We, users and developers of DMI, have learned quite a bit from these varied efforts. In the linked pages, several users share some of their reflections, strategies, and discoveries.

Using DMI in Various Formats: Given the particular needs and demands at different sites, people have experimented with a variety of formats in terms of the length and frequency of meetings. On this page, users offer comments on the formats of their seminars.

Seminar meeting for three hours after school biweekly: "Meeting every other week gave both the participants and the facilitators some time to reflect on the work we were doing. As a facilitator who was at the same time a classroom teacher, it was important for me to have the week between to prepare with my co-facilitators and to reflect on our participants' work. The participants commented that meeting every other week made it seem more manageable in terms of the amount of time they spent in class. They also liked that they had time to try things they had been thinking about."

Seminar meeting for three hours on Saturdays biweekly: "To my surprise, the teachers don't complain about spending their Saturday mornings this way. In contrast to after-school seminars, they arrive rested and refreshed. We'll meet approximately every other week throughout the year to cover both modules."
Seminar meeting for 90 minutes weekly: "We chose this format to accommodate child-care issues for the teachers in our school. Having weekly meetings for an entire year seems to maintain a high level of 'mindfulness' about math, individual commitment, and openness to the work of reform; it also offers an invaluable cohesiveness to the group and to our shared purpose. The drawback is that the case discussions and the math activities that support examination of the cases do not take place in the same session."

Seminar meeting for 90 minutes biweekly: "This format encourages teachers who are involved in numerous after school activities or who have young children at home to be able to do DMI. One disadvantage is that it spreads each session over a month, and I'm now feeling the need to move more quickly. It's also not possible to do both modules in one school year unless you add some sessions at other times."

Each module covered in four full-day workshops: "Teachers who participated in the four full-day meetings explained that, because of child-care issues and other concerns, they would have been unable to attend after-school meetings. They said they liked the feeling of shedding their teaching responsibilities in the morning, to have the experience of being the student all day. And they liked the intensity. Issues to think about include how to cover teachers' classrooms and arranging agendas (such as what to do with the interview, writing episodes, etc.)."

Summer institute, 10 full days "We scheduled sessions so that two afternoons were spent conducting interviews. Teachers were told ahead of time which days they would be interviewing children and were asked to make an appointment with a child. They also had long lunch breaks with enough time to reread the cases we would be discussing in the afternoon. Although we had to sacrifice the assignments to bring in student work and write cases-teachers could not be testing out new ideas in their classrooms as the seminar progressed-the concentration and immersion into the work was a positive."

Summer school course, 210 minutes, 2 times a week: "I taught a summer school course that met twice a week for three and a half hours, four weeks (though it should have been longer). I didn't follow DMI exactly, but we worked through all of Modules 1 and 2. Students read 2 or 3 cases for each class and, as much as possible, I posed problems for the class to try before they were asked to read about what the children did. I also selected particular focus questions for students to read, think, and write about before the case discussions."

Semester course for undergraduates: "Due to the structure of the college schedule, DMI for undergraduates met for 28, 75 minute sessions--twice a week for 14 weeks. Working our way through both modules, there was a reading assignment and a portfolio assignment due each week. In addition, each student was paired with a local teacher, who is working to reform his or her own practice, to facilitate access to classrooms and students."

DMI offers a form of professional development that departs from those generally available in school districts. For that reason, many people who have set up DMI seminars have realized that they need to employ a variety of strategies to communicate what DMI is about and to enlist the support of school administrators. Christopher Fraley, from the Lake Washington School District in WA, describes below the strategies he employed and shares the text of messages he sent to potential participants, principals, staff developers, and other administrators.

First, I try to keep my supervisors and principals as informed as possible before I start the recruiting process. District administrators and principals can sometimes be a road block to effective recruiting if they are not "on board." However, they can also be tremendous assets in getting the word out if they are "on board."

Second, I try to connect the goals of the program to the goals of the district and/or building. I try to show that this experience will support teachers in "going" forward with the organization's goals. If administrators are convinced of this, they can often be your best recruiters (outside of teachers who have been through the seminar) because they are in contact with teachers and may know who would be really interested.

