Advice from DMI Users
DMI was produced, it has been used at various sites around the country.
The contexts in which the DMI projects have been set up, the backgrounds
of facilitators, the constituencies participating at each site, and the
formats in which the materials are used, are all quite varied.
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
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 weekly: "I really like this format. It's very intense,
yet enough time between sessions that participants have time to reflect,
try things in classrooms, etc. Even when we had two weeks between sessions
(because of vacation), participants still admitted to doing the work the
Using DMI in Various Formats
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."
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."
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.)."
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."
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
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."
Recruiting for DMI
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
Message to Principals
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:
- think through the
major ideas of K-6 mathematics,
- examine how children
develop these ideas,
- determine what
constitutes a teaching practice that supports children's development
into powerful mathematical thinkers, and
- 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.
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:
Please let me know if
you have any questions. My phone number is 425-702-3253.
- Send to me (Christopher
Fraley) the name of a teacher for your guaranteed spot by SEPTEMBER
- 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.
(At this point, I
provided a description of the seminar.)
B) Reminder message
with sharing of those who have committed
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
*** 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
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.
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
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
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.
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
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.
(At this point, I
enclosed a detailed description of the seminar.)
E) a Message for
the commitment of participating teachers
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.
(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
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
Responding to Teachers' Writing
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.
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.
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
- 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.)
hope that I will learn more about HOW to ask questions so I can push
a child's thinking.
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.
- Asking questions
that encourage more analysis of an issue raised.
tried one of the math activities with my students, too.
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?
- Providing encouragement
(I want to encourage teachers in their struggle but not trivialize it
or distract them from it.)
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.
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
- Suggesting resources
would also like to find some materials to support what I am doing
in my classroom.
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.
- Expressing the
ideas that their writing had simulated in my own thinking.
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.
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.
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.
I am pondering this idea of "recipe." To me (I have
very basic cooking skills), a recipe implies a set way
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
and because learning is dynamic. I guess you could
think of this as a gourmet mathematics class.
Adapting DMI for an Undergraduate Course
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
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
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.
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.
Support for Cultivating
Inquiry into the Teaching of Mathematics has been provided by the Noyce
For more information,
SummerMath for Teachers
Mount Holyoke College
South Hadley, MA 01075-1441
413) 538-2071 FAX: (413) 538-2002
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
- 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
- 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.