service and in-service teachers with experience and specific training in critical thinking skills. The essential concepts in elementary mathematics curricula can be augmented to include and cultivate critical thinking skills that have tremendous ramifications for future leaders and for those who move on to more technical training. A sample problem, along with pre-service teacher responses, is used here to show the necessity and importance of this kind of training as the responses show clear evidence of a certain naivete on the part of these college level students. The responses do show evidence of budding social conscience in the students, but the level of expertise in critical thinking is not at all sophisticated. We discuss and explore the implications of our approach.

This study is part of a larger project that investigated the impact of instructional practices on university students learning in pre-calculus classes. The students in one section consistently outperformed the students in the other sections and led us to investigate whether discursive practices used by the instructor promoted student learning. The theoretical framework is based on Truxaw's and Sfards' ideas about the functions of discourse and the subsequent development of mathematical ideas. Qualitative analysis using constant comparative and matrix methods indicates that how an instructor invites student participation and responds to students' comments determines the function of the discourse.

This chapter identifies issues of status that often arise in the classroom. These issues are difficult to deal with and teachers often lack the necessary tools. A professional development (PD) program for K-12 math teachers attempted to address these issues and train teachers how to recognize and to deal with them, by, teaching them about community agreements, group roles and protocols for use in small groups. After two years of a PD program, teacher participants reported seeing positive results among their students. The quiet students were speaking up, while the domineering students were learning to allow equal time for all students to have a voice.

Discourse in mathematics classrooms is surprisingly complex and both student and teacher mathematical discourse contain distinct, identifiable elements. Student discourse is necessarily focused on understanding concepts and solving mathematical problems. Teacher discourse contains some of these same elements, but when examined critically it gives rise to major distinctions. Teacher discourse is directed at improving student understanding and also the logistics of the classroom, and thus is often meta-mathematical in nature. We shine a light on the tactics teachers use which are part of meta-mathematical discourse such as re-voicing, redirecting, questioning, and clarifying. Contrasts are explored between student and teacher discourse.

Spring term of 2005, three Web-assisted undergraduate mathematics courses were taught at the University of Idaho: Math 235 Mathematics for Elementary Teachers 1; Math 236 Mathematics for Elementary Teachers II; and Math 391 Modern Geometry. While the content of these courses differ, they share common goals: to foster a deep understanding of critical mathematical content; to train students in the use of computer-based modeling and analysis technologies; and to promote the development of mathematical communication and collaboration concepts, skills, and dispositions. Outside of regular class periods, students participated in an ongoing asynchronous mathematical dialogue using the Idaho Virtual Campus Discussion Tool. The structure of this dialogue was analyzed using graph theoretic methods associated with social network analysis. These findings were compared to student achievement data and the results used to answer the question, “In Web-assisted undergraduate mathematics courses, how is the structure of asynchronous communication related to student achievement?”