Cultivating a Math Coaching Practice: A Guide for K–8 Math Educators
 Chapters
 Front Matter
 Back Matter

 Introduction
 Chapter 1: Observing, Studying, Analyzing, Planning: Preparing to Coach
 Case: Moving between Models
 Math Activity: Models, Fractions, and Percents
 Focus Questions Activity
 Chapter 2: Discerning and Responding: Coaching in Real Time
 Case: Analyzing Multiplication
 Math Activity: How do you Know?
 Focus Questions Activity
 Chapter 3: Strategic Coaching: GoalCentered Modeling in the Classroom
 Case: “It's 30 Less and 90 More”: A Case about Listening to Children's Ideas
 Math Activity: Angles and Angle Measurement
 Chapter 4: Reaching a New Teacher: Math as the Conduit
 Case: A Case of Coaching: Multiplication and Division Journal Entries
 Math Activity: Looking beneath the Surface
 Focus Questions Activity
 Chapter 5: Preparing for Thoughtful Dialogue
 Transcript 1
 Transcript 2
 Transcript 3
 Chapter 6: Purposeful Planning and Facilitation
 Case: Coaching in a Group: Moving from 1:1 to 1:?
 Focus Questions Activity
 Planning Activity: Facilitating Group Learning
 Chapter 7: Refining and Reimagining One's Coaching Practice
 Case: Learning about Counting, Learning about Coaching
 Focus Questions Activity
 Planning Activity: Cultivating Collaborative Study
 Chapter 8: Cultivating Relationships with Administrators and other Leadership Colleagues
 Case: Crafting an Invitation: Shifting from Isolation to Inclusion
 Focus Questions Activity
 Planning Activity: Considering Collaboration and Communication
 Chapter 9: Taking the Lead as a Teacher of Teachers
 Case: Encountering Venus
 Math Activity: Exploring Story Problems
 Focus Questions Activity
 Chapter 10: Maintaining a Focus on Mathematics
 Case: Struggling to Keep Math at the Center
 Focus Questions Activity
 Chapter 11: Framing the Connection between Coach and Teacher Goals
 Case: Unsatisfied in the Seminar
 Focus Questions Activity
 Planning Activity: Meeting the Challenge
 Chapter 12: Examining the Role of Authority in Coaching
 Case: Claiming Authority
 Focus Questions Activity
 Planning Activity: Two Posters Activity

Dedication
This book is dedicated to Lily, Lila, and Chloe—the three wonders of my universe.
Copyright
Copyright © 2009 by Corwin
All rights reserved. When forms and sample documents are included, their use is authorized only by educators, local school sites, and/or noncommercial or nonprofit entities that have purchased the book. Except for that usage, no part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.
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Library of Congress CataloginginPublication Data
Morse, Amy.
Cultivating a math coaching practice: a guide for K8 math educators/Amy Morse. p. cm.
“A joint publication with the Education Development Center.”
ISBN 9781412971058 (cloth)
ISBN 9781412971065 (pbk.)
1. Mathematics—Study and teaching (Elementary school) 2. Mathematics—Study and teaching (Middle school) 3. Mathematics teachers—Training of. I. Title.
QA11.2.M67 2009
372.7—dc22
2009004461
This book is printed on acidfree paper.
09 10 11 12 13 10 9 8 7 6 5 4 3 2 1
Acquisitions Editor: Dan Alpert
Associate Editor: Megan Bedell
Production Editor: Cassandra Margaret Seibel
Copy Editor: Codi Bowman
Typesetter: C&M Digitals (P) Ltd.
Proofreader: Wendy Jo Dymond
Cover Designer: Scott Van Atta
Foreword
Cultivating a Math Coaching Practice: A Guide for K–8 Math Educators is a book by, for, and about coaches and their supervisors. Amy Morse—who, in the 1990s, worked as a math coach for an urban school district and since then has supported the professional development of math coaches in various school systems throughout the country—recognized a growing need and has created this book to provide it.
Many schools are setting ambitious goals for mathematics instruction. Rather than exclusively focusing on speedy and accurate calculation, teachers are now being asked to structure activities and facilitate discussions that help students make connections among mathematical concepts, various types of representations, and realworld contexts. In addition to the content of number and operations, young, elementary students are to delve into the study of geometry, data, and early algebraic concepts. There is much greater emphasis on mathematical understanding in several mathematical realms, in addition to computational fluency.
To achieve a practice that embodies these ambitious goals, teachers must develop a qualitatively different and significantly richer understanding of mathematics than most now possess. They must come to see learning as the result of students' active efforts to make things comprehensible for themselves, and they must come to recognize common places where children need to stop and work through conceptual issues implicated in the mathematics to be learned. Teachers must acquire new pedagogical habits and routines, and where school systems have adopted new mathematics programs to align with new goals, they must learn to use a new curriculum effectively. The demands on teachers are considerable.
Even where strong professional development programs are in place, translating new insights into classroom practice is a difficult, and often a frustrating, process. The many demands on teachers' time, conflicting district policies, and the pressure of standardized tests may all reduce the actual effect that professional development has on instruction. Implementation efforts may be put off indefinitely. Initial efforts that do not meet with instant success (the norm rather than the exception) are often abandoned. A more profound and longerlasting impact can be realized when programs integrate classroom support with professional development.
In recognition of this, many school systems have created the position of mathematics coach or mathematics specialist. The mathematics coach provides regular classroom consultation offering support as changes are introduced into the classroom, sustaining teachers' learning in the context that matters most.
However, with a new role established in the school system, many new questions arise: When coaches enter a classroom, what do they look for and how do they interact with teachers and students? What and how do coaches communicate with teachers before and after lessons? What are the responsibilities of mathematics coaches? What authority do they have? With whom do they negotiate? And most important, what must a coach understand to coach effectively?
Cultivating a Math Coaching Practice is an important resource for coaches and their supervisors who ponder such questions. Filled with cases written by mathematics coaches, it provides images of coaches in classrooms, coaches in meetings with teachers, coaches in discussion with principals, and coaches sitting alone working on issues of their practice. And it suggests some answers.
First, coaches must understand that the practice of coaching requires thoughtful reflection and continued learning. They must be prepared to learn mathematics, analyze student thinking, examine curriculum, and consider alternative teacher moves.
Second, coaches must work on developing strategic moves for teacher learning: How do you bring teachers' attention to mathematics content, to significant aspects of student thinking, and to the impact of their actions in the classroom? How do you help teachers develop the skills and habits of preparing for a lesson, of eliciting student thinking, and of analyzing student work?
Third, coaches must navigate the administrative structures of a school. What structures are in place that provide teachers and coaches the opportunity to reflect together? What niches can be used to open up a space for communication between teacher and coach? How can a coach negotiate with school administrators to align structures and policies more closely with the goals for mathematics instruction?
Cultivating a Math Coaching Practice provides a mechanism to initiate the processes of learning, reflecting, and navigating required for effective coaching. That is, this book is material for the professional development of mathematics coaches and brings readers right into the heart of the central issues of practice. For new coaches, it will help them prepare for the situations they are about to confront. For experienced coaches, it will support them in deepening their practice, by helping them become more effective in their work. For coaches who meet together, Cultivating a Math Coaching Practice can support the development of a culture of collegial inquiry, sharing and discussing dilemmas that arise in their work. For those who read the book on their own, it provides a connection to the community of coaches engaged in this important enterprise.
Acknowledgments
My heartfelt thanks go to the teachers, math coaches, math specialists, and math leaders across the country who have contributed to and helped shape all parts of this book. I especially appreciate the contributions of the Boston Public School elementary coaches with whom I spent many “growing” years and whose inquisitiveness, thoughtfulness, and tenacity I admire. To the generous and brave coachauthors from across the country whose work is represented in these chapters and whose writing has kept me company over the past three years, I thank you.
Linda Davenport, Liz Sweeney, Lisa Y., Carol, Arusha, Francesca, Nicole, Marisa, Nancy, Megan K–P, Virginia Bastable, Angie Stephens, and Kay Caruso, your contributions sing loud and clear. Deborah Schifter, Polly Wagner, and Jan Rook, it is your encouragement and your many kindnesses—and the wisdom you so freely share—that makes all the difference.
I am deeply indebted to the many years of support from ExxonMobil Foundation for the projects focused on cultivating case writing as a tool for professional development for coaching. This book also represents work supported by a National Science Foundation grant ESI010228. Any opinions expressed herein are those of the author and do not necessarily represent the views of either Foundation.
Publisher's AcknowledgmentsCorwin gratefully acknowledges the contributions of the following reviewers:
Carl Ameen, Math Coach Silvio Conte Community School Pittsfiled, MA
Beth Skipper, PreK–3 Math Coach
National Council of Teachers of Mathematics (NCTM)
Jenny Sue Flannagan, Director of Martinson Center for Math
Virginia Beach City Public Schools, VA
Elizabeth Marquez, Math Assessment Specialist
Educational Testing Services
Elizabeth Alvarez, Citywide Math Coach
Colman Elementary School Chicago, IL
Kathryn Chval, Assistant Professor of Mathematics Education
University of Missouri
About the Author
Amy Morse is a national leader in designing and providing professional development for math coaches and district leaders responsible for coaching programs. In 1997, Amy joined Education Development Center (EDC) where, as a project director, she works on teacher and coach development programs. In this capacity, she worked for many years with the Boston Public Schools Elementary Math Department to design and facilitate the professional development for the district's elementary math coaches. At present, she directs Professional Development for Math Coaching institutes for EDC's Math Leadership program (http://www.mathleadership.org). Amy travels extensively, working as a consultant to school districts in various stages of implementing a math coaching strategy and is the director of the Center for the Professional Development of Coaching and Math Leadership at EDC.

Facilitation Guide
Dear Facilitator,
The Facilitation Guide is designed to support you as you lead these professional development sessions. In each chapter of the guide, Session Goals, Case Description, and Session Overview sections provide images of what the session is about and how it unfolds. In addition, the Facilitation Notes include indepth descriptions of each agenda activity for each session and comments about materials to prepare. The Facilitation Guide was written as a companion document; in this way, it describes comments participants might make in wholegroup discussions, details of math ideas participants might offer in a session, and effective responses from facilitators. The notes also include insights about trajectories of coaching practice, important ideas to emphasize, and commentary about coaching that should provide useful background for making facilitation decisions in the session.
Successful facilitation of the cases and related activities is predicated on the idea that facilitators carefully prepare for each session by reading the case, doing the math activity or planning activity, and by considering your own responses to each of the focus questions. Some facilitators have found that investing the time to actually write out responses to focus questions, taking notes on the math activities, and, then, keeping a journal of what participants say and do in the session, provides a detailed and personal resource.
At some point, you might ask the coaches in your group to consider writing their own cases for group discussion. In a structured professional development setting, coaches often take turns offering cases; individual coaches might share two cases during a school year. One way to begin is to ask the group to reflect on cases from these materials that have been influential and to talk together about what makes a case effective. It is helpful to work with the coaches to compare a variety of types of cases, for instance Case 1, Case 2, Case 5, and Case 10. Ask coaches to consider the similarities between the cases, the particulars that make these cases opportunities for learning, and the specifics of what makes a case readable and clear. Coaches will discover that recording specific dialogue, considering and representing the mathematics of students' ideas, and writing honestly about puzzles of practice and considerations for next steps are elements of productive cases. It is important to respect those who are written about and to protect these stories of teaching and coaching practice. For this reason, make a habit of using pseudonyms for teacher, school, and student names—even in your own coaching group. A noteworthy feature (and reward) of case writing and sharing in a group is the high value the group puts on confidentiality and trusting relationships that allow for open discussion. Discussing these norms before sharing matters. Facilitators are in a position to protect writers by doing two things: One is to ask for their cases ahead of the meeting so that you can create focus questions that will center discussion on the most salient features of the case and the second is to collect the copies of shared cases before the close of the session. And remember that, for many, writing and exposing one's practice takes courage.
I continue to be fascinated by stories of coaching and am deeply connected to issues of facilitation. I welcome the opportunity to hear from you if you have questions, ideas, or advice—or a case to share about your own experiences!
amorse@edc.org
Chapter 1 Facilitation Notes: Observing, Studying, Analyzing, and Planning: Preparing to Coach2.5Hour Session
Session Goals Examine representations of rational numbers
 Explore relationships between fractions and percents
 Consider what it means to prepare to coach—examining classroom mathematics and students' ideas to build content knowledge
 Strategize about using school structures, or pressing for structures, that provide opportunities for coaching and for teacher learning
Case Description: Moving between ModelsThe Moving Between Models case offers an example of a reflective practitioner who studies the curriculum, the central mathematical idea in the classroom, the students' work as they engage in a new concept, and the implications for her coaching. Lila, the author, wrote the case midway into her second year as an elementary school coach. Specifically, she raises questions, both mathematical and pedagogical, about the relationship between fractions and percents and the issues that influence children's reasoning as they begin to work with ways to represent rational numbers.
The coach uses her case writing to help her learn more about the mathematics she encounters in the classroom and, at the same time, to help her analyze her next coaching moves and supports for the teacher.
The case offers an interesting perspective on the role of coaching. All coaching does not happen in the moment; certainly, there is a significant amount of preparation coaches undertake for individual classroom visits and meetings or workshops. The case raises a question for discussion: What is the nature of preparing to coach?
Session OverviewParticipants begin the session by reading the case. Before a case discussion, they next engage in smallgroup work on a series of math problems designed to highlight the complex thinking involved in establishing relationships between fractions and percents. The activity is followed by a facilitatorled, wholegroup discussion, during which participants share their models for comparing fractional amounts and for describing percents. The discussion is an interactive one with participants and the facilitator both asking questions and exploring the variety of models created in small groups. At this point, the participants rejoin their small groups to work on a Focus Questions Activity designed to highlight both the student thinking and the coach's perspectives on the classroom visits. During the ensuing wholegroup discussion, participants will have an opportunity to discuss the coach's role through the eyes of the coachauthor and to think together about the implications for their own coaching and developing practices.
Materials Needed for the Session Create a chart of the agenda and the Session Goals
 Provide graph paper, plain paper, colored pencils, and counters or small cubes
 Provide chart paper and colored markers for displaying math work and participant ideas
Session Agenda Introduction: 5 minutes
 Case Reading: 20 minutes
 Math Activity in Small Groups and WholeGroup Discussion: 70 minutes
 Focus Questions Activity and WholeGroup Discussion: 55 minutes
Introduction
Introduce the session by describing the agenda, briefly reviewing the Session Goals and the context for the Moving Between Models case. Explain that this case is an example of journal type of case where the coachauthor uses writing as a way of analyzing her work and considering next steps.
Case Reading
Reading the case is the first activity in the session. Encourage participants to write questions or comments in the margins and note sentences or paragraphs that resonate for or confuse them. A participant once remarked, “A case is a piece of work to do, not just a story to read!” This is a helpful analogy to offer the group.
Math Activity and WholeGroup Discussion
Before a case discussion, participants work in small groups to solve a series of math problems designed to highlight relationships between fractions and percents.
The facilitation of the math discussion is an opportunity to support coaches' consideration of expressing parts of a whole, in a variety of ways, and the models that represent these amounts. The focus of the discussion is also aimed at developing an appreciation for the trajectory of these ideas and for the complex thinking that elementary students—and teachers—engage in when challenged to express fraction amounts as related to 100. Questions participants will consider include:
 How does one's understanding and visual images of 3/16 map onto a percent model? How are these models related?
 What sort of thinking do children need to do in order to ground themselves in this new form of expressing an amount?
These issues are central to the discussions of both the small and whole groups.
The math activity consists of three word problems. The directions ask individual participants to “think through these problems and draw models on paper,” thus encouraging coaches to consider the representations and visual models that help support their understanding of the problems. It is important for the facilitator to circulate and encourage individuals to draw and share their thinking in these smallgroup discussions.
Note: You may find it useful to ask participants to solve only Question 1, discuss it for a brief moment in their groups, and then have the wholegroup discussion with models displayed before you move on to Question 2.
