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Variations in Coaching Knowledge and Practice That Explain Elementary and Middle School Mathematics Teacher Change

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Title: Variations in Coaching Knowledge and Practice That Explain Elementary and Middle School Mathematics Teacher Change
Language: English
Authors: Yopp, David A., Burroughs, Elizabeth A., Sutton, John T., Greenwood, Mark C.
Source: Journal of Mathematics Teacher Education. Feb 2019 22(1):5-36.
Availability: Springer. Available from: Springer Nature. 233 Spring Street, New York, NY 10013. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-348-4505; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 32
Publication Date: 2019
Sponsoring Agency: National Science Foundation (NSF)
Contract Number: 0918326
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Middle Schools
Secondary Education
Junior High Schools
Descriptors: Elementary School Teachers, Middle School Teachers, Coaching (Performance), Mathematics Instruction, Teacher Attitudes, Educational Improvement, Knowledge Base for Teaching, Pedagogical Content Knowledge, Teaching Methods, Self Efficacy, Educational Change, Self Evaluation (Individuals), Correlation
DOI: 10.1007/s10857-017-9373-3
ISSN: 1386-4416
Abstract: This study investigated relationships between changes in certain types of coaching knowledge and practices among mathematics classroom coaches and how these explain changes in the attitudes, knowledge, and practice of the teachers they coach. Participants in this study were 51 school-based mathematics classroom coaches in the USA and 180 of the teachers whom they coached between 2009 and 2014. The participating coaches were recruited from schools that hired their own coaches independently from this research project. This study found evidence that improvements in coaches' use of practices recommended by particular coaching models are related to improvements in teachers' mathematical knowledge for teaching. The study also found that improvements in coaches' self-assessment of their own coaching skills are related to improvements in teachers' mathematics content knowledge for teaching, mathematics teaching practices, and attitudes about self-efficacy for teaching mathematics. The study did not detect relationships between changes in coaches' mathematics knowledge and changes in teachers' knowledge or practices.
Abstractor: As Provided
Entry Date: 2019
Accession Number: EJ1204494
Database: ERIC
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  Value: <anid>AN0134564965;oih01feb.19;2019Feb08.10:56;v2.2.500</anid> <title id="AN0134564965-1">Variations in coaching knowledge and practice that explain elementary and middle school mathematics teacher change </title> <p>This study investigated relationships between changes in certain types of coaching knowledge and practices among mathematics classroom coaches and how these explain changes in the attitudes, knowledge, and practice of the teachers they coach. Participants in this study were 51 school-based mathematics classroom coaches in the USA and 180 of the teachers whom they coached between 2009 and 2014. The participating coaches were recruited from schools that hired their own coaches independently from this research project. This study found evidence that improvements in coaches' use of practices recommended by particular coaching models are related to improvements in teachers' mathematical knowledge for teaching. The study also found that improvements in coaches' self-assessment of their own coaching skills are related to improvements in teachers' mathematics content knowledge for teaching, mathematics teaching practices, and attitudes about self-efficacy for teaching mathematics. The study did not detect relationships between changes in coaches' mathematics knowledge and changes in teachers' knowledge or practices.</p> <p>Keywords: Classroom coaching; Professional development; Mathematics education; Mentoring; Teacher knowledge; Teacher practice</p> <hd id="AN0134564965-2">Introduction</hd> <p>Classroom coaching is a professional development method in which a coach works in the classroom of a teacher to improve teaching practice. Many schools use coaching in an effort to improve student outcomes. There is growing empirical evidence that coaching can be a component of effective professional development aimed at improving teacher practice (Biancarosa and Bryk 2011; Powell and Diamond 2011; Neuman and Wright 2010; Ramey et al. 2011) and student achievement (Campbell and Malkus 2011; Biancarosa and Bryk 2011; Powell and Diamond 2011; Ramey et al. 2011). A natural question is whether particular types of coaching knowledge and practice can explain the effectiveness of coaching programs, where coaching effectiveness is measured by improvements in the attitudes, knowledge, or practices of the teachers being coached. This is a particularly important question for schools that hire coaches from a pool of teachers and then allocate resources to develop coaching knowledge, skills, and practices among those coaches.</p> <p>There have been few studies that investigate how a coach's knowledge explains coaching effectiveness. All of the studies cited above used coaches who were trained in both teaching and coaching and do not attempt to isolate types of knowledge and practices that explain a coach's effectiveness. Campbell and Malkus (2011) found that schools' use of highly trained mathematics specialists, whose responsibilities included coaching, had a modest positive effect on student achievement. Prior to their assignment as specialists, each of these specialists had obtained a master's degree focused on both school mathematics teaching and leadership. That study lays a strong foundation for further investigation. While it was comprehensive, the Campbell and Malkus study did not include an examination of the nature and extent of how variation in the coaches' knowledge influenced teaching or student outcomes, and it did not address coaching as a school-based initiative in which schools hired their own mathematics coaches.</p> <p>Establishing relationships between changes in a coach's knowledge, skills, or practices over time and teacher change is critical for both research and practice in classroom coaching. Researchers must know what types of knowledge, skills, and practices explain changes in teacher knowledge and practices in order to develop studies that demonstrate causal effects. Once relationships are identified, researchers can develop experiments to determine the strength of those relationships.</p> <p>Examining Mathematics Coaching (EMC) is a project that empirically studied the hypotheses suggested by existing coaching literature that a coach's knowledge, skills, and practices explain coaching effectiveness. In this study, we sought to identify how changes in school-based mathematics coaches' knowledge and practices over time explain coaches' effectiveness. Because of the size of this study, we did not attempt to address the question completely. Instead, we focused on the particular types of mathematical and coaching knowledge, skills, and practices that were practical to examine in the study. Stated as a research goal, we sought to explore how changes in coach variables explain changes in teacher variables.</p> <hd id="AN0134564965-3">Theoretical framework</hd> <p></p> <hd id="AN0134564965-4">Coach responsibilities and associated knowledge and skills</hd> <p>Coaching knowledge, skills, and practices is a construct that has not been widely studied empirically. There are, however, a number of published coaching texts that offer suggestions about the knowledge, skills, and practices that effective coaches should hold. Many authors offer recommendations for coaching knowledge, skills, and practices in numerous coaching models and texts, some of which emerged after the onset of the EMC project.</p> <p>In order to develop a project vision of effective coaching knowledge and practices, we had to make choices among the available coaching models and texts. When this study began, we identified four coaching texts that were discussed and featured regularly at mathematics education conferences such the National Council of Teachers of Mathematics Annual Meetings and Exposition, the National Council of Supervisors of Mathematics Annual Conference, and the Association of Mathematics Teacher Educators' Annual Conference. These are <emph>cognitive coaching</emph> (Costa and Garmston 2002); <emph>instructional coaching</emph> (Knight 2007); <emph>content</emph>-<emph>focused coaching</emph> (West and Staub 2003); and <emph>a guide to mathematics coaching</emph> (Hull et al. 2009). Many of the study participants, though not all, were either familiar with at least one of these texts or had attended trainings associated with one of these texts. Some of the participants had developed their coaching programs around recommendations made in at least one of these texts.</p> <p>These four texts can differ in the knowledge, skills, and practices they suggest for coaches. Sometimes these texts present models for coaching in scenarios that are different from the contexts of the participants in the study. Despite the differences among these texts and possible mismatches between particular coaching models and school contexts, we chose to focus only on these four coaching texts (and the coaching methods they describe). This guided the development of the EMC study and its instruments.</p> <p>There are numerous commonalities among the four texts and their models for coaching, which also influenced our choices and focus in the study. All four of these coaching texts address a coach's interaction with teachers about content. Cognitive coaching relies heavily on coaches' use of reflective questions to encourage teachers to refine their professional knowledge bases (Costa and Garmston 2002). Instructional coaching suggests that coaches use structured co-planning to help teachers make connections among concepts (Knight 2007). Content-focused coaching features a coach who takes a direct approach to improving teacher content knowledge by explicitly illustrating important content for the teacher (West and Staub 2003). In mathematics coaching, coaches interweave information about instructional strategies and content knowledge in coaching sessions with teachers (Hull et al. 2009). Within all of the texts are underlying assumptions about a knowledge base or skill set for asking questions that are challenging enough to bring about teacher change. Some of the differences in how these texts recommend addressing teachers' understandings of content result from their different assumptions about the knowledge base of the coach.