Third, I try to make it seem like this seminar is a sought after thing… that if principals and/or teachers don't act soon they may miss out. Deadlines, mention of waiting lists, and the sharing of who has already committed often help promote this sense of urgency.

Below, I have included examples of A) an informational message to principals, B) a reminder message with a list of those who have committed, C) a message to staff-development personnel (they have contacts with teachers and have insights that I do not), D) a letter of invitation to teachers, E) a message asking for the commitment of participating teachers, and F) a recruitment up-date for administrators.

A) Informational Message to Principals

Dear Principal,

Here is an exciting opportunity to build capacity in your building for improving the mathematics performance of students by developing and supporting a potential teacher-leader in mathematics education.

I am offering a seminar/workshop for elementary teachers (K-6) called "Developing Mathematical Ideas." This course will help teachers to:

1. think through the major ideas of K-6 mathematics,
2. examine how children develop these ideas,
3. determine what constitutes a teaching practice that supports children's development into powerful mathematical thinkers, and
4. understand what effective student-centered instruction looks and feels like in the classroom.

Two critical components of any effort to improve student performance in mathematics are: 1) to strengthen teachers' understanding of the mathematics they are being asked to teach, and 2) to modify or improve the effectiveness of teaching practices.

This seminar/workshop supports teachers in both of these areas.

I AM GUARANTEEING EACH ELEMENTARY BUILDING ONE SLOT IN THIS YEAR'S SEMINAR. This seminar may be a useful piece of support for those buildings considering a focus on mathematics. Some buildings may want this seminar for some or all of their teachers as a part of next year's implementation plan.

I am encouraging each building to find one teacher who will commit to this seminar. This teacher will be able to help your building determine if this kind of training would benefit more of your staff.

Since this seminar is consistent with the district's profile goals, the 3 profile-incentive days could be used to help pay for some of a teacher's time. Teachers can also earn 4 college credits or 40 clock hours.

A detailed description of the seminar is enclosed and is included at the end of this quickmail. A letter of invitation for teachers is also enclosed and included at the end of this quickmail.

*** IMPORTANT! ***
Timeline and Procedural Information:

• Send to me (Christopher Fraley) the name of a teacher for your guaranteed spot by SEPTEMBER 26th.
• If more than one teacher would like to participate, send me all their names. The one you list first will be given the guaranteed spot and the others will be put on a waiting list.
• On September 26th, I will open up the seminar to any interested teacher on a "first come, first serve" basis. Those teachers on the waiting list will be given first priority.

Sincerely,

Christopher Fraley Staff Development

(At this point, I provided a description of the seminar.)

B) Reminder message with sharing of those who have committed

Hello,

Today (9/26) is the last day that I will be reserving a spot in the "Developing Mathematical Ideas" K-6 seminar for a teacher in your building. So far, we have representatives from 10 different buildings. If I do not hear from you today, I will assume that no teacher will be representing your building. This afternoon, I will be opening up the seminar for those teachers on the waiting list.

The following is the information that I sent to you last week (9/17) about this seminar/workshop.

C) Message to Staff Development personnel

Below are the dates for the Developing Mathematical Ideas Seminar/Workshop.

But first, please read the following carefully.

* As you know, one of our focus filters is "promoting building capacity by developing teacher leaders." Developing Mathematical Ideas will be a great opportunity for teachers who have potential for taking on some leadership in their buildings. I am particularly interested in recruiting teachers who could become future facilitators of this workshop.

** Please encourage teachers who you think, given an opportunity like this, could really increase the capacity of their building. Or give their names to me and I can give them a call.

I hope that each of you will be able to participate. I believe that this experience can be a powerful way for us to move math forward in our district. As you know, there is a strong interest in this area throughout the elementary buildings.

*** Lois has ok'd the offering of this course. So the time frame will be as follows:

First: A letter will be sent out to all principals by tomorrow morning. It will provide an overview of the seminar and explain that I am guaranteeing their building one spot if I receive the commitment of one of their teachers by September 25.

