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Does the choice of observation instrument make a difference in the feedback and ratings that teachers receive? This study explores how lessons are rated differentially across various observation instruments. To investigate this question, ten randomly selected mathematics lessons were rated using six different observation instruments. Overall scores varied little across instruments. Our analyses indicate that differences in scores can be attributed to what we call instrumental occlusion, instrumental emphasis, and element density. This article concludes with implications for the selection and use of observation instruments in school settings.

Too often, high school algebra students encounter topics in a discrete, disconnected fashion. In the authors' first years of teaching, they found students to be adept at solving problems they were currently studying; for instance, they could solve quadratics during a quadratics unit. Too often, however, students struggled with that same content once they moved onto new topics in the same course. For instance, the quadratic formula, a routine algorithm in October, was quickly forgotten by spring, when exponential functions were the focus. As the authors gained confidence and experience as teachers, they retooled their instruction to encourage students to make connections across topics throughout the school year. In this article, they share an investigation that exemplifies this style of practice, a rich task designed for second-year algebra students, one that connects polynomial and exponential functions in a way that fosters inquiry, discovery, and problem-solving tenacity. More specifically, they explore the Quadratic to a Quadratic (Q2Q) task, a low-threshold, high-ceiling task that encourages students to use a variety of problem-solving strategies as they find solutions to a nonroutine algebraic equation.

Successful teacher preparation programs provide learning experiences that help teacher candidates make the shift from "student" to "teacher." In this paper we present research on the implementation of a process for providing candidates such experiences. Utilizing the Mathematics as Teacher Heuristic (MATH) process, prospective high school mathematics teachers explore rich problems by solving the task, analyzing samples of student work, designing a solution key, and modifying the task. We use their engagement in these explorations and reflections on the process to analyse the development of candidates' Content Knowledge and Pedagogical Content Knowledge.

There is a dearth of research on the mechanisms for preservice teachers' development of the pedagogical knowledge necessary for effective use of such technologies. We explored the emergent Technological Pedagogical and Content Knowledge (TPACK) (Niess 2005, 2006, 2007) of a group of secondary mathematics preservice teachers in a methods course as they designed and implemented technology-rich teaching materials in field settings. Participant surveys and collected assignments were analyzed through the lens of the TPACK framework. The data were also analyzed to examine the trajectory of the participants' beliefs about the appropriate role of advanced digital technologies in mathematics. The results indicate that the participants' understanding of technology shifted from viewing technology as a tool for reinforcement into viewing technology as a tool for developing student understanding. Collected data supports the notion that preservice teacher TPACK development is closely related to a shift in identity from "learners of mathematics" to "teachers of mathematics". In a class where advanced digital technologies were used extensively as a catalyst for promoting inquiry-based learning, preservice teachers retained a great deal of skepticism about the appropriateness of using technology in concept development roles, despite their confidence that they can incorporate technology into their future teaching. (Contains 5 figures.)

The use of technology to create multiple representations of a concept has become one of the significant instructional environments that the National Council of Teachers of Mathematics (2000) suggests strongly for mathematics teachers to consider. One example of this type of environment is educational software with linked multiple representations. An activity for both linked and semi-linked versions of multi-representational software which was used in a dissertation study is presented along with two ninth grade algebra students' responses in order to provide an example of possible uses and effects of semi-linked and linked computer software in mathematics classrooms. It was also aimed to make connections between practice and research. The conclusion of this study was that semi-linked representations could be as effective as linked representations and that there was a role for each in different situations, at different levels, and with different mathematical concepts. (Contains 3 figures and 3 tables.)

Sustaining engagement with a mathematics task is not a novel suggestion for effective mathematics teaching. "Principles and Standards for School Mathematics" (2000) specified that "students need to know that a challenging problem will take some time and that perseverance is an important aspect of the problem-solving process and of doing mathematics" (NCTM 2000, p. 186). Yet there is often an expectation in mathematics class that solutions should come quickly, and teachers work against a norm that has equated fast responses with proficiency in mathematics. The Common Core's Standards for Mathematical Practice (SMP) (CCSSI 2010) have brought renewed attention to the importance of perseverance in mathematical problem solving. In this article, we offer a set of classroom practices that foster students' perseverance in problem solving.

This study is based on a Task Design and Analysis activity from a year-long professional development program. The activity was designed to increase teacher growth in several areas, including knowledge of mathematics, understanding of students' cognitive activity, knowledge of good questions, and ability to develop and improve high quality tasks. The study collected data from 30 classroom teachers. Results include teachers' initial understanding of the task, teachers' interpretations and awareness of the strategies their students used to solve the task, and the pattern of revised tasks. The process and findings of the study will expand teacher educators' understanding of structured approaches to link an inservice PD with teachers' work context as well as develop a better analytic framework within which teachers and teacher educators analyze student work.

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This study investigated the views of Turkish prospective secondary mathematics teachers on the use of advanced calculators with CAS in algebra instruction. An open-ended questionnaire and group interviews revealed prospective teachers' views and beliefs about when and why they prefer three possible uses of CAS--black box, white box, or Symbolic Math Guide (SMG). The results showed that participants mainly preferred the white box methods and especially SMG to the black box method. They suggested that while the black box method could be used after students mastered the skills, the general white box method and SMG could be used to teach symbolic manipulation.

