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OCR for page 77
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The second day of the workshop focused on two examples from the United States of
the use of teaching practice as a medium for professional development. Both sessions
dealt with the use of records of practice records of what teachers do as they teach as
a way to discuss mathematics teaching. Deborah Ball and Hyman Bass discussed the
use of classroom video and addressed the questions
· How do observations of what teachers do in the act of teaching enable teachers to
learn mathematics?
· How do such observations enable teachers to learn how to teach the mathematics
they need to teach?
Margaret Smith presented two written cases of classroom teaching and addressed the
questions
How can cases designed to investigate teaching and learning be a site for learning
about teaching?
What does it mean for teachers to use the study of others' practice to learn mathematics
and about teaching mathematics?
Professional Development Through Records of Instruction
Deborah Loewenherg Ball, Professor, University of Michigan
Hyman Bass, Professor, University of Michigan
Professional Development Through Written Cases
Margaret S. Smith, Assistant Professor, University of Pittsha~rgh
OCR for page 78
OCR for page 79
~11
~~-
~-
Deborah Loewenberg Ball, University of Michigan
Hyman Bass, University of Michigan
Ball: Our focus today wall be on profes-
sional development using what we call,
"records of practice." Like "lesson study,"
this is a form of professional development
that uses practice to learn about teaching.
However, while lesson study engages
teachers in examining their own practice
and in the practice of colleagues, the work
we wall be discussing involves examining
records of practice. What is a record of
practice? It is a detailed documentation of
leaching andlearning. Examples might
be videotapes, either segments from
lessons or whole lessons; written cases of
teaching and learning; students' written
work from classrooms; transcripts of
classroom discussions; teachers' notes
and lesson plans. The point is that these
are documents taken directly from
teaching and learning of mathematics,
without an analysis, which enable teachers
to look at practice (Erectly, together with
other teachers.
WHY USE PROFESSIONAL
DEVELOPMENT IN RECORDS OF
PRACTICE?
Some of the reasons for using records
of practice are intended to address a
number of problems in our professional
(levelopment and teacher education
system. For example, recor(ls of practice
provide a common context for teachers to
work on teaching. Teachers in the United
States (lo not usually have opportunities to
work (Erectly on teaching and learning
with other teachers. When they (lo meet
with other teachers, often all they do is
tell each other about their work or work
on something else like a new technique or
a new curriculum or some mathematics.
But rarely do they have the sorts of
opportunities that we saw teachers in
Japan have regularly. Recor(ls of practice
provide a common opportunity to study
teaching anti learning.
A second advantage is that records of
practice provide a way for professional
development to be grounded in practice
so that the problems and issues that
teachers work on are directly connected
to the work of teaching. Sometimes
teachers learn from professional develop-
ment experiences but are then unable to
use that knowledge in their teaching.
Records of practice provide an opportu-
nity for teachers to learn knowledge as
they would need to use it with students.
Therefore, one compelling reason to use
records of practice is to ensure that the
OCR for page 80
knowledge that teachers generate as they
work is both useful for practice and usable
· ~
In practice.
Records of practice allow professional
development leaders and teacher educa-
tors to select particular problems of
mathematics or of the teaching and
learning of mathematics. A teacher
educator could select a specific challenge
of teaching mathematics and then select
records that provide an occasion for
teachers to consider that challenge. This
is different from discussing one another's
teaching where the problems that arise
are dependent on what teachers happen to
bring up. Using records of practice allows
a teacher educator to design work around
a particular problem of practice.
Another benefit of using records of
practice is that teacher educators can
provide opportunities for teachers to see
practices, problems, or issues that they
have not seen yet in their own practice.
They might see, for example, children
discussing mathematics in a way that their
own students do not yet know how to do.
So it allows teacher educators to expose
teachers to issues beyond teachers' own
individual classrooms.
Working with records of practice can
also develop teachers' abilities to learn
from their own practice, to learn to look
more carefully at student work, to learn to
listen more attentively to students' talk, to
analyze mathematical tasks in ways they
have not done before. One thing that
shruck several of us while watching lesson
stu(ly was how skilled many Japanese
teachers are in the discussion, analysis,
and study of practice. Records of practice
provide opportunities for teacher educa-
tors and teachers to develop some of
those skills anti capabilities, not just of
teaching, but of the study of teaching.
And finally, records of practice can
enable teachers to talk safely about
problems of teaching and learning,
because the teaching and learning that
they are looking at is not their own and
not their colleagues. There can be more
freedom to raise hard questions or to
consider problems without the worry of
being polite or not hurting someone else's
professional pri(le.
