
How Should FifthGrade Mathematics Teachers Start the School Year? Relations Between Teacher–Student Interactions and Mathematics Instruction Over One Yearby Holland W. Banse, Timothy W. Curby, Natalia A. Palacios & Sara E. RimmKaufman  2018 Background: Teaching is comprised of interconnected practices. Some practices are domain neutral (DN), or independent of a content area. Examples of DN practices include emotional and instructional support and classroom organization. Others are domain specific (DS), or content dependent. Within a mathematics context, examples of DS practices include mathematical discourse, tasks, and coherent lessons. Purpose: Using extant fifthgrade teacher observation data, we investigate the following questions: (1) Do quality DN practices at the start of a fifthgrade school year relate to higher DS practices at the end of the year? (2) Do early, quality DS practices relate to later, higher use of DN practices? Specifically, we investigate relations between emotional support and mathematical discourse, instructional support and mathematical tasks, and classroom organization and mathematical coherence. Research Design: We use an autoregressive, crosslagged structural equation model with three time points from a single academic year. Findings/Results: Results indicated that early levels of high emotional support and classroom organization were associated with later high levels of mathematical discourse and coherence, respectively. Early implementation of demanding tasks was associated with later, higher instructional support. Recommendations: We discuss implications for professional development and practice. Specifically, we suggest that teachers and instructional coaches consider how DN and DS practices relate to each other in order to boost teachers’ effectiveness. Instructional quality is determined by a complexity of interwoven practices. Research has identified both domainneutral (DN) and domainspecific (DS) practices that comprise quality instruction (i.e., Danielson, 2007; Pianta, La Paro, & Hamre, 2008a; Sawada et al., 2002; Seidel & Shavelson, 2007; Walkowiak, Berry, Meyer, RimmKaufman, & Ottmar, 2014). DN practices are broad practices that describe interactions between students and teachers and do not depend on a content area. For example, DN practices refer to how effectively teachers manage their classrooms, the quality of thinking that teachers encourage, and the level of sensitivity that teachers display toward students (Pianta et al., 2008a). DS practices are contentdependent practices that promote mastery of a content area, such as mathematics. The mathematics instructional practices advocated by the Common Core State Standards Initiative (CCSSI, 2010) can be characterized as DS practices. They include multiple representations of a mathematical idea, discourse about mathematical ideas, and rigorous mathematical tasks. Is it useful to draw a distinction between DN and DS practices? Some might argue that all practices depend on the content being taught. However, a teacher scaffolding a student’s understanding may use openended questioning as a strategy during both history and science lessons. We argue that, although the content of DN practices will vary with subject area, the processes by which a teacher engages a DN practice (e.g., persistent questioning, showing regard for students’ perspectives, facilitating efficient transitions) can occur across subject areas (Pianta et al., 2008a). We discuss this distinction further in the sections below. The following example illustrates how DN and DS practices can cooccur. Imagine an elementary mathematics educator teaching students to identify the properties of quadrilaterals. The teacher employs a mathematicsspecific practice by presenting students with a task of classifying shapes as quadrilaterals. The teacher then circulates and notices a student is struggling with how to classify a trapezoid. The teacher then scaffolds the students’ understanding by asking followup questions (e.g., “Does this shape have any right angles?”), a practice that can be used across all content areas. The teacher also employs contentneutral classroom management practices by redirecting offtask students and praising students who are ontask. Next, the teacher wants to understand how successful the scaffolding has been, and so facilitates a mathematicsspecific discussion, asking students to reason about why some shapes are quadrilaterals. Since a public display of thinking can be intimidating for students, the teacher cultivates a warm and respectful atmosphere that encourages participation. It is clear that DN and DS practices are used in tandem; as is apparent in this example, they frequently overlap each other within a single lesson. Indeed, it is difficult to appreciate their contribution to each other unless these relations are examined over the course of a school year. DN and DS practices have been examined separately (e.g., Pianta et al., 2008a; Sawada et al., 2002), although few if any studies have systematically examined the relations between the practices over an academic year. An important question remains unanswered: Do DN practices and DS mathematics practices set the stage for each other over the course of a school year, and if so, in which direction? For instance, in the example above, will the teacher’s early cultivation of a warm and supportive classroom environment at the start of the year help to facilitate mathematical discussion as the year progresses, or is the converse true, or both? The purpose of this study is to examine, over a year, whether bidirectional associations exist between DN and DS practices. It is important to note that the relations between these two types of practices are not necessarily causal. Rather, early implementation of a DN practice may be associated with better use of DS practices over the course of a school year and vice versa. This study bears implications for teachers’ professional development and practice. Teachers, principals, and instructional coaches can use teacher observations to improve practice by employing both DN and DS observational measures and examining relations across DN and DS measures. The development of one type of practice may, in some way, help to move the development of another practice. Examining how relations across DN and DS practices play out over time can help teachers learn how to leverage both practices in order to improve their overall mathematics instruction. We provide concrete examples of how teachers can leverage these relationships at the conclusion of this paper. ELEMENTS OF INSTRUCTIONAL QUALITY There exists an ongoing attempt to understand the properties of quality instruction. With regard to DS practices, Lampert (1990) discussed practices for encouraging students to become active mathematicians, such as asking them to pose hypotheses about mathematical problemsolving. In another example, Pianta and colleagues (2008b) found that the DN emotional climate of the classroom predicted fifth graders’ mathematics and literacy outcomes. Given the empirical relevance of both practice types, the Measures of Effective Teaching (MET) Project included DN and DS measures of teacher quality (e.g., Danielson, 2007; Grossman et al., 2010; Pianta et al., 