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Joel Amidon, Daniel Chazan, Dana Grosser-Clarkson and Elizabeth Fleming

This article explores the ways in which a teacher educator uses digital technology to create a virtual field placement as a way to blur the boundaries between a university methods course and teacher candidates' field placements. After describing his goals for the course, the teacher educator provides a description of three LessonSketch experiences his teacher candidates complete in this virtual field placement site and how these experiences create opportunities for teacher candidates to learn to teach mathematics. The design process and choices of these virtual field placement experiences are explored via interviews with the first author. Reflecting on these LessonSketch experiences, all of the authors then explore affordances of virtual and hybrid placements as resources for supplementing real placements and bridging theory/practice divides in teacher education.

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Hortensia Soto-Johnson, Michele Iiams, April Hoffmeister, Barbara Boschmans and Todd Oberg

When teaching mathematics content courses designed for preservice elementary teachers, we often meet resistance when we ask our students to learn mathematics conceptually; they consider us out of touch with what works in the classroom. Preservice elementary teachers question the need for conceptual understanding, feeling certain that elementary students cannot and will not understand mathematics in this way.

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Jeff Suzuki

The NCTM standards for courses designed to prepare students for the “new calculus” include using “all appropriate calculator and computer technologies,” as well as promoting “the use of experimentation and conjecturing” (NCTM 2000, p. 289). Both these goals can be met in the following experiment to find the area under a curve. The experiment is a modification of a classroom exercise that I conducted in an introductory college calculus course for nonmajors.

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Richard P. Giles

Edited by Joseph N. Payne and William C. Lowry

In a mathematics course designed for prospective teachers at the University of Illinois, the students were required to design and build a project that could be used in a high-school classroom to illustrate or demonstrate some mathematical concept. Since earlier in the semester we had discussed the growing emphasis that is being given in high schools to the study of symbolic logic, I decided to build an electrical device that would graphically represent the truth table and the various operations which can be performed on propositions.

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Wilbur H. Dutton

Teachers' understanding of basic arithmetical concepts is closely associated with the ability to present these concepts in classrooms. Numerous studies have been made to show the amount of understanding of arithmetic possessed by elementaryschool teachers and students preparing to become teachers. Relatively little study has been made of changes made in students' understanding of arithmetical concepts as they progress through the courses designed to teach these processes. This study deals with measuring students' changes in understanding arithmetical concepts before and after completing a lower division mathematics course for elementary teachers.

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Michael D. Steele and Amy F. Hillen

In the majority of secondary mathematics teacher preparation programs, the work of learning mathematics and the work of learning to teach mathematics are separated, leaving open the question of when and how teachers integrate their knowledge of content and pedagogy. We present a model for a content-focused methods course, which systematically develops a slice of mathematics content in the context of typical methods course activities. Three design principles are posited that undergird the design of such a course, addressing the nature of the mathematics content, the sequencing and design of activities, and the ways in which the course addresses the needs of diverse learners. Data from an instantiation of one such course is presented to illustrate the ways in which the course design framed teachers' opportunities to learn about both content and pedagogy.

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Jill Adler and Zain Davis

This article describes an investigation into mathematics for teaching in current teacher education practice in South Africa. The study focuses on formal evaluative events across mathematics teacher education courses in a range of institutions. Its theoretical orientation is informed by Bernstein's educational code theory and the analytic frame builds on Ball and Bass' notion of “unpacking” in the mathematical work of teaching. The analysis of formal evaluative events reveals that across the range of courses, and particularly mathematics courses designed specifically for teachers, compression or abbreviation (in contrast to unpacking) of mathematical ideas is dominant. The article offers theoretical and practical explanations for why this might be so, as well as avenues for further research.

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Lewis B. Smith

Guided discovery lessons may lead to unexpected discoveries as we11 as to those that were intended by their teachers. The following account is based upon the experiences of a small group of advanced college students in a course designed to fulfill the spirit of the Nuffield Foundation film, I Do and I Understand. The first part of this article describes a lesson brought by a student with a specific goal—to enable the learner to predict the height of an unknown right triangle when certain other information iS provided. Because of a prevailing spirit of inquiry additional available discoveries were made. The reader will realize that the experiences described here involve not only ratios and proportions but also similarity, primes, composites, and common factors.

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Daniel F. McGaffrey, Laura S. Hamilton, Brian M. Stecher, Stephen P. Klein, Delia Bugliari and Abby Robyn

A number of recent efforts to improve mathematics instruction have focused on professional development activities designed to promote instruction that is consistent with professional standards such as those published by the National Council of Teachers of Mathematics. This paper describes the results of a study investigating the degree to which teachers' use of instructional practices aligned with these reforms is related to improved student achievement, after controlling for student background characteristics and prior achievement. In particular we focus on the effects of curriculum on the relationship between instructional practices and student outcomes. We collected data on tenth-grade students during the 1997–98 academic year. Some students were enrolled in integrated math courses designed to be consistent with the reforms, whereas others took the more traditional algebra and geometry sequence. Use of instructional practices was measured through a teacher questionnaire, and student achievement was measured using both the multiple-choice and open-ended components of the Stanford achievement tests. Use of standards-based or reform practices was positively related to achievement on both tests for students in integrated math courses, whereas use of reform practices was unrelated to achievement in the more traditional algebra and geometry courses. These results suggest that changes to instructional practices may need to be coupled with changes in curriculum to realize effects on student achievement.

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Caitlin Riegel and Maritza M. Branker

Technology has been linked to increased student motivation in the twenty-first-century classroom (Rau, Gao, and Wu 2008). In addition to engaging students by using a familiar medium to present content, technology allows educators an opportunity to focus on reaching a deeper conceptual understanding. Specifically, educators teaching AP Calculus, a course designed to provide the same content as a college level calculus course, can use technology to promote understanding of material that high school standards do not mention as crucial, but are nonetheless considered fundamental and included in college textbooks. If AP students do not achieve a thorough conceptual understanding of content considered fundamental to calculus, the result may be a piecemeal view of the subject, a lessened appreciation for its applications, and a lack of preparation for postsecondary mathematics education (Bressoud 2004). Instructional technology can help teachers present material to meet the standards outlined by The College Board (2015), as well as provide students with content knowledge that will prepare them to meet collegiate expectations. When instructional technologies are used as contemporary tools and resources “aimed at deepening students' understanding of content” (Drijvers et al. 2010; Zazkis and Nunez 2015, p. 126) they can also increase deductive thinking by communicating, demonstrating, and explaining advanced conceptual material. Furthermore, instructional technology can connect with current pedagogy that emphasizes technology in education (ISTE 2008) when used as blended learning tools that teach “some fraction of the content through online sources” and implementing “non-lecture based activities” (Zazkis and Nunez 2015, p. 126). In all, the use of technology in calculus can provide a more holistic view of the mathematics without sacrificing class time needed to meet the standards.