Second: On Sept 26, I will open it up to all teachers on a "first come" basis. It is IMPORTANT that I have the name of potential teachers prior to this time so that I can ensure that they have a place.

Third: If there are not enough teachers signed up by October 3rd, then I will open the workshop up to secondary teachers.

Please let me know if you have any suggestions for or concerns about this plan or the dates (see below).

The dates are:

October 20 and 27 November 24 December 8 and 15 January 12 February 9 and 23 March 9 and 23 and 30 April 20 and 27 May 11 and 18 June 8

Please respond as soon as possible if you see any problems or if you have any concerns. I need to get this project rolling.

Thank you for your help and suggestions, Christopher

D) A letter of invitation to teachers

Dear Colleague,

I am excited to invite you to participate in a seminar/ workshop for K-6 teachers called "Developing Mathematical Ideas." This seminar will be an opportunity: 1) to explore the major ideas of K-6 mathematics, 2) to analyze how students' thinking about these ideas develop and change as they progress through the grade levels, and 3) to determine what constitutes a teaching practice that supports children's development into powerful mathematical thinkers.

The seminar is organized around a series of cases written by teachers in which they describe classroom events during the teaching of mathematics. These brief narratives capture both student dialogue and the teacher's analysis and questions. These cases are supported by videotapes of classroom sessions and student interviews as well as related research.

Participants in this seminar will discuss these cases and videotapes to better understand what students' thinking reveals about their understanding of mathematics and how their understandings can be used to inform teaching decisions.

These cases will be grouped under the topics of "Building a System of Tens" and "Making Meaning for Operations." Participants will consider ideas of the base ten number system from learning to count to operating with multi-digit numbers and working with decimals. They will also consider the strategies students use to solve problems in real contexts and the actions and situations that students use to model the four basic operations.

Seminar assignments will include reading cases and research articles as well as writing short papers. Participants will also be responsible for keeping a portfolio.

Each session will be held at the Resource Center in the Sammamish Room from 4:15 - 7:15 PM. The dates for the sessions are:

October 20 and 27 November 24 December 8 and 15 January 12 February 9 and 23 March 9, 23, and 30 April 20 and 27 May 11 and May 18 June 8

Participants may choose to earn 4 credits or 40 clock hours.

Cost $10.00 Materials Fee; $35.00 / 1 continuing education credit; $1.50 / 1 CEU (continuing education unit = 10 clock hours)

You may register by sending me your name, school, and grade taught via quickmail.

Please let me know if you have any questions or if you would like a more complete description of the seminar.

Sincerely,

Christopher Fraley, Staff Development

(At this point, I enclosed a detailed description of the seminar.)

E) a Message for the commitment of participating teachers

Hello Everyone!

Welcome to the Developing Mathematical Ideas Seminar. You or your principal sent your name to me and I am happy to inform you that you made it into the seminar. I look forward to meeting each one of you on October 20th.

You should receive the seminar casebook and your first assignment (a short one) by October 15. A materials fee of $10 will be due at our first meeting and you may sign up for credits or clock hours at that time.

It is very important that you are able to come to each seminar session and that you complete each assignment. Each session builds on the previous one. Your participation in the activities, the assignments, and the discussions will impact the quality of our experience.

Since there are teachers on a waiting list who would like participate, please review the seminar summary below. If you decide that you are unable to participate, please let me know by Thursday afternoon (October 2). If I do not here from you by then, I will assume that you are fully committed to this seminar.

I believe that you will find this seminar to be a great experience.

Let me know if you have any questions or comments.

Christopher

(At this point, I again provide them a detailed summary of the seminar.)

F) A recruitment up-date for administrators (to keep them informed and "on board").

The participant roster for the Developing Mathematical Ideas seminar has been finalized. There are 36 teachers and one principal participating. These teachers represent 19 different elementary buildings as well as every grade level K through 6.