</p> <p>Not all of the texts make the same assumptions about an effective coach's knowledge of mathematics. Instructional coaching (Knight 2007) and cognitive coaching (Costa and Garmston 2002) make no assumption that the coach is more knowledgeable about the content than the teacher being coached. In contrast, content-focused coaching (West and Staub 2003) and the mathematics coaching model (Hull et al. 2009) assume that the coach has a high level of content knowledge and is more experienced than the teacher being coached. Content-focused coaching acknowledges that a coach might work with a teacher who has higher content knowledge than the coach, but this model does assume that the coach has more teaching experience than the teacher being coached. When coaches address teacher content knowledge in mathematics, this can include both a teacher's own mathematics content knowledge and a teacher's understanding of students' mathematics content knowledge (Sutton et al. 2011).</p> <p>The four coaching texts are most similar in their emphasis on promoting professional relationships. All four texts acknowledge that a coach must be a leader for change but at the same time establish collegial rapport with teachers. Instructional coaching and mathematics coaching assert that coaches should have knowledge of adult learning. They suggest that a coach should facilitate coaching sessions in a manner that attends to personal relationships with teachers, while ensuring that these relationships are based on professional objectives, such as instructional improvement. In these models, the coach is an agent of change in the school.</p> <p>There is a tension between building rapport and facilitating teacher change. Positive, professional relationships should be developed around mathematics, mathematics teaching, and student learning (West and Staub 2003), but at the same time the relationships must allow for the coach to have difficult conversations with teachers that may cause some cognitive disequilibrium (Knight 2007). The texts suggest that focusing on student learning and using student work in coaching sessions are ways to prevent a habit of criticizing teachers on superficial aspects of particular teaching practices.</p> <p>Coaches are encouraged to work with the principal to establish a clear, shared vision for mathematics instruction in schools (Knight 2007; West and Staub 2003). Coaches are also encouraged to give feedback to the principal about the school's progress toward meeting its vision for mathematics (West and Staub 2003). There is potential for tension in the coach-teacher relationship due to the coach's position as a school leader. The texts suggest that coaches should position themselves in a partnering, trusting relationship (Hull et al. 2009) with teachers so that the coach is not seen as an evaluator or the "ears and eyes" of the principal.</p> <p>Content-focused coaching also suggests that coaches should diagnose teachers' stated and unstated needs (West and Staub 2003). Mathematics coaching asserts that it is a coach's responsibility to maintain and share best-practices research and be knowledgeable about data acquisition and analysis and its use to guide coaching conversations and school improvement (Hull et al. 2009). The texts also suggest that it is important for coaches to plan lessons with teachers (Hull et al. 2009; Knight 2007; West and Staub 2003). Cognitive coaching emphasizes that coaches should use reflective questions as a critical tool in coaching conversations (Costa and Garmston 2002). The texts also suggest that coaches should be concerned with teachers' beliefs about effective mathematics instruction (Knight 2007; West and Staub 2003).</p> <p>Although the preceding description does not completely capture the extensive recommendations for coaches found in the texts we have cited, these are the features of coaching that we focused on in our study and in our coaching professional development.</p> <hd id="AN0134564965-5">Mathematics knowledge for teaching</hd> <p>The majority of the mathematics-specific coaching textbooks referenced assert that the coach should possess a deep understanding of the mathematics taught by the coached teachers. Our construct of mathematics knowledge for teaching and its measurement aligns with that expressed by Hill et al. (2004) and Ball et al. (2008). Underlying this construct is the notion that there exists mathematics knowledge that is unique to the profession of mathematics teaching. To assist teachers in leveraging that knowledge when planning and delivering lessons, coaches must possess that knowledge as well. When a teacher lacks mathematics teaching knowledge in a particular area, a coach might need to facilitate more mathematics learning.</p> <hd id="AN0134564965-6">Coaching knowledge and practice</hd> <p>Coaching texts suggest a collection of knowledge and practices for effective coaching. We had difficulty distinguishing between knowledge that an individual holds as a view or belief about coaching and knowledge that is evident through the coach's reported practices. We agreed that there is a distinction between these two types of knowledge. Some readers may interpret the texts' recommendations as a collection of practices that an effective coach should possess. We do not necessarily disagree with this perspective. We do, however, take the position that knowledge of the practices recommended in prominent coaching texts is a knowledge domain.</p> <p>Knowledge of recommendations in coaching texts does not guarantee that a coach leverages that knowledge in developing a coaching strategy. A coach may know what practice is recommended, but for whatever reason choose not to implement that practice. A coach's knowledge of recommendations of the literature may or may not explain teacher change. The EMC project began with an intent to measure coaching knowledge as a predictor of teacher change; however, the instrument we developed and tested ultimately measured coaches' reported practices and the degree to which these practices aligned with recommendations from our selected coaching literature. This instrument will be discussed in more detail in the "Methods" section.</p> <hd id="AN0134564965-7">Coaching skills</hd> <p>We sought to develop a construct of coaching effectiveness unique from other constructs of coaching practice. A coach might know of a recommendation from a coaching text and implement that recommendation but not feel skillful in the implementation of the practice.</p> <p>In this construct, we also sought to include more practices that are specific to mathematics teaching, such as coaching teachers on incorporating mathematical problem solving into their lessons. The coaching practices construct is not specific to mathematics given that some of the coaching recommendations arose from coaching programs that are not content specific (e.g., Costa and Garmston 2002; Knight 2007). We also sought to develop this construct so that it was unique from our construct of mathematics knowledge for teaching. We focused on skills that coaching texts identify as important: building professional relationships; coaching for particular teaching strategies, both those that could be considered general and those that are specific to mathematics content; and coaching for mathematics content. The coaching skills inventory (CSI) instrument that resulted after field-testing and analysis is available in "Appendix"; more details about this instrument are in the "Methods" section.</p> <hd id="AN0134564965-8">Coaching intensity</hd> <p>Studies that examine contextual factors that influence a coach's effectiveness are also just emerging. Ramey et al. (2011) reported that teachers who participated in "dense coaching" (20 consecutive days) showed greater gains in their use of effective instructional reading practices when compared to a "low density" coaching group. Biancarosa and Bryk (2011) reported that teachers who participated in above average amounts of coaching implemented targeted practices with greater frequency. Campbell and Malkus (2011) collected data on daily activities of the mathematics specialists in their study, including time spent in direct contact with teachers, but they did not use these data to model individual teacher change. Campbell and Malkus noted that as the coaching activities of the specialists decreased and were replaced by other duties, the positive effects of the specialists on student achievement diminished.</p> <hd id="AN0134564965-9">Types of teacher change</hd> <p>Based on the definitions of coaching knowledge defined through the modified Delphi study of national experts, coaches were provided training on research-based, high-leverage practices (Ball 2012) and growth mindsets (Dweck 2006) to help teachers shift in ways that improved student achievement. These practices included student discourse, small group work, and worthwhile mathematical tasks. Questions remain open about ways in which coaches can impact teachers' knowledge and practice of the content they teach, and few studies have attempted to isolate the effect. One study that examined the effects of coaching on teacher knowledge of early language literacy was unable to detect improvements in any of their groups, including those who received coursework or coaching (Neuman and Wright 2010).</p> <hd id="AN0134564965-10">Data sources</hd> <p>The EMC Project recruited coaches and teachers of grades K-8 to participate throughout the 5 years of the project (2009-2014). The data set as analyzed includes a total of 51 school-based coaches and 180 coached teachers from 28 districts across eight states (see Table 1). Because information was being collected throughout each year and aligned across coaches and teachers and some participants left the study at different times, the details of the measurements used are complicated to summarize. In other words, teachers and coaches came and went during the study, as one would expect, so we developed rules about what measurements and from whom would be appropriate for our repeated measures analysis.