A Guide to Mathematics Coaching: Processes for Increasing Student Achievement


Ted H. Hull, Don S. Balka & Ruth Harbin Miles

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  • Chapters
  • Front Matter
  • Back Matter
  • Subject Index
  • Part I: Preparing the Foundation

    Part II: Mathematics Coaching Model

    Part III: Continuing the Work

  • Copyright

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    List of Figures


    Mathematics is the new literacy. Studies show that achievement in mathematics is strongly linked to career opportunities and personal economic success (National Mathematics Advisory Panel, 2008). To ensure bright futures for all students, mathematics learning, understanding, and achievement must increase. Under pressure to improve mathematics achievement, district leaders, campus leaders, and mathematics teachers are vulnerable to grasping at quick fix ideas that may waste financial resources and time. Unfortunately, there is no magic elixir or silver bullet. Increasing mathematics achievement is a rigorous and challenging goal that will take concerted effort and time, yet it can be reached.

    Mathematics coaching shows promise as an effective method of changing teacher practice and improving student achievement. Mathematics coaches are positioned to see and impact the daily teaching practice of classroom teachers. Yet they also are positioned to see the much broader picture of what is happening schoolwide and at the district level. They are able to view the content organization and delivery both horizontally and vertically. They are positioned to be an important component in improving student learning. However, the roles, responsibilities, and duties of mathematics coaches are not always clear to coaches and the teachers who work with them.

    If mathematics coaches turn to current research to better understand their roles and to determine a direction for their work, they will be disappointed. The research that is available focuses primarily upon either principals and effective leadership strategies or generalized content coaching recommendations. The guidance and suggestions offered are not specific to mathematics, nor are they presented as part of a developmental process that has been thoughtfully designed.

    Confusion about their roles and the lack of a developmental process are key challenges faced by mathematics coaches. These challenges are exacerbated if school leaders have launched a coaching initiative with little preparation or planning for success through supportive internal structures. Coaches, formerly strong and effective classroom teachers, occupy a unique place in the education system, a place that is sometimes defined by what they are not. Coaches are not classroom teachers, supervisors, coordinators, or administrators. Even more, they may not actually “belong” to any particular school, nor do they “belong” to a central office. In all likelihood, messages about what they are not have already been made vividly clear to coaches in both obvious and subtle ways. What has failed to be communicated is who mathematics coaches are and what they can do. Instead, many coaches have been left to their own devices to figure out their job—where to work, who to work with, what to do, and how to actually increase student learning.

    Effective coaching requires a range of skills, strategies, techniques, and practices. It is much more than any one method, such as conducting demonstration lessons for classroom teachers to emulate behind closed doors. Coaches tackle the necessary work of removing the walls that isolate teachers and initiating processes of productive collaboration. The good news is that professional, adult contact is valued and appreciated by most teachers. They want and need feedback on their instructional performance if they are to be effective, and they seek to constantly improve. Rather than offering feedback that is primarily evaluative, mathematics coaches offer supportive, nurturing, and practical guidance. Providing this support and information is a job that is tailor-made for coaches. Even so, entering classrooms and working with teachers poses challenges. Coaches may wonder “How can I build trust with my teachers? How do I get them to welcome me into their classrooms? How do I provide meaningful feedback? What is my responsibility in working with reluctant or difficult teachers?”

    While countless books on effective leadership have been published in recent years, most of these address leaders in line authority positions, such as principals and superintendents—those with the power to hire or fire, reward or punish, and promote or demote. Other staff members, such as coordinators and supervisors, have lesser decision-making authority. Mathematics coaches hold a unique position as leaders. Their primary responsibility is to work with classroom teachers to increase student learning by implementing decisions made by leaders in line authority positions. Mathematics coaches influence instruction by building trusting relationships, challenging ineffective instructional practices, supporting teachers as they learn new practices, providing meaningful and focused feedback, and implementing manageable, effective improvement processes. As any practicing mathematics coach knows, the role is complex and challenging.