One of my goals for this year is to determine if this kind of training should become one of the resources that buildings can use to more effectively implement the mathematics portion of the framework.

Below is the information I sent out to principals regarding the seminar. I sent this to you earlier in the year, so you do not need to read it unless you want to refresh your understanding of what this seminar is about.

If anyone made it this far, you deserve an award. I hope you may find something of use in this.

Christopher

Several DMI users have reported that responding to teachers' writing is a new task for them, one that requires new skills and ways of thinking about giving feedback. In the messages below, we hear from Keith Cochran of Clark County School District (which includes Las Vegas) in NV and from Christopher Fraley of Lake Washington School District (near Seattle) in WA. Keith discusses his general thoughts about the experience and value of responding to teachers' writing. Christopher offers his reflections on his first set of responses and is quite explicit about the strategies he used.

  From Keith:

To respond or not to respond, that is the question. Although I believe it's not really the question at all, because if we value the participants writing the portfolio assignments, I believe we have an obligation to respond. The real question is, how do we do it? As my colleague and co-facilitator, Debbie Hodler, and I sat down to talk about this before the seminar began, we decided we wanted to respond individually to each participant. We also decided to share the duty, so that we each took half the portfolios, and then would switch the next time. That way we got to read everyone's portfolio.

Reading and responding to the portfolios can be a difficult process. Finding the time to do this is hard enough. It's also difficult to know how to respond. You want the response to be meaningful, but do you challenge beliefs? How far to you challenge people? How much encouragement do people need? While I believed responding was important, and tried to do my best in making meaningful comments, I just wasn't sure if the process was having any impact.

After we finished Module 1, I met with a focus group to discuss how the seminar went. Two of the participants mentioned that comments written by Debbie and me had a definite and direct impact on the work they did with their children. These two were very explicit about the power of specific responses, and the rest of the group, though they talked in general terms, agreed that it was important to receive our messages. It was very exciting to hear this, and I realized that responding to the portfolio assignments was worth every agonizing minute we spent.

From Christopher:

I have just finished responding to teachers' reflections on their expectations for the seminar. I found that my attempts to give meaningful feedback on what they shared really forced me to think more deeply about mathematics, the goals of this seminar, and student learning.

Initially, I wasn't sure what I was trying to accomplish with my responses. I knew that I wanted to acknowledge their thinking; I wanted them to know that someone was listening and taking their comments and ideas seriously. I also wanted to encourage them to think more deeply about what they were saying. I ended up responding to their writing in at least five different ways. Sometimes I responded in two or three of these ways to the same paper.

The following provides a brief description of each way with examples to illustrate the approach. These examples are excerpts taken from the teachers' reflections and my comments.

Approaches with Examples:

1. Suggesting that they keep a record in their portfolio of insights they gain over the course of the seminar to a particular expectation or issue that they mention. (I want teachers to see their portfolio as a tool for focusing on and remembering important insights to the issues that concern them.)

   Teacher: I hope that I will learn more about HOW to ask questions so I can push a child's thinking.

   Response: In your portfolio, you may want to keep a record of the insights that you gain from this seminar experience with regard to questioning. I would be very interested in reading what you discover.

2. Asking questions that encourage more analysis of an issue raised.

   Teacher: I tried one of the math activities with my students, too.

   Response: How did the activity you tried with your students go? Were they able to articulate the different kinds of strategies that they used? What did their actions and statement reveal about their understandings? Did you notice any evidence that the activity increased their understanding of how numbers can be built and broken apart?

3. Providing encouragement (I want to encourage teachers in their struggle but not trivialize it or distract them from it.)

   Teacher: The video was very good, showing a real classroom tackling problems and explaining their understanding. I felt inadequate though, in my current teaching, doubting that my students could have performed as well as the second graders on the tape.

   Response: You mention some feelings of inadequacy with regard to what you saw in the first video. I can definitely relate to that feeling. Changes in your students' ability to think and explain their thinking will not happen over night. Give yourself permission to be a learner. Start small. Take one step at a time. After trying a new approach or strategy, think about what you did and how the students did and make some revisions for the next time. You also have your teammates from this seminar to talk things over with and get ideas from. I think you will be pleasantly surprised to see the changes that do occur over the course of this year.