</p> <p>Examining mathematics coaching (EMC) project participation, years 1-5</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="2">Participant type</th><th align="left" colspan="2">Year 1</th><th align="left" colspan="2">Year 2</th><th align="left" colspan="2">Year 3</th><th align="left" colspan="2">Year 4</th><th align="left" colspan="2">Year 5</th></tr><tr><th align="left">Data set</th><th align="left">Participants</th><th align="left">Data set</th><th align="left">Participants</th><th align="left">Data set</th><th align="left">Participants</th><th align="left">Data set</th><th align="left">Participants</th><th align="left">Data set</th><th align="left">Participants</th></tr></thead><tbody><tr><td align="left">Coaches<sup>a,c</sup></td><td align="char" char=".">47</td><td align="char" char=".">61</td><td align="char" char=".">50</td><td align="char" char=".">56</td><td align="char" char=".">49</td><td align="char" char=".">56</td><td align="char" char=".">48</td><td align="char" char=".">55</td><td align="char" char=".">45</td><td align="char" char=".">52</td></tr><tr><td align="left">Teachers<sup>b,c</sup></td><td align="char" char=".">125</td><td align="char" char=".">174</td><td align="char" char=".">146</td><td align="char" char=".">165</td><td align="char" char=".">139</td><td align="char" char=".">157</td><td align="char" char=".">133</td><td align="char" char=".">151</td><td align="char" char=".">121</td><td align="char" char=".">152</td></tr></tbody></table> </ephtml> </p> <p> <sups>a</sups>Coaches in the data set as analyzed (<emph>n</emph> = 51) were added as late as Year 2, and some coaches included in the data set withdrew from the project during Years 1-4</p> <p> <sups>b</sups>Teachers in the data set as analyzed (<emph>n</emph> = 180) were added as late as the start of Year 4 to replace any teachers who withdrew from the project during Years 1-4. This was done to maintain three teachers per coach whenever possible</p> <p> <sups>c</sups>Coaches and teachers worked in 28 districts across Colorado, Georgia, Idaho, Montana, Nebraska, North Dakota, Washington, and Wisconsin</p> <p>The rules for measurements to be included in the data set we analyzed are that (<reflink idref="bib1" id="ref1">1</reflink>) the participants (coach and teacher) were both measured in a given year, (<reflink idref="bib2" id="ref2">2</reflink>) only measurements from coaches who entered in the first two years of the study were included. Rule 2 restricted the analysis to data from coaches who were randomly allocated to treatments groups. The counts of teachers and coaches being analyzed per year along with additions and subtractions are provided in Table 1.</p> <p>Our participants possessed a wide variety of backgrounds, certifications, and experience (see Table 2). Our coaches reported an average of 12.6 years of teaching experience and an average of 11.6-year experience in teaching mathematics. The coaches reported an average of 2.1 years of experience coaching in grades K-8 and 1.7 years coaching mathematics in grades K-8. Our teachers reported an average 9.2 years of teaching experience and an average of 8.4 years teaching mathematics. These data were collected for each participant in the first year that the participant entered the project.</p> <p>Characteristics reported by participants</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left">Characteristic reported</th><th align="left">Number of coach participants meeting each characteristic (<italic>n</italic> = 51)</th><th align="left">% of coach participants<sup>a</sup></th><th align="left">Number of teacher participants meeting each characteristic (<italic>n</italic> = 180)</th><th align="left">% of teacher participants<sup>a</sup></th></tr></thead><tbody><tr><td align="left">Highest degree: Bachelor's</td><td align="char" char=".">12</td><td align="char" char=".">24</td><td align="char" char=".">104</td><td align="char" char=".">22</td></tr><tr><td align="left">Highest degree: Master's</td><td align="char" char=".">37</td><td align="char" char=".">71</td><td align="char" char=".">72</td><td align="char" char=".">40</td></tr><tr><td align="left">Highest degree: Other</td><td align="char" char=".">1</td><td align="char" char=".">2</td><td align="char" char=".">4</td><td align="char" char=".">2</td></tr><tr><td align="left">Number certified in elementary teaching</td><td align="char" char=".">45</td><td align="char" char=".">88</td><td align="char" char=".">170</td><td align="char" char=".">94</td></tr><tr><td align="left">Number certified in middle school teaching</td><td align="char" char=".">25</td><td align="char" char=".">49</td><td align="char" char=".">82</td><td align="char" char=".">46</td></tr><tr><td align="left">Number certified as secondary teachers</td><td align="char" char=".">13</td><td align="char" char=".">25</td><td align="char" char=".">16</td><td align="char" char=".">9</td></tr><tr><td align="left">Number holding endorsement in mathematics</td><td align="char" char=".">11</td><td align="char" char=".">24</td><td align="char" char=".">16</td><td align="char" char=".">9</td></tr></tbody></table> </ephtml> </p> <p> <sups>a</sups>Percents do not total to 100 due to number of respondents per category, ability to answer more than one choice, and rounding. Results for subjects in analyzed longitudinal data set only</p> <p>In the USA, where the study took place, the notion of "qualified to teach mathematics" is complicated. In Idaho and Montana, for example, an elementary teaching certificate is certification to teach in kindergarten through grade 8, and a teacher can be considered qualified to teach mathematics with these credentials. Teachers can be considered highly qualified to teach mathematics through a variety of experiences such as additional mathematics courses, successful completion of certain assessments, and alternative certification programs. Thus, a differentiation among teacher qualifications beyond what is presented in Table 2 could be confusing and misleading. Therefore, in terms our coaches' and teachers' knowledge of mathematics for teaching, we look more to the results of the mathematics knowledge for teaching (MKT) assessments.</p> <p>The sample of districts in this study is geographically diverse. District-level demographics are more appropriate to the study than school-level demographics because many of the coaches coached in multiple schools within a district. Twelve of the districts are described as urban or suburban, five are in remote towns, and 11 are rural districts (US Department of Education 2012). Our districts are less culturally and racially diverse than perhaps some districts in major cities, but some diversity is present. Table 3 further summarizes the participating districts' demographics.</p> <p>Summary of demographics for the 28 EMC participating districts</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="2">Characteristic reported</th><th align="left" colspan="4">Percentage of students in the district meeting each characteristic</th></tr><tr><th align="left">0-24%</th><th align="left">25-49%</th><th align="left">50-74%</th><th align="left">75-100%</th></tr></thead><tbody><tr><td align="left">Eligibility for free or reduced lunch<sup>a</sup></td><td align="char" char=".">2</td><td align="char" char=".">12</td><td align="char" char=".">6</td><td align="char" char=".">7</td></tr><tr><td align="left">Hispanic<sup>b</sup></td><td align="char" char=".">26</td><td align="char" char=".">1</td><td align="char" char=".">0</td><td align="char" char=".">1</td></tr><tr><td align="left">Native American</td><td align="char" char=".">21</td><td align="char" char=".">1</td><td align="char" char=".">1</td><td align="char" char=".">5</td></tr><tr><td align="left">Black</td><td align="char" char=".">27</td><td align="char" char=".">0</td><td align="char" char=".">0</td><td align="char" char=".">1</td></tr></tbody></table> </ephtml> </p> <p> <sups>a</sups>Data supplied by districts or state departments of education. Not available for one rural district (data suppressed for privacy reasons)</p> <p> <sups>b</sups>Ethnicity data from US Department of Education (2012)</p> <p>EMC collected data on coach and teacher knowledge and practices for 5 consecutive years, and here we report our analysis of this 5-year study. Coaches received professional development experiences, designed and presented by project staff, in both coaching and mathematics for teaching at different times over the duration of the project. At the beginning of the study, coaches were randomly assigned to one of two cohorts. These cohorts received professional development from the project in different orders and at different times: "PD Cohort 1" received mathematics professional development in Year 1 and coaching professional development in Year 3, while "PD Cohort 2" received coaching professional development in Year 2 and mathematics professional development in Year 4.</p> <p>Our purposes were not to study the effects on the professional development on coaches and the teachers they coached specifically. Instead, by focusing the professional development on the constructs we defined in our Theoretical Framework and varying the professional development among cohorts, we intended to create variation in the data that might not otherwise exist. We assumed that if coaching knowledge, skills, and practices as we defined and measured them was highly impactful, then variation in the coaches' measures could explain changes in the teachers' measures. In the study reported here, the professional development effects and the random assignment of coaches to cohorts are not incorporated into our statistical models. (However, we include a report on the effects of our professional development as part of the conclusion to this study in explanation of our findings.)</p> <p>Because some participants, both coaches and teachers, withdrew from the project due to factors that had nothing to do with the study (e.g., change of job), some data are missing, but our models proceed under the assumption that these data are missing completely at random. New coach enrollments were allowed as late as the start of Year 2 within the existing random allocation structure. Teacher replacements were allowed through the start of Year 4. See Table 1 for more details on the study's sample sizes.</p> <p>Explanatory measures (coach variables) include assessments ofCoaches completed the MKT and CSI at the beginning of the project, then completed the MKT, CSI, and CPS at the end of Year 1. They again completed the MKT, CSI, and CPS at the start of Years 2, 3, 4, and 5. This structure ensured that we had two Year 1, pre-treatment measures for coaches followed by measures of the knowledge and practices coaches brought to each year's coaching activities.</p> <p></p> <ulist> <item> coaches' mathematics knowledge for teaching, as measured by the Mathematics Knowledge for Teaching (MKT) number and operation (K-5 or 6-8) instrument (Hill et al. 2005, 2008);</item> <p></p> <item> coaches' alignment with the coaching practices suggested in our chosen coaching recommendations, as measured by the Coaching Practices Survey (CPS), a project-developed instrument based on the common coaching recommendations found in Costa and Garmston (2002), Knight (2007), West and Staub (2003), Hull et al. (2009), and Sutton et al. (2011);</item> <p></p> <item> coaches' self-assessment of their mathematics coaching skills, as measured by the Coaching Skills Inventory (CSI), a project-developed instrument aligned with coaching practices expressed in our selected coaching texts and practices expressed on the project's classroom observation protocol; and</item> <p></p> <item> the number of minutes a coach spent in pre- or post-lesson conferences each year with each teacher.</item> </ulist> <p>Response measures (teacher variables) include assessments ofTeachers completed the MKT and TS at the beginning of the project, then for a second time at the end of Year 1. They again completed the MKT and TS at the end of Years 2, 3, 4, and 5. This structure ensured that we had two first-year measures of teacher knowledge and beliefs followed by measures of these attributes at the end of each coaching year. EMC staff observed each teacher annually in Years 1-5 using the ITCOP.