The first math problem gives coaches an entry place, and their resulting drawings will serve as the markers of the conceptual distance between the representations and ideas of early fraction work and the math ideas embedded in the final problem. Prepare for the wholegroup discussion by moving from group to group during the math activity and noting the range of participants' models for “5/8 of the horses” described in the first problem. You will likely find a variety of models ranging from a randomly arranged set of 8 symbols with 5 looped into a collection to a rectangle or circle representing a unit of 8 with 5 sections shaded. In the whole group, when you discuss the participants' responses to Problem 1, scribe or ask participants to come up and draw the range of models for 5/8. Keep these drawings posted so that you can refer back to them as you are bringing the math discussion to a close. It is useful to raise the idea that the model that makes use of a single unit where 5 portions are shaded requires a developed sense of organization than the one that shows 8 horses drawn with 5 crossed off. We begin here to appreciate a child's developing sense of number and reasoning. Question 2 steps up the inquiry for the group. If we now have a sense of how to describe the fraction 5/8, how do we express 5/8 as a percent? What does it look like? Participants who are less experienced with array or number line models may find it challenging to tackle creating the drawings and visual models for this activity. As the participants work through this problem, again make note of models and drawings that display a variety of approaches for the wholegroup discussion. Participants are likely to create grids of 10 × 10 and shade in in several different ways.
Figure FG1.1The coach who drew this representation (Figure FG1.1) described the way she used her model to help her solve the problem.
I used 50 as a landmark for . So here I filled in the 50 squares and knew that took care of . If I filled in 25 more squares that would be , so I divided 25 by 2 and got 12 squares and one more half of a square. That makes equal to 62 and percent.
Another participant drew an array of 100 but thought about it area differently (Figure FG1.2).
I had to make this arrangement of 100 squares equal to 8 horses. I cut the last two columns (with the shaded Xs) off and redistributed them evenly across the first 8. That meant that in each of the new 8 columns there are now rows. Then, I could calculate that 5 columns contain a total of squares or !
One participant simply cut the array of 100 into 4 quadrants, shaded 2 quadrants and of another, thus illustrating 25% + 25% = 50% or and adding another squares brings the total to 62.5%.
Figure FG1.2Another participant drew a circle, divided it into quarters, and then marked 8 even sections. She shaded one quarter () and labeled it 25% with a J, another quarter with a J, and one more . She then added all the J sections to equal 62.5%.
You will likely see number line models illustrating 0 to 100 and use of 25 and 50 as landmarks again relating to and as a way of illustrating a comparison between and . Another is a line labeled 0 through . A participant created a chart below this model that read: , , and . She wrote, “If jumpers, then jumpers which means 50% + 12.5% = 62.5%. If line number models are not explored in participants' work, it may be useful to suggest that participants try one.
One participant created a table representing ratios of horses to jumpers. She explained,
I kept doubling to a certain point, and then I knew I would be over 100 so I just added what I knew about 32 horses to the 64 horses, and then divided by 2 what I knew about 8 horses.
These models are all valuable for exploring the group's ideas. A selection of the different models that come out of the group's work should be displayed one at a time so that the whole group has a chance to consider the ideas inherent in each model, where the is, where the are, and how the coach who worked with the model used it as a tool for solving the problem.
Note: It is important to not only discuss the relationship between to 62.5% but also to have participants see the relationships between the models themselves.
As you bring this discussion to a close, refer back to the original drawings of and ask people to comment on the distance between students' working knowledge at the stage where describing “a part of one whole” is perceived using fractions and to where the discussion rests now with regard to the translation of this same fraction into percents. You will want coaches to leave with an appreciation for this complex work as well as with new ideas about the significant value of using—and discussing—visual models and representations. Participants should now have multiple representations to reflect on, more content knowledge to consider, and be ready to engage in the inquiry about the potential for student learning.
Question 3 is for a coach or small group of coaches to consider on their own. Inevitably, the time allotted and the focus of attention is on the first two problems so Question 3 gives individual participants and chance to further an investigation. The coaches' math work in Questions 1 and 2, and the ensuing discussion, is sufficient for the purpose of this case set activity.
Focus Questions Activity and Case Discussion
The focus questions are designed to help coaches engage in discussion both about the mathematics children wrestle with and the pedagogical implications for teaching. Because participants have had a chance to work through the related math activities for their own content learning, they are situated now to have considered discussion about the analytical questions and insights Lila raises in her case. In small groups, coaches discuss a series of questions with focused attention on young students' development of fraction and percent models and ideas with a more finetuned lens on the issues.
Have participants reference line numbers in the smalland wholegroup discussions. Referring to line numbers and then waiting until everyone locates the passage, helps keep participants accountable to what is actually in the case and not slip into a discussion that can become overly general. As you circulate among small groups, encourage coaches to take the time to consider each student's response in Question 1. You might suggest that each participant write out her response to Question 2 before discussing it openly in the small group. Ask individuals to sketch a list of ideas in response to Question 4 and to take time to write down their questions and ideas with regard to future work for Amari and her classmates in Question 5. It is important that coaches learn to, and are encouraged to, give themselves time to articulate their thoughts and bring words to their confusions or questions.
Considering the logic and reasoning beneath student responses helps coaches to focus on where students are in their thinking, as opposed to dismissing “wrong” ideas or underappreciating potentially robust ones. In the wholegroup discussion, as participants are talking about the ideas present at the beginning of this classroom episode, remind them that these are ideas likely to be present in any fifth grade class. Not only is this discussion an opportunity to consider carefully the students Lila writes about, it is also an opening for coaches to enrich their understanding, in each of their own coaching contexts, of how students, naïve about these ideas, will likely respond.
Following the thread of Amari's thinking offers insights into the mathematical issues. On one hand, Amari has quite sturdy ideas about factors of 42 and how they are related to each other. In fact, her statement that generalizes this notion is not only appropriate; it is also an indicator of how Amari thinks. One might interpret her remarks throughout the case as based on the premise that mathematics is ultimately about ideas and about making sense of them. Asking coaches to comment on the generalizable nature of Amari's idea—even applying algebraic notation—illustrates the type of important reasoning work that Amari can do.
But we read on to find Amari in a perplexed state. Teddy, Amari, and their classmates offer images of students sorting through ideas. At this stage, Lila leaves off from writing about the classroom and begins to puzzle out how models influence children's ideas of fractions and of percents. From line 82 on, Lila considers what it means to understand and apply the model of “6 people out of 10 people wearing stripes” to a model of “60 shaded squares in an array of 100.” Reading her question as, “What does it take to really see the link between the two?” we see she is asking what students need to understand, what models are useful images, and how students work with the ideas to come to reason and successfully apply these ideas.
While coaches may not yet (as Lila herself has not) come to conclusions, the point of the case discussion is to open up these questions to the whole group and to consider hypotheses and possible next steps based on this session experience and the opportunity for coaches to share and consider multiple perspectives.
Participants need an opportunity to talk about the implications of this case for their coaching. They have had a chance to discuss the mathematics of the case and to puzzle through the questions the author has raised; now, it is appropriate for the facilitator to help the group consider their new insights and what these mean to their own practices. Support this consideration by asking the group to note the ways the coach author considers her own next steps. Ask the participants to consider structures and opportunities afforded in their own schools. Ask, “If you have ideas and questions about how children come to learn and engage in the mathematical ideas we've discussed, how does this get incorporated into your own coaching?”
Save a few minutes at the end of the session for participants to respond to exit card questions. You might ask, “How was it to do math with colleagues? What did you learn about the way you work and the ideas you have?” Ask coaches to describe one way the case and case discussions have influenced their thinking.
Chapter 2 Facilitation Notes: Discerning and Responding: Coaching in Real Time2Hour Session
Session Goals Analyze multiplication and using numbers of groups to compare amounts
 Consider the use of gradelevel meetings to support teachers' content knowledge
 Explore coach decision making and improvisational moves in the role
 Examine the role of coach and what it means to navigate a “teacher of teachers” role
Case Description: Analyzing MultiplicationLisa, the coachauthor, first encounters a fourth grade classroom where the students are no longer working on the school curriculum; rather, they spend the math class hour working on drill and practice. She listens carefully to the teacher who expresses her dismay at her students' work and the tensions she feels about the upcoming state test. Her concerns drive her to revert to work that focuses on solving many problems with little, if any, emphasis on explaining or exploring reasoning and development of strategies for solving problems. Rather than intervene, Lisa takes it all in as she weighs her coaching choices. Lisa presents an image of a coach who sees her role in multiple dimensions. She describes a teachers' meeting that, through skillful coaching, responds to multiple issues. The case sheds light on how a coach perceives her role with respect to teachers' learning.
Session OverviewThe session begins with a math activity, “How Do I Know?” Participants work through the same ideas that both students and teachers investigate in the case. The activity and ensuing discussion help prepare coaches to focus on the strategies described in the case as the coach navigates a single coaching day.
Participants read the case and discuss focus questions in small groups. Next, in the whole group, coaches explore such concepts as preparation for coaching, drawing on resources for teacher professional development, using school structures to enact coaching goals, and examining the role of the coach.
In discussing how Lisa considers her authority and her responsibilities, the coaches have a chance to consider why she makes particular choices and to analyze her decision making for the way it supports teacher learning. In this way, coaches are afforded opportunities to enrich their own ideas about ways to move in coaching and, too, to hear multiple interpretations from their coach colleagues. It is important for participants to explore these images of coach work; it is through analyzing and discussing these coaching stories that participants are able to reflect on and enrich their own images of their role.
Materials for the Session Create a chart of the agenda and the Session Goals
 Provide graph paper, plain paper, and colored pencils for exploring the math problems
 Obtain chart paper and different colored markers to display participants' math work
Session Agenda Math Activity in Small Groups: 15 minutes
 WholeGroup Math Discussion: 20 minutes
 Case Reading: 20 minutes
 Focus Questions Activity and WholeGroup Discussion: 65 minutes
Math Activity
Before coming together to discuss the case, participants work in small groups to solve a series of math problems designed to highlight important ideas the case writer raises. In addition to writing descriptive sentences, encourage participants to draw models that illustrate their reasoning. Explain that the purpose of the activity is for participants to analyze these problems and to consider the mathematical logic of their answers. There are several ways groups might work on problems. For participants with little experience working on adult math content, it can be useful for participants to work on one problem on their own and then discuss their ideas in small groups before they move to the next problem. This way, they bring new ideas and approaches to each of the successive problems. For more experienced participants, working on the problems alone before discussing them is one way they might challenge themselves to consider solving the problems in more than one way. Encourage discussion and collaboration in the small groups.
This collection of problems give participants an opportunity to consider the way the size of groups and the numbers of groups affect the total amount. As they work through the problems, make note of models and drawings that display a variety of approaches in preparation for facilitating the wholegroup discussion. Participants might use multiples of 10 to approximate total amounts. They may draw arrays such as 312 × 10 versus 300 × 9 to compare two different amounts. If coaches have not explored diagrams (models or story problems) for these problems, encourage them to find the size of the total based on an array model. You might encourage a participant to draw a model that shows the solution such that no calculation is necessary. While some participants might use arrays to compare size, one might draw arrays of just the “missing” or “extra” amounts and compare these. If participants are not aware of open arrays, the wholegroup discussion might be an opportunity to explore that idea. Some participants might use a chart or a graph to describe how amounts grow based on the number and size of groups. These models are all valuable for exploring the participants' ideas.
As you observe the groups working, listen for ideas or questions you can choose to highlight in the wholegroup discussion. While these problems lend themselves to considering groups and numbers in a group, models participants use to describe the operation may lead to a broader discussion of how students' understand—and come to understand—multiplication. Some examples of participants' thinking that hold promise for discussion may be similar to the following:
I enjoyed using my ideas to address multiplication concepts. My big “aha” came when I could see how these activities might help kids explore number sense (distributing, decomposing, landmarks.) For Question 2, I explored using representations a bit differently:
I didn't really care about the answer, just the numerical relationships. Looks a bit algebraic too, don't you think?
Another participant offered:
Considering the 13 × 62 problem, I thought about building towers of 13. I first knew that 13 × 3 is 39. So then 13 × 30 would be 390. That meant that double that many (or 60 towers) would be over 700 already. I solved the others in different ways, but this was my way into to the first one.
WholeGroup Discussion
The wholegroup discussion is an opportunity to support participants' investigations into multiplication and the relationships between groups and sizes of groups. The purpose of the discussion is also aimed at helping participants develop an appreciation for the power of arrays as expressions of multiplication and to examine models and other drawings by their colleagues.
Ask participants to describe the logic used to solve the first two problems. It will be evident that participants use what they know to decide on the approximate size of 13 × 62 and 39 × 22. Ask what ideas students would rely on to solve the problems. If participants understand that the numbers can be broken into separate problems and added together such as (10 × 62) + (3 × 62), they might use this approach to help them appreciate the size of 13 × 62. If no one has used array models to compare sizes of, perhaps, 40 × 22, 39 × 20, and 39 × 22 in Question 2, it would be important to ask the group to take a moment and try using an array model to “see” the closer amount. Bringing to the discussion the logic and the strategy behind participants' responses for Questions 3 and 4 will round out the wholegroup sharing.
The smallgroup and wholegroup discussions should leave participants with an appreciation for the power of models and representations of multiplication, as well as, new ideas about how to compare sizes of factor pairs. Participants should now have multiple representations to reflect on and more content knowledge with which to consider the potential for student and teacher learning. Questions participants will consider in the following small and whole groups include “What sort of thinking do coaches, teachers, and students need to do to ground themselves in multiplication and how it operates?” and “What models make sense as ways to express multiplication?”
Case Reading
Reading the case is the second activity in this session. Encourage participants to write questions or comments in the margins and note sentences or paragraphs that resonate for or confuse them. Sometimes participants appreciate reading the focus questions ahead of reading the case. Explain that this is an option.
Focus Questions Activity and WholeGroup Discussion
The case discussion focuses on two aspects of the writing, the mathematical ideas of the students' and the teachers' and the implications for coaching. In small groups, coaches discuss a series of questions designed to further explore multiplication and ways students understand the ideas of groups. Reference to line numbers in both the smallgroup and wholegroup support discussions that are grounded in the case and issues particular to the work at hand. As you circulate among the groups, listen for ways the participants discuss Tyrone's reasoning in lines 53–59. Appreciating his thinking and considering the mathematical ideas his logic rests on is an important type of discussion for participants to engage in.
In addition, the coaches focus on the teachers' work. They consider Ms. Birt's hypothesis that “the estimate will be closer if the factor changes by a smaller number.” Lisa, the coach in the case, gives the teachers an assignment to sort through Ms. Birt's hypothesis; now, the coaches have an opportunity to do this together in their small groups. The discussion in the whole group should also include reference to why Lisa was pleased with Ms. Birt's statement even though it doesn't prove to be true in all cases.
Finally, coaches are asked to work together to relate the mathematics of the students' work to future learning and to consider the kinds of work young students have done to be able to explore problems such as these. Listening to the participants ideas about this will offer you a window on how well articulated the connections are for your group.
The second part of the focus questions draws attention to the coach decisions and moves throughout the day. The exercise of exploring Lisa's ideas about Mrs. Martin's shift in pedagogy and content in her classroom both highlights ways coaches can think about what's beneath resistance or lack of confidence and ways one might attend to Mrs. Martin's needs. It also provides an opportunity to highlight the importance of careful consideration for teachers' behavior and concerns. Moving on to discuss the moves Lisa makes, in particular, will give participants an important chance to imagine a coach day and appreciate the judgments a coach makes in the moment. The discussion of how Lisa might characterize her role and how her perception of her responsibilities is likely to drive the decisions she makes is a most important conversation for participants and a big idea in coming to understand what coaches do and on what basis they choose the next steps in their work. It is in this conversation that participants have yet one more opportunity to examine the coaching role and the ways to leverage that role for teacher learning.
Chapter 3 Facilitation Notes: Strategic Coaching: GoalCentered Modeling in the Classroom2Hour Session
Session Goals Investigate defining “angle” and measuring angles in polygonal shapes
 Explore reframing coaching goals and strategies to align with new insights about teachers' beliefs about teaching and learning
 Analyze variations of “modeling” as a coaching strategy
 Analyze students' mathematical thinking from a fourth grade geometry class
 Learn about the design of collaborative coachteacher study with the math class as the primary site
Case Description: “It's 30 Less and 90 More”: A Case about Listening to Children's IdeasThe case, written by an experienced elementary and middle school coach, describes teacher and coach interactions based on fourth grade geometry ideas and students' work in class. The coach negotiates the teacher's requests and expectations and considers her own goals for the teacher's learning. As she works to frame a coaching experience that will get at the heart of important ideas of teaching practice, in the case, she lays out her own beliefs about guiding principles of teaching and learning. It is from these beliefs that she devises a structure for her collaboration with the teacher in the geometry class. The case presents an opportunity to “peer in” closely at student work and to analyze the logic of students' ideas as they explore angles and measurement. The case also stands as an example of strategic coaching, by capturing an image of core elements of the role as they are enacted in a single math class.
