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.

### Anne Garrison Wilhelm and Sungyeun Kim

### Melissa Boston, Jonathan Bostic, Kristin Lesseig and Milan Sherman

In this article, we provide information to assist mathematics teacher educators in selecting classroom observation tools. We review three classroom observation tools: (1) the Reform-Oriented Teaching Observation Protocol (RTOP); (2) the Instructional Quality Assessment (IQA) in Mathematics; and (3) the Mathematical Quality of Instruction (MQI). We begin by describing each tool and providing examples of research studies or program evaluations using each tool. We then look across tools to identify each tool's specific focus, and we discuss how the features of each tool (and the protocol for its use) might serve as affordances or constraints in relation to the goals, purposes, and resources of a specific investigation. We close the article with suggestions for how each tool might be used by mathematics teacher educators to support teachers' learning and instructional change.

### Ayanna D. Perry

These instructional qualities have the potential to make a positive impact on student engagement during classroom discussions.

### Arthur J. Reynolds and Herbert J. Walberg

A structural model of mathematics achievement and attitude was tested with a national probability sample of 3,116 young adolescents from the Longitudinal Study of American Youth using structural modeling. A three-wave longitudinal design incorporated data from students, teachers, and parents to construct a prespecified theoretical model of mathematics achievement and mathematics attitude. A revised model provided a better fit than the original specification and than regression models used in 23 previous studies. Supported in cross-validation, the model revealed a complexity of direct and indirect effects not apparent in previous studies. Prior achievement and home environment influenced subsequent achievement most powerfully; motivation, exposure to extramural reading media, peer environment, and instructional exposure also had significant influences on achievement. Previous attitude had the most powerful influence on subsequent attitude, although the direct effects of instructional quality and the indirect effects of motivation and home environment were also notable. Appropriate teacher use of instructional time, thorough textbook coverage, and daily introduction of new material, although educationally alterable, are themselves influenced by previous student achievement. Similarly, instructional practices are significant alterable influences on mathematics attitudes, but such practices are themselves influenced by students' initial attitudes.

### Amber G. Candela, Melissa D. Boston and Juli K. Dixon

Instructional Quality . Bloomington, IN : Solution Tree . Cai , Jinfa , John C. Moyer , Ning Wang , Stephen Hwang , Bikai Nie , and Tammy Garber . 2013 . “ Mathematical Problem Posing as a Measure of Curricular Effect on Students' Learning

### Nickolaus A. Ortiz and Trina J. Davis

. “Using Culturally Relevant Pedagogy and Social Justice to Understand Mathematics Instructional Quality in an Urban Context.” In African American Students in Urban Schools: Critical Issues and Solutions for Achievement , edited by James Moore and

### S. Asli Özgün-Koca, Jennifer M. Lewis, and Thomas Edwards

been shown to predict their students’ success, teachers’ MKT has been shown to be linked to student achievement and instructional quality ( Hatisaru & Erbas, 2017 ; Hill et al., 2005 , 2008 , 2011 ; Hoover et al., 2016 ). Mathematical Knowledge for