</p> <p></p> <ulist> <item> teachers' mathematics knowledge for teaching, as measured by the MKT number and operation (K-5 or 6-8) instrument;</item> <p></p> <item> teachers' classroom practice, as measured by the 7-category ordered capsule response found on the Horizon Research 2003 version of the Inside the Classroom Observation Protocol, or ITCOP (see Weiss et al. 2003, for the instrument and a description of its use, and Horizon Research Inc. 2000, for validity and reliability estimates); and</item> <p></p> <item> teachers' self-efficacy for teaching mathematics, as measured by the Teacher Survey (TS), a project-developed instrument.</item> </ulist> <p>We examined the relationship between the variations in the coaches' scores (explanatory variables) and the associated changes across time in the teachers' scores (response variables). In our analysis, we used the "centering" of variables in order to reduce the variability across means. Quantitative predictor variables that vary over time contain two types of variation: variation among participants and variation over time. In order to estimate the effects of each type of variation, we calculated the mean over time for the teacher or coach, depending on whether the explanatory variable was measured on the teacher or coach, and used that as one predictor, and then subtracted this from the original variable to create the "centered" predictor (see Gelman 2008, for alternatives to this centering procedure). This allows the effects of differences among teachers and coaches on that predictor, the coach or teacher mean, to be separated from the effects of variation over time on that predictor, the centered predictor variable.</p> <p>We use hierarchical models to account for the hierarchical nature of the study, with repeated measures on coaches and teachers assigned to coaches. Other control variables include the professional development cohort to which the project randomly assigned the participant, whether or not a coach received outside (i.e., not provided by EMC) coaching and/or mathematics professional development in a given year, and whether or not a teacher received outside mathematics professional development in a given year.</p> <hd id="AN0134564965-11">Methods</hd> <p></p> <hd id="AN0134564965-12">Measures</hd> <p></p> <hd id="AN0134564965-13">Coaching skills inventory (CSI)</hd> <p>The CSI is a project-developed instrument designed to measure a mathematics coach's perspective on his or her own level of effectiveness or confidence with various coaching responsibilities. The inventory has 20 items measured on a 5-point Likert scale, with a higher rating indicating a higher level of self-reported effectiveness. (Example item: <emph>How effective do you feel coaching teachers on creating an environment where students listen to one another?</emph>)</p> <p>Exploratory factor analysis (varimax rotation) revealed three categories, which we labeled (a) mathematics content and mathematics-specific pedagogy, (b) student-centered pedagogy, and (c) building coaching relationships. For these factors, internal reliability scales were high (Cronbach's α = 0.935, 0.932, and 0.822, respectively). A single score for a participant is extracted from the instrument by averaging the averages of the scales (factors). The full CSI and the results of the factor analysis are available in "Appendix."</p> <p>CSI scores were used in the models as explanatory variables in two ways: the averages across the 5 time points for each coach (CSImean) and the yearly variation around the mean for each coach (CSIcentered). CSIcentered scores explained gains in some teacher variables across years.</p> <hd id="AN0134564965-14">Coaching practices survey (CPS)</hd> <p>The CPS is a project-developed instrument designed to capture the extent to which a coach's knowledge of effective coaching practices aligns with particular coaching texts (i.e., Costa and Garmston 2002; Knight 2007; Hull et al. 2009; West and Staub 2003). The full CPS instrument is available in "Appendix."</p> <p>The instrument contains both 7-point Likert scale items and selected response items for a total of 20 items. Each item was coded according to whether or not the response conformed with assertions found in coaching texts. For example, the item <emph>I work with principals or other administrators to form a clear message to teachers about effective mathematics instruction</emph> was rated by participants on a 7-point Likert scale from <emph>strongly disagree</emph> to <emph>strongly agree</emph>. A coach with extensive knowledge of these coaching texts knows that there can be tension between the role of a coach as confidant for teachers and the role of a coach as a school mathematics leader and a possible member of a school's leadership team. Some texts caution against reporting to administrators about particular teachers' actions (e.g., Hull et al. 2009). Yet a coach with extensive understanding of these particular texts knows it is recommended that coaches take a leadership role in developing a school vision for mathematics, and that the principal plays a critical role in communicating and supporting this vision (e.g., Knight 2007). A knowledgeable coach might have personal experiences that preclude the coach from strongly agreeing with this statement (e.g., a principal who is not supportive of standards-based instruction), but a coach who has knowledge about coaching texts knows which side of the fence to be on. Thus, coaches who rated this item as "5" (1 category above neutral) or higher (agree) were coded as reporting practices that conform to those recommended in the texts. Coaches who rated this item as "1," "2," or "3" (disagreeing) or "4" (neutral) were coded as expressing views that do not conform to the literature. Table 4 offers a subset of the items from the CPS to illustrate types of items on the instrument.</p> <p>The CPS was developed and validated through the following process. First, EMC researchers developed a pilot instrument that expressed the practices recommended in our selected coaching texts. Responses to the pilot instrument from 191 coaches throughout the USA, none of whom were participants in the EMC Project, were used to retain a collection of 39 items. This was a sample of convenience created through an open e-mail invitation sent to a network of mathematics coaches and mathematics coaching experts that included a link to the survey. More details on the development of this instrument are available in Greenwood and Jesse (2014).</p> <p>Briefly, a tetrachoric correlation matrix (Revelle 2014) for the 39 binary items was developed to explore the underlying dimensions in the items. Scree plot and associated parallel analysis suggested that the items contained 5 or more dimensions. In a maximum likelihood factor analysis, 20 items were associated with the first factor (loadings greater than 0.4) and were retained for further exploration. This collection of 20 items exhibits good, but not excellent, internal consistency (Cronbach's α estimated at 0.81).</p> <p>Item response theory (IRT) 1- and 2-parameter logistic (1PL and 2PL) models were considered to develop a scoring model for the CPS responses on the 20 identified items. These items were all relatively easy for the pilot participants whose responses we used to develop the scoring model (the most difficult having 65% conforming responses and the least difficult having 96% conforming responses). A 1PL model was found to be adequate with <emph>p</emph> = 0.06 when tested relative to the 2PL model. There was no evidence of a lack of fit of the 1PL model from a bootstrap test (<emph>p</emph> = 0.556), where the null hypothesis was of model adequacy with the 1PL model. Additionally, there was no evidence of multidimensionality, with <emph>p</emph> = 0.204 from 999 bootstrap samples using methods described in Rizopoulos (2006). The estimated 1PL scoring model was then applied to the longitudinal responses of the study participants.</p> <p>Our next concern was whether or not the collection of 20 items is a useful and valid assessment of the EMC coaches. Figure 1 demonstrates that the CPS scores of the EMC coaches at the pretest appear to be about 1 standard deviation lower than the pilot sample (mean of 0 by construction) and increased over time in the study. We modeled the longitudinal CPS responses using a 2-level hierarchical linear model over 5 years, assessing for evidence of any differences based on the timing of professional development between PD Cohorts 1 and 2, controlling for whether or not a coach had received outside mathematics professional development and whether or not a coach had received outside coaching professional development in a given year. There was no evidence of a time-by-PD cohort interaction (<emph>F</emph>(<reflink idref="bib4" id="ref3">4</reflink>,<reflink idref="bib189" id="ref4">189</reflink>) = 0.66, <emph>p</emph> = 0.62). With the interaction removed from the model, there was strong evidence for changes over time (<emph>F</emph>(<reflink idref="bib4" id="ref5">4</reflink>,<reflink idref="bib193" id="ref6">193</reflink>) = 8.8, <emph>p</emph> < 0.0001) and impacts of outside coaching professional development (<emph>F</emph>(<reflink idref="bib1" id="ref7">1</reflink>,<reflink idref="bib193" id="ref8">193</reflink>) = 10.1, <emph>p</emph> = 0.002), but no evidence of a difference in the two PD cohorts (<emph>F</emph>(<reflink idref="bib1" id="ref9">1</reflink>,<reflink idref="bib49" id="ref10">49</reflink>) = 2.1, <emph>p</emph> = 0.153) or based on having had outside mathematics training (<emph>F</emph>(<reflink idref="bib1" id="ref11">1</reflink>,<reflink idref="bib193" id="ref12">193</reflink>) = 1.96, <emph>p</emph> = 0.163). In this model, the coaches in Year 5 had mean CPS scores estimated to be 0.55 standard deviations (SD) higher than in Year 1 (95% CI 0.347-0.754), and the coaches who had outside coaching training in the prior year had a mean CPS score that is 0.25 SDs higher than coaches who had no outside training in the prior year (95% CI 0.094-0.400). The detected effects of outside coaching professional development and time on the EMC participants' longitudinal CPS scores suggest that our collection of 20 items has predictive validity and that these items are useful for detecting changes in coaches' conformance to coaching texts and for relating scores on this instrument to teacher-level changes.Boxplots with means (<emph>circles</emph>) and 95% confidence intervals for the CPS scores by year of study, 1-5. The pilot data set was scored to have a mean of 0, and the study participants mean increases from 1 standard deviation below 0-0.5 standard deviations below 0</p> <p>PHOTO (COLOR)</p> <p>Coaches' annual CPS scores were used as explanatory variables in two ways in the statistical models below: the averages across the 5 years for each coach (CPSmean) and the yearly variation around the mean for each coach (CPScentered). We explain further below that the CPSmean scores had negative relationships with all teacher measures (though there was only limited evidence for the effects with <emph>p</emph> values either slightly or much over 0.05), but the variation in CPS scores (CPScentered) explained gains in one teacher response across years.