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Michele B. Carney, Jonathan L. Brendefur, Gwyneth R. Hughes and Keith Thiede

As mathematics teacher educators, it is imperative that we have high-quality tools that conceptualize and operationalize mathematics instruction for large-scale examination. We first describe existing instructional practice survey scales, including their conceptualization of practice and related validity evidence. We then present the framework and initial validity evidence for our mathematics instructional practice survey. Survey participants were inservice teachers in a statewide mandated mathematics professional development course. Statistical analyses indicate the items measure two constructs: social-constructivist and transmission-based instructional practice. Of particular interest is the result that these two constructs were negligibly correlated. This is in contrast to the generally accepted notion that social-constructivist and transmission-based instructional practices are the two polar ends of a single construct for describing instructional practice.

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Amanda Milewski and Sharon Strickland

In this article we examine an analytical framework generated by secondary mathematics teachers for tracking changes to their own instructional practices across time. We describe the journey of this group of teachers through professional development focused on improving instructional practice. In the midst of that experience, teachers struggled to find an analytical tool to examine one another's practices of responding to students' mathematical ideas and ultimately overcame this problem by considering the practitioner literature and their own experiences. We also describe how we adapted the framework to investigate its use for detecting shifts in teachers' practices, sharing findings obtained from its use. Lastly, we argue for this type of collaborative work with teachers as a means to develop common language for instructional practice.

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Anne Garrison Wilhelm and Sungyeun Kim

One crucial question for researchers who study teachers' classroom practice is how to maximize information about what is happening in classrooms while minimizing costs. This report extends prior studies of the reliability of the Instructional Quality Assessment (IQA), a widely used classroom observation toolkit, and offers insight into the often asked question: “What is the number of observations required to reliably measure a teacher's instructional practice using the IQA?” We found that in some situations, as few as three observations are needed to reliably measure a teacher's instructional practice using the IQA. However, that result depends on a variety of other 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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Jane O. Swafford, Graham A. Jones and Carol A. Thornton

This study examined the effects on instruction of an intervention program designed to enhance teachers' knowledge of geometry and their knowledge of research on student cognition in geometry. Forty-nine middle-grade (4-8) teachers participated in a 4-week program consisting of a content course in geometry and a research seminar on van Hiele theory. The pretest and posttest results showed significant gains in content knowledge and in van Hiele level. The analysis of a lesson-plan task revealed a significant shift in goals and expectations to the next higher van Hiele level. Follow-up observations of 8 teachers found marked changes in what was taught, how it was taught, and the characteristics teachers displayed. Teachers attributed these changes to increased geometrical content knowledge and research-based knowledge of student cognition.

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Jinfa Cai, Anne Morris, Charles Hohensee, Stephen Hwang, Victoria Robison and James Hiebert

In our March editorial (Cai et al., 2018), we considered the problem of isolation in the work of teachers and researchers. In particular, we proposed ways to take advantage of emerging technological resources, such as online archives of student data linked to instructional activities and indexed by learning goals, to produce a professional knowledge base (Cai et al., 2017b, 2018). This proposal would refashion our conceptions of the nature and collection of data so that teachers, researchers, and teacher-researcher partnerships could benefit from the accumulated learning of ordinarily isolated groups. Although we have discussed the general parameters for such a system in previous editorials, in this editorial, we present a potential mechanism for accumulating learning into a professional knowledge base, a mechanism that involves collaboration between multiple teacher-researcher partnerships. To illustrate our ideas, we return once again to the collaboration between fourth-grade teacher Mr. Lovemath and mathematics education researcher Ms. Research, who are mentioned in our previous editorials(Cai et al., 2017a, 2017b).

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Patricia F. Campbell

Over the last fifteen years, much research has investigated children's learning of mathematics. This research indicates that when permitted, children frequently devise approaches to solve problems that are distinct from those typically used by adults.

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Bilge Yurekli, Mary Kay Stein, Richard Correnti and Zahid Kisa

). Recent reviews of research ( Cross Francis, Rapacki, & Eker, 2015 ; Fives & Buehl, 2012 ) have shown that teachers’ enactment of instructional practices aligned with reform initiatives is substantially shaped by their beliefs regarding how important

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Dawn M. Woods and Anne Garrison Wilhelm

implementation of complex tasks ( Munter, 2015 ; Wilhelm, 2014 ), improvement in teachers’ instructional practices over time ( Munter & Correnti, 2017 ), and teachers’ decisions about their careers ( Hammerness, 2008 ). Teachers’ instructional visions are ever

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not capable of engaging in rigorous and challenging mathematics Consistently implementing research-informed, equity-based instructional practices Providing students with targeted additional instructional time and other instructional supports on the