2008a). By including multiple measures, the MET Project acknowledged that a “range of indicators” is necessary in order to provide teachers with “actionable data on specific strengths and weaknesses” (Kane & Cantrell, 2010, p. 5). DN ELEMENTS OF INSTRUCTIONAL QUALITY Three DN elements of instructional quality are often identified in the literature: instructional and emotional support and classroom organization (Pianta, Belsky, Vandergrift, Houts, & Morrison, 2008b; Pianta & Hamre, 2009). These practices are described as “teacherchild interactions likely to contribute positively to students’ development [and] that are presumed to be important to students’ academic and/or social outcomes” (Pianta & Hamre, 2009, p. 112). Some of these practices may appear to be content specific, especially instructional support. However, the common elements of instructional support can appear across content areas. Perhaps a teacher tends to use advanced language modeling more during literacy instruction, and openended questions more during a science lesson. Although the frequency and quality of these behaviors vary depending on whether the teacher is teaching a science or literacy lesson, the behaviors can still occur in either context. Instructional support describes how teachers develop concepts, offer quality feedback to students, and provide a languagerich environment. At its best, instructional support provides students with exposure to and interaction with higherorder ideas that challenge their current thinking (Taylor, Pearson, Peterson, & Rodriguez, 2003). Teacher scaffolding, which extends students’ ideas, targets misconceptions, and promotes deeper understanding, should then follow (Vygotsky, 1978). Throughout the instructional process, teachers should model appropriate academic language that concurrently supports students’ language development alongside their academic development (Catts, Fey, Tomblin, & Zhang, 2002). Beyond the facets of classroom instruction, attachment theory and selfdetermination theory provide insights into the roots of emotional support. Students must be able to perceive their teacher as a secure emotional base, from which they can safely explore new material (Hamre & Pianta, 2001; Pianta, Nimetz, & Bennett, 1997). Moreover, students should experience a sense of relatedness, autonomy, and capability within their classrooms, so that they are motivated to master new material (Ryan & Deci, 2000). Classroom organization refers to the quality of behavior management and instructional learning formats, as well as amount of instructional productivity within the classroom. Teachers must establish and maintain physical and behavioral routines at the start of the year, in order to promote desirable school behaviors and outcomes from students (Emmer & Stough, 2001; RimmKaufman et al., 2014). Facilitating quality learning formats by creating opportunities for student involvement, incorporating a variety of modalities, and stating clear learning objectives may serve to capture students’ attention (Evertson, Anderson, Anderson, & Brophy, 1980). For example, a kindergarten teacher might add another modality to a vocabulary lesson by encouraging students to sign the new words, to keep students thoroughly engaged. DN practices have been shown to progress throughout a school year. For instance, teachers who recognize the importance of high emotional support in order to engage students’ attention and foster positive relationships with students may implement high levels of emotional support throughout the year (Klem & Connell, 2004). Similarly, teachers who want to cultivate productive classrooms may proactively institute physical, instructional, and behavioral routines that promote productivity over the year (Cameron, Connor, & Morrison, 2005). Curby, RimmKaufman, & Abry (2013), using the same data source analyzed in the present paper, examined how DN practices develop over time, and determined that emotional support and instructional support formed a bidirectional association. Higher levels of emotional support early in the year were related to later, higher instructional support, and higher levels of instructional support earlier in the year were related to later, higher emotional support after accounting for their concurrent associations. This study supports the notion that earlier DN elements may influence the presence of later DN elements, and thus indicates the complexity of how teaching may play out over the course of the year. The present study builds on this work by not just examining DN elements over a school year, but also incorporating mathematicsspecific elements. DS Elements of Mathematics Instructional Quality DN elements of instruction can be found broadly across content areas—how these practices are used will vary depending on the content area. A challenging assignment—or task—in a fifthgrade social studies class could involve researching and writing a report on a civil rights hero. However, a challenging fifthgrade mathematical task could involve figuring out how to share 15 brownies evenly among 12 people. Both tasks require problem solving and allow for multiple approaches; simultaneously, both are specific to their content. Concrete examples of these DS practices, along with explanations of why these are mathematicsspecific practices, can be found in Table 1.
Both the National Council of Teachers of Mathematics (NCTM) and the Common Core advocate for multiple elements of quality mathematics instruction, including discourse, tasks, coherence, and representations. Discourse refers to the communication of mathematical ideas between teachers and peers, as well as among peers. Discourse is encouraged as an instructional context in which students offer their own explanations and justifications, so that others may critique or affirm their ideas (CCSSI, 2010; NCTM, 2000). Ideally, during mathematical discourse knowledge is expanded and shared as a class (Wenger, 1998). The presence of a discourse community, in which students understand the conventions and language associated with explaining, defending, and evaluating mathematical ideas, allows discourse to be a primary method of mathematical instruction (Yackel & Cobb, 1996). Teacher guidance can direct the formation of a mathematics discourse community, so that conceptual understanding of mathematical topics is developed through discourse (HufferdAckles, Fuson, & Sherin, 2004). A second mathematicsspecific practice is the inclusion of cognitively demanding tasks. Cognitively demanding tasks are exercises in which students actively engage their mathematical understanding in order to solve mathematical problems (Stein, Grover, & Henningsen, 1996). A quality task provides opportunities for students to consider multiple strategies in authentic contexts—for example, how to maximize the perimeter of their fencedin playground given a set area (CCSSI, 2010; NCTM, 2000). Teachers should provide scaffolding that sustains the tasks’ high level of cognitive demand (Stein, Engle, Smith, & Hughes, 2008), and tasks should promote conceptual problemsolving, rather than rote utilization of mathematical procedures (Hiebert & Stigler, 2000). Tasks have great impact in the presence of a coherent lesson. Coherence has two components: first, lessons