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Two broad categories of instructional practices, (a) explicitly attending to concepts and (b) fostering students' opportunities to struggle, have been consistently linked to improving students' mathematical learning and achievement. In this article, we describe an effort to build these practices into a framework that is useful for a diverse set of professional development (PD) offerings. We describe three examples of how the framework is used to support teacher learning and classroom instructional practice: a state-mandated course, lesson studies, and a large-scale teacher-researcher alliance. Initial findings suggest that consistently emphasizing this framework provides both content and structural guidance during PD development and gives coherence and focus to teachers' PD experiences.

The authors present a new classroom-tested lesson that was designed to engage students in the joy of mathematical inquiry through a card game, Frustration, while building number sense, understanding of uncertainty, statistical reasoning, and discourse skills. The purpose in developing and sharing these lessons is to highlight how games of chance can offer rich opportunities to teach probability and statistics for conceptual understanding in the middle grades. The lessons described allow students to explore probability with nonroutine inquiry and discourse, and intentionally included ways to use the teaching practices of Explicit Attention to Concepts and Student Opportunities to Struggle to further encourage students' understanding. These are difficult topics to fit into already packed curricula, so the multiple versions of the lesson are included to be useful to a wide range of teachers.

Using a cluster randomized trial, we investigated the effects of mathematics professional development on student achievement in elementary schools. Pairs of schools were matched on prior mathematics; percentage of students receiving free or reduced-price lunch and percentage of limited English proficient students were randomly assigned to two groups. One received professional development on mathematics that focuses on diagnosing student thinking and the other received professional development on formative assessment. Student-level achievement data (pretest and posttest performance on a standardized test of mathematics) were nested within grades and schools and analyzed using mixed-effects multi-level linear modeling. The results indicate students in schools that were provided the mathematics professional development had greater mathematics achievement than students in schools in a control condition. The effect held across grade levels.

Teachers need ways to efficiently assess students' cognitive understanding. One promising approach involves easily adapted and administered item types that yield quantitative scores that can be interpreted in terms of whether or not students likely possess key understandings. This study illustrates an approach to analyzing response process validity evidence from item types for assessing two important aspects of proportional reasoning. Data include results from an interview protocol used with 33 middle school students to compare their responses to prototypical item types to their conceptions of composed unit and multiplicative comparison. The findings provide validity evidence in support of the score interpretations for the item types but also detail important item specifications and caveats. Discussion includes recommendations for extending the research for examining response process validity evidence in support of claims related to cognitive interpretations of scores for other key mathematical conceptions.

This paper outlines a University-School District partnership with the intent to increase the number of middle grades mathematics and science teachers. This externally funded initiative includes onsite, authentically situated professional development for pre- and in-service teachers at three different urban, low-socioeconomic schools with a majority Hispanic population of students. Program objectives include increasing mathematics and science content knowledge, increasing self-efficacy in teaching math and science, building and incorporating a success-driven school culture and infrastructure to increase student performance in a well-articulated, scalable and transformable model. Program components include site based common planning times, STEM Thursdays where science and mathematics lessons are practiced and refined, authentic summer research opportunities for pre- and in-service teachers to work with scientists and university faculty, teacher certification workshops and a mentoring model that includes program graduates and pre-service teachers. First year results show that the program had a positive impact on the teachers' self-efficacy and outcome expectancy as their scores significantly increased after participation in the project. Key elements in the model included (1) a strong partnership between a school district and institution of higher education, (2) a unique collaboration between mathematics and science educators and scientists, pre-and in-service teachers, (3) a professional development and mentoring program designed around the school district's adopted course of study and the NSES, (4) the integration of community resources, (5) a partnership with preservice and inservice teachers and district administrators with science and mathematics higher education faculty, (6) the development of teacher leaders, and (7) a comprehensive evaluation program.