Many of these advantages apply also to
lesson study. One thing we might discuss
together later is how using records of
practice is similar to and different from
lesson stu(ly.
POSSIBLE PROBLEMS IN USING
RECORDS OF PRACTICE FOR
PROFESSIONAL DEVELOPMENT
There are also problems in using
records of practice. Because the material
is not from a teacher's own classroom, it
might not be relevant. Teachers might
say, "T (lon't have this problem; these
students are not like my students; this
classroom is not like my classroom." So
teacher educators may face problems of
making sure that the work seems relevant
to teachers. Similarly, it may not seem
interesting to teachers when the problems
they witness or stu(ly are not their own.
There are problems in (leveloping good
records of practice. Not every videotape
is suitable for stu(ly. Not every lesson is
provocative for teachers' learning. Not all
examples of chil(lren's work are equally
useful in professional (development.
Gathering, cataloging, examining, anti
becoming familiar with high-quality
records of practice is a problem.
We have learned that for this work to be
profitable for teachers, the tasks that
teachers work on with these materials
makes a (lifference. This is no (lifferent
from the knowledge that the mathematical
tasks that chil(lren work on make a
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS
OF PRACTICE
OCR for page 81
difference for their opportunities to learn
mathematics. The same is true for
teachers' learning; not all tasks are
equally useful in professional develop-
ment. Quite often in our early work we
neglected to frame tasks at all and
thought that simply by looking at video-
tapes, teachers would learn. It matters
what the task is.
In the United States we do not have, in
general, highly developed norms for the
study of teaching. Teachers are not used
to discussing, analyzing, or closely
probing teaching. Working with recor(ls
of practice means that we also have to
work on developing a culture and a set of
norms for work of that kind. One chal-
lenge that we face is to move from a habit
of evaluating and judging teaching to
analyzingit. Yesterday we saw many
examples of Japanese teachers and of
ourselves as workshop participants-
closely analyzing the teachers' decisions,
the nature of a task, a child's contribution.
That sort of work, that kind of analysis is
different from saying this was a good
lesson, this was a smart child. Learning
to do this kind of analysis is part of what
we have to develop in the U.S. culture of
teaching.
At the same time, too much analysis
can move very far away from teaching.
Teachers are not just researchers. They
must act with students. In work that
analyzes records of practice, it is impor-
tant to maintain a balance between analytic
work and practice and to strive for the
development of knowledge usable for
teachers' work. There is the challenge,
like the challenge of working with chil-
dren, of bringing sessions to closure so
that teachers go away with knowledge and
ideas that make them feel the work has
been useful and they have something to
take with them to their own classrooms.
DEVELOPING OPPORTUNITIES FOR
TEACHERS' LEARNING USING
RECORDS OF PRACTICE
Interestingly, the work using records of
practice has much the same structure as
the structure for lesson study: prepara-
tion, enactment, and analysis.
Preparation Phase: Design.
In (leveloping opportunities for teachers'
learning, there is a phase of (resign for the
teacher educator that inclu(les asking
such questions as, What is the goal or
purpose for teachers' learning? The
preparation phase includes designing a
task that teachers would work on together
using records of practice and designing
the enactment of that task in a session
with teachers. It includes selecting
resources to support that learning. For
example, what sorts of records of practice
~ ~ ~ ~ ~ =~ . . ~ . . ~
WOU1d nelp' what other materials mlgnt
be needed? Might teachers need the
curriculum materials on which chil(lren
were working in order to interpret a
videotape? This process looks very much
like the design work for teaching math-
ematics to children.
Enactment Phase: Facilitation.
There is the complicate(1 process of
enacting the work with teachers. What
(foes it take to enact useful, constructive,
productive sessions with teachers where
analysis of teaching anti learning are the
subject? These are some of the tasks
involved in this phase of the work: setting
purposes; posing the task or problem to
be worked on; organizing how time gets
spent; attending to teachers' engagement
anti learning, to teachers' i(leas anti
lifficulties.
Anti there is the work of processing the
(liscussions, sharing the work from the
sessions, and developing ideas that
everyone takes from the work. This
involves keeping the work groun(le(1 in
PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
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mathematics, so that teachers have
opportunities to develop content knowI-
edge. Keeping the work connected
deeply to concrete materials of practice
means learning to use evidence for
statements that are made about teaching
and learning. It includes learning how to
generalize from studying particular
examples and forming more general ideas
that can be useful in classrooms other
than the one being studied.
Analysis Phase: Reflection and Design
of Next Steps.