are well structured, so that mathematical concepts build logically upon one another (Hiebert et al., 2005); and second, lessons are accurate. Wellstructured lessons have clear mathematical objectives and progress smoothly, with apparent beginning, middle, and ending segments (Cai, Ding, & Wang, 2014). Wellstructured lessons also have minimal external disruptions to learning, such as offtopic conversations, while appropriately addressing student interruptions related to mathematics learning (Cai et al., 2013; Hiebert et al., 2005). Accurate lessons mean that students are exposed to correct information, and misconceptions are addressed. Accurate lessons may be more possible when teachers have strong conceptual mathematical knowledge for teaching, and thus are able to redirect students’ misconceptions (Ball & Bass, 2000; Hill et al., 2008). Representations refer to student and teacher use of manipulatives, images, and mathematical symbols. Students can use representations as tools when problem solving, so that they fully express their conceptual understanding (e.g., Hammer, Sherin, & Kolpakowski, 1991). It is important to note that although research and theory indicate that some DN and DS practices may pair together (which will be discussed below), neither theory nor empirical work provides insight into which DN practices pair well with representations. Like the DN practices of instructional support, emotional support, and classroom organization, use of DS practices at the start of a school year may contribute to their continued use. For instance, teachers who encourage students to engage in mathematical discourse from the start of the year may promote the formation of a sophisticated discourse community as the year progresses (HufferdAckles et al., 2004). THEORETICAL MODEL Theory and practice in mathematics education call attention to how some DN and DS practices are coupled more than others. To examine the potential for bidirectional relations between DN and DS practices, we chose three primary DN constructs (emotional support, instructional support, and classroom organization) and reflected upon theory and practice in mathematics to identify likely pairings. In doing so, we identified three potential DN and DS pairings: emotional support and discourse, instructional support and tasks, and classroom organization and coherence. Owing to a lack of theoretical and research support for mathematical representations relating to any of the DN practices, and vice versa, we have not included representations in our theoretical model. A broad outline of these possible relationships is displayed in Figure 1. Two types of relationships are displayed in this figure. First, both DN and DS practices continue over time (the DNàDN and DSàDS relationships). Second, both DN and DS practices hypothetically relate to each other’s improvement over the course of a school year (the bidirectional relationships; DNàDS and DSàDN at time points 1, 2, and 3). As interplay between these practices may exist concurrently, we also include correlations between DN and DS practices at each time point. While other potential relations among these practices (e.g., instructional support and discourse) are possible, the selected pairs bear the greatest theoretical and empirical support, which is detailed in the following section. Figure 1. Potential Longitudinal and Bidirectional Relationships Between DS and DN Practices. Concurrent relations between DN and DS practices are also theorized, although removed from the model to reduce complexity. EMOTIONAL SUPPORT AND DISCOURSE Emotional Support for Discourse Teachers who foster emotionally supportive environments may signal encouraging messages from the first day of school (Patrick et al., 2003). These messages include sensitivity to students’ personal development and genuine interest in students’ learning. Teachers in emotionally supportive classrooms consistently uphold these messages, both in words and actions (Patrick et al., 2003). Students subsequently may respond positively. Turner and colleagues (2003) found that, over time, students in emotionally supportive classrooms reported fewer avoidance strategies during mathematics instruction, including selfhandicapping, disruptive behaviors, and avoiding helpseeking. In particular, students in classrooms with both high instructional support and motivational discourse reported low negative affect and less selfhandicapping. The establishment of an emotionally supportive classroom early in the school year thus may provide a foundation for later mathematical discourse. Students who feel emotionally supported by their teachers may engage in fewer avoidance behaviors. As a result, teachers’ emotional support at the start of the school year may create a foundation for better mathematical discourse by the end of the school year. Discourse for Emotional Support There is some evidence that discourse is related to the development of emotional support during a mathematics lesson. For instance, teachers who facilitate high levels of discourse may frame discourse around issues of interest to students, including culturally relevant examples or social justice issues (Dominguez, 2011; Imm & Stylianou, 2012). Framing discourse around these issues indicates teacher sensitivity to students’ lives. Moreover, teachers who understand the intent of mathematical discourse may be less likely to punish students for errors, or view themselves as the sole mathematical authority (Imm & Stylianou, 2012). Instead, misconceptions are perceived as opportunities to refine students’ understanding, and students are encouraged to adopt identities as mathematicians. As a result, discourse becomes the context in which students gradually feel safe to share their ideas. High levels of discourse implementation at the start of the school year may thus positively direct the development of emotional support within the classroom. INSTRUCTIONAL SUPPORT AND TASKS Instructional Support for Tasks Cognitively demanding tasks that offer opportunities for students to problem solve and forge connections across concepts and contexts are more meaningful when the teacher offers instructional support (Stein et al., 2008). How the implementation of instructional supports and tasks relate to each other over time is less apparent. Pakarinen and colleagues (2011) found evidence that offering high instructional support, such as scaffolding and promoting higherorder thinking skills, provided kindergarten students with the competence to grapple with mathematical tasks, resulting in less task avoidance and higher achievement. Instructional support provides students with the mathematical prowess they need to handle cognitively demanding tasks, so that they are able to problem solve instead of relying on procedures. Teachers who initially provide higher levels of instructional support may be able to increase the cognitive rigor of mathematical tasks over the year. Tasks for Instructional Support Conversely, the presence of cognitively demanding tasks at the start of the year may be associated with later instructional support. When teachers implement cognitively demanding tasks early on, there is a greater need for them to provide instructional supports. Implementing instructional supports