Session OverviewParticipants will begin the session with a smallgroup math activity exploring the central geometry themes of the case. Followed by a discussion about angles and measurement, the group reads the case and engages in a smallgroup Focus Questions Activity. The focus questions are designed so that coaches explore the same math ideas as they emerge in a classroom setting, examine the teacher's practice, and consider the implications for Tina's coaching. A wholegroup discussion then centers on two things: the student's ideas that may still need to be explored and more clearly understood and the ways the coach negotiates her role in this context. This wholegroup discussion also offers coaches a chance to air their own thoughts about modeling in classrooms. This common coaching strategy is portrayed in a new way in this case and the coach's skillful use provides a learning opportunity for novice and experienced coaches.
Materials for the Session Create a poster of the Session Goals
 Provide sets of plastic polygonal shapes or prepare sets of paper polygon shapes (see Figure 3.12 in the book)
 Supply plain paper, colored pencils, and graph paper
 Use chart paper and different colored markers to scribe participants' ideas
Session Agenda Introduction: 5 minutes
 SmallGroup Math Activity: 15 minutes
 WholeGroup Math Discussion: 20 minutes
 Case Reading: 20 minutes
 SmallGroup Focus Questions Activity: 30 minutes
 WholeGroup Case Discussion: 30 minutes
Introduction
Begin the session by offering a brief description of the agenda and the learning goals for the session. Set the stage for the math activity by explaining that these same ideas will be addressed in the students' work explored in this case about a fourth grade geometry class.
Mathematics Activity
Participants work in pairs or small groups to explore 90degree angles. Next, they choose two shapes and determine the exterior angles of each of the shapes based on information and measuring ideas from their work on the first problem. For the final problem, participants will determine the sums of both the interior and exterior angles of shapes and make summative comments about what they discover. As always, giving yourself time to explore with the shapes and, too, anticipating participants' responses to the activity ahead of the session will enhance your facilitation of the math activity.
WholeGroup Math Discussion
In almost every group, there will be participants with very little experience investigating angles and angle measures. Some will only have a memorized definition of what constitutes an angle, unable to rely on handson experience. A participant might offer, “When two lines meet each other, they form an angle” or “An angle is a measure of a turn.” While these definitions can be further refined (the figure formed by two line segments sharing the same endpoint is one version of a fleshed out definition), asking a participant to demonstrate “a turn” can be illuminating for many. It is useful for this wholegroup discussion to be a very visual one, supported by participants' drawings on the board so that the whole group is focused on the very same space. In one group, a facilitator asked a coach to draw an angle on the board. She next asked the group, “Where is the angle?” What appeared to be a simple question brought up confusions about the relationship of the lines, the site of actual measure, and others that were a part of that groups' new learning about angles. It is reassuring to participants to honor and sort through these confusions as they are directly related to the issue the coach brings to light for the teacher with whom she's working in this case. Having conversations about the way we perceive, or have to adjust our perceptions, of angles and measurement is the point of this math discussion. Some terms may come up such as “reflex angle” or “interior” and “exterior angle.” For the purposes of this discussion, a reflex angle is an angle greater than 180 degrees but less than 360 degrees. An interior angle is one on the inside a shape; these polygon shapes have one interior angle per vertex. If you add the measure of an interior angle and the contrasting exterior angle formed by one side of the polygon and a line extended from the adjacent side, together these two angles measure 180 degrees.
Reading the Case and Focus Questions
Participants will need at least 20 minutes to read the case carefully. Let the group know that the case discussion will focus on ideas from these three points of view. Remind them to separate three different lines of thinking: the coach's, the teacher's, and the students'. Ask participants, explicitly, to focus on sorting out the thinking of individual students and to underline or note sections where their own or the students' confusions surface.
Focus Questions Activity
The focus questions are designed to offer participants four lenses: understanding the students, understanding the teacher, coaching goals, and, finally, strategic opportunities for learning. The focus questions, discussed in small groups, provide time for participants to practice listening to and discerning early geometric ideas and to consider how to approach a debriefing discussion with the teacher after the lesson.
Case Discussion
Begin by asking participants to focus in on the students in the case. Ask how the small groups sorted out Bobby's confusion by looking at Figures 3.7 and 3.8 in the book. What hypotheses do they have about his question, “Aren't they both the same?”
In this discussion, participants will likely note that it took some time to make sense of the two sets of shapes. Some participants consider the triangles similar enough that Bobby cannot discern a difference between the two. Others offer that Bobby cannot distinguish a 90degree angle formed by the rhombus and the triangle in Figure 3.8 or that he is looking in the exterior angle formed to the left of the rhombus between the rhombus and the triangle, which indeed, are the same. One participant pointed out, “Just looking at the shapes and comparing Figures 3.7 and 3.8 so carefully makes me think of the kids who I don't pay close enough attention to. Because it says in lines 190–196 that Bobby “hadn't touched the shapes much,” I would have assumed that that's why Bobby says they're the same and not because he's genuinely engaged in this as a question.”
You might ask what tools the students are using to make decisions about the measurement of angles. Kelan, for instance, uses arithmetic by adding two 45degree angles and justifies by saying that each is half of 90. At the same time, some children (line 200) decide that the slim brown rhombus is 30 degrees because three same shapes create a 90degree angle. Others are using horizontal lines as a tool. In the case of Chloe and Emerald, the horizontal line becomes an essential tool for discerning that “it's 30 less and 90 more.”
After a discussion of the students' ideas, ask the group to consider, “What are the decisions that Tina is making in this case? Where are the decision points for the coach?” In sorting out these questions, participants can follow the ways the coach continues to evaluate the opportunities she has for coaching.
It is important to make the point that Tina is gathering information about the teacher with whom she's working, discerning learning goals for the teacher, and evaluating the scenarios and the resources she has to make progress. In this case, the coach suggests up front that the teacher spends more time on his math “teaching” and less on his students' “learning.” The coach uses this insight to highlight a tenet of her coaching—that she could spend more time on coaching strategies and behaviors but if they are not linked to the teacher's beliefs and ideas and knowledge, then what's the effect of the coaching? This case provides an example of a coach bringing the teacher's attention to the mathematics his students are working on and the ways they are engaging with it.
Ask the group how the coach chooses to respond based on what she's hearing from Mr. Gallagher. Ask them to use line numbers to point out the coach's moves and to discuss why she's making them. Participants often raise the issue of modeling; the teacher has asked the coach to model teaching. It is useful for coaches to have an opportunity to discuss this strategy for coaching. How does the coach in this case work with this idea? In what ways do the participants in the group understand the benefits of modeling? Do they use modeling for a particular focus? Under what circumstances is modeling helpful and when it is it, perhaps, detrimental? How do individuals in the group make choices about modeling? As the coach in this case explains, she steps through every window of opportunity to align her goals with the situation at hand and the developing interest on the part of the teacher. What does it mean to “step through windows of opportunity” in coaching?
Ask participants to talk about what Mr. Gallagher learned through the experience in his classroom. What would they have hoped for? Some participants might talk about individual students by noting, for instance, that Mr. Gallagher has a new appreciation for Chloe and how important it is that coaches bring to light students who might otherwise recede from the teacher's view. Others may point out lines 242–252, where Mr. Gallagher raises a new insight about moving about the class from student to student but not making mathematical connections between students' ideas, nor using this as information gathering to support a plan for a wholegroup discussion. This insight is an important one to cover in a discussion. If a participant doesn't raise it, you might ask how participants think about the issues Mr. Gallagher raises in this section. Ask, “What work would Mr. Gallagher have to do to make the connections between students' mathematical ideas?” Or ask, “What work would he have to do to conceive of a purposeful wholegroup discussion?” and, “If we agree that these are of high value in a teacher's math practice, what are the implications for coaching with these ideas in mind?”
Save time for exit card writing at the end of this session. Ask participants to reflect on what it was like to sort out these students' ideas; one thing they learned, in particular, from the coachauthor; and to describe how this insight will affect their coaching practice.
Chapter 4 Facilitation Notes: Reaching a New Teacher: Math as the Conduit2Hour Session
Session Goals Investigate ideas about division and explore related story problems
 Examine the logic in student thinking
 Consider coaching implications for engaging a hesitant teacher in the study of her students' ideas and a reexamination of the gradelevel mathematics
 Explore written communication as a tool for coaching
Case Description: A Case of Coaching: Multiplication and Division Journal EntriesA Case of Coaching is a set of journal entries written by a secondyear coach as she works to cultivate a successful coachteacher relationship with a new teacher. During her visit to the fourth grade math class, the coach, Ellie, is surprised to discover that her own ideas about division are less than solid. In preparation to coach well, the author sets about studying types of division and sorting through related story problems. She decides to use this example of a selfdirected investigation as an entrée to a collaborative study with the classroom teacher and, thus, begins a series of written communications that help form the basis for their working relationship.
Session OverviewThe session begins with a small group math activity focused on division and multiplication problems similar to the ones the case writer poses for her own investigation. Participants next work to uncover the logic in a fourth grader's response to a daily number routine, the same response that stumps the coach as she describes in her journal. A wholegroup discussion of the math follows during which the point is made that as a coach one has a new vantage point from which to study classroom mathematics and that continuing to learn math from this perspective is a facet of preparing to coach well. Participants read the case and work together in small groups on a Focus Questions Activity designed to highlight mathematical issues, coaching issues, and considerations of beliefs and attitudes of hesitant or resistant teachers. The session ends with a wholegroup discussion.
Materials for the Session Create a chart of session goals
 Provide about 150 connecting cubes per pair for the focus questions and math activities
 Obtain chart paper and different colored markers for displaying coaches' math work
Session Agenda Introduction: 5 minutes
 Math Activity in Small Groups: 15 minutes
 WholeGroup Math Discussion: 20 minutes
 Case Reading: 20 minutes
 Focus Questions Activity in Small Groups: 30 minutes
 WholeGroup Case Discussion: 30 minutes
Introduction
Begin the session by briefly describing the session goals and the agenda. Set the stage for the case reading by describing, in just a few words, the context for the coachauthored case. Let the participants know that before anything else happens, they will have a chance to dig into a math activity designed to surface ideas directly related to those described in the case.
Math Activity
Participants begin the session by working to solve a series of math problems. Remind the participants to talk for a few minutes about ways they most successfully work on math in group settings. Remind them to listen for each other's ideas and questions.
This math activity provides an opportunity for participants to work through the logic of three division and multiplication problems. In small groups, participants consider the reasons why the answers are correct or incorrect. In this way, the coaches considers students' ideas about breaking numbers apart in hundreds, tens, and ones to multiply and divide, just as students have learned to do with addition and subtraction. The first two problems are different versions of 48 ÷ 4. In one case, 48 is separated by 10s and 1s into 40 and 8. Then, 40 and 8 are each divided by 4. The results, 10 and 2, are then added together for a total of 12.
A participant might think of it this way: I want to put 48 cookies in bags of 4. First, I take 40 cookies and find that I can fill 10 bags with 4 cookies. Then I take the other 8 cookies and fill 2 more bags. I know that 10 bags and 2 bags gives me 12 bags with 4 cookies in each.
Another might act out the problem with cubes. She takes 40 cubes and 8 cubes and arranges them in two arrays each with one dimension of 4. She counts off 10 sets of 4 and 2 more sets of 4 and adds them for 12 sets of 4. For Part B, the coach follows the problem by separating 48 into two amounts, and this time, separates the divisor as well. 48 becomes 24 + 24, and each 24 is divided by 2. For the total, the next step is described as 12 + 12 = 24.
In working to figure out why the first strategy works and the second doesn't, the participants meet up against solutions that fourth grade students might offer while, at the same time, investigating both physical models and story contexts.
Next, participants examine a fourth grade student's idea that 17 × 17 = 149. In considering the student's logic—and this theme of breaking numbers apart and operating on them—participants again run into typical confusions about multiplying. In both cases, ask participants to draw pictures, use the cubes, and to consider story problems to illustrate their thinking. In the whole group, use several models to support the discussion. Ask the participants to consider Martina's response. Her comment highlights the idea that examining the operations and the similarities and differences between them is particularly useful work for students, just as it is for adults.
Case Reading
Encourage participants to write questions or comments in the margins and note sentences or paragraphs that are confusing or that resonate for them. This case carefully takes the reader through a thorough examination of the coach's ideas about division. More than just underlining or noting important sections of the case, press readers to get out paper and pencil and work the mathematics alongside the author.
Focus Questions Activity
The focus questions help participants dig into issues regarding; developing trusting and productive relationships for coaching; explorations of division and multiplication through the eyes of the students, the teacher, and the coach; and ways to consider the next steps for the teacher's learning.
Remind participants to reference line numbers for clarity in building their discussions. As you circulate among the groups, listen for ways the participants engage in this deliberate exploration of multiplication and division. These explorations will help you get a picture of how your participants understand these math issues for themselves. Listen for ways participants are responding to the coach's considerations of building trusting relationships, “telling,” and resistance to change. The coach in this case makes an interesting and somewhat unusual decision to share her math exploration quite explicitly with the teacher. Participants will have a variety of opinions about this move.
There is a tendency for people to react and respond to the math issues too quickly; for this reason, you might need to press people to take their thinking to a deeper level. One way to do this is to be explicit about this as you introduce the activity. You can say that sometimes we just don't press ourselves to investigate questions as thoroughly as we might, and in this way, we miss going beyond what we already know. Then ask the group to pay explicit attention to whether their own group has fully investigated each of the focus questions. You might hear participants say, “Division means groups of 17 and this isn't groups of 17,” and then move on or “I'd do multiplication arrays with this student so he can see 17 × 17 isn't 149.” These responses are along the lines of solutions and are not yet representative of investigations into the mathematical issues. Suggestions, such as to write a story with a descriptive context for the math problem, draw arrays to explore multiplication, and to apply the same strategies using different sets of numbers, can offer participants different ways to work through the math of the case.
WholeGroup Discussion
Begin your discussion by asking participants to take a few moments to consider lines 214–246 where the coach describes her work on the idea of sharing. She concludes, “So what I know about division and sharing tells me that this is not sharing the cards evenly between 17 friends.” Ask the group to sort this out. What has the coach figured out?
Next, explore the coach's reasoning behind the lines 247–261. Ask participants to demonstrate this line of thinking with cubes. How does the visual image shift in playing out the coach's work in lines 262–274?
In a discussion of these math explorations, one small group of participants shared the following problem context to illustrate their ideas: “Imagine 149 baseball cards in one room, and you send in 17 kids to share them. You could have 100 cards on one table and 49 cards on another. That shows you can break up the number you are dividing into. But if you think of 100 cards in one room and 49 in another, and then you send 10 kids to one room and 7 into another. Then 10 kids get 10 cards and 7 kids get 7 cards; that's not the same.”
The forays into sharing and partitioning explored in the case and focus questions open up useful ways of examining what Ellie, the teacher, and the students are wrestling with in learning about division. The coach's work is accessible and thought provoking. She has taken just a few student ideas from class and played them out in depth, both for her own learning and, also, so that she can join this teacher in an exploration of her students' thinking. You will want to step aside of the math at this point and explicitly make note that this work Ellie is doing represents one interesting and purposeful image of preparing to coach. As we also see in the chapter Moving Between Models, coaches prepare for their workday by examining the mathematics for themselves. It may be supportive, as the facilitator of the group, to acknowledge that learning how to prepare as a coach takes some time, but in many respects, it is not so different from the fundamentals of preparing for classroom teaching. Doing the math ahead of working in class—or with teachers—is a norm of good teaching. Coaches learn that seeing math activities and listening carefully to classroom discussions, coupled with paying close attention to students' and teachers' ideas from this new vantage point of a coach, often raises new questions, new insights, and new ways of thinking that are worthy of exploration.
The second aspect of Ellie's case deals with sorting out the pathways toward a relationship built on trust and a mathematical focus. Participants have a chance to talk about the issues of resistance the coach describes and to consider the choices Ellie makes in establishing a relationship with a teacher who might otherwise have remained isolated.
Sometimes, participants disagree with Ellie's choice to offer the teacher her letter. They ask, “Why would she want to tell all she knows about division?” or “Why not let the teacher figure it out for herself?” and “Hasn't she done just what we ask teachers not to do with their students?” And yet others might find Ellie's exploration helpful; one participant said, “The way Ellie laid out her thinking was helpful to me. She did things I never would have thought of doing. I still needed to do my own thinking and work alongside hers.”
After discussing the mathematics of the case, this can be a useful debate. Not only will the conversation afford you insights into your participants' ideas, but it also gives the group a chance to try on different versions of the case and to make decisions based on thoughtful dialogue. There is, of course, no one right move, and Ellie's choices are not meant to showcase “the way to do it.” The more important questions are, “How do you nurture teachers' curiosity about their students?” and “How do Ellie's coaching moves relate to her earlier reflections on ‘cracking open’ one's practice?” In the final portion of the group discussion, turn the group's attention to their own considerations of practice. Ask, “Given this scenario, what are the implications for coaching?” “What is it that you would want this teacher to learn?” and “Consider some ways of working with this teacher; in what ways are your next steps related to your goals for her learning?”