</p> <p>The responses analyzed on the CPS and the CSI are based on Likert scales and could change if different resolution scales or different wording of items had been used. We made every attempt to generate valid and reliable questions and used scales that we felt were appropriate to the questions being asked. We did not detect any issues with the scales as used by the respondents but there is the potential for more study of ways to improve these instruments.</p> <hd id="AN0134564965-15">Mathematics knowledge for teaching (MKT)</hd> <p>The MKT instrument was created by the Learning Mathematics for Teaching Project at the University of Michigan (Hill et al. 2005, 2008) using an IRT model to produce estimated latent ability scores related to mathematics knowledge for teaching. This instrument was selected for this study because of its validity and reliability (Hill et al. 2004) and its alignment with content that K-8 teachers teach (Ball et al. 2008). Hill et al. (2008) discussed the merits of using the MKT for this type of study. Teachers and coaches in the project completed the grade-band version of the MKT (K-5 or 6-8) that best aligned with their teaching assignment or coaching assignment on the date they entered the project. Participants alternated between two versions of the MKT at their level of instruction, with the initial version randomly assigned. The scores on the two versions were equated using the original data sets used to develop the instruments. The MKT IRT scores from each year of the study were used for teachers as response variables. The Coach MKT IRT scores were used in the models as explanatory variables in two ways: the averages across the 5 time points for each coach (CMKTmean) and the yearly variation around the mean for each coach (CMKTcentered). As we explain further below, we found minimal evidence of relationships between coaches' MKT and teachers' MKT.</p> <hd id="AN0134564965-16">Inside the classroom observation protocol (ITCOP)</hd> <p>The ITCOP prompts the observer to rate a lesson in four domains: design, implementation, mathematics content, and classroom culture. After rating these domains, the rater gives an overall capsule rating of the lesson quality and impact. For this study, we used only the capsule rating.</p> <p>The intention of the capsule rating is to record the rater's overall assessment of instruction and lesson quality and impact. The lesson is rated from 1 to 5, with "1" being ineffective instruction, "2" being elements of effective instruction, "3" being beginning stages of effective instruction, "4" being highly accomplished effective instruction, and "5" being exemplary instruction. Capsule ratings of "3" are then refined as either "Low 3," "Solid 3," or "High 3." These responses were converted into a 7-level Likert scale to respect this ranking of the intermediate responses.</p> <p>The ITCOP capsule rating, as developed by Weiss et al. (2003), included the Low, Solid, and High components to add discrimination to the differences in this part of the scale. We used the scale as Weiss et al. presented it, with 7 possible ratings. We placed strong emphasis on calibrating all the steps on this scale across the raters used over the time of the study. We believe these additional steps on the ordinal scale enhance the resolution of the instrument to discern differences in these intermediate level ratings.</p> <p>Data from the ITCOP were collected by 12 observers who underwent annual reliability training to standardize the ITCOP observations. Videos of non-EMC classroom lessons used in reliability training in Year 1 and Year 2 were incorporated into the Year 5 training to guard against observer "drift." Each year in training, project observers exhibited a high degree of reliability (between 80 and 100% agreement within 1 point of median), and we found no evidence of drift on observer scores between Years 1 and 2 and Year 5 (100% interrater and cross-year agreement within 1 point of median). The observers were then allocated to observe all teachers in the study each spring based on geographical efficiency, which tended to confound the 28 districts and 12 observers. The observers did not have knowledge of the teachers' MKT scores, and for the vast majority, the observers were previously unacquainted with the teachers they observed, other than encounters during a brief, formal orientation to the project that occurred in 2009, at the outset of the project and months before the initial observations. Thus, the observers had little existing knowledge of the teachers they observed, thereby reducing bias in the observations. The ITCOP measurements involved pre-arranged classroom visits for lessons related to number sense or algebraic reasoning that took place between March and May each year.</p> <hd id="AN0134564965-17">Teacher survey (TS)</hd> <p>The TS, a 38-item instrument using 8-point scaled responses, is designed to measure a teacher's overall feelings of efficacy as a professional mathematics teacher. The instrument includes a diverse collection of items that capture related factors such as feelings of personal level of preparedness, anxiety, and self-efficacy for teaching mathematics and levels of engagement and comfort with peers in mathematics teaching and reflection activities. Exploratory factor analysis revealed six factors: (a) preparedness to teach mathematics, (b) anxiety for teaching mathematics, (c) efficacy for support of mathematics teaching, (d) efficacy for ability to teach mathematics, (e) efficacy for confidence in teaching mathematics, and (f) engagement in mathematics activities with peer teachers. Internal reliability scores (Cronbach's α) are high for each factor (0.933, 0.944, 0.899, 0.894, 0.882, and 0.846, respectively). All six traits were combined to create an overall score from this instrument with a moderately high internal reliability (Cronbach's α of 0.84) for the combination of the items. This high Cronbach α score supports our assertion that the items are internally consistent, and it supports our decision to combine the six factor scores. Thus, the average teacher responses on the items loading on each factor were averaged to create a single score to use as a response variable for each teacher in each year. We performed factor analysis to affirm dimensions within the instrument so they each equally contributed to the creation of a single score, not to create separate scores.</p> <hd id="AN0134564965-18">Coaching intensity (Cint)</hd> <p>Developing a measure of coaching intensity for the study's context presented many challenges. The coaching texts we selected describe a coaching cycle that includes a pre-lesson conference, a lesson observation or model lesson, and a post-lesson reflection conference. Participating coaches agreed to follow this cyclic model in their coaching practice, and our coaching professional development emphasized this cycle as a central component of coaching practice. However, documenting exactly what went on in these coaching cycles was beyond the project's resources, and there is inconsistency in what constitutes "intense" coaching among the coaching texts.</p> <p>For example, while all the texts recommend including a "lesson" component in the coaching cycle, there is considerable variation among the texts about what a coach is to do during the lesson. Knight (2007) recommends that coaches model lessons, but Hull et al. (2009) discourage the practice. Hull et al. (2009) present the concern that modeling lessons might inadvertently suggest to the teacher that the lesson is one that only the coach can deliver. Some authors suggest data collection is the primary role of the coach during the lesson. However, West and Staub (2003) include a video of a coach interrupting a teacher's lesson to ask students questions. This coach is using this opportunity to coach the teacher by illustrating that certain types of student thinking are being overlooked by the teacher.</p> <p>There is also variation among recommendations for coaching practice during pre- and post-lesson conferencing. Costa and Garmston (2002) encourage reflective questions and discourage direct advice and explicit critiques of the teacher. West and Staub (2003) include a video that illustrates a coach offering explicit advice to a teacher and critiques of a teacher's lesson plan.</p> <p>In the end, we decided that there was too much variation among our chosen coaching texts to quantify coaching practices during the coaching cycle. We chose to educate the participating coaches on the variety of recommendations made by these texts, and to let the coaches decide what types of practices were most appropriate for their teachers. We chose to measure only the amount of time spent in conferencing with the teacher outside of the actual lesson. Time spent with the teacher during the lesson was not included in the measure because the variation among recommended practices (e.g., lesson observation only, team teaching, modeling, interjecting during "coachable moments") seemed far too great to compare across teachers and sessions.</p> <p>The coaching intensity variable (total number of minutes per year of coaching in pre-lesson and post-lesson conferences for each teacher) is decomposed and used in the model in two ways at the teacher level: the CIntMean, which averaged the minutes per teacher across years, and the CIntCentered, which is the variation around that mean for each teacher for each project year. (Note that these values also contain variation at the coach level, although the variable is focused on the variation over time in the minutes spent by the pair in coaching sessions and the average for that pair.) Both of these variables are used in each model to control for differences in the contact time that each teacher received on average (CIntMean) and the variation around that mean for each teacher (CIntCentered).