during tasks requires teachers to actively anticipate, monitor, and connect students’ ideas as they problem solve (Stein et al., 2008). Without teacher guidance alongside tasks, students may gradually disengage from the task, and the cognitive demand of the task may decline (Henningsen & Stein, 1997). As a result, the early implementation of cognitively demanding tasks may relate to later instructional support, in the form of ongoing scaffolding and feedback during problemsolving tasks. CLASSROOM ORGANIZATION AND COHERENCE Classroom Organization for Coherence In Hiebert and colleagues’ (2005) videotape study of classrooms from different countries participating in the Trends in International Math and Science Study (TIMSS), American classrooms were found to be likely to experience disruptions during mathematics lessons. These disruptions included classroom management issues, offtopic conversations, and disorganized transitions (Hiebert et al., 2005). In addition, teachers did not always present mathematical concepts accurately, or in a logical progression. As a result, lessons seemed disjointed. Thus, lesson coherence may positively relate to the early presence of strong classroom organization. Classrooms with higher levels of organization may allow for increased lesson coherence, as physical, instructional, and behavioral structures will be in place that limit disruptions and improve lesson flow. Moreover, the lack of disruptions may free teachers to focus on the accurate presentation of their content, which may also improve lesson coherence. Coherence for Classroom Organization The conceptual association from coherence to classroom organization is not as supported by theoretical or research evidence. However, we posit that teachers who implement wellstructured, accurate mathematical lessons early in the academic year may see fewer disruptions to classroom organization over time. In essence, we suggest that welldefined lessons are associated with greater student engagement and, ultimately, with better classroom organization. Students who are deeply engaged in mathematical learning may be less likely to disrupt lessons, such that classroom transitions and routines occur smoothly. RESEARCH QUESTIONS This study expands the current body of literature on DN and DS teaching practices by examining the classroom implementation of neutral and mathematicsspecific practices over a school year. We adopt a conceptual framework that considers the extent to which one practice early in the year is associated with another practice later in the year. Our research questions are as follows: (1) How does the early implementation of domainneutral practices, such as emotional support, instructional support, and classroom organization, relate to the later implementation of domainspecific practices, including mathematical discourse, mathematical tasks, and mathematical coherence? (2) How does the early implementation of domainspecific practices relate to the later implementation of domainneutral practices? We hypothesize bidirectional relations between emotional support and discourse, as well as instructional support and highquality tasks. As there is a lack of theoretical and empirical evidence demonstrating that early coherence is related to later classroom organization, we hypothesize a unidirectional trend, in which early classroom organization leads to later mathematical coherence. METHODS Participants The present study used extant data from a randomized control trial, the Responsive Classroom Efficacy Study (RimmKaufman et al., 2014). Twentyfour schools in a MidAtlantic school district were chosen to participate based on their interest in adopting the RC approach. The RC approach is a social and emotional learning intervention that offers teachers training on practices designed to support students’ academic and social development (Northeast Foundation for Children [NEFC], 2007). This approach places equal emphasis on students’ social and academic growth (for a further description of the RC approach and related research, see RimmKaufman et al., 2014). The focus of the present study is on fifthgrade mathematics teachers. The teacher sample was comprised of 59 fifthgrade teachers. Of those teachers, 45 (76%) were female, 51 (86%) were Caucasian, four (5%) were African American, two (3%) were Hispanic/Latino, and one (2%) was Asian. Over 63% held master’s degrees, and on average, teachers were 39 years old with 12 years of teaching experience. These teachers worked in ethnically and socioeconomically diverse schools (26% of students qualified for free/reduced price lunch; 43% White, 24% Hispanic, 17% Asian, 11% Black, 7% Other). Procedure and Design We videotaped teachers for one hour during three observation windows during the fall, winter, and spring (Window 1: late September to late November; Window 2: late November to midFebruary; and Window 3: late February to late April). For each observation window, teachers taught one full mathematics lesson. We used the Classroom Assessment Scoring System (CLASS) to measure DN practices. These observations of mathematics lessons would occur over a 60minute period, broken into 15minute sampling segments. Segments took place during the first 15 minutes of the lesson, as well as from minute 30 to minute 45. During the observational segments, observers took notes on classroom interactions, focusing primarily on the teacher. After each observational segment, the observer derived numerical ratings for all of the CLASS dimensions (described in further detail below) based on the observers’ notes, knowledge of dimension definitions, and dimension markers. After each observational segment, the CLASS observational cycle began again for the next segment. Domain scores for each segment were created by then averaging across the various dimensions, and domain scores per day were averaged across the two segments (Pianta et al., 2008a). We used the MathematicsScan (MSCAN) (Berry, RimmKaufman, Ottmar, Walkowiak, & Merritt, 2010) observations to measure mathematicsspecific practices. These observations occurred during a full mathematics lesson. The observations were broken into two 30minute segments. Coders observed the lesson for the first segment and took notes on the practices they observed. After the first segment, coders assigned “soft codes” that served as a record of the first half of the lesson. Coders then observed and took notes for the final half of the lesson. Once the lesson was finished, coders assigned final codes based on the coding guides (Berry et al., 2010). MEASURES The Class Assessment Scoring System (CLASS) Ten dimensions of classroom quality were rated on a 1 to 7 Likert scale, and then aggregated into three empirically validated domains at each time point (Hamre, Pianta, Mashburn, & Downer, 2007): Emotional Support, Classroom Organization, and Instructional Support. Emotional Support contains four dimensions measuring the quality of teacher–student relationships: Positive Climate, Negative Climate (reversed), Regard for Student Perspectives, and Teacher Sensitivity (α = .70–.79 for the present study). Positive Climate is indicated by the presence of positive relationships, affect, and communication, as well as respectful interactions among teachers and