4. Suggesting resources

   Teacher: I would also like to find some materials to support what I am doing in my classroom.

   Response: I just received the first grade book from "Investigations in Number, Data, and Space" which focuses on addition and subtraction. I would be interested to have you look it over and get your opinion of it. I am curious to know if you think that these materials would provide some of the support that you are looking for.

5. Expressing the ideas that their writing had simulated in my own thinking.

   Teacher: Sharing ideas (with table group) also helped with my confidence because I received such supportive comments from my teammates, and I felt as though my ideas were accepted and validated.

   Response: When I think of the times of my greatest insecurity, it is usually in contexts when I am isolated or being negatively challenged by someone. There is nothing like having others to receive encouragement from and to bounce ideas off. As teachers, we need to find more ways to provide constructive support for each other.

   Teacher: I am hoping that this course will help me to begin to make sense of (mathematics in the classroom). If I cannot come away with a recipe, I hope I will at least have the tools to create one.

   Response: I am pondering this idea of "recipe." To me (I have very basic cooking skills), a recipe implies a set way… a reliable way to put ingredients and equipment together so that I end up with something edible. However, for someone with more sophisticated skills than I, a recipe is more of a general guide from which to build. A chef will take his or her understanding of spices, temperatures, textures, and so on to create and mold a dish into something unique and in a sense artistic. The "flavor" of this seminar is more in line with this second view of "recipe." We will be developing some basic tools and general guidelines for thinking about mathematics, instruction, and learning. We will be developing skills for understanding mathematics and student thinking about mathematics. How we use these tools and skills ultimately depends on our students, our understanding of our students' thinking, and our understanding of mathematics. I think what I am saying is that effective instruction in mathematics is dynamic because mathematics is dynamic… because our students are dynamic… and because learning is dynamic. I guess you could think of this as a gourmet mathematics class.
   

SummerMath for Teachers has offered a semester-long DMI class for undergraduate students at Mount Holyoke College since the spring of 1997. For the first two years, the class meet twice a week for 14 weeks with each class session meeting for 75 minutes. ( For the 1998-1999 academic year the undergraduate DMI class became part of the project Sustaining a Community of Inquiry which followed the format described in the agendas--meeting once a week for three hours. )

Even though DMI had been designed for work with practicing teachers, we followed the curriculum closely, modifying the timing suggested in the agendas but not the activities themselves. Since we were meeting twice a week, we broke each session suggested in the guide into two components; often doing the case discussion at one session and a mathematics activity at the next. Each week the assignments included a reading from the casebook and a written portfolio assignment as suggested in the facilitators' guide.

Each student was paired with a local teacher, who has worked with SMT, to provide access to classrooms for the portfolio assignments. The only modification made to the portfolio assignments was the elimination of those requiring the collection and analysis of student work.

As we worked our way through the two seminars-Building a System of Tens and Making Meaning of Operations-the primary focus of one class each week was a case discussion. The second class was structured to include the related activities included in the curriculum. Early in the semester, case discussions were combined with mental math activities; but during the second half of the semester, we sometimes discussed the written and video cases together. Math activities usually required an entire class session, as students engaged in understanding the mathematics for themselves and explored the use of manipulatives with which they were unfamiliar.

Each semester the students enrolled in this course are of varied backgrounds. Some had an interest in becoming elementary school teachers, some were interested in teaching high school math or science, and some were interested in exploring mathematics for themselves.

Through their DMI experience, undergraduates developed a sense of mathematics as a logically interconnected body of ideas to be explored; learned to view learning as a process of concept construction and refinement of ideas; learned to listen to, follow, and analyze children's thinking; and developed a stance of inquiry toward learning and teaching. An additional outcome was a changed view of teaching; the students began to speak of teaching as a thinking profession.

This course is now a regular offering of the Department of Psychology and Education at Mount Holyoke College.