Career-switchers who are returning to university for training as future elementary school teachers join an important and increasing group of adult learners of mathematics. These graduate preservice elementary teachers often place a high value on learning mathematics because of its prominent role in their prospective careers, but their learning often requires overcoming histories of personal struggles with mathematics. This inquiry used a mixed methods designed and focused on ten participants' perceptions of their experiences as learners of mathematics, anxiety regarding mathematics, and graduate mathematics coursework. All of the participants left professional careers to join a full-time graduate teacher preparation program in New York City. Bridging research from adult education and teacher education, we analysed interview transcripts, mathematics anxiety measures, and survey data using mixed methods both qualitative and quantitative research approaches. The findings show that, starting in adolescence, and continuing into adulthood, participants described negative experiences with exam performance, tracking, and instruction during formal school mathematics. Participants perceived these struggles as contributing to patterns of avoiding mathematics coursework and experiencing moderate-to-high levels of test mathematics anxiety. In spite of, or perhaps partly because of, their personal struggles with mathematics, the participants described urgently wanting additional high-quality mathematics preparation focused on upper-elementary content, manipulative-based pedagogy, and current elementary mathematics curricula.

The purpose of this study was to investigate implementation of an early intervention mastery activity during the first two weeks of college algebra and precalculus courses at a large U.S. public university. Statistical modeling of (N = 935) students' performance in the courses, including a logistic regression model of pass/fail course achievement with students' high school rank, ACT Mathematics scores, and performance on the intervention as explanatory variables, suggested significant independent differences in course performance across performance levels on the early mastery activity. An evaluation of diagnostic validity for the model yielded a 19% false negative rate (predicted to fail the course, but passed) and a 7% false positive rate (students predicted to pass the course, but failed), suggesting the early mastery activity, when combined with admissions indicators of mathematics readiness, may be useful in better identifying students at risk of failing their first university mathematics course. This strategy, which also yields information for focused intervention efforts, is currently being explored through a campus-wide advising tool at the research site.

Research-based guidelines for learning variation exist (e.g., Franklin et al., 2007; Garfield, delMas, & Chance, 2007), but little is known about how teachers plan to teach standard deviation, or how these plans align with recent recommendations. In this article, we survey lesson plans designed by inservice and preservice secondary mathematical teachers. We report on the accuracy, technology usage, and visual representations in the lesson plans. We consider how many elements are used, the level of conceptual development, and the mathematical nature. Findings support differences between preservice and master's level students in education, as well as a tendency by in-service teachers to teach in alignment with prior learning experiences, despite professional development. Implications for teacher education and curricular development are offered.

In this paper, we present findings from a preliminary analysis of transcript data in the High School Longitudinal Study (HSLS:09), a large-scale longitudinal investigation of academic achievement among U.S. high school students. Using proportional flow diagrams of course-taking patterns, we illustrate differences in calculus completion associated with non-malleable student characteristics such as ethnicity, gender, and socioeconomic status (SES), as well as malleable student characteristics, such as knowledge of mathematics in ninth grade, the level of mathematics course they take in ninth grade, and self-efficacy. Confirming and extending findings from prior literature, we conclude that "tracks" through high school mathematics curriculum, together with students' ethnicity, SES, and self-efficacy, converge as important factors associated with which of the approximately 19% of high school students complete calculus in high school.

This manuscript presents findings from a study about the knowledge for and planned teaching of standard deviation. We investigate how understanding variance as an unbiased (inferential) estimator--not just a descriptive statistic for the variation (spread) in data--is related to teachers' instruction regarding standard deviation, particularly around the issue of division by n-1. In this regard, the study contributes to our understanding about how knowledge of mathematics beyond the current instructional level, what we refer to as nonlocal mathematics, becomes important for teaching. The findings indicate that acquired knowledge of nonlocal mathematics can play a role in altering teachers' planned instructional approaches in terms of student activity and cognitive demand in their instruction.