One facilitator of this case wrote, “The nextsteps question can be difficult for participants to answer, especially when you are asking the coaches to make connections between the next steps they suggest and the goals they have for the teacher—in light of what the teacher does or doesn't yet understand. Even though we talk about using student work to figure out what to do next, it isn't that easy or that commonly done. We do not have routines to follow that help us determine next steps.”
While participants may have some difficulty articulating appropriate next steps, you will likely also sense that it is agreed among the group that these are important questions to consider as a rule in coaching. They are questions that help coaches consider their own work with goals in mind. At the time, as the facilitator, it may feel less than satisfying that participants struggle with these issues and responses. The power of the questions is in the consideration of them rather than in participants' answers conjured up in the moment. Pressing on questions such as these establishes a habit of appropriate norms for coaching, new ways of thinking that can be incorporated in a coach repertoire over time.
Chapter 5 Facilitation Notes: Preparing for Thoughtful Dialogue2Hour Session
Session Goals Consider implications for coaching, collaborative relationships, and teacherlearning based on ideas about considered versus reactive coaching
 Examine a classroom transcript for mathematical ideas and teacherstudent interaction
 Explore number sense and addition strategies in the primary grades
Case Description: Considered CoachingThe Considered Coaching case examines one coach's responses to a math exchange between the teacher and her students in a second grade classroom. Rosa, the coach, has set out to more fully understand ideas about number sense. As a way of capturing these ideas as they play out in math class, Rosa, with permission from the teacher, brings her laptop to class and transcribes a brief round of mental math. As she types the students' and the teacher's dialogue in one column on the screen, she also records her reactions to what's happening in class in a second column. The title word “considered” becomes central in discussions about the transcript and the implications for coaching. Rosa, herself, explores the difference between her immediate take on the students' ideas about number sense and the teacher's facilitation and the later, more reflective and substantially more useful exploration of the transcript as she prepares for a thoughtful dialogue with the classroom teacher.
Session OverviewThe session begins with a mentalmath activity where coaches engage in a mentalmath routine similar to the one described in Rosa's transcripts. The deliberate facilitation of this routine is participantfocused with opportunities for the facilitator to draw out coaches' math ideas. Participants then work with Transcript 1 to read the teacher and student dialogue from the second grade classroom Rosa observed. Coaches write their own responses, questions, and insights about the classroom discourse on the right hand column of the transcript. The transcript activity is followed by a wholegroup discussion exploring the math ideas and student thinking in the class. Next, participants read and reflect on the coach's considered commentary as she writes out her responses later that night. This, too, is followed by a wholegroup conversation. Next, coaches read about the way Rosa first responded to the classroom discourse as she scribed in real time. This inthemoment commentary leads to an important wholegroup discussion of the difference between reactive coaching and considered coaching—the main theme of the session.
Materials for the Session Obtain chart paper to scribe coaches' strategies and ideas during the mentalmath activity
Session Agenda Introduction: 5 minutes
 MentalMath Activity: 10 minutes
 Read Transcript 1, Respond to Steps 1 and 2: 30 minutes
 WholeGroup Discussion: 15 minutes
 Read Transcript 2, Respond to Step 3, and WholeGroup Discussion: 40 minutes
 Read Transcript 3: 5 minutes
 Final WholeGroup Discussion: 15 minutes
Introduction
Begin this twohour session by describing the agenda. Explain to participants that this case presents a unique version of case writing in that the coach, Rosa, with permission from the teacher, brought her laptop to school and transcribed the mentalmath portion of a second grade math activity early in the school year.
Before class started, Rosa had set up a template with a grid on the left where she would type in the teacher and student dialogue and a grid on the right where she would record her reactions, in the moment, as the discussion ensued (such as participants will find in Transcript 1). Rosa had explained that this would help her capture what happened in the class and that the two of them could use the notes in their subsequent discussion.
After school, Rosa took the transcript home to review in preparation for a teacher debrief the next day (and, too, as a way to consider more carefully her own inthemoment interpretation of the dialogue). That evening, Rosa carefully studied the classroom dialogue and replaced her own initial inclass reactions with more reflective responses and questions (see Transcript 2).
Let participants know that they will first be working with a transcript of just the teacher and student dialogue so that they might have the same opportunity as Rosa did in interpreting the math ideas and the ensuing discussion as it plays out in class (Transcript 1). Later in the session, participants will have opportunities to consider the implications of the other transcript versions. But first, this session begins with a brief mentalmath activity.
Math Activity
Through this math activity, participants will familiarize themselves with the mentalmath routine, referred to as Math of the Day in the Considered Coaching case. As the case describes, the second grade students are asked to suggest a variety of arithmetic problems resulting in the same total, 17. Another function of this first activity is that, as the facilitator, you can model seminar discourse that focuses on the mathematical sense of participants' choices of numbers and operations to create the total and that highlights relationships between these math ideas.
Ask participants to create number sentences that equal 67. Then give the group a few minutes to consider the possibilities that you will scribe for everyone to examine. In the following snippet from a Considered Coaching session, we follow a facilitator as she capitalizes on the number sense and relationship ideas that arise in the mentalmath routine:
Participant 1: 20 + 20 + 20 + 7 = 67. Facilitator: Can you say tell us something about the way you were thinking about 67? Participant 1: Well, for some reason, 60 always conjures up three 20s for me. I never see it as 40 and 20 more. It might be because of the way I interpret an hour; I think in 20minute increments. So I made the problem into three 20s and just added on the 7 to make 67. Participant 2: I said, 50 + 10 + 7. Facilitator: And so were you thinking along the lines of Participant 1? I mean, first thinking about how you know 60? Participant 2: In a way I was, but my first thought was that doubledigit numbers are always either below 50 or above 50—well unless it's 50, so I added the amount on to 50. Facilitator: Okay, let's see if we have other ideas out there. Participant 3: I used subtraction. 100 − 40 +7. Facilitator: It is interesting how, so far, in thinking about 67, the 7 doesn't get much play. Participant 3: Yes, I was thinking about how to make 60 and, sort of, who cares about the 7! I was thinking more like Participant 2 though, than Participant 1. I was thinking about how far away 60 was from 100. Facilitator: Oh, you mean that Participant 2 was considering the distance between 60 and 50, and you were using 100 as a landmark? The dialogue goes on for just a bit more, including a point where the facilitator asks the rest of the participants to comment on how the ideas of the next two participants' are related either to each other or to the previous problems. This facilitation strategy is markedly different from scribing a simple list of responses and sorting out each one distinct from the others; it is clear that the facilitator is working to highlight the ways the participants are considering the number system, operating on numbers, and for the sense of the number 67 as in distance from 50 or 100, equal groups that add to 60, and considering 67 as a group of 10s and 1s. Mental math can be mined for a variety of uses including as practice in flexible thinking and for solidifying number facts. It is usually a brief, tenminute exploration. The point here is that it can also, even in just a few short minutes, reinforce for students that sharing their thinking and articulating their ideas is of high value to their teacher and to their classmates' learning and that there is something to be gleaned from even a quick dissection of relationships between numbers and operations.
While you will want to move on to the exploration of the transcripts without further debrief of this mentalmath exchange, it may be that the facilitation of it will serve as a marker for later wholegroup discussion. For that reason, post the scribed equations so that they are available for reference later in the session.
Case Reading
Participants should now refer to Transcript 1 and Step 1. Ask participants to read through this transcript twice before writing comments in the blank grid. Let them know that there are two streams of thought that they will want to keep in mind but follow separately. The first is to consider just the students' math talk. The second is to pay attention to the studentteacher dialogue. Some participants may find it hard to toggle between the two; it is more effective to consider one line of thinking and record one's comments about the students' ideas and then go back and work on the interactions between the teacher and the students. Participants should work on this activity on their own before coming together in groups of three for smallgroup discussion of the Step 2 questions.
WholeGroup Discussion
The case discussion focuses on two aspects of the writing: (1) the mathematical ideas of the students' and (2) the teacher and the implications for coaching. In this first pass at the transcript, participants will have recorded a range of questions, assumptions, and hypotheses about the students' use of numbers and operation. Most often, the participants will make note of the different ways students use addition to make 10 and the use of doubles. The subtraction problems offer similar ideas, starting with using 10s and + x, − x. Encourage participants to first discuss the students' work. Ask them to look at and discuss the students' math as a series of ideas. Suggest that they consider the relationships between one student's idea and another's. Ask, “What do participants notice about the student work in this first month of second grade?”
The discussion then shifts to the teacherstudent dialogue. Before you begin, remind the participants that it is important to assume the teacher is working on new skills in her practice; the author has written about this. It is sometimes difficult for participants new to this kind of investigation to focus on what analyzing the case can offer their own practice. Thinking carefully about what the teacher is working to do in this studentteacher exchange is one way to learn more about what can be gained by asking students to participate in mentalmath activities. The considered look at the dialogue is meant to be a way of pinpointing places that will ultimately prompt useful conversation between the teacher and the coach. In other words, you may need to actively steer the conversation toward factual commentary about what participants see happening and help them refrain from negative judging talk about the teacher's responses to students. The point of the discussion is to simply read and note what's going on in the snippet of a classroom activity. Next, they will have a chance to view the same from the coach's perspective.
Read Transcript 2 and SmallGroup Discussion
Now, the small group has a chance to read the transcript that includes the coach's reflective thinking. In Transcript 2, the coach writes down ideas and questions that will inform her coaching. She knows that carefully examining the transcript will help her to prepare in thoughtful ways for a debrief discussion with the teacher. This reflection activity is mean to help Rosa make decisions about her own coaching moves.
Give the participants about 30 minutes to read and discuss the focus questions related to Transcript 2. It is helpful to be explicit that participants reread the classroom transcript for Transcript 2, not just the coach's reflections. Remind them to note particular places in the dialogue and in the coach's reflective writing they would like to raise with in their small groups. Circulate as the participants are discussing the focus questions related to Rosa's reflective thinking.
WholeGroup Discussion
The smallgroup work is followed by a brief, 10minute wholegroup discussion. You might jumpstart a group discussion by simply asking participants what stood out for them in Rosa's commentary or by asking what they are learning from the coach's reflections.
Read Transcript 3 and SmallGroup Discussion
Participants will find, again in Transcript 3, the full transcript from the classroom, yet this time the transcript is accompanied by the coach's initial reactions recorded as she was in class observing the Math of the Day activity. You will likely hear some laughter as participants are reading and recognizing the increasingly exasperated tone of the coach's commentary in this very reactive mode. Because we know the coach later went home and reflected carefully about the class and saved herself from responding to the teacher with her initial reactions, we can find some relief in this reading. What is exasperating to Rosa in class becomes data to study with respectful consideration after class.
Final WholeGroup Discussion
Rosa has generously contributed this very honest collection of writing as a way of supporting a critical idea in coaching. The final wholegroup discussion can reiterate the importance of considered versus reactive coaching. In a considered approach, the coach responds to the teacher with learning goals at the forefront of the conversation. In reactive mode, the emotions are at the forefront, and often the coach's own immediate reactions blur the line between coaching and criticism. It is important to assure participants that considered coaching is cultivated over time, that the goal for coaching practice is that a considered mode is the norm even when there is little time for reflection. Going into classrooms with a framework of goals already in mind supports a considered coaching response. These transcripts offer one powerful way of highlighting for math specialists and administrators the importance of both exposing one's practice—even in diary form—as a way of critically examining the ways one might develop a considered and thoughtful, mathematically focused stance. So often the coaching role is held as that of the judging, critical expert. Without a focus on teaching and learning on the part of both the coach and the teacher, we risk falling into the trap of simply reacting. Again, it seems that cultivating a norm of sharing issues of practice, sharing new mathematical insights and questions, and collaborating to refine and strengthen both areas of knowledge are important features of learning to coach.
Chapter 6 Facilitation Notes: Purposeful Planning and Facilitation2.5Hour Session
Session Goals Explore the purposes and potential for gradelevel or teacherteam meetings
 Strategize about creating meaningful agendas for group meetings
 Analyze coaching strategies for responding to teachers' needs and issues while facilitating focused discussion on issues of mathematics
 Sort out factors that help determine an appropriate balance of sharing responsibility for the agenda and facilitation of teacher meetings
Case Description: Coaching in a Group: Moving from 1:1 to 1:?In the case, Coaching in a Group: Moving From 1:1 to 1:?, the coachauthor describes her struggles as she branches out from onetoone coaching to coaching groups of teachers in team meeting structures. This novice and yet deeply selfreflective coach describes her attempts to design agendas that matter and facilitation that captivates groups of teachers at a meaningful level. Chloe's case moves between descriptions of two group meetings, the planning and preparation for each meeting, and her reflections on her coaching. As she struggles, she also gains new insights that help her develop a stronger coaching practice. Novice coaches and experienced coaches will easily relate to the complexities of planning and purposefully facilitating group meetings described in Chloe's case.
Session OverviewThe session begins with a brief introduction to the agenda and the session's goals. The participants read the case and focus questions before engaging in small group discussions. Responding to these questions helps participants follow Chloe's story and analyze the dilemmas she poses. The Focus Questions Activity also calls on coaches to articulate questions for Chloe that would help her consider coaching strategies—just as they would pose questions for teachers that would help teachers uncover new ideas for themselves. Through a Planning Activity and group discussions, both novice and experienced coaches sort out goals, agendas, and issues regarding the engagement of groups of teachers with whom they work.
Materials for the Session Create a poster of the Session Goals.
 Participants will need chart paper and markers for the Planning Activity
Session Agenda Introduction: 5 minutes
 Reading the Case and the Focus Questions: 10 minutes
 SmallGroup Focus Questions Activity: 35 minutes
 WholeGroup Discussion: 30 minutes
 SmallGroup Planning Activity: 40 minutes
 WholeGroup Discussion: 30 minutes
Introduction
Begin the session by offering a brief description of the agenda and the learning goals of the session. Set the stage for the case by describing, in just a few words, the context of the case written by Chloe.
Reading the Case and the Focus Questions
Let participants know they have the choice to read the Focus Questions before reading the case. For some, acquainting themselves with the questions helps center their attention on issues that will be discussed in small groups. Others may still choose to read the case first and then peruse the focus questions quietly while they wait for their group to begin discussion. Reading both should take about 10 minutes.
Focus Questions Activity
As participants are discussing the focus questions, it is important to move around the room and listen carefully to the small groups talk. Joining by pulling up a chair and simply listening for a few minutes to individual's responses to these questions may offer insights into the degree of comfort your own group has in planning for and facilitating group meetings at their own schools. You may hear some disdain for Chloe's efforts, so it may help to remind participants that Chloe is a beginner and is clearly working hard to improve her practice. You may also hear many sympathetic or “I've been there” kinds of responses to the frank descriptions of her struggles. It is not uncommon that coaches have more confidence in their oneonone work with teachers than in their ability to plan for and facilitate the discussions of teachers in a group. For many, these meetings may be the first time they are responsible for a whole group's learning—that is, for a group of adults. And, indeed, for many, these same adults were only recently their rolealike colleagues. It is not a trivial matter to take responsibility for moving a whole group forward, nor for juggling the issues that arise simply because of past—or current—relationships with members of the group.
With regard to Question 4, some might find themselves stuck on the assessment issue Angela raises in the case. While you, or the coaches in the group, may agree or disagree with Angela's point of view, Chloe decides not to move the teachers in this direction. For the purpose of this session, participants might have to agree or disagree with Angela (and Chloe) and move on to discussing Jolene's work.
Listen for the way participants attend to Question 5. There are two separate parts to this question. One part is to craft questions for Chloe that will help her come to her own conclusions. The other is to consider what one might hope for Chloe to be learning. It's a good coaching skill to know how to ask questions that will help the other person uncover their own intentions or directions. And it is useful for each participant to consider how she might refine her thinking if she were in Chloe's position.