</p> <p>Coaching intensity varied among the participating coaches across the years. In Year 1, some coaches were either just beginning coaching or were to begin in the following year, and they may have reported 0 coaching sessions for that first year. Other coaches began the project coaching frequently during the early years of EMC, but then due to changes in job responsibilities or school restructuring, their coaching sessions were reduced or eliminated as the project progressed. This phenomenon can be seen in the subset of the data that follows the participating coaching pairs from Year 1 to Year 5 (coach self-reported data). In Year 1, 61 of 125 coaching pairs participated in 4 or more coaching sessions in mathematics; 92 pairs spent more than 60 min in pre- or post-lesson conferencing (inclusive "or") about mathematics; and 9 pairs participated in no coaching sessions in mathematics. In Year 2, the rates improved: 112 of 146 coaches reported 4 or more coaching sessions with an EMC teacher; 112 coaches reported more than 60 min in pre- or post-lesson conferences (inclusive) with an EMC teacher; and only 7 pairs participated in no coaching sessions in mathematics with an EMC teacher. In both Years 3 and 4, we saw decreases in the number of coaches reporting 4 or more coaching sessions relative to the sample size (e.g., <emph>n</emph> = 139 coaching pairs in Year 3). In Year 5, 64 of 121 coaching pairs reported 4 or more sessions with an EMC teacher; 66 coaching pairs reported more than 60 min in pre- or post-lesson conferences (inclusive) with an EMC teacher; and 37 pairs reported no coaching sessions in mathematics with an EMC teacher.</p> <p>Data about coaching frequency and the nature of their coaching activities were self-reported by coaches and compared with independent reporting by the teachers they coached. The sample had a high degree of reliability among coaches and teachers when responding to questions about whether or not coaching occurred in a given year: only 6 of 138 teachers reported no coaching sessions when the coach reported that sessions occurred. When asked about the exact number of coaching sessions, the teachers and coaches' responses differed slightly. For example, in Year 3, 59% of the coach and teacher pairs reported numbers of mathematics coaching sessions that were no more than 1 apart, and 73% reported numbers of mathematics coaching sessions that were no more than 2 apart. Coaches were asked to keep careful logs of their coaching sessions with EMC teachers, but the teachers were not ask to do so. Teacher reporting was likely based on their memory of a given year. This may explain the discrepancies. Nevertheless, the data provide some confidence in the reliability of the participating coaches' reported data.</p> <p>We also collected data on the nature of these sessions. In Year 3, for example, 71% of the coaches reported that more than half or all of their coaching sessions used a pre-lesson conference; 77% reported that more than half or all used a post-lesson conference; and 79% reported that more than half or all used a lesson observation. Sixty-four percent reported that none of their sessions involved lesson modeling, and 25% reported using modeling no more than twice.</p> <hd id="AN0134564965-19">Other measures</hd> <p>We acknowledge that teacher and coach measures were likely influenced by experiences not related to this study, such as professional development from other projects. Attempting to accurately quantify these "outside" influences was beyond the scope of the project. There can be substantial differences in content, quality, and intensity among professional development experiences, and collecting data about quality from the many participants would be expensive, impractical, and perhaps impossible. Despite these challenges, we felt it was important to incorporate some of these influences into our statistical model.</p> <p>Three variables related to outside training (Coach's Outside Coaching PD, Coach's Outside Mathematics PD, and Teacher's Outside Mathematics PD) were binned into two categories—having had none or having had any in the prior 12 months—without attempting to differentiate among the quality of professional development experiences. This categorization was used because most were heavily right-skewed and could lead to influential observations in the models. For the outside coach professional development, responses for number of hours of professional development were obtained for the periods before EMC started and during each 12-month segment of the study. For the outside mathematics professional development for teachers and coaches, all responses referenced the prior 12 months, with questions concerning the number of hours of mathematics professional development and number of courses. Because of some potential overlap between courses and professional development (e.g., a teacher may have received course credit for a professional development experience), the two sets of responses were combined to create a variable that provided for the presence of either a mathematics course or professional development for each coach and teacher in the prior 12 months. A description of all variables used in the models is available in Table 5.</p> <p>Each longitudinal model for teacher-level repeated measures includes random intercepts for the teacher and coach (2 nested random intercepts). This is generally a 3-level hierarchical framework for the design, except for the rare case where teacher and coach pairings changed (e.g., due to a school district's personnel reassignments), which can create crossing with the same teacher occurring with two different coaches. As will be illustrated in the next section, the models are discussed using hierarchical modeling notation, except that the design is not completely hierarchical with some teachers crossing coaches. Preliminary models also contained a district effect, but the complexity of the model relative to the sample size suggested a simplified version of the hierarchical structure to allow more information to be available for estimation of other, more interesting components in the model.</p> <hd id="AN0134564965-20">The model</hd> <p>Our research goal was to explore how changes in coach variables explain changes in teacher variables, and we needed unique statistical approaches to do so. We were not studying whether higher scores on coaching measures correlate with higher scores on teacher measures, which is the goal in some studies that use hierarchical linear models (see Campbell et al. 2014, for an example of this type of usage in the context of teacher knowledge and student achievement). Instead, because the study's context involved coaches hired by schools without explicit coaching credentials, we were interested to know whether increases in certain types of knowledge, skills, and practices improve these coaches' effectiveness. We expected that our findings would then enable us to make recommendations to schools about how to choose the types of professional development to offer coaches in environments where funding is limited.</p> <p>In order to separate the impacts of differences between coaches and variation in an individual coach's knowledge over time, two versions of quantitative explanatory variables (predictors) were considered. We decomposed the variability in the predictors into differences between coaches or teachers (aggregated to the mean for each coach on the variables) and variability over time for each coach (using the "centered" versions of the variables for each coach or teacher). Predictive variables were "centered" by subtracting the mean for each coach or teacher from their raw values (that vary over time) on the variables. This created a measure of how each coach's individual knowledge and the coaching intensity varied across years.</p> <p>The centered versions of the variables are Level 1 predictors (they vary over time), while the means for the coaches are Level 3 predictors, which do not vary over time or by teacher. The mean coaching intensity is a Level 2 predictor since it varies by teacher. Coaching measures from the CPS (coaching practices), CSI (coaching skills), and MKT (mathematics knowledge) were all used in both their aggregated and centered versions in the models. The aggregated versions explored the potential for differences in the coaches or time with teachers to explain differences in the teachers. The centered versions explored whether the variation in individual coaches or coaching intensity over time explains the variation in their corresponding teachers over time.</p> <p>We employed linear mixed models (Pinheiro and Bates 2000; Singer and Willett 2003; Bickel 2007) to fit all the multilevel hierarchical models, estimated using the lme4 package (Bates et al. 2014) in the statistical software R (R Core Team 2016). Cumulative probit mixed models were used for the ITCOP capsule response (Agresti 2010) and were estimated using the ordinal package (Christensen 2012).</p> <p>The 3-level model for each response variable <emph>y</emph> for the <emph>i</emph>th measurement on the <emph>j</emph>th teacher for the <emph>k</emph>th coach is:</p> <p>Level 1 (variation in teachers over time): yijk=βjk+β2CoachingIntensityCenteredijk+β4CPScenteredijk+β6CSIcenteredijk+β8CMKTcenteredijk+β9CoachPDijk+β10CoachOutsideMathijk+β11TeacherOutsideMathijk+εijk</p> <p>Level 2 (teacher level): <emph>β</emph><subs><emph>jk</emph></subs> = <emph>β</emph><subs>0<emph>k</emph></subs> + <emph>β</emph><subs>1</subs><emph>CoachingIntensityMean</emph><subs><emph>jk</emph></subs> + <emph>γ</emph><subs><emph>jk</emph></subs></p> <p>Level 3 (coach level): β0k=β00+β3CPSmeank+β5CSImeank+β7CMKTmeank+γk, where the random effects in the model include the coach effect, γk∼N0,σCoach2, the teacher effect conditional on the coach effect, γjk∼N0,σTeacher2, and the residual random error for each time of observation of each teacher of εijk∼N(0,σε2). The notation using 0 s as subscripts on the intercept coefficients is used to indicate that it is fixed at a particular level with <emph>β</emph><subs><emph>jk</emph></subs> varying at the teacher level (Level 2), <emph>β</emph><subs>0<emph>k</emph></subs> varying at the coach level (Level 3), and <emph>β</emph><subs>00</subs> the parameter for the overall intercept. All of the predictors are standardized as suggested in Gelman (2008) to provide estimated effects on the response scale for a 2-standard-deviation change in each predictor variable. The outside training variables are coded as −0.5 for absence of training and 0.5 for presence of training to provide similar interpretation scale for their effects.</p> <p>The ordinal mixed model for the response variable ITCOP modifies the previous model to incorporate a probit link between the probabilities and the responses and estimated thresholds among the 7 categories that modify the intercept for the model. Because of the similarity of reported results to those from linear mixed models, further details are omitted but are available upon request.