    Yet coaches can successfully initiate effective mathematics teaching and focus on critical coaching actions that will close the achievement gap and lead to equity. Engaging teachers in the processes of clarifying and aligning the curriculum and planning and teaching effective lessons will help transform mathematics classrooms and ensure equity.

    This book was written to offer manageable, practical advice to mathematics content coaches in order to clarify, define, and fulfill their very important role. The primary audiences for this book are mathematics coaches; those who work with or supervise mathematics coaches, such as principals and central office personnel; and those making districtwide or school decisions about mathematics coaching. A developmental, sequential process for engaging in the work of mathematics coaching is presented. The process provides coaches with a leverage point—a place from which to begin the work with teachers and have an immediate impact on student achievement, thereby providing opportunities for more effective coaching contact in the future. The information presented here is intended to help clarify expectations and to increase the likelihood that mathematics coaching will have the desired impact on student learning.

    Each chapter builds on the guidance and recommendations of the previous chapter toward a high-quality mathematics program. This process incorporates the National Council of Teachers of Mathematics' principles of equity, curriculum, teaching, learning, and assessment as described in Principles and Standards for School Mathematics (2000) as well as the leadership principles advocated by the National Council of Supervisors of Mathematics and published in The PRIME Leadership Framework: Principles and Indicators for Mathematics Education Leaders (2008).

    The knowledge, skills, and complementing actions that are addressed throughout the chapters of the book are separated into three parts:

    • Preparing the Foundation
    • Mathematics Coaching Model
    • Continuing the Work
    Chapter Descriptions

    Chapter 1 defines a mathematics coach as an individual who is well versed in mathematics content and pedagogy and who works directly with classroom teachers to improve student learning of mathematics. In order to fulfill this role, mathematics coaches need to possess certain characteristics, including knowledge of adult learning styles and strategies, group dynamics, and social norms. They must understand data acquisition, analysis, interpretation, and application. And they need to know about confidentiality and trust-building.

    In Chapter 2, “Bridging From the Present to the Future,” we show coaches how to assess the current state of schools and classrooms on an improvement continuum, and we offer a framework for the development of a high-quality mathematics program.

    The content of Chapter 3, “Building Rapport With Teachers,” provides practical advice to help coaches establish and build trust and rapport with classroom teachers. Coaching scenarios demonstrate how coaches can respond effectively in a variety of situations. Examples are drawn from real-life situations that coaches are likely to encounter.

    Chapter 4, “Focusing on the Curriculum,” provides a process by which coaches may initiate working with teachers in a nonthreatening way that directly and positively impacts student learning and achievement. Ensuring that all students have the opportunity to learn is an important part of the work of coaches and a critical point in moving toward equity.

    In Chapter 5, “Implementing the Curriculum as Designed,” coaches are shown how to gather data about the extent to which teachers are implementing the mathematics curriculum and how to provide feedback to classroom teachers. An aligned curriculum forms a solid foundation and leads to the next step in the process, planning and coteaching lessons.

    Chapter 6, “Planning and Coteaching Lessons,” focuses on the implementation of effective classroom instructional strategies to increase student learning. By helping teachers plan effective lessons, coaches are able both to assess what teachers are currently able to do and to encourage adjustment in lessons, so instruction is more research based and provides for active student participation. As lesson planning with teachers continues, it becomes quite natural for coaches to become coteachers for selected lessons. This joint planning reduces the stress levels and purposely moves coaches into classrooms.

    Chapter 7, “Making Student Thinking Visible,” provides strategies for helping teachers move from a traditional, less effective model of mathematics instruction to an alternative method of instruction that incorporates high-effect, research-based instructional strategies and helps teachers better understand student thinking and respond accordingly.

    “Analyzing and Reflecting on Lessons,” discussed in Chapter 8, follows from collaborative planning and coteaching. Coaches initiate a lesson analysis process by discussing their part of the lesson, thereby modeling the process of reflecting upon student actions and evidence of student learning.