WholeGroup Discussion
As you facilitate the discussion, keep in mind that you will have another group discussion to end this session after the Planning Activity. In fielding participants' responses and questions at this point, you can make decisions about whether to follow through with an idea or ask participants to “hold on to that thought” for the final wholegroup discussion. Start off this discussion by asking participants to make explicit what happens in the case, how they understand Chloe's questions, and to analyze how she goes about planning and facilitating the teacher meetings. This time provides an opportunity to stay centered on the ways Chloe chooses to move in her coaching. Asking participants to share the questions they articulated for Chloe (Question 5) will let the whole group hear alternative ways of asking questions meant to support another's learning. Ask the group to keep refining the questions they've crafted until they feel satisfied that these will help reveal for Chloe what has gone well and what her next steps are—questions that help her reflect and refine her thinking. Assure the group that the final discussion will provide time for coaches to talk about their own experiences and how they intend to prepare, or refine their preparation, for coaching responsibilities that move beyond 1:1.
The Planning Activity
The first section of this activity is a reflective writing assignment. The task is for each participant to explore two different points of view—their own and the teachers.' Keep in mind the goal is that, in the end, everyone should be clear about the overall purpose and goal of group meetings. This goaloriented planning, designed with the teachers' beliefs and ideas as well as the coach's agenda in mind, is not a straightforward aspect of coaching practice. If the facilitator of a group meeting is the only one with a clear idea of the agenda, such as we see unfolding in Chloe's case, the participants in that group are left responding to each move she makes. If the participants have a point of view that is out of synch with the facilitator, or if the facilitator is unaware of the participants' beliefs and points of view, this can lead to meetings cloudy with mixed messages and little substance.
Take stock of the level of experience your group has in facilitating group meetings. If your group is very new to, or naïve about, coaching, they may find it more difficult to think about appropriate agendas for group meetings that will help teachers learn or reflect in meaningful ways. If your group includes experienced coaches, provide opportunities for these coaches to share what they have learned about designing and facilitating successful group meetings. And yet we see that even experienced coaches may still struggle with defining the purpose of meetings, designing useful activities or explorations, and helping teachers think more deeply about the mathematics they teach, or the students' ideas in the classroom.
The goal of this planning activity is for participants to have the opportunity to work with colleagues to sort out issues regarding designing agendas for groups and ways of considering the implications for facilitating with teachers' beliefs and expectations in mind—before the group meeting. In addition, it is important to press the idea that taking into consideration longand shortterm goals for the school in every opportunity where math specialists,' or coaches, have a group's attention is a strategic aspect of coaching. One goal of this session is to help participants come to appreciate the complexity of negotiating teachers' expectations, designing appropriate agendas, and determining coaching goals for teachers' learning in wholegroup meetings.
In considering the work of Question 4, think ahead about whether you want to assign each small group a specific question to chart (so that you are assured a poster about each) or whether you want small groups to choose the question from the bulleted list. The second option offers groups the chance to create a poster about what they have the most confidence, the most questions, or that has generated the most discussion in their small group. Ask participants to hang their posters so that the whole group can do a gallery walk and move around the room to read each one.
WholeGroup Discussion
After everyone has reviewed the charts, begin the discussion by asking those who charted the same question, or bullet, to talk to the whole group about the ideas—and questions—their small group discussions raised. Then ask the rest of the participants to add new ideas or perspectives from their own planning activity conversations. In turn, continue to move the discussion back and forth between the ideas offered by those who created the posters and the perspectives and ideas offered by the rest of the participants.
After the posters have been discussed, ask participants to take five minutes to write about the ideas raised in this case and in this session's activities. Ask them to describe at least one specific way the discussions today will influence the planning and facilitation of their school group meetings coming up. If you have time, ask for volunteers to share from their writing.
Chapter 7 Facilitation Notes: Refining and Reimagining One's Coaching Practice2Hour Session
Session Goals Analyze young students' work and math talk as they analyze young students' work and math talk as they engage in counting
 Examine stages of making sense of counting and representing “a count”
 Consider coaching strategies for helping all gradelevel teachers develop a finegrained picture and a big picture of their students' mathematical learning
 Explore effective coaching preparation for the facilitation of prebrief and debrief meetings with teachers
 Consider purposes for wholegroup student discussions
Case Description: Learning about Counting, Learning about CoachingIn Learning About Counting, Learning About Coaching, the coachauthor describes a Kindergarten classroom observation, interviews with several students as they engage in counting activities, and her subsequent questions about effective coaching debriefs. Elaine, a coach with past experience as a high school math teacher, becomes intrigued with the varied strategies for counting in Kindergarten and, over the course of one classroom visit, she is bursting with new questions and a real appreciation for the children's effort involved in sorting out counting as an idea. She describes the ensuing teacher debrief, which she soon realizes falls far short its potential. As she reflects on the classroom experience and her facilitation of the teacher debrief meeting, she gains new insights about the questions she might have asked and about the teacher and coach learning she is aiming for.
Session OverviewThe central focus of this session is a case that, on the surface, is about young children's counting strategies and what is involved in learning about counting. And yet, it is most assuredly a case about coaching dilemmas and strategies applicable to any grade level. Participants begin the session by reading the case and analyzing the children's ideas and to make sense of counting. Through focus question discussions in small groups, coaches also work through the dilemma of facilitation the author poses in her case. In the Planning Activity, coaches spend time planning teacher study groups focused on student learning of specific mathematical topics. They work together to help each other refine their ideas, get clear about the structures available to support such studies, and to discuss ways of measuring the outcomes, or evaluation, of the work. The session ends with a wholegroup discussion, during which coaches share and elaborate on coaching plans and insights from the session.
Materials for the Session Create a chart of the session goals
 Coaches will need coach journals or writing materials for the Planning Activity
Session Agenda Introduction: 5 minutes
 Case and Focus Questions Reading: 15 minutes
 SmallGroup Focus Questions Activity: 25 minutes
 WholeGroup Discussion: 25 minutes
 SmallGroup Planning Activity: 30 minutes
 WholeGroup Discussion: 20 minutes
Introduction
Briefly review the goals of the session and the agenda. Describe, in just a few words, the context for this case of young children's thinking about counting. Be sure to tell participants that while they may or may not coach teachers of Kindergarten children, this case serves as an example of what it means to take a very close look at student thinking and, also, that the facilitation issues that emerge relate to coaching in any gradelevel classroom. As such, this case is less about the specific math content and more about the implications for teaching and coaching.
Case Reading
Sometimes, participants find that reading the focus questions offers a preview and a lens with which to focus their attention on a new case. For coaches who have less experience with very young children, suchasthose described in the case, it may be particularly helpful to have them preview focus questions and the various ways of orienting oneself to the topics that will be discussed in this session. Participants will need about 10 minutes to read the case.
Focus Questions Activity—Small Groups
In this Focus Questions Activity, participants are asked to write brief responses to Questions 1 and 2 on their own before engaging in group discussion. This will give each coach a chance to practice looking at the “fine grain” of the Rafael and Catherine's math talk and engagement with the counting tasks. Peering in closely is a stance coaches develop with practice and is a central idea in highlighting and understanding students' ideas. As we develop strong images of how students engage with specific math ideas, we also develop a keen attention to important work and talk that is all around us in the classroom.
By the time participants have moved to Question 4, they should have a list of their own ideas about young children's counting ideas and strategies and will have added those Elaine noted in her case. Some participants might talk in terms of developmental or more or less sophisticated ideas. While there are many ways to make sense of children's counting processes, it is important to remember that this is, indeed, simply what learning to count looks like. You might find yourself reminding participants that while it is useful to see a trajectory of ideas, it is much less useful to jump to conclusions about students or categorize those who get it and those who don't.
Question 4 asks participants to suggest ways to facilitate the children's group meeting at the end of the counting activity time. Listen for the types of ideas the participants offer. What do your participants want to learn from the students? What do they want the students to hear from each other? What might they ask or how might they begin the classroom discussion? Listening carefully and reflecting on your participants' responses may give you some insights into the ways coaches are supporting the teachers with whom they work. Elaine's writing about the descriptive version of a sharing session is certainly not unique to the case of this teacher's classroom. This is a common issue in math teaching practice and coaches encounter sharing meetings that are quite similar in nature, not just in Kindergarten but in all grade levels. Understanding your own participants' notions of how to run an endofclass math discussion will help you press for analysis of these ideas in the wholegroup discussion.
Question 6 comes full circle from the teacher's approach to an endofclass discussion to the coach, Elaine's, approach to her own meeting. These are very important parallel issues to be examined. Just as the teacher must learn how to use the wholegroup time as an extension of the class learning time so must the coach learn to use teacher debrief discussions as opportunities to extend the teachers' learning. Just as the teacher needs to carefully plan for the final discussion—and participants might discuss how—the coach also needs to think in terms of a learning agenda for debrief meetings.
WholeGroup Discussion
The wholegroup discussion will cover a number of topics. It will be useful to explore and analyze the students' actions and ideas in class. Another topic to raise is the issue of wholegroup sharing in a class of very young children, and yet another is to explore the coach's moves and how these moves might be refined so that the teachers in this scenario could learn more.
Begin the wholegroup discussion by asking participants to contribute to a list of how the students were thinking about and attending to the task of counting and recording a total. You might ask one participant from the small group to record a group's response on chart paper as you accumulate a list. Do not hesitate to step in and ask for clarification if a charted idea has not been fully articulated or to ask a question that helps the participant elaborate on her point. Having a participant chart the responses gives you the opportunity to step back and ask for these clarifications.
When the group has a final list that captures a collection of young children's counting ideas, strategies, and approaches to the task, check it to see that the “naïve” approaches are included. It's common to see a range of counting—disorganized and inaccurate, physically organized and inaccurate or accurate, a miscount total represented by drawings of an even different number objects, deliberate onetoone correspondence, and repeat counting of even a small number of objects. You might see tally marks accurately representing the number of objects counted and papers that reveal little about what they child knows. These can all be represented as ways that children approach counting. Ask participants what they learn from the list. Ask what questions the list raises. Spend time listening to the ways the participants discuss the list; listen for a sense of curiosity, for an energy about wanting to know more. In creating this list, the participants have acted out one portion of what might have been a more satisfying teacher meeting for the case writer and coach, Elaine. As she has written in her reflection, she missed the opportunity; this list creating and discussion is one image of what got lost.
What does a coach raise with a teacher to help her move from wholegroup discussions from descriptive sharing to inquisitive sharing? Ask the participants to consider how the description of the students' sharing circle compares to the teachers' experience of the debrief discussion. You might ask the participants to turn to a partner and talk about the similarities and differences and then come back to the whole group so that these ideas can become part of the larger discussion. Consider the small group discussions you listened to; now is the time to raise questions about what you heard.
Ask participants to suggest what might be a productive agenda for this debrief meeting. What might the coach want teachers to learn, to think about, to become curious about? Ask the participants to suggest, very specifically, how they might begin the debrief meeting with an agenda in mind.
Planning Activity—Small Groups
The point of this Planning Activity is to press participants to take their coaching out of the solo arena and into a context of group work. In this way, coaches are not working entirely in the background, surfacing to coach, and then retreating to a place of selfreflection. Instead, they are helping to build a community of teachers learning together. In fact, the coach may appear more overtly a learner when working with a study group, and this is a useful perspective for teachers to have about coaching and what it entails. It may be that teachers will more readily see that they are able to contribute both individually and as a group toward a collective understanding about a math topic and the ways children come to learn the math ideas.
The Planning Activity has two parts: one that participants do on their own and one that is a smallgroup sharing of new ideas about working with teachers. Each portion should take about 15 minutes, though you can make adjustments if the group finishes writing earlier. In that case, you might want to have them move to the smallgroup listening and sharing portion. If you have time, ask the participants to spend just a few more minutes going back to their writing and adding new ideas or writing notes that reflect refinement of their earlier written responses.
WholeGroup Discussion
The whole group discussion also has two parts; the first is a sharing about the Planning Activity, and the second is a wrapup discussion of the entire session.
The talk generated by the Planning Activity smallgroup discussions can be shared for the whole group. Given what you have been hearing as you moved from group to group and the amount of time before the session closes, you might choose different versions of facilitation. You might want the whole group to hear just briefly from each small group and then move directly to a session wrapup. Asking each small group to share two ideas of value that came from their activity work and from listening carefully to each other will give every one a chance to hear new ideas. You might find that this is enough for now; let participants know that continuing to refine their plans—or move right into them at school—is something the whole group can revisit when you come together at the next session. Or, if you have time, you might chart some of the ideas generated by the planning activity and ask the participants to share more details. This is particularly helpful if a group came up with an idea or plan for collaborative work with teachers that is closely aligned with your own goals for the coach group. It would be strategic for all participants to consider the plan together. Participants might express a concern about strategies for organizing collaborative work. This may merit attention in the wholegroup discussion. Ask participants who are strong strategic thinkers to contribute. It is important for the group not to get bogged down in complaints or discouragement about “having enough time,” but rather help participants accept that structures may not yet be in place (or that they may yet have discovered how to take advantage of what is in place) and that part of coaching is learning how to build these possibilities.
The wholegroup time is also an opportunity to discuss how a collaborative study of students' ideas might further one's coaching goals. Ask participants to think carefully about what steps each coach will need to take to get such a collaborative study off the ground, what it might take to maintain the focus of the work, and to support a specific and important coaching goal. These questions are important to ask; they press on the participants' sense of responsibility and authority in creating goals and agendas for work with teachers.
It is not the case that even with a great plan a coach can turn around and execute the plan with immediate success. But it is important to understand that one aspect of coaching is learning how to reframe existing structures, building relationships with those who can influence how structures for learning could play out, and cultivating partnerships that will actively promote math study and learning in the school. In each of these areas, we can imagine that effective coaching is not always possible if the coach perceives his or her role as a sole interventionist. Thinking long term, considering collaboration, creating opportunities, and determining and maintaining an important focus are all complex aspects of the coaching role.
Take a few minutes to end with exit card writing. Ask participants to describe one new idea and one next step generated by the session.
Chapter 8 Facilitation Notes: Cultivating Relationships with Administrators and other Leadership Colleagues2Hour Session
Session Goals Explore ideas of mathematical reasoning and mathematical discourse
 Counter the notion that the coach is on his or her own in creating a vision for schoolwide success in math and in facilitating teacher learning
 Discern learning agendas for teachers and for school communities
 Explore strategies for, and the design of, principal professional development
 Craft an invitation to a principal or other school or district leaders to collaborate with an eye toward moving math education agendas forward
Case Description: Crafting an Invitation: Shifting from Isolation to InclusionCrafting an Invitation is composed of a draft of a letter the coachauthor, Seth, wants to share with his colleagues for their feedback. Seth has written the letter for several reasons, most importantly to invite the principal in as an effective partner toward the goal of developing a strong math program in the school. Seth also uses this letter as a way of framing his coaching goals such that they connect through the lens of the principal. Finally, the letter stands as a strategically crafted professional development tool for the principal's learning. In the case, Seth lays out two masterfully articulate lists of vital aspects of the math program and coaching. First, he describes, in bulleted form, the successes of the current math program. Next, he meticulously describes the lenses he takes to the work he does, the teaching practices he observes, and the student learning environment as it is enacted in the school.
Session OverviewThe session begins with a brief introduction of the goals and the context for the case, Crafting an Invitation: Shifting From Isolation to Inclusion. Participants will read the case and, in small groups, respond to focus questions that surface the knowledge for coaching Seth articulates in his case. The Focus Questions Activity is followed by a wholegroup discussion aimed at highlighting the goals Seth has for his own practice as well as the goals he has for teachers and students. Coaches will analyze Seth's letter for the coaching strategies he describes and the ways he carefully structured his “invitation” to the principal. After a wholegroup discussion of the letter, coaches will work, individually and in pairs, on a Planning Activity designed to help them articulate their own goals and craft an invitation to a collaborator. The session ends with a wholegroup discussion of the insights and next steps brought to light through the activity.
Materials for the Session Create a chart of the goals for the session and have extra chart paper and markers for the session
 Participants will need coach journals or other writing materials for the Planning Activity
Session Agenda Introduction: 5 minutes
 Case Reading: 15 minutes
 Focus Questions Activity: 20 minutes
 WholeGroup Case Discussion: 20 minutes
 Planning Activity and WholeGroup Discussion: 60 minutes
Introduction
Begin the session by offering a brief description of the agenda and the learning goals of the session. Describe, in just a few words, the context for this case of a letter to a principal written by the school coach, Seth.
Reading the Case and Focus Questions
Participants will spend about 30 minutes reading the case, exploring the focus questions, and discussing their reflections in small groups. While Questions 1 and 2 ask participants to describe the goals Seth has for students and teachers, you can help participants avoid overgeneralizing by letting them know, upfront, that this might seem to be a simplistic question to answer. (Some might simply say Seth wants students to talk in class and to think that they have fully answered the first focus question.) Press on this point by saying that you want each participant to really dig through the case for the many places where Seth lays out an agenda or a goal for teachers and students. Examples of these are sprinkled throughout the case and are worth looking carefully to uncover.