students. Negative Climate is marked by negative affect, disrespectful interactions, punitive control, and extreme negativity (e.g., physical punishment). Teacher Sensitivity is characterized by teachers’ awareness of and responsiveness to students’ concerns, as well as students’ willingness to seek the teacher for comfort. Regard for Student Perspectives refers to how often and well the teacher appropriately incorporates students’ interests, needs, and ideas into the instructional day. Classroom Organization is defined by three dimensions: Behavior Management, Productivity, and Instructional Learning Formats (α = .50–.69 for the present study). Behavior Management refers to how clearly teachers set expectations for student behavior, as well as how proactively they avoid or redirect misbehaviors. Productivity is defined by how efficiently classroom routines, transitions, and materials promote student learning, as well as how often students are engaged in learning in comparison to other activities. Instructional Learning Formats refers to the diversity of classroom activities, contexts, and materials, as well as the clarity of learning objectives, that are available to students. Instructional Support is similarly comprised of three dimensions: Quality of Feedback, Concept Development, and Language Modeling (α = .79–.88 for the present study). Quality of Feedback depends on how well teachers scaffold students, prompt for information, and encourage students to persevere in their understanding. Concept Development refers to how thoroughly concepts are explored, how well they are assimilated into previous knowledge, and how deeply they are connected to students’ lives and interests. Language Modeling describes how often students and teachers engage in conversations, how frequently the teacher utilizes advanced language, and how often the teacher prompts students to extend their speech, or utilizes selftalk or parallel talk to describe students’ ideas or actions. Interrater reliability was calculated after a twoday workshop. Raters coded 10 videos using the 10 dimensions. To be deemed reliable, raters’ codes had to be within one point of master codes 80% of the time. To ensure high interrater reliability, semimonthly meetings occurred in which raters viewed and coded a randomly selected video. The group would then arrive at agreedupon final codes for the video. Intraclass correlations (ICCs) were used as a measure of interrater reliability. Observers’ independent scores were used to calculate the ICCs, which compare the variance of measured items to the total variance, including rater variance. ICCs measuring consistency ranged from .73 to .85 during an 18month coding period, indicating in this instance that variance of the measured items was mostly due to sources of variances other than the raters (i.e., raters were primarily scoring in agreement with one another). Moreover, CLASS trainers would randomly select two videos scored by a rater, and doublecode the video during each of the observational periods. The percent agreement (within one scale point) between the rater and the trainer was maintained above the 80% threshold. Fifteen CLASS raters were used during this study. The MathematicsScan (MSCAN) The MSCAN is a valid measure of standardsbased mathematics practices, which contains four domains: Tasks, Representations, Discourse, and Coherence (Walkowiak et al., 2014). Each domain is comprised of at least two dimensions, which are individually rated on a sevenpoint scale divided into three sections: low (1–2), medium (3–5), and high (6–7). Anchors describing what low, medium, and high levels of each dimension would resemble were included in the coding guide. Low use of a practice indicated the practice occurred “rarely,” medium use indicated the practice occurred “sometimes,” and high use indicated that the practice occurred “often.” For instance, low use of student explanation meant that students rarely provided explanations, whereas high use of student explanation meant that students often provided explanations. The Tasks domain comprises three dimensions: Cognitive Demand, Connections and Applications, and ProblemSolving (α = .58–.72). Cognitive Demand is characterized by how openended the tasks are, and the depth of thinking prompted by the teacher. Connections and Applications refer to whether the teacher explicitly links the lesson to other mathematical concepts, student experiences, and disciplines, as well as to whether the students are asked to apply mathematics in authentic contexts. Problemsolving describes how thoroughly students engage with the task, and whether they can utilize multiple strategies to solve the task. Two dimensions define Representations: use of Multiple Representations, and students’ Tool Use (α = .63–.84). Multiple Representations is defined by whether multiple types of representations are used within the lesson, as well as how often teachers and students translate across the types of representations. Furthermore, students’ Tool Use depends on whether or not students use physical representations (i.e., manipulatives) during their problemsolving processes, while connecting their tool use to mathematical concepts. Two dimensions similarly comprise Discourse: Explanation and Justification, and Mathematical Discourse Community (α = .62–.81). Explanation and Justification is contingent upon whether students are provided with opportunities to defend their solutions, as well as the depth of mathematical explanations that teachers require from students (i.e., procedural vs. conceptual). Mathematical Discourse Community describes the extent to which the teacher elicits students’ mathematical ideas, how often students reference mathematical thinking during discourse, and how often teacher questioning prioritizes student thinking over correct answers. Lastly, Coherence is also comprised of two dimensions: Lesson Structure and Mathematical Accuracy (α = .59–.77). Lesson Structure refers to how logically lesson components are sequenced, whether mathematical connections are apparent among lesson components, and how well the overall lesson sequence appears to lead students to a deeper understanding of the concept. Mathematical Accuracy describes how correctly teachers present mathematical concepts, the clarity of concept presentation, and how effectively teachers respond to students’ misconceptions.
