Variations in coaching knowledge and practice that explain elementary and middle school mathematics teacher change


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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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.

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Corresponding author

Correspondence to David A. Yopp.

Appendix: Project-developed instruments

Appendix: Project-developed instruments

Coaching practices survey

The following survey is designed to capture information about your practices related to instructional coaching. Please indicate your level of agreement with each of the following statements, from 1 (not at all descriptive of my coaching) to 7 (very descriptive of my coaching).

  Not at all descriptive of my coaching Mostly not descriptive of my coaching More not descriptive than descriptive Equally not descriptive and descriptive More descriptive than not descriptive Mostly descriptive of my coaching Very descriptive of my coaching
1 2 3 4 5 6 7
I collect students’ mathematics work from a teacher’s classroom to guide our coaching conversations        
When decisions about mathematics instruction are being made, I ensure that the decision-makers interpret research literature accurately        
I coach teachers on needs that I observe in the teacher, even when the teacher is unaware of these needs        
I have difficult conversations with teachers, when necessary, about mathematics misconceptions they hold        
I always make sure that coaching conversations with mathematics teachers are grounded in the mathematics content        
I meet with the principal to discuss the school’s vision for mathematics instruction        
I encourage teachers to include, in each lesson they teach, summaries of what students learned or discovered        
I provide feedback to teachers about whether or not the school is meeting its vision for mathematics instruction        
I try to provide the teachers I coach with an understanding of how the mathematics they teach supports learning beyond the grade level they teach        
I ask the principal what he or she believes the mathematics teachers’ needs are        
I encourage the teachers I coach to reflect on similarities and differences among mathematics topics in the curriculum        
I help teachers plan their lessons        
I help teachers identify consistencies and inconsistencies between their own practices and the practices recommended by the National Council of Teachers of Mathematics        
I work with principals or other administrators to form a clear message to teachers about effective mathematics instruction        
I help teachers reflect on discrepancies between espoused beliefs and actual practices        
I reflect on state assessment data to identify curriculum areas that need to be strengthened        
I use student work when coaching mathematics teachers        
I provide feedback to the principal about whether or not the school is meeting its vision for mathematics instruction        
I encourage teachers to set personal improvement goals for mathematics instruction        
When a teacher complains about the school’s vision for mathematics, I ask the teacher about her or his vision for mathematics        

Coaching skills inventory

For each of the following 24 questions, please rate the items on a scale from 1 to 5 based on how effective (or confident) you are with the various coaching functions, with 1 meaning not at all effective (or confident) and 5 meaning very effective (or confident).

  Not at all effective Very effective
1 2 3 4 5
I. Mathematics content and mathematics-specific pedagogy
 1. How effective do you feel coaching teachers on mathematical content?
 2. How effective do you feel coaching teachers on mathematics-specific pedagogy? (Examples of mathematics-specific pedagogy include but are not limited to incorporating inquiry, discovery or investigative mathematics into lessons, and incorporating problem solving and conceptual understanding into lessons.)
 3. How confident are you with the mathematics taught at the grade levels that you coach?
 4. How confident are you with the mathematical reasoning behind mathematics taught at the grade levels that you coach, meaning the understanding of why we teach it, how it relates to other mathematics topics, and why it is valid?
 5. How effective do you feel coaching teachers on number sense and computation topics relevant to their classrooms?
 6. How effective do you feel coaching teachers on creating and using mathematical applications and connections for/in their mathematics classes?
 7. How effective do you feel coaching teachers on incorporating mathematics conceptual understanding into their lessons?
 8. How effective do you feel coaching teachers on incorporating genuine mathematical problem solving into their lessons?
 9. How effective do you feel coaching teachers on incorporating investigative, inquiry-based, or discovery-based mathematics learning into their lessons?
 10. How effective do you feel coaching teachers on engaging students in mathematical abstraction or sense-making?
II. Student-centered pedagogy coaching
 11. How effective do you feel coaching teachers on general (not necessarily mathematics-specific) pedagogy? (Examples of general pedagogy include but are not limited to engaging students, use of questioning strategies, use of cooperative learning, and classroom management.)
 12. How effective do you feel coaching teachers on encouraging student participation?
 13. How effective do you feel coaching teachers on using strategies to increase student collaboration or dialogue among students?
 14. How effective do you feel coaching teachers on creating an environment where students listen to one another?
 15. How effective do you feel coaching teachers on the use of cooperative learning?
 16. How effective do you feel coaching teachers on classroom management?
III. Building coaching relationships
 17. How effective do you feel observing lessons and giving teachers feedback?
 18. How effective do you feel creating environments where teachers reflect openly on their instructional practices?
 19. How effective do you feel helping teachers set goals and objectives aimed at improving their instruction?
 20. How effective do you feel creating an environment of open discussion and constructive criticism with teachers?

See Exhibit 1.

Exhibit 1 Coaching skills inventory factor structure

Internal Reliability. Internal reliability of the scales on the CSI, as presented in Exhibit 2, reveals a high level of reliability for each of the three scales.

Exhibit 2 Reliability analysis for the CSI

Descriptive Statistics from the EMC CSI Data Set. Means and standard deviations for the three scales derived from the CSI are presented in Exhibit 3.

Exhibit 3 Means and standard deviations for scale items on the CSI (N = 61)

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Yopp, D.A., Burroughs, E.A., Sutton, J.T. et al. Variations in coaching knowledge and practice that explain elementary and middle school mathematics teacher change. J Math Teacher Educ 22, 5–36 (2019).

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  • Classroom coaching
  • Professional development
  • Mathematics education
  • Mentoring
  • Teacher knowledge
  • Teacher practice