Participants will spend time going through this wellcrafted letter to highlight the structure within it. The coach has strategically used this letter as a professional development tool for his principal; Seth has not only determined a learning goal for the teaching staff but learning goals for the principal too. Zeroing in on the structure of the letter will help participants see just how strategic and thoughtful Seth has been in crafting this invitation. And also, as participants discuss their ideas about Seth's image of coaching, it is the intention that they see Seth as moving beyond carrying the charge of change and shifting practice as the definition of a role he plays out on his own. Rather, in this case, one sees Seth moving to bring everyone—coach, principal, and teachers—together in meeting the mathlearning goals for the students in the school.
As you circulate in the small groups, you will likely note that participants have reactions about a variety of ideas raised through this case. It might be that this is the first opportunity these math leaders have to discuss the necessary support of the school principal.
WholeGroup Discussion
Participants are often quite taken by the lists Seth describes and the way his ideas compare to ones they might have listed. (Many participants, both new and experienced coaches, express admiration for how clear Seth is about his coaching work.) Some participants remark that they are seldom asked to articulate their goals. They might also add that if they were asked, they would not be able to describe them as articulately as Seth does in his letter. As the facilitator, you may know the very contexts in which the participants are coaching—in fact you may know the principals and school leaders with whom the participants work. This may effect what you want participants to gain from the discussion. You could ask groups to talk together for just two minutes and decide the most important issue that came up in their small group; these ideas, by table group, can then become the list of topics to discuss.
It may require some effort for the whole group to stick with discussing the specifics of Seth's case instead of veering off into issues that pertain only to their local district contexts (the curriculum pacing guide or district assessments, for instance, might be a point of tension and the group wants to talk about this instead of what it means to set goals for a community, how one determines the goals, and how one articulates these ideas). To that end, to fully explore issues of mathematical reasoning, you might ask specifically what Seth feels teachers need to work through or engage in. It is useful to bring participants attention to Seth's idea of mathematical reasoning as the central goal, with math discourse in classrooms in service of that goal. Ask participants to reflect on the purpose of classroom discourse and the difference between dialogue (or student talk in classrooms) and discourse. As some participants have noted, there is more to discourse than simply assuring that students “have a voice” or that students are heard in classrooms. Mathematical discourse, as Seth is describing it, is about building ideas by listening to each other and analyzing what one hears and measuring that against what one already knows, by learning in the context of listening, contributing, and making sense.
Planning Activity: Considering Collaboration and Communication
This thinking, writing, and planning activity consists of a series of structured tasks to take on alone and in pair discussions. The backandforth structure provides time for the participants' ideas and writing to develop—and with each interaction, become more clearly articulated—and the invitation to a colleague more effectively laid out.
Let the participants know that the Planning Activity is equally focused on all three elements: thinking, planning, and the actual writing of a letter of invitation. In introducing the Planning Activity, tell participants that this is an opportunity to think both about their coaching goals and potential collaborations with those goals in mind—at a school level, or a personal level, as goals for a specific teacher, or for a principal. To construct or discern goals to ground this activity, coaches should consider questions such as the following:
 What's complex about what the teachers are learning or struggling with and how might coaching support that learning?
 What do the students need that is not yet a part of the norm of teaching practice in the school?
 How refined are specific aspects of math teaching (such as facilitating classroom discourse, initiating important math tasks, and the like)?
 What mathematics content needs to be addressed with teachers that will support more robust teaching and learning in classrooms?
As the facilitator, you may have a sense of other questions you would like participants to consider as they work to define appropriate goals for their coaching work. It might be helpful to participants to post a chart of these reflection questions before participants begin the writing activity.
Activity Description
On Your Own
Participants will spend 10 minutes on the first portion of the Planning Activity reviewing the case and thinking about the coach's goals and the structure of the letter. These issues will have also have been addressed during the focus question portion of session. The idea is that by reacquainting with the case, the participants will have one structure (Seth's) to work from as they write. Their task now is to think through a coaching goal for themselves that might be more successfully reached with the collaborative help of another. Make it clear that a participant does not need to choose a principal as the collaborator for this activity. It may be that a teacher leader's support would enhance the coach's work. Or it may be that a coach would want to choose another coach who could form a collaborative effort across schools.
Participants should be encouraged to work on their own and to aim to write as clear an articulation of their goal as possible. An important aspect of the participants' thinking in this activity is exploring a justification that will support the appropriateness of the chosen goal for their particular site. Another is to choose a collaborator whose strategic participation will make a difference in achieving the goal.
In Pairs
At this juncture, participants meet with a partner. Each person has several minutes to—as clearly as they can—describe their coaching goal, why it's important, and whom they have chosen to invite in. The idea is for each person to simply describe these things as carefully as they can to the other. Each person will listen with real care without discussion. Tell the group this is a time to “have the ear of a partner who cares that you are becoming more comfortable with the articulation of your ideas.”
On Your Own
Next, participants spend 10 minutes writing an invitation to collaborate to a carefully selected teacher, administrator, or other whose role, authority, or knowledge will lend the work some heft. In some sessions, there will be participants who are reluctant to actually sit and write a letter. They might complain that they are too tired late in the day (if you have a session late in the afternoon, there is some empathy for weariness!) or that the task seems too burdensome. Stay the course, and specifically press these participants to push through their resistance and see what comes of writing. If you insist, you will find that in almost every instance, the participant is surprised at the depth of what they write and the way the writing lends a measure of reality to actually extending an invitation and opening up the coaching arena to a larger group.
In Pairs
Partners take turns describing new ideas about collaboration that were brought to light through this writing activity. Plans may have unfolded, challenges may have emerged, or goals may have been further refined. Give participants 10 minutes to take turns talking to each other about these new ideas. If you have time in your session, at this point, you might choose to have participants write one sentence on chart paper that describes the goal, one sentence that articulates the importance of that goal, and the person they are choosing to invite in. Again, if there is time, you could also ask for a sentence that captures the next steps. These chart papers can be viewed by everyone during a brief break, and then discussed in the whole group.
WholeGroup Discussion
The Planning Activity is followed by a wholegroup discussion of articulated goals and of next steps for how coaches will initiate and followthrough on these collaborations. You might start the discussion by asking participants what they learned by working on the activity. Ask how their goal shifted in their thinking as they worked toward articulating it and in writing an invitation for someone to join them in it. It might be useful to explore the reason for participants' goals. (Why that one? What importance does it hold?) As the facilitator, you may find that there is a range of purposeful thinking regarding the “why.” Are the participants thinking strategically about important issues in schools? Ask how the goal ultimately supports overall goals for the learning community.
Another aspect of the group discussion is to explore the next steps. Ask participants to describe the plans generated by this activity and to specifically name some next steps they have in mind. Ask coaches to describe which aspects of their work will be affected by the work and thinking that they have done today. Ask participants to think back to the ways the group discussions and the talking in small groups and with a partner supported the deeperlevel thinking they were able to experience. In that sense, you will reinforce the collaborative nature of the session.
End the session by reserving a few minutes for exit card writing. You might ask participants to describe a new and important idea heard from a colleague during the session. Ask participants to describe one next step for the coming week or month.
Chapter 9 Facilitation Notes: Taking the Lead as a Teacher of Teachers4Hour Session or Two 2Hour Sessions
Session Goals Analyze models for subtraction and students' ideas about the operation
 Learn about gathering data on teacher beliefs and math knowledge to design appropriately challenging assignments and “study opportunities” for teachers
 Analyze coaching questions that support teachers' construction of new ideas about teaching and student learning
 Examine teacher, coach, and student exchanges in a classroom for students with special needs
 Explore images of the coach role as a one who actively teaches teachers
Case Description: Encountering VenusThis longer case, written by an experienced math coach, describes a coaching context that unfolds over a series of days. It offers a poignant and intelligent example of a coach who is unafraid to teach a “willing, but far from novice” teacher by giving her homework assignments and setting up opportunities for the teacher to examine and reveal her practice. The case follows the story of the introductory meeting with Venus (a teacher of students with special needs), a subsequent math interview with her students, debriefs with the teacher and the coach, and the collaboration that develops between the two.
Session OverviewBecause Encountering Venus is the longest case in the series, giving participants ample time to read the case, participate in math activities, and discussions of focus questions will require four hours. However, because of the way the case is structured, it is possible to separate this session into two separate twohour sessions. The agenda describes the twohour sessions and notes for a single fourhour session. (If using the Encountering Venus chapter in a fullday professional development session with coaches, separate the two twohour sessions with a lunch or other break.)
The twohour periods are described as Session 1 and Session 2. Session 1 begins with a math activity exploring story problems and specific types of word problems. The math content of the case centers on subtraction, and this operation is explored in ways that may be new for teachers and for coaches. Ample time is allotted for the math exploration before participants read Part 1 of the case and discuss it. A wholegroup discussion follows before the group reads Part 2 of the case. Next, coaches work on a Focus Questions Activity, again meeting in their small groups to analyze the next section of the case. The whole group will convene again to explore these first interactions between the coach and the teacher.
When used for two separate professional development sessions, begin Session 2 with a brief review of the previous readings. Otherwise, the session starts with participants reading Part 3 of the case, Investigating With Venus, followed by smallgroup discussions and a wholegroup discussion. Next, participants read Part 4 of the case, Debriefing With Venus and, again, move into the smallgroup discussions focused on the final teacher and coach meeting described in the case. A wholegroup conversation wraps up the session.
Materials for the Session Create a chart of the Session Goals
 Provide counting cubes for exploring math problems and chart paper and colored markers for displaying participants' math work for group discussion.
Session Agendas Session 1: 120 Minutes Math Activity in Small Groups: 25 minutes
 WholeGroup Discussion: 20 minutes
 Case Reading—Part 1, First Conversation, and SmallGroup Discussion: 20 minutes
 WholeGroup Discussion: 15 minutes
 Case Reading—Part 2, You Mean You're Not Going to Tell Me? and Focus Questions Activity in Small Groups: 25 minutes
 WholeGroup Discussion: 15 minutes
Mathematics Activity
Session 1 begins with a math activity. Participants, in groups of three, work through a set of story problems designed to highlight a collection of math ideas such as relationships between subtraction and addition, the role sets and subsets play in reasoning through the stories and influence solution processes, and models of subtractiontype problems.
These four story problems represent a variety of problem types. To explore this notion fully, the directions ask participants to use manipulatives and/or drawings to model each story context and then to create a number sentence that matches the problem. As you circulate among the groups, encourage individuals to actually draw sketches and use objects to show the ways they are thinking about the situations and to share their ideas in the small group. Sometimes participants, new to working alongside other adults on elementary arithmetic problems, will work to expand their thinking when they hear you validate their developing ideas about problems that appear so easily solved. Simply listening in, asking clarifying questions, or helping participants compare their models are ways to help participants go deeper with confidence. Keep an eye out for a variety of models, questions, and insights raised in the groups.
WholeGroup Discussion
Once participants have had a chance to work on the story problems and to discuss their ideas, begin a wholegroup discussion by asking for volunteers to come up to the board and share their models for individual problems. For each, ask participants for alternative ideas or models. Overall, aim to help participants compare the problems and the operations used to solve the problem.
Some participants will draw number lines; some will use pictures or model the problem with linking cubes. Some will cross off (or separate) cubes starting at the end of the line of cubes and some will start at the beginning. Some will add up, and some will subtract chunks of numbers. Ask participants to comment and make note of the similarities and differences between these approaches to the problems.
Ask the whole group to share visual images for what's happening in the problems that are not yet represented in the shared drawings. Some participants welcome explicit directions to practice constructing mental images. One participant offered, “I just don't think that way. My math is cut and dried—I'm not a visual thinker.” The facilitator then suggested that conjuring up mental images of how the numbers operate as a useful way to engage with problems may simply be a new idea and that practicing this way of thinking can be help develop new insights. The facilitator's comment left open the possibility that one can develop skills of visualization that have been previously untapped and helped dispel the notion that we are necessarily limited to our usual ways of thinking.
Some participants have difficulty making sense of the operation used to solve the second problem:
Ella keeps candies in her desk drawer for those chocolate emergencies! This week she had a real craving for sugar and ate a lot of candies. If she ate 18 candies during the week and by Friday afternoon ended up with 17, how many candies had Ella started with?
Most participants will view this collection of problems as subtraction types whether they solve the problems with addition or subtraction. “This candy problem feels different. It's just pure adding. There's no subtraction in here,” claimed one participant. Another countered that she had set up her number sentence as ___ − 17 = 18. Comments such as these will generate discussion that, again, seem to unsettle the group; this time with regard to a previously held notion of subtraction.
At some point, it will be clear that many people used addition to solve the same problems they perceive as subtraction problems. This poses a challenge to the group—just what is subtraction if one uses addition? Let participants know that the teacher and the students in the case pose the same question and that as they read and discuss the case, they should keep this question in mind.
Case Reading and Related Focus Questions
Let participants know they have about 10 minutes to read Part 1 of the case and to respond in small groups to the accompanying focus questions. Reading Part 1 of Encountering Venus is the second activity in the session. Encourage participants to write questions or comments in the margins and note sentences or paragraphs that resonate for or confuse them. If a participant finishes reading before the rest of the group, suggest that they highlight line numbers related to the focus questions as a way to prepare for the smallgroup discussion.
In groups of three or four, participants briefly discuss the questions designed to orient the readers to the issues in the case. In reading Part 1, we see a relationship being established between the coach and the teacher. Participants take a few moments to discuss how the dialogue unfolds and how Gloria and Venus are setting up a collaboration.
WholeGroup Discussion
The wholegroup discussion turns the reader's attention to the coach and teacher interactions. Ask the group, “What's going on between the coach and the teacher so far in this case?” The group might consider variations of, “What is the teacher saying?” “How is the coach interacting?” and, “What seems to be contributing to the developing collaboration?” Remind participants that referencing line numbers in both smalland wholegroup helps ground discussions in the details of the case.
In a discussion about Part 1, one participant offered, “Gloria is really direct. Look at line 38; she just goes right for it! I'm not sure I zero in so quickly.” Another participant concurred, noting that Gloria did go right to the mathematics by offering a math problem for the teacher to solve in just the first minutes of their first sitdown discussion. The facilitator recognized these observations as an opportunity to press on what Gloria was doing—and aiming for—in this first substantive discussion with Venus. The facilitator asked, “So what happens as a result of Gloria's direct approach? What do we learn about Gloria and about Venus?”
It is true that Venus is forthcoming, and we know that not all teachers are, especially at the first teacher and coach meeting. Gloria does, however, say that Venus is working hard; participants might need to hear that again. It is especially important to note that Venus might have just said, “My students can't really solve these really simple problems” and Gloria could have chosen to empathize and express friendly acknowledgment that Venus has challenging students. Instead, as we see consistently throughout the case, Gloria takes statements like these as entry places for coaching work that supports a learning agenda for Venus—a learning agenda based on math content for student understanding, developing an ear for student thinking, and considering students' ideas—while at the same time, working to deepen Venus' understanding of the content. It might be useful to point out what Gloria didn't do at this stage.
In this, and other parts of the Venus case, the parallels between teacher and student learning are useful to consider. For instance, in Part 1, Venus says she offers a key words strategy to which Gloria responds, “I always include the wrong key words to throw them off.” At this stage, neither the teacher nor the students are sure what operations to use, and so in this way, Gloria is coaching on a variety of levels.
Setting participants up to analyze the case by looking carefully at Gloria's moves and the ways (and on what basis) she begins to establish a learning partnership with Venus will support a lens for the overall experience of working with the Venus case and materials. As the participants continue their work throughout both Sessions 1 and 2, paying careful attention to Gloria's coaching and evolving learning agenda for Venus will offer important images of the potential for coaching.
Case Reading and Related Focus Questions Discussion
Participants read Part 2, You Mean You're Not Going to Tell Me? and again work in small groups to respond to related focus questions. If you have time in your session, you might give a few more minutes to this reading period so participants can take a short break before they begin their smallgroup discussion.
This section of the case introduces readers to Phil and Bruce, two seventh grade boys who are currently struggling with mathematics. The focus questions draw participants' attention to the students' confusions and to Venus's interactions with the boys. Smallgroup discussions will draw reader's attention to the differences in the ways Gloria interacts with the students and Venus's responses to their work and their questions. This section of the case also offers an image of coaching in lines 133–155 that are at the heart of Gloria's pedagogical skills.
WholeGroup Discussion
Bring the groups back together for a wholegroup discussion focusing on several fronts. The participants will have a chance to consider the math Bruce and Phil struggle with and the implications for learning based on the ways Venus and Gloria interact with the two students. In addition, the discussion will highlight the way Gloria elicits Venus' ideas about her own teaching.