And of course there is the phase of
analyzing and reflecting on how the
sessions with teachers work.
EXAMPLES OF USING
RECORDS OF PRACTICE IN
PROFESSIONAL DEVELOPMENT
We will take brief tours of two (lifferent
examples of this work. There are many
such examples in the United States, and
although this is not the main form of
professional development at this point, it
is also not rare. Deborah Schifter
(Schifter et al., 1998), Ed Silver (Stein et
al., 1999), and Alice Gill (American
Federation of Teachers, 2000), among
many others, have all engaged in this kind
of work.
So together, we will (liscuss some of the
work we have been doing to develop
approaches to the study of practice.
These have some resemblance to lesson
study but are also different from it. We
have to examine how these are different,
how these are similar, anti to learn
together ways that we might join some of
the special features of each.
The example we would like to share
draws on work that we have been design-
ing over the past ten years with several of
our colleagues at the University of Michi-
gan. We have been working with a very
large collection of records from two
classrooms one grade three (8 year
olds), and one grade five (~1 year olds)
across an entire school year. We collected
videotapes every day in these classrooms
for a whole school year. We also collecte(1
all of the chil(lren's work, all the tests they
took, all the materials they used. This
includes a detailed teacher's journal with
indications of what the teacher expected
over a range of lessons. We have been
designing materials and experiences for
teachers' learning that draw from this
very rich collection of records. What
follows is one short exposure to the sort
of work we do with teachers using this
material.
THE PROBLEM OF THE DAY
Imagine you are a group of teachers.
The problem on which we are going to
work is that of (resigning anti enacting
mathematical work at the beginning of a
school year, actually the fourth day of
class. In many schools in the United
States, teachers get entirely new groups of
students at the beginning of every school
year. Those chil(lren have often been in
many schools anti have not worke(1
together before. The school where ~ was
teaching, for example, was very mobile:
Chil(lren move(1 in anti out all the time.
My school ha(1 the a(l(litional challenge of
serving an international community anti
many of my students (li(1 not speak
English language anti cultural (liffer-
ences increase(1 the complexities of
bringing the students together. But, even
without cross-cultural anti multilingual
considerations, classroom teachers
throughout the Unite(1 States must take
into account the fact that many chil(lren
come from different schools with different
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
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past experiences and may never have
done the sort of work in which the teacher
aims to engage them. Teachers must find
out what their students know how to do,
and they must begin to teach them the
curriculum. They must also teach them
the ways of working that the teacher aims
to use during that school year.
Before watching the classroom video,
there are two kinds of questions we want
you to consider.
CONSIDERATIONS FOR THE
DESIGN AND ENACTMENT OF
MATHEMATICAL WORK AT THE
BEGINNING OF THE SCHOOL YEAR
Bass: Place yourself in the position of
having to do this work which faces the
teacher at the beginning of the school
year. What are some of the consider-
ations in the design of such a lesson, the
enactment of such a lesson? What are the
things you want the students to be doing?
With what things should the teacher be
concerned? What would the teacher want
to find out in these early lessons? What
are the problems that should be on the
teacher's mind at this stage?
U.S. Participant: At the beginning of the
school year, my most important goal is
creating a mathematical culture, to get the
students asking questions, making
conjectures. ~ don't worry so much about
particular content, starting the textbook
or anything like that. So ~ have sort of
favorite activities that ~ know are very
engaging, that bring up a lot of ideas, and
that bring up a lot of questions. ~ start the
year, to set the stage for how they are to
act for the rest of the year.
U.S. Participant: ~ think with the student
population that Deborah Ball described,
the language issue is one that is especially
critical for the teacher to both be under-
stood and to be sensitive to understanding
what the children are saying. So sensitivity
to language is important.
U.S. Participant: For me, it's developing
a culture of people being respectful to
each other in these conversations. The
content is one thing, but the social dynam-
ics of listening, appreciating each other's
ideas is very important to develop in the
beginning of the school year.
U.S. Participant: ~ think one of the
things that ~ like to do at the beginning of
the year is to give students an opportunity
to let me understand some of what they
know and how they are used to working.
Are they used to responding to a question,
or are they simply looking for something
the teacher says to say back?
CONSIDERATIONS OF THE
MATHEMATICAL TASK AND ITS USE
Bass: Let us move now to another aspect
and consider the mathematical work itself.
You were given a homework assignment
(Appendix G) that includes the math-
ematical task you will see worked on in
this lesson. There are questions about
the capacity of this task to support the
considerations you brought up about
doing serious mathematical work, estab-
lishing norms for communication with
each other, learning and showing respect
for other students. What are the kinds of
questions you would pose in the enact-
ment of the three-coin problem (Box I) ?