Raters were trained during a fourday workshop using videos, readings, and discussions related to the nine MSCAN dimensions. Coders reached 80% agreement (within one scale point) for six consecutive tapes before they began coding. Biweekly drift tests helped coders to maintain reliability throughout the study (ICCs measuring interrater reliability > .94). Twenty percent of tapes were doublecoded, and overall reliability was .83. Coders used videotapes of mathematical lessons for the present study, and seven MSCAN coders were used. DATA ANALYSIS Descriptive statistics as well as correlations were computed between study variables. Research questions were addressed by running a series of crosslagged autoregressive models within a structural equation modeling (SEM) framework using AMOS v 19.0 (Arbuckle, 2010). Given the small sample size, teachers were not nested in schools. Each variable is an aggregate of the various subdimensions of each DN or DS practice, respectively. Each set of associations between two variables involved running four different nested models. These four nested models were compared through a chisquared change test. Specifically, analyses identified significant chisquared changes relative to the changes in degrees of freedom. A general crosslagged autoregressive model is presented in Figure 1. The first step involved is computing the unconditional model, which then served as the basis for comparison with the other three models. In the unconditional model, each set of variables was used to predict itself (i.e., the autoregressive ‘a’ and ‘b’ paths in the model) and acrossdomain errors are allowed to correlate (not shown). Next, unidirectional lagged associations between variables were added to the model. For example, Emotional Support at times 1 and 2 was tested to see if it predicted Discourse at times 2 and 3 (‘L1’ in Figure 1). Then, the opposite unidirectional effect was tested. For example, Discourse at times 1 and 2 was tested to see if it predicted Emotional Support at times 2 and 3 (‘L2’ in Figure 1). Each unidirectional model was compared to the unconditional model to determine which provided a better fit. Finally, a bidirectional model was tested that included both unidirectional associations (includes ‘L1’ and ‘L2’). The bidirectional model was compared to the best fitting unidirectional model to see if it provided a better fit. Both autoregressive and crosslagged parameters were constrained to be equivalent over time. The same methodology was used to test each of three sets of associations: emotional support and discourse, classroom organization and coherence, and instructional support and tasks. Concurrent relations between DN and DS practices were included in the model; however, as these were not related to the research questions of interest, they are not presented. To account for potential bias associated with treatment condition, we included intervention status as a covariate for each practice at each time point. To determine a bestfitting model, we examined traditional SEM fit indices (Kline, 2005). We used both chisquare difference tests as well as the root mean square error of approximation (RMSEA) and comparative fit index (CFI). RMSEA values of .10 or lower and CFI values of .90 or higher were considered indicators of best fit. Similarly, significant differences resulting from the chisquare difference test served as an indication that the inclusion of additional paths resulted in a better fit. We compared fit indices and chisquare differences across all four models to determine which model provided the best fit. RESULTS Correlations and descriptive statistics are provided in Table 2. Several results require mention. Random assignment to RC condition related positively to Coherence at Time 3 (r = .30) and Tasks at Time 2 (r = .31). CLASS and MSCAN domains tended to show higher relations concurrently (ranging from .17 to .56) than across time (ranging from .09 to .46). Specifically, correlations between emotional support and discourse were r^{2} = .26 at times 1, 2, and 3; correlations between coherence and classroom organization were r^{2} = .24 at time 1 and r^{2} = .17 at times 2 and 3; and correlations between instructional support and tasks were r^{2} = .56 at time 1 and r^{2} = .29 at times 2 and 3. These are residual correlations between the errors of these variables at times 1, 2, and 3. The correlation at time 1 residualizes the effect of whether or not the teacher participated in the intervention. At times 2 and 3, the residualized correlations account for three elements: whether or not the teacher participated in the intervention, the prior time point, and the cross lag. The correlations at times 2 and 3 are constrained to be equivalent. Table 2. Correlations and Descriptive Statistics On average, CLASS domains evidenced highmoderate levels of stability for emotional support and classroom organization and lowmoderate levels of stability for instructional support. Among the MSCAN domains, discourse was the most stable over time. Other domains of MSCAN were rated in the low to lowmoderate range. FINAL MODELS See Table 3 for a listing of the models with the corresponding chisquared difference. Below, final models for each research question are presented and these final models are summarized in Figure 2. Table 3. Models with Corresponding Chisquare Change Note. c^{2} refers to the chisquare value, df refers to the degrees of freedom, and Dc^{2} and Ddf refer to the difference in chisquare value and degrees of freedom, respectively, between the unconditional and selected models. Figure 2. Final Crosslagged Trends Using Unstandardized Betas across DN and DS Practices. A. Emotional Support and Discourse (RMSEA = .07; CFI = .93); B. Classroom Organization and Coherence (RMSEA = .00; CFI = 1.00); and C. Instructional Support and Tasks (RMSEA = .04; CFI = .98). Concurrent relations are not shown in the model in order to reduce complexity. Intervention status was included as a covariate at each time point (1 = intervention, 0 = control). Emotional Support and Discourse The unidirectional model that included positive associations from earlier emotional support to later discourse (b = 0.43, p < .01) proved to be the bestfitting model (Figure 2a; RMSEA = .07, CFI = .93). In other words, during math, teachers who had higher levels of emotional support earlier in the year were found to have higher levels of discourse later in the year. From these models, we can estimate the stability of emotional support (after taking into account any crosslags). Emotional support appeared to be moderately stable throughout the year (b = .47, p < .001), whereas discourse was less stable throughout the year (b = .19, p = .04). Classroom Organization and Coherence The bestfitting model was identified as the unidirectional model with paths from earlier classroom organization to later coherence (b = .40, p < .01) (Figure 2c; RMSEA = 0, CFI = 1). In this model, the stability of classroom organization was also low moderate (b = 0.27, p < .01). Coherence was not stable (b = –0.01, p = .96), after taking into account the crosslags. Thus, higher levels of classroom organization at the prior time point were related to higher levels of mathematical coherence later on in the school year. Instructional Support and Tasks The bestfitting model that emerged was the unidirectional model with paths from earlier tasks to later instructional support (b = .27, p < .01) (Figure 2d; RMSEA = .04; CFI = .98). Instructional support was unstable after taking into account the crosslag (b = .04, p = .72), while tasks had moderate stability (b = .31, p < .001). In short, higherquality tasks at the prior time point were associated with higher levels of Instructional Support at later time points. DISCUSSION Three findings emerge from our analyses of lagged relationships between three DN practices (instructional and emotional support, and classroom management) and DS practices (mathematical tasks, mathematical discourse, and mathematical coherence). Two of the findings indicate that the early implementation of DN practices related to later use of DS practices, specifically, emotional support to mathematical discourse, and classroom organization to mathematical coherence. One finding indicates that a DS practice related to the later use of a DN tool: mathematical tasks to instructional support. There was no evidence of bidirectionality across DN and DS practices over the school year. Below we discuss what these findings mean specifically. We also provide concrete examples of how these relations played out in teachers’ classrooms in Table 4.