Begin by asking participants to describe how they interpret what is happening mathematically for Bruce and Phil. Then you can move the discussion to the ways Gloria and Venus interact with the students. As participants offer comments on types of moves Gloria and Venus make, remind everyone that backing up general comments with the line numbers is an important way to stay focused on what is actually happening in this case. Participants will note the ways Gloria and Venus are beginning to partner in their observations and interactions with the two students such as in lines 97–103 and 105–112. If not, ask them to characterize the relationship of Gloria and Venus in the case and insist on line number evidence. In following the back and forth between Venus teaching and Gloria teaching, participants will see Venus' questioning skills emerging. In lines 108, 117, and 128, Venus asks questions that, indeed, are targeted at both focusing the students' attention on their own work and on eliciting students' ideas as a path toward understanding her students. In contrast to the directive teaching she has described to Gloria in Part 1, Venus is engaging in the students' learning just as Gloria has engaged in Venus's learning.
Ask participants to take a moment and underline each of Gloria's remarks in this section. Then, ask for two volunteers who will read aloud the part of Venus and the part of Gloria in lines 133–155. Now open the wholegroup discussion for a consideration of the nature of Gloria's questions and what these questions afford. What do both Gloria and Venus learn in this exchange, and what does it say about a stance of coaching? Parts 1 and 2 end with Gloria assigning a specific homework task for Venus as they gear up to continue learning about the students and to dig into a study of subtraction.
As you wrap up the discussion for this session, ask participants to comment on how this case helps them think about interactions with teachers in their own coaching, and in particular, how they might be thinking about the goals they have in mind for the teachers with whom they work.
Session 2 Agenda—120 Minutes Review: Part 1 of the case Encountering Venus and Part 2, You Mean You're Not Going to Tell Me? (Skip this review for a single fourhour session, instead use the time for a brief break.)
 Case Reading—Part 3 of the case, Investigating With Venus: 25 minutes
 Focus Questions Activity in Small Groups: 35 minutes
 WholeGroup Discussion: 15 minutes
 Case Reading—Part 4, Debriefing With Venus and Focus Questions Activity: 30 minutes
 WholeGroup Discussion and WrapUp Activity: 30 minutes
Case Reading
In this second session, begin by offering participants a chance to reread and review the issues highlighted in Parts 1 and 2. The smallgroup discussion that follows will narrow in on Part 3, Investigating With Venus, where Venus describes her next steps with Bruce and Phil. She recounts the math problems she wrote for them (based on the homework assignment Gloria had set up at the end of Part 2) and describes the ways the two boys tackled the problem. Again, in this section of the case, Gloria presents new perspectives on ways of thinking about subtraction strategies and describes a visit to Venus's classroom and the ideas raised by Phil and Bruce's classmates.
Focus Questions Activity and SmallGroup Discussion
A discussion of these focus questions helps participants move between the mathematical ideas raised in the class and the coaching moves Gloria makes that influence the new type of classroom experience the students have. Bring the participants back to their own work in Session 1 on issues of subtraction. These themes offer learning on three levels for participants. In analyzing the student thinking, they pay close attention to the ways Bruce, Phil, Karen, and Natalie—and likely students they have in their own classes—understand subtraction. In considering Gloria's moves in the classroom and the stance with which she poses questions of both the students and of Venus, participants have a lens on effective coaching in process. And again, participants will continue to develop insights into the mathematics for themselves.
WholeGroup Discussion
Begin by asking participants to discuss the ideas and reasoning offered by students in Venus' classroom. Ask, “How do these students view subtraction?” Remind individuals to refer specifically to students by name or by line number. In this section of the case, we now see Bruce and Phil contributing to a discussion about the operation. Bruce sees that you can add up; he suggests that he'd “go 2+? = 7. It's an addition problem.” It is interesting to note the difference of this kind of engagement with ideas about math given his previous experiences as Venus described them in Part 1 (lines 19–23). The group might discuss the difference in the characteristics of this lesson. What is it that Venus is doing that matters? What's the nature of Gloria's interventions and contributions with regard to the class and with regard to Venus?
Next, turn the group's attention to lines 291–315. Ask participants to talk about the question, “What do all the different types have in common?” You might refer the group back to the story problems in the math activity they solved prior reading the case.
After the group discusses this question for a few minutes, ask participants to spend a few minutes writing their own subtraction problems. Ask them to consider a range of problems and to discuss, back within their small group, the ways their problems are related to subtraction. This activity is similar to the assignment Venus assigns the students at the end of class.
Case Reading
The participants now read the last section of the Venus case, Part 4, Debriefing With Venus. In this section, Gloria describes a quick debrief with Venus after class. Inherent in the case, but perhaps most transparent in Part 4, is a portrayal of coaching as making thoughtful decisions about when to be directive or explicit, when to reflect together, and when to push and probe. Gloria is equally assertive in her questioning as she is in providing explicit directions or explanations.
Focus Questions Activity in Small Groups
The focus of the final reading and discussion will be on Gloria's coaching and the way her perception of the role, coupled with her beliefs about learning, influenced her moves, her questions, and Venus's learning. Participants should be encouraged to focus first on the dialogue between Gloria and Venus in Part 4. As you circulate among the groups, you might ask participants to underline or circle line numbers where Gloria is asking questions, where she is offering advice, and where she is giving answers. A next step for their discussion is to consider the implications of Gloria's moves for Venus' learning.
WholeGroup Discussion
In this last discussion of the Venus case, the participants will explore the focus question, “If we can say that a coach makes moves based on her understanding of the mathematics, the students' ideas, the teachers' ideas, and her beliefs about both learning and her role in it, what do we learn from Gloria about coaching?”
Ask the group to take a few minutes to go back through all sections of the case and choose two excerpts that capture a particularly significant coaching move. Give the group a few minutes to do this and then solicit from about five participants just the line numbers they chose without comment. Write these up on the board and give yourself and the group a few more minutes to find these sections and to read them. Then ask the same volunteers to share why they chose these excerpts. Invite the whole group to either comment on these same excerpts or to offer a new one for the group to consider.
One participant chose lines 205–235. She said, “Right here, Gloria is doing this thing where she dances back and forth between being really explicit in her answers, and then instead of being explicit in responses, she asks explicit questions. So she answers Venus' question about negative numbers and even gives an example of how Venus could think about it. But in the next turn, she could have answered Venus's question, ‘What's going on with Phil?’ and instead she asks her very direct types of questions. It's so interesting to me that she constantly makes these decisions about what Venus can take in directly and what Venus should consider for herself.”
Another offered lines 272–305. She said, “I think this is really important here. Reading this part of the case, Gloria describes places where she and Venus are both interacting with the students, and you really see how they have a partnership going. Gloria could have just come in and modeled the lesson and then ask Venus what she thought of it. And I've done that before. But this strikes me as not only a really solid back and forth kind of coteaching, but you also hear how Gloria respects Venus. She says in line 287, ‘Venus takes the reins.’ And then a couple lines down, Gloria says she's asked the kids way too many questions and that ‘Venus picks up the important one and repeats it.’ So often I think, either we coaches or even the teachers think of us as ‘the knowers’ and the teachers as the ones who ‘don't know.’ I mean I get that Gloria really knows what's she's doing, but she also participates with Venus in all this.”
Comments like these will reveal what the participants think is important about this case and also what they are learning about coaching in their analysis of it. Take some time to elicit from the group what they consider salient to their own work and focus on a reflective discussion. Before you close the session, ask participants to spend a few minutes writing an extended exit card about how the discussions about coaching relate to decisions and learning agendas they have in their own work.
Chapter 10 Facilitation Notes: Maintaining a Focus on Mathematics2Hour Session
Session Goals Learning about coaching and facilitating with a focus on mathematics
 Considering successful classroom debriefs
 Navigating complaints and other obstacles to learning in group settings
 Considering successful strategies for including the principal in professional development settings
Case Description: Struggling to Keep Math at the CenterThe case Struggling to Keep Math at the Center describes a coach's efforts to facilitate productive and mathcentered dialogue with teachers and with the principal. In her writing, the coach, Ivy, describes two different teacher team meetings; one she deems a failure, the other a success. Sorting out her reflections and ideas as she writes, Ivy uses case writing as a tool for analyzing and affirming the features of a successful classroom debrief. In addition, she explores strategies for maintaining an effective facilitation stance even in the face of potentially undermining distractions. This case reminds readers of the importance, and the complexities, of “keeping the math” in math coaching.
Session OverviewParticipants first engage in a mentalmath activity related to the math ideas in the case and structured to go beyond sharing strategies to opening up ideas about the operation of subtraction. Once the group is oriented to the math ideas in the case, the facilitator describes the goals of the session and the context for the case. Before reading the case, participants respond to a set of writing prompts designed to surface facilitation and decision making in their current coaching contexts. Reading Struggling to Keep Math at the Center next, participants are reminded of an important principle of effective coaching: mathematics and students' ideas are central to learning in postobservation discussions. A Focus Questions Activity, during which participants discuss the differences between issues raised in the case, is followed by a wholegroup discussion.
Materials for the Session Prepare two charts: one that describes the session goals and one that lists the two writing prompts (see the Writing Activity and Case Reading section)
 Provide extra chart paper for recording mentalmath strategies and the participant responses to Focus Question 2
 Provide counters or small cubes for exploring the math problems and the focus questions
 Provide writing materials, or a coaching journal, for the Writing Activity
Session Agenda WholeGroup MentalMath Activity: 25 minutes
 Writing Activity and Case Reading: 30 minutes
 SmallGroup Focus Questions Activity: 30 minutes
 WholeGroup Discussion: 35 minutes
Math Activity
Begin the session by jumping in to a mentalmath activity with a subtraction problem before describing the session goals or the agenda. As you write the problem on the board or chart paper, ask participants to mentally solve 143–87. In addition to calculating the answer, explain to participants that you are going to ask them to share “images” of the problem: either a story or visual images that illustrate or help them in solving the problem. Give the group a few minutes to think, and then ask for volunteers to share their ideas.
In one group, a coach named Cheryl responded to the mentalmath problem by saying, “At first I think of it being in the 60s because of the 14 and the 8, but then I check on the ones and see that the 7 is higher than a 3. I immediately switch to a 50 and see the difference between 7 and 3 as 6. Then I know the answer is 56. I know,” she said a bit apologetically, “it sounds like it takes forever, but it's really efficient for me.” While the group took a minute or so to consider the problem from Cheryl's point of view, the facilitator asked them to also think about how this description might be explored on a number line. The participants spent some time, with one volunteer illustrating their ideas on the chart paper, to sort out the model of the distance between 87 and 143 following the logic that Cheryl had offered.
Participants will come up with a variety of methods; some will be very familiar to the others in the group and some, perhaps, unexpected. Engage participants in sharing descriptions of their strategies and, most importantly, ask individuals to go further by including an analysis of how they “see” the math solution unfolding. Let the group know that it will be helpful to also look at how the strategies, problems, and visual models might be related. This more elaborated version of a mentalmath activity will enhance the participants' view of subtraction and the variety of ways the operation can be illustrated or modeled. Also, discussing each other's points of view and looking for relationships helps the coaches appreciate what it takes to facilitate in ways that reach beneath the surface of a problem.
Writing Activity and Case Reading
Move from the Math Activity to a brief description of the session goals and the agenda. Next, before reading the case, participants will spend a few minutes reflecting on their own recent coaching experiences as facilitators. Explain to the group that the writing exercise is intended to bring to mind two particular issues embedded in the work of facilitation: connecting to an important focus and engaging participants with their own ideas at the center. Each coach should spend 5 to 10 minutes responding to the following writing prompts that are posted on chart paper:
 Describe two recent coaching experiences when you were responsible for facilitating the discussion. Describe the purpose of these meetings.
 Describe, in some detail, the goals you had in mind. What sorts of things—questions, ideas, or insights—did you think through to support your goals as you were facilitating the discussions in these meetings?
Let participants know that they should consider writing about facilitation in any of the contexts within the scope of their work. They can choose to write about a meeting with a single teacher or with a group in a gradelevel meeting or at a math workshop. Tell the group that the writing is intended to help them connect on a personal level with the case they will be reading; Ivy, the coach, describes the details of two meetings in much the same way. Explain that the purpose of this reflective writing is to describe the real experiences as they unfolded—whether the meetings met the coach's expectations or not. Decide ahead of time if you are going to collect this writing and let participants know.
When you sense that participants are ready to begin reading, end the writing time and let the group know you will refer back to their writing in a wholegroup discussion later in the session. Distribute the focus questions for their smallgroup discussion. If anyone finishes reading the case before the rest of their small group members, they might go back to their written reflection and add new thoughts.
Focus Questions in Small Groups
Participants will spend about 30 minutes in their small groups of four discussing the focus questions. You might circulate with your own copy of the case and ask participants what evidence they have found and what ideas they have about what helps Ivy make her coaching decisions. If participants are responding generally to this, your question will prompt a more indepth look. There are several questions that require paying attention to students' logic. In Question 4, participants are encouraged to consider how students might approach a subtraction story problem. Having a set of about 20 cubes or other counting objects at each table will support an exploration of this problem.
Case Discussion
Before leading the case discussion, review the case for your own responses to Questions 3 and 7. It is useful to have your own sense of Ivy's stance as a coach. In this way, you can prepare to follow two lines of thinking, one is to briefly consider the math in the case and the other, then, is to follow the decision making process of the coach.
The case discussion can begin with a question that helps illustrate Kwame's idea and the logic behind laying out two piles of cubes when solving 13 − 5. Ask, “What story problem might easily prompt a young child to place a pile of 13 and a pile of 5 on the table to solve this subtraction problem?”
Next, move on to consider the decisions that underlie daytoday work in coaching. In this case, we hear quite clearly Ivy's frustration with the pace of progress at the school. Coach participants will relate to these feelings. One coach said that her greatest challenge is to deal with her own patience; “I know that I am there as a support and as a teacher which should, by definition, mean that I am a patient person. But I'm not always. On some days, I just want to say, ‘Come on now. Can't you see we are getting nowhere with the complaints and the roadblocks? Don't you care as much as I do?’” Another participant nodded her head and said, “I feel like this is such an urgent agenda that means all of us have to dig in, and sometimes, I don't know what to say when I feel stymied in the conversation. I know what Ivy is talking about, and I feel it a lot at my school.”
In facilitating these discussions, it seems easy to get into the position of sympathizing or commiserating. In the end, though, the conversation needs to help coaches build their frameworks for coaching principles. What principles drive the participants' decision making when times are tough or the conversations are unproductive? On what do they rely as they press for more accountability or more engagement? Toggling back and forth between Ivy's assertion that “math at the center” is what matters for her, and participants' central ideas might be a possible discussion for your group. If the group has less experience, it is particularly important that they begin to define the principles upon which they make choices and decisions. Sometimes, as we see in Ivy's case, positioning oneself to press forward, based on a firmly held belief and principle regarding deepening math teaching practice, is what will make the difference between work that is unproductive and work that is generative.
Some participants note that the teachers may not know what is expected of them at such meetings. One math specialist pointed out, “If you are just told what the meeting's about, then how do you know how to behave in such a new way?” If no one suggests this in the session, you can raise this as a thought to consider. Ask what work the participants do, very consciously, to help teachers and principals appreciate the nature of the meetings—the cultural norms coaches are striving for—as they facilitate. Ask how teachers and principals understand the nature of the collaborations coaches are working to build.
Before you bring the discussion to a close, ask the participants to share, in pairs, the gist of their previous writing activity. Ask them, in light of the wholegroup discussion, how they are thinking about the ideas in their writing. Before the session closes, ask participants to write an exit card responding to a question such as, “How does the idea of math at the center play out in your daytoday work? What new ideas do you have about facilitating based on principles of coaching and learning?”
Chapter 11 Facilitation Notes: Framing the Connection between Coach and Teacher Goals2Hour Session
Session Goals Explore facilitation as an aspect of coaching
 Learn about setting clear learning goals for participants
 Consider the intersection of participants' ideas and beliefs with the goals of professional development
 Learn to balance learning goals, participants' ideas, and participants' perceived needs
Case Description: Unsatisfied in the SeminarThe coachauthor, Bonita, describes her efforts to facilitate meaningful and challenging math professional development for teachers. She describes her struggles in reaching two resistant teachers and in connecting the seminar to their beliefs and concerns. Bonita includes specific examples of her correspondence between the teachers and her efforts to reframe the professional development goals to meet the teachers' perceived needs and so that the teachers can appreciate and connect to the purpose and direction of the seminar. In this way, Bonita's case also highlights the use of written correspondence as a useful coaching strategy.