</p> <hd id="AN0134564965-21">Results</hd> <p>Our findings suggest that improvements over time in coaches' self-assessment of mathematics coaching skills (CSI) were positively related to all three of the teacher response variables. As coaches' assessment of their mathematics coaching skills increased, so did teachers' mathematics knowledge (MKT), teachers' reports of self-efficacy (TS), and teachers' use of standards-based practices (ITCOP). These effects were also detected when we analyzed subsets of the data, covering just the first 3 and then the first 4 years. This indicates to us that positive changes in coach measures that explain changes in teachers' knowledge and practice can be detected on a shorter term than 5 years. Improvements in coaches' alignment with the recommendations of particular coaching texts (CPS) were related to increases in teachers' mathematics knowledge for teaching number and operation (MKT). We found no evidence that changes in coaches' MKT scores explained variation in teacher practices.</p> <p>In terms of control variables, we found no evidence that coaches' outside professional development in coaching knowledge was positively related to changes in the teacher measures, even though many of the participating coaches reported having considerable amounts of such training. Teachers' outside professional development in mathematics was related to increases in teachers' self-efficacy (TS) and improvements in teachers' use of standards-based practices (ITCOP).</p> <hd id="AN0134564965-22">Coaching variables that explain changes in preparedness, anxiety, and self-efficacy for teach...</hd> <p>We saw improvements in teachers' TS scores (measuring teachers' preparedness, anxiety, and self-efficacy for teaching mathematics) over the course of the project, as shown in Fig. 2. Our question is not whether the TS scores went up, nor simply whether TS scores correlate with coach measures, but instead whether changes in the coaches' measures explain those changes.Boxplots with means (<emph>circles</emph>) and 95% confidence intervals for the TS scores by year of study, 1-5</p> <p>PHOTO (COLOR)</p> <p>Table 6 displays the results of our model for how changes in coach measures explain changes in teachers' TS ratings. In the model for the TS total scores, there was strong evidence of a relationship between CSIcentered and TS scores (<emph>p</emph> = 0.0001). For a 2-SD change in CSIcentered, the mean TS was estimated to change by 0.174 points (SE = 0.044). There was also strong evidence of an effect of outside mathematics training on the TS scores (<emph>p</emph> = 0.0001), with a teacher who had outside professional development in a given year having an estimated mean TS score that was higher by 0.21 points (SE = 0.053). In data sets up to Years 3 and 4, there was similar evidence of effects of both the CSIcentered and teacher outside professional development being related to changes in TS scores, which provides additional confidence in the strength of the findings. There was moderate evidence of an effect of the mean coaching intensity (<emph>p</emph> = 0.026), with higher coaching intensity related to higher TS scores, on average. None of the other effects were suggested as being important in the model except for the random effects of teacher and coach, with the intraclass correlation (ICC) of two measurements on the same coach of 0.072 and the ICC of two measurements on the same teacher with the same coach of 0.66.</p> <hd id="AN0134564965-23">Coaching variables that explain changes in classroom practice</hd> <p>We saw improvements in teachers' ITCOP scores over the course of the project (Fig. 3) with the median score moving from a 3-5 on the 7-point scale. Again, our primary question is whether changes in the coaches' measures explain those changes in teachers' scores.Boxplots with means (<emph>circles</emph>) and 95% confidence intervals for the ITCOP scores by year of study, 1-5. Original 7-point Likert scale converted to numerical scores to make this plot consistent with other results</p> <p>PHOTO (COLOR)</p> <p>Table 7 displays the results of our model for how changes in coach measures explain changes in teachers' ITCOP ratings across the 5-year study. In Year 5, the evidence was quite strong for a CSIcentered effect (<emph>p</emph> < 0.0001) and strong for a Teacher's Outside Mathematics PD effect (<emph>p</emph> = 0.0486). The CSIcentered effect was detected as important in analyses using a subset of the data up to Years 3 and 4, and the Teacher's Outside Mathematics PD effect was detected as important in analyses using a subset of the data up to Year 4. None of the other effects were suggested as being important in the model except for the random effects of teacher and coach, with ICCs for two measurements of the same coach of 0.044 and two measurements of the same teacher of 0.46. We found no evidence that changes in the CPS score over time were related to changes in the ITCOP responses (<emph>p</emph> = 0.32).</p> <hd id="AN0134564965-24">Coaching variables that explain changes in teacher mathematics content knowledge</hd> <p>The teachers in the study began the project with average MKT scores just slightly above the mean of the nationally normed sample (the mean of the participants was 0.107, with a standard deviation of 0.962). We saw improvements in teachers' MKT scores over the course of the project, ending 0.433 standard deviations above average as seen in Fig. 4. Once again, we are concerned with whether or not changes in the coaches' measures explain these changes to teachers' scores.Boxplots with means (<emph>circles</emph>) and 95% confidence intervals for the teacher MKT scores by year of study, 1-5, MKT units are standard deviations above or below average from original MKT instrument construction</p> <p>PHOTO (COLOR)</p> <p>Table 8 displays the results of our model for how changes in coach measures explain changes in teachers' MKT scores. CPScentered scores (<emph>p</emph> = 0.002) and CSIcentered scores (<emph>p</emph> = 0.015) showed strong evidence of relationships with changes in teacher MKT scores, both exhibiting positive effects. For a 2-SD change in the CPS variation over time, the MKT score was estimated to increase on average by 0.13 standard deviations of MKT. For the same increase in the CSI variation over time, the teacher MKT mean was estimated to increase by 0.1 on the MKT scale. These relationships were also detected in our Year 3 and Year 4 analyses, indicating that this effect of coaching can be seen in a 3-year time span. None of the other effects were suggested as being important in the model except for the random effects of teacher and coach, which estimated the ICC of two measurements of the same coach of 0.072 and two measurements of the same teacher with the same coach of 0.556.</p> <p>Marginal evidence was found for a relationship with CPSmean (<emph>p</emph> = 0.054) and even weaker evidence for CMKTmean (<emph>p</emph> = 0.090) in the same model that controls for a variety of outside influences. While there was some evidence from the CMKT mean effect (CMKTmean) that high-scoring coaches were associated with high-scoring teachers, which could represent an undetected school effect or a recruitment bias (since coaches selected the teachers they would coach for this project), these <emph>p</emph> values provided limited evidence of these effects.</p> <p>We detected some evidence of an effect of CPSmean on the teacher MKT responses, with a <emph>p</emph> value of 0.054 and the estimated relationships negative. There was limited multicollinearity in the predictors in the model, so that was not a likely cause of the reversal of the effect from expectations. We found this puzzling. This could reflect limitations in our instrument and the possibility that sufficient variation was not observed. Or, perhaps, coaches with a better understanding of coaching texts (high CPS) tended to select teachers with lower MKT (e.g., struggling teachers). Such a hypothesis is supported by the fact that improvements in CPS (CPScentered) corresponded to improved average teacher MKT scores, although, based on the data, explanations of why CPSmean might be negatively related to teacher MKT scores are purely speculative.</p> <hd id="AN0134564965-25">Intensity</hd> <p>We were surprised that our analysis did not detect a relationship between our construct of coaching intensity and most of the teacher variables. Because our primary analysis defined intensity as the sum of total annual minutes in a pre- or post-lesson conference, we were worried that we may have missed an effect. Our qualitative approaches in the study, which included examination of data collected through interviews and focus groups, closely followed methods explicated by Miles et al. (2014). Through this analysis, which identified common themes using data displays and matrices, we found evidence that some coaches were working with teachers in a variety of ways that did not align with the EMC definition of coaching. For example, some coaches were holding larger-audience professional development sessions and informal follow-ups, and some coaches were at times "coaching" the teacher during the lesson observation with neither a formal pre-lesson conference nor a post-lesson conference. These coaches would ask the teacher questions and offer suggestions as the students performed individual or group work.</p> <p>As a post hoc analysis, we replaced our measure of coaching intensity with a measure of the number of coaching sessions in mathematics reported and reran the analysis as described above. We then replaced this measure with the number of coaching sessions in mathematics that were reported as involving number and operation. In both cases, the overall picture did not change. The relationships discussed above remained intact, and these alternative measures of intensity did not explain changes in teacher measures. We had no way of keeping track of the other types of professional development, such as larger group trainings, so these additional impacts were not considered.</p> <hd id="AN0134564965-26">EMC coaching professional development</hd> <p>We exercise caution in discussing the effects of the EMC coaching professional development on participating coaches' effectiveness. Our professional development was designed to ensure variation in participating coaches' knowledge and practice measures across years. Our professional development was not designed for a traditional causal study.</p> <p>Despite those qualifications, we feel it is important that we report on the effects of our professional development that might inform future studies. With one exception, we were unable to detect increases in participating coaches' measures when compared to the "cross-over" control cohort. While the cohorts always exhibited higher mean scores after receiving the professional development course, both in mathematics and in coaching, the comparison cohort did as well. Simply stated, in general, all of the participating coaches became more knowledgeable and felt more skillful over time.