Cultivating Inquiry into the Teaching of Mathematics (CITM)
Using DMI to support the Work of Teacher-Educators

In the fall of 1996, SummerMath for Teachers began experimenting with the DMI materials to determine how they could support discussion among teacher-educators. We developed graduate-level courses using not only the DMI casebooks, but also drawing on materials in the DMI facilitators' guide--specifically, Maxine's Journal and Two Portraits of Change--and articles from What's Happening in Math Class?, an anthology of teacher narratives with teacher educators' commentaries edited by Deborah Schifter and published by Teachers' College Press.

In the 1996-1997 program, CITM participants worked through Building a System of Tens; in 1997-1998, the focus was Making Meaning for Operations. Participants met for a total sixteen three-hour meetings each year. While CITM offered four graduate credits for each year's of participation, the work was spread out across two years or four semesters. The first four sessions of each semester were devoted to following the DMI agendas as described in the facilitators' guide. In the remaining eight sessions each year, CITM participants took on the view of a teacher educator by working with Maxine's Journal, Two Portraits of Change essay, and articles from What's Happening in Math Class?.

In the same way that the cases from the DMI casebooks present situated images of the mathematics classroom for teachers to consider, Maxine's Journal offers teacher educators opportunities to reflect on the issues they face as they work to help teachers extend their mathematical understandings and to develop new notions of teaching and learning. Excerpts from Two Portraits of Change served as a basis for activities in which CITM participants "practiced" writing responses to DMI teacher portfolio assignments. Through discussions about their responses, the teacher-leaders could articulate their goals and consider how best to communicate those goals to the teachers with whom they work. The teacher-written narratives in What's Happening in Math Class? provided a mechanism to examine classroom instruction in mathematics; the teacher educator commentaries raised issues teacher educators face as they work in collaboration with teachers.

Twenty-six educators enrolled in the CITM program in 1996. They included classroom teachers released from daily classroom duties to serve as mentor teachers, school system administrators charged with supporting mathematics education reform in their districts, teachers responsible for organizing staff development, and classroom teachers who wanted an opportunity to explore mathematics learning for themselves. There were also twenty-six participants in the second year of the program, eleven of the original educators and fifteen participants new to the program.

For more information, please contact:

Comments from CITM teachers
As a result of their work in CITM, many participants wrote about the way their ideas about how to work with teachers had changed. In particular, they wrote their desire to change the way they interact with the teachers with whom they work. The kinds of shifts they described mirror the changes the reform movement suggests for classroom teachers in relationship to their students. For instance, CITM participants were redefining their role-- away from being perceived as an expert who told the answers --and to an image of their work as a mentor or coach who listened hard and asked provocative questions. A common thread in their comments is the desire to create an atmosphere in which genuine curiosity and intellectual inquiry is valued.

• Reflecting on the process of inquiry as modeled in this course has made me more reflective in my work. I am trying new methods for interaction with teachers that, more than ever before, honor the process of inquiry and allow teachers a more constructivist way to improve the teaching and learning in our schools.

• Being able to share our thoughts and ideas in a relaxed, professional manner was a delight.. . .. How can we get the American public to become aware of and accept this legitimate need?

• If my colleagues are to embrace a new way of thinking about the teaching of math, they need the opportunity to think it through themselves-to wrestle with it until it makes sense to them.

• Although I've always listened to the words people are saying to me, this was my year to truly practice "just" listening carefully. That is, I'm trying to perfect the technique of not thinking ahead to a solution or a response. [This experience] made me discover I've been trying to supply answers to the teachers. When I force myself to be a sounding board it never ceases to amaze me...the teachers are able to solve the problems themselves.

• I let go of the notion that I had to have all of the answers or that I had to know more than [the teacher I am working with] and began to accept that our discussions didn't have to have closure. We could leave with questions.

• My views on working with others has changed from being a dispenser of information to a facilitator in helping teachers look at, try, and discuss mathematics to build a schoolwide vision of our mathematics program.
  

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