How would you pose the questions for
the work on this task? How and when
would you do so? What kinds of student
response might you anticipate? What
kinds of responses would you want to
PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
OCR for page 84
r.~.e
~ ~ i-1 ! 1 [~ I ~ 1
I have pennies (1 ¢), nickels (5¢), and Jimes (10¢) in my pocket.
If I pull three coins out, what amount of money could I haven
SOURCE: NCTM (1989).
elicit? And what elaboration of the task
might you want to be prepared to do in
case it turns out to be too difficult or too
easy? How can you be ready to incorpo-
rate responses from the students in the
development of the lesson?
This task does involve some serious
mathematical work. How do you recon-
cile that with the fact that this is a very
diverse class? There are large differences
in background and in background knowI-
edge, in the students' sense of how to
work with each other. What is the suit-
ability of this kind of a task, given that
kind of diversity, and so many unknowns?
U.S. Participant: It's a good task in that
it has many entry levels. For example, the
task could be set up in such a way that
you could find one answer or many
answers. And the ultimate part of the
task, ~ would imagine for third grade, is
considering how do you know you have
them all? You might start by asking
students in the whole group to think of
one way, one amount of money they could
have, and start to collect the ways. And
then allow the students to go off, perhaps
in pairs or by themselves, giving them
choices about how they want to work
initially. This woul(1 allow you to observe
who works in(livi(lually, who works with a
partner, who works with a small group,
who knows who, who doesn't all of those
social issues. Anti who has confi(lence as
they begin to solve the problem? Who
takes it further?
U.S. Participant In a class where
children have many languages, ~ might
take this problem and indicate that the
pennies are worth one; the nickels, five;
and the dimes, ten. ~ would not be certain
that a multilingual class woul(1 un(lerstan
the value of those coins. But ~ think it's a
good task to get them to begin to think
about things.
U.S. Participant ~ guess I'm thinking
just a little differently about it. Actually,
approach this at the beginning of the
course in methods of teaching elementary
school mathematics. ~ use a problem that
~ have airea(ly tried out, perhaps many
times, and have (levelope(1 a (letaile(1
lesson plan. So I'm familiar with the kinds
of responses that ~ woul(1 likely get from
the students and also how ~ woul(1 (teal
with those responses.
U.S. Participant: ~ woul(1 also have jars
of money with the coins, for those stu-
(lents who felt they wanted them.
U.S. Participant ~ (lon't teach elemen-
tary school, so I'm not quite sure if my
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
OCR for page 85
concern would be valid, but listening to
what Deborah Ball said about the kinds of
students that she worked with, one of the
concerns that ~ would have with this
particular problem is the fact that there
are many answers. And many children
might not be used to this idea of a math
problem having several solutions. ~ think
that's something we want children to
understand, but is that too big of a prob-
lem to bring up on the very first three or
four days of the school year? ~ might
consider actually softening that fact by
maybe putting this into a game context or
something that they see players playing,
and whoever gets the most wins. Some-
thing where there are many solutions but
where that doesn't become an initial
focus.
Japanese Participant: It's important to
know what students have learned in the
previous years. What is the level of
knowledge of those children in your
classroom? ~ think that kind of context is
a major concern for Japanese teachers.
Japanese Participant ~ think the three-
coin problem is too difficult for the
elementary school student. How do you
position this problem in the context of
classroom teaching?
U.S. Participant: ~ did teach third grade
most of my career, and ~ do think it is
suitable for third graders. The problem
would give me as a teacher a sense of
whether or not these students were able
to approach a problem in an organized
manner. This is important to me as a
teacher because if they can't, then ~ need
to do some things to help them organize
their thinking.
Bass: Your questions indicate where you
want to speculate a little bit. When you
watch the enactment in the class, think
about how to reevaluate that concern.
One thing that is often emphasized is the
importance of the teacher doing the math
of the lesson prior to the lesson. The
math is typically elementary, but the
insoles of it often involve intricacies and
complications that vitally affect the
instruction. That was part of the intention,
in fact, of having you do the homework: It
would allow you to be inside the territory
where the children are working.
EXAMINATION OF THE RECORDS
OF PRACTICE
Ball: A lesson plan (Appendix G) starts
with the problem and provides an explana-
tion of what the purposes are for the class.
There were three purposes that ~ had as
the teacher. The first was to develop
students' habits of searching out multiple
solutions and establishing whether all
solutions to a problem have been found.
