Understanding, Questioning, and Representing Mathematics: What Makes a Difference in Middle School Classrooms?.

IntroductionTeacher quality has been found to be the single most important factor related to student achievement (Darling- Hammond, 2000; Rice, 2003). Consequently, it is important to understand the nexus of theorized Teaching Quality Measures (TQMs) and classroom enactments of learning goals (Kelly & Lesh, 2000). Numerous researchers (e.g., Goldhaber & Anthony, 2004; Good & Brophy, 2003; Nye, Konstantopoulos, & Hedges, 2004; Wright, Paul, Horn, & Sanders, 1997) have identified and articulated nearly 100 different potential TQMs. Additionally, an agreement as to which measures best indicate teacher quality and which TQMs researchers and practitioners should use remains unclear. Some of the popular measures researchers commonly use to gauge teacher quality are subject knowledge, pedagogical content knowledge, years of experience, level of certification, and degree level (cf. Wayne & Youngs, 2003). Each of these measures, however, is more about who the teacher is rather than what the teacher does in the classroom. The aim of this study was to examine what teachers are actually doing during instructional time and which aspects make a difference in student learning of mathematics.Applicability and interpretability of TQMs are essential to conceptual development, so TQMs devoid of meaningful content expectations in classroom enactments are meaningless to mathematics education research (Le et al., 2006). The three TQMs used in this study were adapted from the American Association for the Advancement of Science (AAAS) Curriculum Quality Measures of the middle grades mathematics textbook analysis (AAAS, 2006). The measures of probing for student understanding, encouraging curiosity and questioning, and using accurate representational forms were deemed essential for this study due to their importance in contributing to student mathematical conceptual understanding (Goldin & Shteingold, 2001; Hiebert & Carpenter, 1992; National Council of Teachers of Mathematics [NCTM], 2000; Weiss & Pasley, 2004; Wood, Cobb, & Yackel, 2000). Although there are a number of varied ways to measure teacher quality, these TQMs, a sampling from the domain, were chosen because of their ties to reform-based curricula used by the teachers in this study and common to the lessons observed. We examine these three TQMs and discuss in detail below how they were promising indicators of student success.Theoretical Framework for Teacher Quality MeasuresBreaking Ranks in the Middle (NASSP, 2006), Turning Points 2000 (Jackson & Davis, 2000), and the National Forum's Schools to Watch criteria (2007) are consistent in reporting that quality teaching makes a difference in creating successful middle schools (Andrews, Caskey, & Anfara, 2007). Further, classrooms should be places where students and teachers are engaged in active, purposeful learning, characterized by students at the center of the learning process; the curriculum is challenging, exploratory, integrative, and relevant, and classroom discussion is encouraged and supported by educators who incorporate multiple learning and teaching approaches that meet the needs of students (National Middle School Association, 2010).Probing for Student Understanding as a TQMAlthough teachers have consistently been observed doing most of the talking in classrooms, improved levels of classroom discourse can be achieved through effective questioning (Cazden, 2001). One of the most common teacher questioning approaches is the IRE sequence where the teacher initiates the sequence with a question, a student or students respond(s), and then the teacher evaluates the response (Mehan, 1979). Edwards and Mercer (1987) suggested an alternative to this sequence whereby the teacher's evaluation of the student's response is replaced with, or added to by, an expansion on the student's response. The differences between these types of interactional sequences are profound. …