    By this stage of the process, coaches will have established professional working relationships and a high degree of trust and rapport with teachers. This degree of rapport and trust lays the groundwork for “Charting Long-Term Progress,” discussed in Chapter 9. Coaches, in order to stay focused on critical elements of change, draw upon trend data. Teachers are developing a true sense of efficacy and can see a relationship between their actions in the classroom and student learning. Teachers also are better able to consider areas for improvement. This connection and sense of efficacy are vitally important to sustaining change initiatives, and they complete the Mathematics Coaching Model.

    In Chapter 10, “Working Within the Education System,” we examine how mathematics coaches build collegial partners and groups for planning, analyzing, and reflecting, actively assisting teachers and leaders in establishing both formal and informal networks.

    The book ends with Chapter 11, where coaches are given strategies and encouragement to continue “Sustaining Momentum” for change. This chapter examines processes that support adult change and institutional change. We discuss the reality of fluctuations in change: spurts of improvement followed by brief periods of stagnation or even decline. We encourage coaches to focus on the bigger picture of trend changes rather than daily fluctuations of individual behaviors.

    The end of the book is actually a beginning, since the process described is highly recursive. We encourage coaches to repeat the process, cycling through the activities and recommendations, at increasingly higher levels of performance. We challenge coaches to critically analyze where they are in terms of communication and understanding about their job, to think about the rapport they have established and where more attention is needed, and to use trend data to determine which areas of the mathematics curriculum are still misaligned. Coaches may do the same for planning, teaching, and analyzing. The improvement cycle does not end, but spirals upward.

    The recommendations and actions contained within this book reside within the control of mathematics coaches and may, therefore, be immediately implemented. Through their positive, supportive actions, mathematics coaches can bring hope and encouragement to classroom teachers and play a critical role in advancing mathematics achievement for all students.


    Corwin gratefully acknowledges the contributions of the following reviewers:

    • Cheryl Avalos, Mathematics Consultant, Hacienda Heights, CA

    • Joyce Fischer, Assistant Professor of Mathematics, Texas State University, San Marcos, TX

    • Diane Kinch, Secondary Mathematics Specialist, Pomona Unified School District, Pomona, CA

    • Renée Peoples, K–5 Mathematics Coach, Swain County Schools, Bryson City, NC

    • Gail Underwood, Mathematics Coach, Grant Elementary School & Field Elementary School, Columbia, MO

    • Lois Williams, Regional Faculty, Mary Baldwin College, Staunton, VA

    About the Authors

    Ted H. Hull completed 32 years of service in public education before retiring and opening Hull Educational Consulting. He served as a mathematics teacher, K–12 mathematics coordinator, middle school principal, and director of curriculum and instruction as well as project director for the Charles A. Dana Center at the University of Texas in Austin. While at the University of Texas (2001–2005) he directed the research project Transforming Schools: Moving From Low-Achieving to High-Performing Learning Communities. As part of the project, Ted worked directly with district leaders, school administrators, and teachers in Arkansas, Oklahoma, Louisiana, and Texas to develop instructional leadership skills and implement effective mathematics instruction.

    Ted is a regular presenter at local, state, and national meetings. He has written numerous articles for the newsletter of the National Council of Supervisors of Mathematics (NCSM), including “Understanding the Six Steps of Implementation: Engagement by an Internal or External Facilitator” (2005) and “Leadership Equity: Moving Professional Development Into the Classroom” (2005) as well as “Manager to Instructional Leader” (2007) for the NCSM Journal of Mathematics Education Leadership. He published “Teacher Input Into Classroom Visits: Customized Classroom Visit Form” in the Texas Mathematics Teacher (2006). Ted was also a contributing author for publications from the Charles A. Dana Center: Mathematics Standards in the Classroom: Resources for Grades 6–8 (2002) and Middle School Mathematics Assessments: Proportional Reasoning (2004). He is an active member of the Texas Association of Supervisors of Mathematics and served on the NCSM board of directors as regional director for Southern 2. Ted lives with his wife, Susan, in Pflugerville, Texas.