Handheld graphing technologies have fundamentally been utilized in the teaching and learning of many secondary school and college mathematics concepts. In order to observe the effects of these new kinds of technologies on students' learning of mathematics, their teachers need to decide whether it is reasonable to do, learn, and teach mathematics with a calculator, and moreover, if graphing calculators are worth using in their classrooms as an instructional tool. With this aim in mind, this study utilized two conceptual frameworks--the possible roles of advanced calculators in mathematics teaching and learning, as well as technological pedagogical content knowledge--to investigate the views of preservice secondary mathematics teachers on the use of calculators in mathematics instruction. Open-ended survey questionnaire and group interviews were the main data collection methods for this study. Results revealed preservice teachers' views on the use of handheld technology in mathematics instruction and the roles that they assigned to the calculators such as a visualization tool, a transformational tool, a computational tool or a discovery tool. Their beliefs about advantages and disadvantages of using graphing calculators and when and why they would prefer to use graphing calculators were discussed. (Contains 2 tables and 1 figure.)

Teacher educators and researchers have studied the improvement of teacher education programs so that pre-service teachers may be well prepared. Research has shown that not only teachers' knowledge but also their beliefs have major influence on their approach to teaching. This present research examined the mathematics and science pre-service teachers' beliefs and perceptions in Turkey. Specifically, the study focused on the student-teachers' perspectives on "effective education" at the end of their subject area dominant program before the student teaching and the changes in those beliefs after having teaching experience. Many pre-service teachers described the teacher-centered environment as effective, and those who supported a student-centered environment discussed difficulties they faced in creating it during student and full-time teaching.

Presents results from a study that uses questionnaires to ask mathematics teachers' (N=64) opinions about mathematics education computer software. Indicates that respondents overwhelmingly favored discovery and remediation as the purpose of using software in mathematics classrooms. Teachers believed that real world situations and multiple representations were the most important attributes of educational mathematics software. (Contains 11 references.) (Author)

Generating mathematical conjectures is an important mathematical habit of mind for preservice teachers to develop so that they can, in turn, help their students to develop this skill. In this paper we present a classroom episode in which preservice mathematics teachers experience conjecturing in the context of a rich, open-ended task. They also reflect, throughout the task, on how they might nurture the generation and exploration of mathematical conjectures with their own students. [ABSTRACT FROM AUTHOR]

A transition from "doer" to "teacher" for prospective teachers requires them to reorient from thinking about how they do mathematics to engaging with students and their work, understanding student representations, and planning instruction accordingly. To scaffold a transition, we developed a fivestep mathematics as teacher heuristic (MATH) model. The study investigated the impact of MATH on the development of teacher candidates' mathematical knowledge for teaching (MKT) and their pedagogical knowledge. Twenty-two preservice secondary mathematics teachers enrolled in a methods course participated in the study. Findings of the study showed that teacher candidates' MKT was engaged as a result of analysis of the student work. While some teacher candidates based subsequent instructional planning work on what they noticed in the student work, others had gaps between what they noticed and their further planning. Teacher candidates' work samples with noticing and noticing with gaps are shared in the results section. [ABSTRACT FROM AUTHOR]

In this technology-oriented age, teachers face daily decisions regarding the use of advanced digital technologies-graphing calculators, dynamic geometry software, blogs, wikis, podcasts and the like-to enhance student mathematical understanding in their classrooms. In this case study, the authors use the Technological, Pedagogical, and Content Knowledge (TPACK) model in conjunction with a five-stage developmental model, which can be used to describe growth in TPACK to describe the initial attempts of a teacher, Jane, to develop TPACK as she learns and attempts to integrate an advanced teaching technology into her classroom, namely the TI-Nspire graphing calculator. The study tracks her struggles to reconcile some traditional beliefs about how students learn with her desire to be responsive to what she perceives as affordances of advanced digital technologies. Main data collection methods were journal writing, observations, document analysis, and interviews. Using the five-stage developmental model, we saw that this experience helped Jane to move among different stages. This study showed that the TPACK model with the five-stage developmental model can be a beneficial tool for researchers to study teachers' professional growth and is also a valuable tool for teachers to reflect on their own growth. [ABSTRACT FROM AUTHOR]

Students' beliefs, attitudes, emotions, and values toward a subject not only influence their interests and achievement, but also their actions such as their choices of course work, major, and career. In this study, middle school, high school, and college students' affective responses and perspectives toward mathematics were observed by using metaphors. The outcomes of the use of metaphors to learn about students' affective domains not only revealed the nature of students' affective domains regarding mathematics, but also students' justifications for those affective perceptions. Therefore, another aim of this paper is to model how teachers and researchers might categorize and interpret students' responses to reflect on not only the nature of students' affective tendencies, but also the reasoning behind them. [ABSTRACT FROM AUTHOR]

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This study investigated the views of Turkish prospective secondary mathematics teachers on the use of advanced calculators with CAS in algebra instruction. An open-ended questionnaire and group interviews revealed prospective teachers’ views and beliefs about when and why they prefer three possible uses of CAS—black box, white box, or Symbolic Math Guide (SMG). The results showed that participants mainly preferred the white box methods and especially SMG to the black box method. They suggested that while the black box method could be used after students mastered the skills, the general white box method and SMG could be used to teach symbolic manipulation. [ABSTRACT FROM AUTHOR]

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What knowledge is needed to teach mathematics with digital technologies? The overarching construct, called technology, pedagogy, and content knowledge (TPACK), has been proposed as the interconnection and intersection of technology, pedagogy, and content knowledge. Mathematics Teacher TPACK Standards offer guidelines for thinking about this construct. A Mathematics Teacher Development Model describes the development of TPACK toward meeting these standards. The standards and model provide structured detail to further the work of various groups. The proposals may guide teachers, researchers, teacher educators, professional development consultants, and school administrators in the development and evaluation of professional development activities, mathematics education programs, and school mathematics programs. [ABSTRACT FROM AUTHOR]