Students are faced with many transitions in their middle school mathematics classes. To build knowledge, skills, and confidence in the key areas of algebra and geometry, students often need to practice using numbers and polygons in a variety of contexts. Teachers also want students to explore ideas from probability and statistics. Teachers know that one way to help students navigate these transitions is to use hands-on activities and manipulatives, but it can be a struggle to find probability and statistics activities that go beyond employing spinners, M&M's®, dice, coins, and bags of marbles. This led the authors to consider geoboards as an alternative manipulative for helping students explore probability. Geoboards, rectangular boards with equally spaced pegs, were invented in the 1950s by Egyptian mathematician Caleb Gattegno. Teachers using geoboards have shared many exciting investigations in geometry and discrete mathematics. Students can begin using geoboards as early as preschool, making shapes by stretching rubber bands or wrapping string around the pegs. Later, teachers use geoboards to help students learn about polygons, transformations (e.g., translations, rotations, reflections), angles, area, perimeter, and countless more topics. Geoboards are simple and inexpensive; free virtual versions are also available. In this article, the authors share an engaging geoboard activity that integrates key middle school concepts found in geometry, algebra, and probability.

Pattern blocks are inexpensive wooden, foam, or plastic manipulatives developed in the 1960s to help students build an understanding of shapes, proportions, equivalence, and fractions (EDC 1968). The colorful collection of basic shapes in classic pattern block kits affords opportunities for amazing puzzle-like problem-solving tasks and for exploring underlying mathematical relationships. These visuals can help build students' conceptual and procedural understandings of rational numbers (Reimer and Moyer-Packenham 2005). To illustrate the accessibility and challenges of problem solving with pattern blocks at the middle school level, the authors of this article present increasingly challenging problems. The problems described progress from constructing shapes with given proportions to creating arrangements with given percentage, fractional, and decimal quantities to building seven completely simplified representations of any given rational number.

The purpose of this study was to identify the degree to which college algebra students' value mathematical skills in their prospective careers. A survey was administered to N = 144 students in 6 college algebra classes at a mid-sized doctoral granting university. Students in half the classes completed a data analysis project, and half of the students planned to major in a business-related degree. Logistic regression suggested that students held mostly positive attitudes about the value of mathematics in their career, with business students expressing more positive attitudes than those reported by non-business students. Unexpectedly, those who completed the data analysis project expressed less positive attitudes on 6 of the 20 survey items.

In this article, the authors share two lessons that incorporate children's literature with the Pythagorean theorem and area to engage preservice teachers (PSTs) mathematically. Sample responses, example texts, and future work are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present findings from a preliminary analysis of transcript data in the High School Longitudinal Study (HSLS:09), a large-scale longitudinal investigation of academic achievement among U.S. high school students. Using proportional flow diagrams of course-taking patterns, we illustrate differences in calculus completion associated with non-malleable student characteristics such as ethnicity, gender, and socioeconomic status (SES), as well as malleable student characteristics, such as knowledge of mathematics in ninth grade, the level of mathematics course they take in ninth grade, and self-efficacy. Confirming and extending findings from prior literature, we conclude that “tracks” through high school mathematics curriculum, together with students' ethnicity, SES, and self-efficacy, converge as important factors associated with which of the approximately 19% of high school students complete calculus in high school. [ABSTRACT FROM AUTHOR]

This paper outlines a University-School District partnership with the intent to increase the number of middle grades mathematics and science teachers. This externally funded initiative includes onsite, authentically situated professional development for pre- and in-service teachers at three different urban, low-socioeconomic schools with a majority Hispanic population of students. Program objectives include increasing mathematics and science content knowledge, increasing self-efficacy in teaching math and science, building and incorporating a success-driven school culture and infrastructure to increase student performance in a well-articulated, scalable and transformable model. Program components include site based common planning times, STEM Thursdays where science and mathematics lessons are practiced and refined, authentic summer research opportunities for pre- and in-service teachers to work with scientists and university faculty, teacher certification workshops and a mentoring model that includes program graduates and pre-service teachers. First year results show that the program had a positive impact on the teachers' self-efficacy and outcome expectancy as their scores significantly increased after participation in the project. Key elements in the model included (1) a strong partnership between a school district and institution of higher education, (2) a unique collaboration between mathematics and science educators and scientists, pre-and in-service teachers, (3) a professional development and mentoring program designed around the school district's adopted course of study and the NSES, (4) the integration of community resources, (5) a partnership with preservice and inservice teachers and district administrators with science and mathematics higher education faculty, (6) the development of teacher leaders, and (7) a comprehensive evaluation program. [ABSTRACT FROM AUTHOR]