    Don S. Balka, a former middle school and high school mathematics teacher, is professor emeritus in the mathematics department at Saint Mary's College, Notre Dame, Indiana. During his career as an educator, Don has presented over 2,000 workshops on the use of manipulatives with elementary and secondary students at national and regional conferences of the National Council of Teachers of Mathematics (NCTM), state mathematics conferences, and inservice training sessions for school districts throughout the United States. In addition, he has taught classes in schools throughout the world, including Ireland, Scotland, England, Saudi Arabia, Italy, Greece, Japan, and the Mariana Islands in the South Pacific. Don has written over 20 books on the use of manipulatives for teaching K–12 mathematics and is a coauthor of the Macmillan K–5 elementary mathematics series, Math Connects. He has served on the boards of directors of the NCTM, NCSM, and the School Science and Mathematics Association. Don resides with his wife, Sharon, in LaPaz, Indiana.

    Ruth Harbin Miles coaches rural, suburban, and inner-city school mathematics teachers. Her professional experience includes coordinating the K–12 mathematics teaching and learning program for the Olathe, Kansas, public schools for over 25 years; teaching mathematics methods courses at Virginia's Mary Baldwin College and at Ottawa, Mid America Nazarene, St. Mary's, and Fort Hays state universities in Kansas; and serving as president of the Kansas Association of Teachers of Mathematics. She represented eight midwestern states on the board of directors for NCSM and has been a copresenter for NCSM's leadership professional development national conferences. Ruth is the coauthor of Walkway to the Future: How to Implement the NCTM Standards (Jansen Publications, 1996) and is one of the writers for NCSM's PRIME Leadership Framework (2008). As co-owner of Happy Mountain Learning, she specializes in developing teachers' content knowledge and strategies for engaging students to meet high standards of achievement in mathematics. Ruth resides with her husband, Samuel, near the Blue Ridge Mountains in Madison, Virginia.

  • Resource A

    Statistical Definitions

    • closed response: A test question with only one correct answer.
    • common assessment: A test item administered at an established time to every student in the same course or grade.
    • criterion-referenced test: An assessment designed to reveal what a student knows, understands, or can do in relation to specific objectives. Criterion-referenced tests are intended to identify strengths and weaknesses in individual students in terms of knowledge or skills.
    • free response: A task with multiple paths to explore and multiple solutions to the answer.
    • multiple choice: A test item with a controlled number of responses from which the student must select an answer.
    • norm-referenced test: A standardized assessment designed to place a student or group of students in rank order compared to other test takers of the same age and grade.
    • open-ended response: A performance task in which students are given a stimulus or prompt and then asked to communicate a response. Tasks may be more or less open depending on how many restrictions or directions are included.
    • reliability: A measure of the constancy of scoring outcomes over time or over many evaluators. A test is considered reliable if the same answers produce the same score no matter when and how the scoring is done.
    • rubric: An established set of criteria for scoring or rating students' tests, portfolios, or performances.
    • validity: A measure of how well an assessment relates to what students are expected to have learned. A valid assessment measures what it is supposed to measure and not some peripheral features.
    Definitions From:
    • Hart, D. (1994). Authentic assessment: A handbook for educators. Menlo Park, CA: Addison-Wesley, pp. 105–114.

    Resource B

    Mapping Principles and Classroom Vision

    A Reflective Guide

    Coaches, in order to move forward, are to occasionally look back. At the beginning of this book, two big ideas were presented—the National Council of Teachers of Mathematics (NCTM) and National Council of Supervisors of Mathematics (NCSM) principles and the vision of the futuristic, high-quality mathematics program as depicted in an effective mathematics classroom. Obviously, these two ideas are highly correlated. Achieving the principles also achieves the vision of the desired classroom. It is now time for mathematics coaches to reflect upon their progress, take stock of successes and misses, establish a new baseline, and prepare to repeat the entire process outlined in this book.

    The job, of course, never ends. There are always things to learn or techniques to perfect. The reflective process is based on actual experience of the coaches, data gathered through various sources, and the professional opinion of the coaches and, perhaps, some of their colleagues.