</p> <p>The single exception is in the CSI scores—the self-reports of coaching skills. Here, we did detect improvements that are tied to the randomly assigned professional development cohort. In Year 3, Cohort 2, which was the first to receive the coaching professional development, improved their CSI scores by 0.28 units on average (<emph>p</emph> = 0.005, SE = 0.1), over their Year 2 scores. This compares to an estimated increase of 0.038 units (<emph>p</emph> = 0.72, SE = 0.105) for Cohort 1, which had yet to receive the coaching professional development. Comparing these changes across cohorts, Cohort 2 improved 0.245 units over Cohort 1 (<emph>p</emph> = 0.089, SE = 0.143), giving moderate support for a cohort effect of the coaching professional development. This finding gives us reason to believe that the coaching professional development supported improvements in some coaching variables and that these improvements can be observed in measures at the teacher level.</p> <hd id="AN0134564965-27">Discussion</hd> <p>We set out to understand how and in what ways variations in coaching knowledge and practice over time explain coaching effectiveness. Our hypothesis was that as individual coaches improved in identified knowledge domains, skills, and practices as defined by the EMC project, these improvements would translate into improvements in coaching effectiveness as defined by the teacher measures.</p> <p>Our analysis supports the hypothesis that improvements in some types of coaching knowledge, skills, and practices can be isolated as explaining coaches' effects on teachers' improvements. As coaches reported improved coaching skills, these higher scores explained improvements in all of the teacher measures: mathematics knowledge, mathematics teaching practices, and self-efficacy. We found evidence that as coaches reported practices that more aligned with the practices recommended by the coaching texts we selected, teachers' MKT scores also improved. We were unable to detect relationships between improvements in coaches' mathematics knowledge and mathematics coaching effectiveness.</p> <p>The EMC study had a variety of limitations that should be considered when reflecting on the findings reported here. A major limitation is the intensity of coaching, which decreased over the course of the project and thereby decreased the sample size. By Year 5, many of the original coach participants were no longer operating as coaches due to district personnel changes beyond our control. This is a consequence of studying school-based coaches hired by their districts. While this approach is in contrast to the approach taken by other coaching studies that have used project-supported coaches, school-based or otherwise, we suggest that this aspect of the EMC study makes our conclusions more robust in that they reflect how coaching is enacted by school districts. We made efforts to take the decreasing coaching intensity into account. We also recognize that the quality of a coach's interactions with teachers might be more predictive of effectiveness than the time spent with teachers, but the EMC study does not address this.</p> <p>A second limitation is the instrumentation used to measure coaching practices and skill. These were project-developed instruments because there were no existing instruments available. Different measures of coaches' practices and skills based on other coaching texts might explain changes in teachers' knowledge, views, and practices. Moreover, the four coaching texts we referenced regarding development of this instrument are not research-based in the sense that it is not a research finding that coaches must follow these models to be effective. Thus, our measures of "conforming" or "not conforming" are in reference to this literature alone. There may be more effective models in the current coaching texts than the ones we selected. Instruments based on coaching models in other coaching texts might produce different findings than the findings reported here.</p> <p>A third limitation is related to the control variables for coach and teacher for outside professional development. We used a binary measure of whether or not the participants had outside professional development in a given year. We had no way of measuring the quality of the professional development experiences. Even though the participating coaches and teachers reported the amount of time spent in outside professional development, including courses, we were uncomfortable using the total number of hours reported in our models, because these experiences may vary significantly in quality and type. A more detailed measure of the quality of professional development might produce different results in a similar study.</p> <p>Finally, we situate this study within what is currently known about coaching as enacted by individual schools that hire their own coaches. Our work offers future researchers empirical evidence for types of coaching knowledge, skills, and practice that can explain coaching effectiveness. As future studies explore causal relationships between coaching knowledge, skills, and practice and teacher change, the EMC results can inform research design by offering evidence on domains that demonstrate relationships to teacher change.</p> <p>We found no evidence that improvements in coaches' MKT scores explained teacher change, and we remain surprised by this result. This non-result may be due to limitations in our research design. But, at least one coaching model relies heavily on teacher reflection (i.e., Costa and Garmston 2002) as the mechanism for teacher improvement, which contrasts with models that present coaches "teaching" mathematics to teachers (e.g., West and Staub 2003). These differences raise questions about the degree to which mathematics knowledge is critical to effective mathematics coaching. We do not assert that a link between improved coach mathematics knowledge and effective coaching does not exist, but we wish to document that we looked for and did not find this link in our study. We find it important to document this discrepancy between what we expected to find and what we did find, in order to avoid contributing to publication bias in results about mathematics classroom coaching.</p> <p>These results suggest that as coaches' self-assessment of efficacy for coaching increases, the teachers with whom they work increase their use of standards-based classroom practices, exhibit greater feelings of self-efficacy, and improve in their mathematics content knowledge. These are important results about the knowledge that mathematics classroom coaches need and leads to further questions about the relationships between content knowledge and coaching performance and between coaching and teaching.</p> <p>Further research is warranted to clarify the result that the project looked for, but did not find, associations between improvements in coaches' and teachers' mathematical content knowledge for teaching. At what level is a coach's content knowledge sufficient?</p> <p>Further research is also warranted concerning the nuanced ways that coaches interact with their teachers that are not formally defined as part of the coaching cycle. Our study highlights that coaching intensity is a complex variable, one not easily quantified because it is so heavily dependent on interpersonal interactions between a coach and a teacher. A post-lesson discussion is an important part of the coaching cycle and is recommended by all coaching texts on which we rely. Is this component an essential part of the coaching cycle? If a coach incorporates that reflection into the next pre-lesson coaching session, is that sufficient?</p> <p>Our results support assumptions that improvements in some types of coaching knowledge, skills, and practices explain some aspects of teacher improvement. Though the sample of coaches was recruited rather than selected at random, the diverse nature of the school districts whose coaches participated in the study, combined with the way that the project studied mathematics coaches in the varied contexts in which coaching is being enacted in those school districts, provides evidence that the results from this study are useful in understanding coaching in varied settings.</p> <hd id="AN0134564965-28">Acknowledgements</hd> <p>This material is based upon work supported by the National Science Foundation under Grant 0918326. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.</p> <hd id="AN0134564965-29">References</hd> <ref id="AN0134564965-30"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Agresti A, Analysis of ordinal categorical data, 2010, 2, Hoboken, NJ, Wiley</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> Ball, D. L. (2012). Tackling inequity by teaching to teach: Focusing on high leverage strategies. Presentation at Fall 2012 CAEP conference, Arlington, VA. 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Greenwood</p> </aug> <nolink nlid="nl1" bibid="bib189" firstref="ref4"></nolink> <nolink nlid="nl2" bibid="bib193" firstref="ref6"></nolink> <nolink nlid="nl3" bibid="bib49" firstref="ref10"></nolink>
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  Data: Springer. Available from: Springer Nature. 233 Spring Street, New York, NY 10013. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-348-4505; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
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  Data: This study investigated relationships between changes in certain types of coaching knowledge and practices among mathematics classroom coaches and how these explain changes in the attitudes, knowledge, and practice of the teachers they coach. Participants in this study were 51 school-based mathematics classroom coaches in the USA and 180 of the teachers whom they coached between 2009 and 2014. The participating coaches were recruited from schools that hired their own coaches independently from this research project. This study found evidence that improvements in coaches' use of practices recommended by particular coaching models are related to improvements in teachers' mathematical knowledge for teaching. The study also found that improvements in coaches' self-assessment of their own coaching skills are related to improvements in teachers' mathematics content knowledge for teaching, mathematics teaching practices, and attitudes about self-efficacy for teaching mathematics. The study did not detect relationships between changes in coaches' mathematics knowledge and changes in teachers' knowledge or practices.
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      – SubjectFull: Elementary School Teachers
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      – TitleFull: Variations in Coaching Knowledge and Practice That Explain Elementary and Middle School Mathematics Teacher Change
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          Dates:
            – D: 01
              M: 02
              Type: published
              Y: 2019
          Identifiers:
            – Type: issn-print
              Value: 1386-4416
          Numbering:
            – Type: volume
              Value: 22
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: Journal of Mathematics Teacher Education
              Type: main
ResultId 1