Session OverviewThe session begins with a brief description of the goals and the context for the case, Unsatisfied in the Seminar. Participants read the case and the focus questions before moving into smallgroup discussions that focus on an analysis of the coaching and facilitation issues Bonita's case highlights. Participants will next engage in a reflective writing and planning activity designed to support the articulation of a series of steps toward a more effective and successful collaboration with a teacher or administrator with whom they are currently “out of synch.” A final facilitatorled discussion follows.
Materials for the Session Create a poster of the Session Goals
 Provide writing materials, or a coaching journal, for the reflective writing during the Planning Activity
Session Agenda Introduction: 5 minutes
 Case Reading and Focus Questions: 15 minutes
 SmallGroup Focus Questions Activity: 25 minutes
 WholeGroup Discussion: 20 minutes
 Planning Activity: 35 minutes
 WholeGroup Discussion: 20 minutes
Introduction
Begin the session by offering a brief description of the agenda and the learning goals of the session. Set the stage for the case by describing, in just a few words, the context of this coaching case written by Bonita.
Case Reading and Focus Questions
Participants should read the focus questions before they read this case. The questions will provide a useful lens with which to read Bonita's professional development facilitator journal and her coaching questions. Let participants know that they have about 15 minutes for reading the Focus Questions and the case.
Focus Questions Activity in Small Groups
Participants work together in small groups to zero in on important issues of facilitation of a seminar—and coaching in general. Preface the smallgroup work by helping make the connections between the participants' coaching work and Bonita's case. While the coaches in your group may not facilitate math seminars, they are likely charged with facilitating workshops, gradelevel meetings, and other professional development experiences where they may find the learning goals are in conflict with the participants' perceived needs. Even though you are asking the group to stay focused on Bonita's experience, this careful analysis of her struggles and decisions will provide new and thoughtful perspectives for coaches to bring to bear on their own facilitation dilemmas.
The first question is aimed at clarifying what sort of math seminar this is and how the goals may differ significantly from previous experiences—and expectations of professional development—that Bonita's and Sylvia's participants have had in the past. Being clear about the seminar goals will help smallgroup participants appreciate the importance of Bonita's dilemma about how to respond. She wants very much for her seminar participants to find satisfaction in learning about students' thinking and in developing their own math content knowledge; she knows that deep engagement with both will offer more robust learning for teachers than dropping her seminar agenda to respond to the more teacher practice issues the participants raise after the first session. Bonita wants the teachers to feel heard and to understand that their questions and beliefs are important, but at the same time, she needs to figure out how to incorporate these issues in such a way that does not diminish the important and, to Bonita's way of thinking, more fundamental, goals of the seminar.
Participants should investigate the issues Bonita raises through the context of this seminar and the beliefs and ideas unearthed in the teachers' homework assignments. Coaches will have a chance to talk about the specific ways Bonita manages to help the participants find purpose in her seminar while “staying the course.” If there is time for the groups to address Question 7, they will have a chance to talk together about the implications of these issues in their own work.
WholeGroup Discussion
Open the wholegroup discussion by asking participants to frame the facilitation issues Bonita grapples with in this case. One response to this question might be that while Bonita appreciates the tenor that change can affect for teachers in her district, she begins to appreciate this much more deeply as the seminar begins. In fact, even as she and her cofacilitator, Sylvia, plan and prepare for the seminar by studying the ideas and the work incorporated in the materials, Bonita describes, “I hadn't realized how much time I would spend thinking about the learners' experience, about what it means to engage someone—and engage people coming from lots of different traditions and beliefs.”
Inexperienced facilitators may not realize the skill involved in incorporating teachers' beliefs into an already focused agenda. Throughout the series of cases in this book, the themes of working at the level of participants' ideas, focusing on mathematics, students' thinking, and analyzing implications for next steps are evident in and explored through coachwritten cases and activities. Bonita's case serves as an example of someone wrestling with the way she perceives the teachers' ideas and wishes, while at the same time, believing that staying the course with the seminar material—even as it doesn't first appear to the teachers to meet their needs—will be the appropriate learning opportunity for them.
One participant pointed out this same excerpt about teachers' beliefs and said, “I spent so much time planning the workshop, it never occurred to me that they would not be engaged or that I would have to fight for their engagement. I was so flustered; it was a flop.” New to a facilitation role, it is easy to get caught in studying the session materials, or designing the workshop, without considering how the materials or plan intersects with current teacher beliefs or experiences. The facilitator thoughtfully responded by acknowledging the similarity between Bonita and Sylvia's preparation and the way teachers new to a math curriculum might prepare. She added, “Inexperienced with new material, it is easy to focus preparation on finding the exact wording, making copies, setting up an agenda, etc. All the while we've forgotten there are actual students involved!”
Bonita believes that by studying students' voices and their work, by reading authentic accounts of classroom experiences written by teachers, and by investigating related math activities, the teachers' concerns will actually be addressed. The effort she makes to frame the seminar so that the teachers feel heard and feel confident that the work is worth engaging in are at the heart of Questions 4, 5, and 6. Ask participants to describe the actions Bonita takes and how these support her goals. It will be important for participants to pinpoint line numbers in the case so that the wholegroup can focus their attention on specifics.
In closing the discussion, ask coaches to talk for a few minutes about when, in their own work, they need to call on facilitation skills similar to those that Bonita describes. Point to the line where Bonita asks, “How do I stay centered when the going gets tough? What do I rely on as the facilitator that will help me stay the course and push through resistance to a place where people want to ‘come to the table’?” Ask the participants on what principles they rely as they move through difficult facilitation experiences. Here, you are not asking necessarily for descriptions of what they do, rather you are asking about the principles of the work that center them. New coaches may not yet be able to answer this question about principles but will be able to answer what they do in these situations. This question can be asked over the course of any of the case sessions in the book. As coaches develop their practice, they come to recognize and articulate the principles that guide their decisions.
Planning Activity in Small Groups
Set up the Planning Activity by assigning small groups of three. Because the first part of the activity requires that each person in the group reflect and talk with the others in the group, limiting the size to three also limits this part to no more than 15 minutes. Launch the activity by describing how the activity will unfold. First, each person in each small group will have a few minutes to reflect before describing the challenging coaching collaboration to the others. As each person talks in the small group, the others listen. It is not a time for offering advice or suggestions; rather this is a time for each person to work at articulating the challenge. Remind participants that there is a great deal they can learn from each other and that this careful listening is one very important way of taking advantage of this time together.
Next, participants respond to the writing prompts. These prompts are designed to give coaches a chance to think carefully about these challenges of facilitation and of relationship building in the context of coaching and fulfilling important mathfocused goals.
The Planning Activity ends with a smallgroup discussion, during which each participant has a chance, once again, to describe his or her thoughts and ideas. Ask groups to pay careful attention to their listening skills and to ask each other questions with the intent of helping the other person think more deeply or productively without resorting to simply offering advice.
WholeGroup Discussion
The wholegroup discussion is fairly brief. Participants have had ample time to discuss Bonita's ideas, to articulate their own and to listen carefully to each other. Ask for sharing about the implications of this case and the discussions in the session on ideas about their own facilitation and decision making in coaching. What came up in the Planning Activity discussions and reflective writing that offered new insights? What did they hear from other participants that helped them think about their own contexts? One of the advantages of coming together in a coach group and discussing ideas at this level is to gain new perspectives from others engaged in similar work and who struggle with similar challenges. Open the conversation so that coaches are encouraged to talk with each other in this wholegroup discussion.
Save a few minutes for participants to write exit cards. You might ask coaches to describe the most important new idea that came from the session and to explain how the session will specifically influence their facilitation practice.
Chapter 12 Facilitation Notes: Examining the Role of Authority in Coaching3Hour Session
Session Goals Cultivate a reflective coaching practice
 Explore what it means to claim the authority to make decisions about the learning needs of other adults and to design experiences with these goals in mind
 Explore the authority we negotiate with others and how to maintain one's authority while not usurping others
 Examine the links between teaching and coaching and analyzing the shift from a teacher of children to a teacher of teachers
 Learn from other coaches' experiences and perspectives
Case Description: Claiming AuthorityClaiming Authority, written by a math coach in her third year of coaching, provides participants with a view of the trajectory of one coach's developing practice. The author describes connections between her developing coaching practice to the early developing years of her classroom teaching. The author tackles the issues of authority in coaching, an important theme that will have resonance for all coaches and teacher leaders. She considers ways she might have moved more quickly toward a stronger practice, including taking advantage of the wisdom of coach colleagues.
Session OverviewThe session begins with a brief introduction of the goals and the context for the case, Claiming Authority. Participants will read the case then move into a smallgroup discussion of the themes and issues the author raises followed by a facilitatorled wholegroup discussion. Next, participants work in new small groups on a Planning Activity. This activity is designed to help coaches examine their own trajectory of learning, to explore their own principles of practice, and to articulate ways they plan to continue refining their coaching practice. The session ends with a wholegroup sharing and discussion focusing on insights about coaching practice and claiming authority in the role.
Materials for the Session You will need enough chart paper and markers for participants to complete the Two Posters Activity and, in the event that you decide to, chart ideas that emerge from the two wholegroup discussions.
 Participants will be in two different groupings for smallgroup activities, thus, writing up a chart with two sets of assigned seats will save transition time.
Session Agenda Introduction: 5 minutes
 Case Reading and Focus Questions: 15 minutes
 Focus Questions Activity in Small Groups: 35 minutes
 WholeGroup Discussion: 30 minutes
 Two Posters Activity: 60 minutes
 WholeGroup Discussion: 35 minutes
Introduction
Begin the session by offering a brief description of the agenda and the learning goals of the session. Set the stage for the case by describing, in just a few words, the context of this coaching case written by Carina.
Reading Case and Focus Questions
Claiming Authority is a rich case where the author compares her developing sense of authority as a coach to the way a similar sense of confidence and authority developed in her first three years of classroom teaching. Suggest to participants that reading Carina's writing will be like reading a journal of a colleague's reflections. Ask them to then review the case by underlining particularly important passages; they will want to easily locate lines that have particular resonance during both smallgroup discussions. For this session, ask participants to peruse the focus questions ahead of reading the case.
Focus Questions Activity in Small Groups
After reading the case, participants will have approximately 35 minutes to discuss the focus questions in small groups. If there is not enough time to get to the last question, let the group know they will have a chance to talk this one through in one of the two wholegroup discussions. The goal of this activity is twofold; the focus questions are designed to help the participants make sense of Carina's developing coaching ideas and the principles that drive her work. The discussions also set participants up to do the Two Posters Activity while using Carina's ideas as a springboard for thinking about their own.
The first focus question points to a powerful passage in the opening of the case. Here, Carina describes her entrée to establishing coaching partnerships with teachers. There are four or five separate statements of coaching moves she made in her new position. This section is also where Carina provides readers with the noteworthy insight that drawing on her “deep respect and admiration for children's thinking” appears to be what enabled her to manage teacher criticism. During one smallgroup discussion in this section, a participant looked at her colleagues and commented, “It's really true. It's not just about what we say or do; it's also about how teachers perceive what I value. I think Carina's saying that, in the end or, in fact, in the beginning, we can do this. We can meet here. We can all come together in this place we all value.”
As you circulate among the groups, unless participants are discussing Question 6, encourage them to stick with the details of the case. It sometimes happens that the case provokes so much about what happens in daytoday coaching that individuals might veer off to discussing mostly their own experiences. If so, they will be missing this opportunity to learn from a very reflective and experienced math coach. It is important that the smallgroup discussions take advantage of learning from these particular sentiments and principles Carina describes. Again, they will have a chance to work on ideas about their own coaching trajectory during the Two Posters Activity.
WholeGroup Discussion
The aim of the wholegroup discussion is to elicit main ideas from Carina's case and offer the participants an opportunity to explore how they understand the issues of cultivating a practice. Ask volunteers to offer line numbers of particularly important sections of the case that they would like the whole group to have a chance to discuss. Ask, “What really stood out for you? Look through the case and note a section that resonates in your own coaching or that points to an issue that you wrestle with in your practice or that you particularly would like to hear discussed in the whole group.”
Give the group a minute to review the case, and then ask for four or five sets of line numbers. Tell participants that you want to get a list of the sections on a chart; discussion will follow once a list is noted. Explain that this way, if there isn't enough time to discuss the whole list at this point, there might be time to go back to these before the end of the day's session. You can then begin the wholegroup discussion by asking all participants (i.e., your question is not aimed at the one who volunteered but rather, opened to everyone) how they understand the section listed first on your chart. Variations of useful framing questions are: Why is the section important in understanding more about coaching? What about this section stands out for you? What seems to be complex about the issue(s) raised in this section? How does this section help us understand the role more clearly?
Two Posters Activity in Small Groups
Before the participants set to work, preface the activity by reminding the group of the norms of these sessions. It will be helpful to have the group talk for a minute about what it means to talk about one's developing practice as honestly as Carina has in her case and about the importance of confidentiality and safety in the group. Assure participants that it is the intention of the group—and an important aspect of your job as a facilitator—to create an atmosphere of trust and a place for open discussion about cultivating a coaching practice.
First, each participant makes his or her own version of the poster reflecting one's own stages of coaching. The second poster can be created either by pairs or by individual participants. As the facilitator, you will want to review this activity and make careful choices about the make up of the small groups and whether the second poster should be a solo or pair activity.
The first poster is a selfreflective piece of work. The participants might represent a mix of very novice to very experienced coaches; they should be assured that the stage of coaching they describe on their posters and the items they list are not compared for speed of trajectory. On the contrary, looking for comparisons should be through a lens of learning more about coaching as a practice, learning about themes across the group, learning about the complexities, and learning from each other's experiences.
The second poster will describe guiding principles, next questions, and ways to continue learning about coaching practice. There should be some consideration for pairing coaches or having each coach create a poster. If your whole group represents a wide range of experience, you may choose to partner the more novice coaches with each other. You might choose to have pairs of coaches discuss and brainstorm ideas, and then ask each coach to make a poster that represents only his or her own principles and questions. The balance is in helping coaches dig deeply to create meaningful posters to share, sometimes more easily accomplished by having a brainstorming partner, while at the same time, allowing each coach to create a very personal list without being influenced by a partner or without deferring to a partner's ideas.
You will need to act as timekeeper. Ask participants to create straightforward posters that clearly display their thinking. (Sometimes people might need to be encouraged to add words to their more pictorial representations!) Remind them that they have one hour to consider and create posters; it will be important to plan their time accordingly. Be sure that participants have 30 minutes to plan and to create the first poster. If any participant has not completed Poster 1 by then, have them move on and spend the rest of the time on Poster 2. As charts are created, group them by first and by second poster and tape them to the wall in a gallery style. When the hour is up, let participants know that it is time to take a break, and then carefully review each poster on the wall. Remind participants that the posters have been created with real care and that each one deserves their thoughtful attention. All participants should then quietly read and review as they do a gallery walk.
WholeGroup Discussion
The final wholegroup discussion is aimed at the ideas revealed in the two posters. These will reveal a variety of perspectives on coaching and themes of coaching practice, a central goal of the activity. Ask participants to describe what's typical about the beginning stages of coaching ideas. Ask the group to compare the themes from participants' first charts to their previous reflections regarding Carina's trajectory. Are they similar? If so, what might that mean? If not, what might account for the difference? What did you learn by looking at your practice through the lens of your developing ideas?
Ask participants to take a few minutes to review the posters that describe, guiding principles, next steps, and ways to continue learning. Take time to discuss the lists of guiding principles. Again, look for commonalities and for new ideas. Choose a wellframed principle from a list and ask participants to comment on the implications of the principle for the work that they do. Ask how this principle is important and in what way their work is more successful or effective when grounded with this principle in mind. Review the lists with an eye toward bringing forward principles that are aligned with a strong coaching vision. You may find that, given the personal and revealing nature of the reflective work participants have been engaged in, they will welcome your reassuring leadership in the facilitation of this discussion of principles and a focus on strengthening and refining practice.
End the discussion by asking for comments on the last section, Ways to Continue Learning. You may have particular structures or strategies in your setting that coaches can use to their advantage. Perhaps coaches want to attend coaching seminars, intend to write (or continue writing) their own cases of practice to share with colleagues, or partner with another coach to observe each other and debrief their work. Look for promising examples such as these, and ask coaches how these ideas might be accomplished.
As the session comes to a close, ask participants to respond to a set of exit card prompts. You might ask coaches to describe the most important new idea that came from the session, to explain how the session will specifically influence their coaching practice, or to articulate one next step in cultivating a coaching practice.
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