Note. The table elucidates how these relations play out over the academic year. The first column describes how teachers enacted a practice early in the year whereas the second column describes how a practice evolved at the end of the year. DOMAINNEUTRAL TO DOMAINSPECIFIC PRACTICES Emotional Support to Discourse The analyses indicate that early emotional support is associated with later mathematical discourse over the course of the year. This finding aligns with earlier mixedmethods work, which demonstrates the need for a warm instructional atmosphere in order for students to actively participate in mathematics instruction (Patrick et al., 2003; Turner et al., 2003). Participating in mathematical discourse is a risky endeavor for a student. If students do not feel secure to participate, then they may prioritize their emotional wellbeing and avoid discourse. When emotional support is available, and students are comfortable sharing their thoughts, they may be more willing to enter into discourse. The finding that early emotional support relates to later mathematical discourse suggests that teachers have to be purposeful in initially establishing emotional support. The relation between discourse and emotional support was not evident in the analysis. As indicated by the correlations among practices, salient aspects of this relation may be concurrent, rather than occurring over the school year. For instance, a teacher could frame mathematical discussion around domains or contexts of interest to students, making discourse more appealing and familiar to students (Imm & Stylianou, 2012). Thus, discourse becomes a context through which emotional support is evident to students. However, once discourse has concluded, the context for emotional support disappears, and the teacher must make different efforts to cement emotional support within the classroom. As a result, mathematical discourse may positively influence emotional support within each time point, but not across time points. Classroom Organization to Coherence Another DN to DS practice upheld in the analyses was the relation between early classroom organization and later mathematical coherence. As stated earlier, there are no theoretical or empirical reasons to expect that coherent mathematics lessons will lead to more organized classrooms. As is indicated in our findings, classroom organization may be related to greater coherence over time, but the pieces of mathematical coherence—lesson structure and mathematical accuracy—cannot support the development of broader classroom organization. We were correct in anticipating that early classroom organization would be associated with later mathematical coherence. In particular, it is likely that teachers who initiate and reinforce strong routines, transitions, and expectations for positive student behavior during mathematics lessons will decrease classroom disruptions and subsequently increase learning productivity (Brophy & Good, 1986; Cameron et al., 2005). As a result, mathematics lessons can maintain a comprehensible structure in which students can perceive relevant links between concepts, and attention is not diverted from mathematical ideas. This finding suggests that classroom routines that prevent offtask behaviors and facilitate productive learning environments need to be established early in the year to support the later development of mathematical coherence. DOMAINSPECIFIC TO DOMAINNEUTRAL PRACTICES Tasks to Later Instructional Support Our findings indicate that the early implementation of tasks is associated with later instructional support over the course of the year, but early instructional support was not associated with later tasks. Instructional support describes teachers’ efforts to scaffold students during cognitively demanding tasks. As teachers demand more rigor from their students by providing more challenging tasks, they are also, as the year progresses, more precisely targeting instruction toward students’ needs. Providing challenging tasks over the course of the year allows teachers to maintain or increase the level of cognitive demand within tasks, while helping students to make meaning of the tasks (Henningsen & Stein, 1997). Previous qualitative research has noted the importance of beginning with cognitively demanding tasks so that students always have opportunities to engage deeply with tasks, and then coupling tasks with extensive teacher support, so that students maintain a high level of engagement with the tasks (Stein & Smith, 1998). Thus, this finding quantitatively strengthens that claim. This relation between an early DS practice and a later DN practice was the only one of its kind. As part of instructional support is scaffolding, this relationship is logical, although anomalous compared to the direction of our other findings. This finding suggests that students need time to grapple with mathematical tasks before teachers can understand precisely the type of instructional support students require to be successful in solving the tasks. Further empirical investigations are required to understand whether this trend holds using different measures and across different content areas. IMPLICATIONS FOR PRACTICE These findings bear important implications for educators and education decisionmakers. Education decisionmakers should encourage the use of both DN and DS observations to support teachers’ professional development. Collection of both types of observational data may provide insight into opportunities for schoolwide professional development or individualized coaching. We provide three scenarios showing how the need for professional development or coaching could play out. For instance, observational data may reveal that the teacher offering the quadrilaterals lesson at the start of this paper needs to prioritize the development of a supportive classroom environment. Instructional coaches could help the teacher come up with strategies to do so, such as exercising sensitivity to students’ identities and forging positive relationships with students from the first day of school. Thus, over time students may feel more confident venturing their mathematical ideas in the presence of an emotionally supportive teacher. This finding is also an important message for teachers in light of the increased call for students to communicate during mathematics lessons (CCSSI, 2010; NCTM, 2000). A similar pattern exists with regard to teachers who wish to improve upon their mathematical instructional support. Recent findings reveal that teachers tend to exhibit low levels of instructional support across content areas, including mathematics (Kane & Staiger, 2012). However, teachers may be able to boost their instructional support by providing and scaffolding tasks that prompt students to problem solve in