    The reflective process, at this point, is intended to focus on the NCTM and NCSM principles of

    • equity,
    • curriculum,
    • teaching,
    • learning, and
    • assessment.

    These principles are combined with the traits of a desired high-quality mathematics program found in Chapter 3, “Bridging the Present and Future.” These traits are

    • teachers are empowered,
    • curriculum is implemented as designed,
    • multiple instructional strategies are employed,
    • students are actively engaged, and
    • assessment is varied.

    Matching the principles to the traits provides the following pairings:

    • Equity is directly related to the empowering of teachers,
    • curriculum is directly related to implementing the curriculum as designed,
    • teaching is directly related to employing multiple instructional strategies,
    • learning is directly related to actively engaging the students, and
    • assessment is directly related to using a variety of assessment methods.
    Equity Related to Teacher Empowerment

    The equity principle is in place when all students are provided the opportunity to learn the appropriate mathematics curriculum. This opportunity is directly related to the content, but it also is related to inclusive instructional techniques that bring students into the learning process. The empowerment of teachers is related to efficacy—the belief that all students can learn meaningful mathematics and the teacher has the ability to actually teach it to the students.

    Evidence: This “can-do” attitude of empowered teachers is demonstrated by their willingness to do whatever it takes for students to stay engaged and to learn. Coaches will no longer hear such statements as “they can't,” or “they won't.” They will hear statements such as “That didn't seem to work too well,” “What else can I try?” and “What did you do?”

    Curriculum Related to Implementation

    The curriculum principle is in place when every teacher is teaching the mathematics content identified by a district and directly aligned to state-mandated standards.

    Evidence: Mathematics coaches will routinely see teachers referencing their curriculum documents while planning. Teachers will be highly conversant with state standards and the organization of district documents.

    Teaching Related to Multiple Strategies

    The teaching principle is in place when teachers are meaningfully using a variety of instructional strategies. Furthermore, the strategies are intentionally selected based upon the content to be taught, the misconceptions students tend to have, and how the strategies help make student thinking visible and overt.

    Evidence: The strategies are included in lesson plans, shared with other teachers, and routinely discussed as to the degree of effectiveness. Strategies are constantly being revised and refined. Mathematics coaches will see evidence of this in coplanning and coteaching sessions. Coaches will hear the conversations at a rich level of analysis during group meetings.

    Learning Related to Student Engagement

    The learning principle is in place when students are active participants in the classroom. Students are talking and discussing, gathering and analyzing data, making predictions, and justifying their thinking.

    Evidence: Teachers are still leading the classroom, but not necessarily from the front of the room, and certainly not by monopolizing the conversation. Coaches will see evidence of this while coteaching. They will also hear evidence during planning sessions and group meetings.

    Assessment Related to Variety

    The assessment principle is in place when teachers select and use a variety of ways in which to determine the degree of student learning. Teachers recognize that students need to demonstrate what they know in order to make the knowledge clearer to the students themselves and clearer to the teacher.

    Evidence: Students routinely show their work and explain their thinking. Formative and summative assessments are in constant use. More importantly, teachers actually used the collected data to inform their instructional practices. When they discuss teaching and teaching strategies, they have evidence of which practices worked and which did not in the given situations. Coaches will see evidence of the data and progress of learning. They will also be involved in helping teachers reflect upon their practices.

    Teacher Beliefs

    Although more difficult perhaps to identify, mathematics coaches are also to seek evidence of changing teacher beliefs. These beliefs, again from Chapter 3, are as follows:

    • All students can learn mathematics.
    • Teachers can teach students.
    • Mathematics is conceptual and developmental.

    These beliefs are intertwined with the principles and with the initial principle of equity. As beliefs change, coaches will hear a change in how teachers talk about the students and the mathematics.

    Evidence: Teachers will take much more ownership and responsibility for learning. Learning mathematics will no longer be attributed to the “math gene” or “good parenting” but rather to classroom instruction the students were given.