authentic contexts. There is a difference between offering a worksheet that tests students’ procedural understanding of finding the area of a square, versus handing students yardsticks and asking them to determine the area of the classroom and explain their answers. Although both exercises address the same skill, the second task contextualizes the problem, provides space for students to use tools, and asks students to articulate their reasoning. If DN and DS measures reveal that teachers in a given school exhibit low levels of instructional support and mathematical tasks, then schoolwide professional development may be required to help teachers learn how to craft and implement quality tasks. Lastly, classroom organization may contribute to the coherence of a mathematical lesson. An organized classroom is desirable not only because it promotes order within the classroom, but it also appears to relate to how smoothly content is delivered. When teachers implement clear expectations for classroom behavior, such as remaining ontopic during a lesson, and efficient transitions, such as handing out papers in a timely fashion, the structure of the lesson may be more conceptually seamless. Moreover, the teacher may be more able to pick up on students’ misconceptions, as well as maintain a clear line of thought throughout the lesson, so that the lesson is accurately delivered. A novice teacher struggling to manage student behaviors may also find that students are struggling to pick up on mathematical content. Individualized coaching may be required to help the teacher maintain order and improve upon the delivery of mathematical content. Finally, principals and instructional coaches need to be aware of how these practices relate to one another when offering professional development to teachers. Focusing solely on contentarea practices, such as standardsbased practices in mathematics instruction, is important but not sufficient. Principals and coaches should consider how elements such as classroom organization are promoting lesson coherence, or if tasks are of a sufficient caliber to foster high levels of instructional support. Moreover, these observations should occur regularly, beginning at the start of the school year and continuing at appropriate intervals, so that teachers have time to receive and incorporate targeted support as the school year progresses. By considering acrosstheyear relations between these practices, coaches and principals may be more able to boost teachers’ instructional effectiveness. LIMITATIONS AND FUTURE DIRECTIONS A primary limitation of this study is that we were only able to collect single instances of observational data at three time points across a single school year. More time points could provide more insight into the nuances of how these domains relate to one another, an advantage that would need to be balanced with the practical constraints (i.e., cost) of frequent data collection. Similarly, we did not collect multiple observations of teachers per time point, and consequently, correlations within practices over time are sometimes low (e.g., classroom organization at times 1 and 3). We note that by using observational measures we are strengthening the construct and criteria validity of our measure compared to studies using only teacherreport data (Kunter & Baumert, 2006). However, the single observation per time point design of this study may be a limitation. Although there is evidence to suggest that the time point at which data is collected is a smaller source of variance relative to other sources (e.g., rater, classroom) (Mashburn, Downer, Rivers, Brackett, & Martinez, 2013), future research should attempt both types of data collection (i.e., more time points and more observations per time point) to ensure strong consistency of measurement over time. We note that lack of stability within measures does not diminish the measure’s ability to predict the use of another measured practice at a future time point. Furthermore, our teacher sample is small (n = 59), and reliabilities of our measures vary across time points. It is possible that these issues contributed to some of the null findings. Another limitation of these analyses is their restriction to one subject domain and one grade level. Future research should investigate whether these trends are upheld across subject areas. For instance, within the new standardsbased science practices (NGSS Lead States, 2013), which promote inquiry and discourse about scientific ideas, are these trends also found? If these trends are apparent in other domains, we will gain a greater understanding of instructional patterns that lead to quality teaching. Future work should also examine whether these trends vary across grade levels, as children’s academic needs vary throughout development, and these patterns may be different in elementary versus secondary school. Similarly, researchers could also consider whether these trends are replicable across all student populations. It is also possible that investigating other relations between DN and DS practices, such as emotional support and tasks, would yield useful results. The coupled practices of interest were selected based on the level of theoretical and empirical support available. Future research should examine other possible relations as new evidence is presented. Quantitative research could also examine how these relations function concurrently. Additionally, future longitudinal, qualitative or mixedmethods research could confirm the existence of these relations between instructional practices over time. Future research could also investigate how the relations between these instructional practices relate to student outcomes. Finally, it is important to acknowledge that DN and DS practices share characteristics—for example, there are overlapping aspects of Tasks and Instructional Support—which complicates our efforts to view these practices as distinct. Our research begins to address these possible similarities by examining the association between related DN and DS practices within the course of a year. CONCLUSION Teaching is a multifaceted endeavor. As a result, efforts to measure and understand teaching require strategies for determining how teaching practices influence one another over a year. The beginning of the school year can be challenging for teachers, who are trying to establish quality instructional contexts. Teachers must decide how to coordinate various instructional practices from the start of the year in order to reach that goal. In the past decade, observational tools have been introduced that permit us to empirically test ideas that are widely held by practitioners. As the current paper demonstrates, these tools allow for a specific and richer understanding of how teachers can orchestrate these practices across the school year to create highquality instructional contexts from the beginning. Acknowledgement The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant No. R305B090002 and Grant No. R305A070063 to the University of Virginia. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education. This material is also based on work supported by the National Science Foundation under Grant No. (DRL0814872). 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