    The Cyclic Nature

    The cycle repeats. Prepared with the above information, mathematics coaches establish a new baseline. Teachers, coaches, principals, schools, and the students have moved from point A to point B on the continuum improvement scale.

    Coaches are to be honest but not overly self-critical. There are many factors to be considered. The educational system is still operating.

    Resource C

    Transparent Communication + Positive Power Actions + Domains of Power:

    Power of Position—Granted by District, Chain of Command
    • Analyzing current conditions—have you been up front with the staff?
    • Making clear expectations—have you actually told the staff what you want done?
    • Sharing power—have you included staff in the decision process?
    • Modeling—have you used the desired behaviors or techniques?
    Power of Knowledge/Expertise—Achievements, Experience
    • Sharing knowledge—have you provided the staff with the data?
    • Sharing rationale—have you explained your reasoning and the basis for your decision, concern, or need?
    • Providing follow-up training—is it needed by group or individual?
    Power of Reward/Recognition—Tacit or Overt
    • Verbal—have you specifically recognized individual staff efforts? (overt)
      • —have you specifically recognized group efforts? (tacit)
    • Written—have you specifically recognized individual staff efforts? (overt)
      • —have you specifically recognized group efforts? (tacit)
    Power of Persuasion—Communication, Solicit Support
    • Conducting one-on-one—have you talked to the individual alone?
    • Conducting small group—have you talked with small groups of staff?
    Power of Peers—Using Leaders, Critical Mass
    • Supporting peer coaching—have you used leaders to bring on or support staff?
    • Building collaborative energy—have you asked small groups to offer recommendations?

    Resource D

    Leadership Styles That Get Results by Daniel Goleman from The Institute for Management Excellence, located at http://www.itstime.com/mar2003.htm.

    Coercive Leadership Style: “Do what i Tell you.”
    • Demands immediate compliance.
    • May be used in a crisis to address a problem or a difficult employee.
    • Has a negative impact on the climate.
    • Power of coercion.
    Pacesetting Leadership Style: “Do as i do, now.”
    • Sets a high standard of performance.
    • Used to get quick results from a highly motivated and competent team.
    • Has a negative impact on the climate.
    • Power of position.
    Affiliative Leadership Style: “People Come First.”
    • Creates harmony and builds emotional bonds.
    • Used to heal rifts in a team or to motivate people during stressful circumstances.
    • Has a positive impact on the climate.
    • Power of reward/recognition.
    Democratic Leadership Style: “What do you think?”
    • Forges consensus through participation.
    • Used to build buy-in or consensus or to get input from valuable employees.
    • Has a positive impact on the climate.
    • Power of peers.
    Coaching Leadership Style: “Try This.”
    • Develops people for the future.
    • Used to help employee improve performance or develop long-term strengths.
    • Has a positive impact on the climate.
    • Power of knowledge/expertise.
    Authoritative Leadership Style: “Come with Me.”
    • Mobilizes people toward a vision.
    • Used when changes require a new vision or a clear direction.
    • Has a strongly positive impact on the climate.
    • Power of persuasion.


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    Williams, B. (1996). Closing the achievement gap: A vision for changing beliefs and practices. Alexandria, VA: Association for Supervision and Curriculum Development.
    York-Barr, J., Sommers, W., Ghere, G., & Montie, J. (2001). Reflective practice to improve schools: An action guide for educators. Thousand Oaks, CA: Corwin.
    Youngs, P., & King, M. (2002). Principal leadership for professional development to build school capacity. Educational Administration Quarterly, 38 (5), 643–670. http://dx.doi.org/10.1177/0013161X02239642
    Zepeda, S. (2004). Instructional leadership for school improvement. Larchmont, NY: Eye on Education.

    Corwin a SAGE Company

    The Corwin logo—a raven striding across an open book—represents the union of courage and learning. Corwin is committed to improving education for all learners by publishing books and other professional development resources for those serving the field of PreK–12 education. By providing practical, hands-on materials, Corwin continues to carry out the promise of its motto: “Helping Educators Do Their Work Better.”

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