This inclu(le(1 (1eveloping students' ability
to produce a mathematical explanation. In
this problem, an explanation for a solution
must establish two things. First, that
three coins were used from among these
three types, anti second, that the amount
of money produced is correct in total.
The second purpose is to communicate
to the students what (loin" mathematics
wall mean this year in this class. For
example, students will learn that math-
ematical work will inclu(le producing
explanations for one's work to the teacher
anti to other students; they will learn to
listen, to critique, anti to use other stu-
dents' ideas; and they wall learn to be
accountable for their own mathematical
ideas.
The third purpose was for the teacher
to begin to learn about the student to
learn, for example, about the students'
PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
OCR for page 86
addition and multiplication skills, their
openness to multiple answers and solu-
tions, the strategies they use for finding
solutions, how they keep track of their
work and of the solutions, how skeptical
they are that they are finished, and how
they go about determining whether they
are done with the problem. How do they
work with concrete materials, in this case
coins? What is their disposition to confer
with other students and to consider
others' ideas?
The rest of the lesson plan outlines the
steps. On the second page is a list of
strategies that ~ knew that students of this
age were likely to use and different
approaches to recording their work that
anticipated them using. This was a
problem that ~ had used many times
before with this age and also with both
younger and older students.
The video includes a segment from the
beginning of the class. The problem was
posed, and there was some discussion
about what the problem meant, including
an example of a solution. Then the
chil(lren worked alone. After about ten
minutes of work they came back into
large group discussion of solutions to the
problem.
(Transcript anc! class description in
Appenclix G}
See the vicleo clip: RecorcIs of
Instruction: Reasoning About Three
Coins at Thirc! Gracle
ANALYSIS OF THE LESSON
Ball: The next phase of our work is to
return to the problem that we're consi(ler-
ing in this session, which has to (lo with
organizing mathematical work in the
beginning of the year in light of a whole
host of considerations that bear on how
one begins a school year. In this lesson
segment we saw how the teacher's
considerations about the beginning of the
year were handled. What came up? How
did the segment correspond to the
teacher's goals and anticipations? What
seemed unexpected either to the teacher
or to you? How did this problem work out
so far? Obviously, we haven't seen the
whole completion of the work. How did
this problem work out so far, given the
teacher's goals? What surprised you?
Was this what you had expected?
Having had an opportunity to examine
one example of a teacher's work in this
problem (lomain of (resigning anti enact-
ing mathematical work at the beginning of
the year, what comments woul(1 you like
to make?
U.S. Participant: The first question the
teacher aske(1 after bringing the class
back together was to predict how many
different solutions there were. Then the
students worked on the problems a little
bit more on their own. But when the class
reconvened as a whole class, the (liscus-
sion was on what amounts can you get.
was wondering what purpose that initial
question served?
U.S. Participant ~ thought the lesson
just came to something of an abrupt end.
was anticipating some request of the
students to think in a more structure
way about the (lifferent solutions they
were generating.
U.S. Participant The thir(1 state(1
purpose for the goal inclu(le(1 a list of the
many things the teacher wants to learn.
Unless the teacher vi(leotape(1 a lesson
anti sat (town afterwards, ~ saw no evi-
(lence or any specific way the teacher
recor(le(lthatinformation. In other
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
OCR for page 87
words, it is nice to ask, 'what strategies
did the children use?," but unless there is
a systematic way to record that, especially
the first week of school, it's very difficult
to keep track of that sort of information.
So I'm not sure how the teacher assessed
that objective.
Japanese Participant: Three coins are
taken out, and students asked what
amount could be made. That was the
situation. However, what was the motiva-
tion in this class? In this situation, the
Japanese teacher would take three coins
out of his pocket and say, for example,
'~his would buy you an apple." Then the
children would think about wanting to
purchase something and would get more
interested in working on the problem. In
the video, the children did not think about
why they needed to solve the problem. It
was just because the teacher said so. This
content would be in the sixth grade in
Japan where they would consider how
many cases, not how much the value
could be. The students would solve it
structurally, by drawing a diagram or
counting each case. The mathematical
content would have a focus on analyzing
the number of distinct combinations.
Bass: One of the advantages of having
the very complete record of what hap-
pened over the entire year is that when we
see something from a small window of a
lesson, we can ask questions about how
the students became acclimated to the
meaning of the coins or how the children
worked with the coins, and we can trace
what happened in the earlier lesson where
the teacher worked on the two-coin
problem. And very much what you were
suggesting a Japanese teacher might do,
happened in that lesson. There was time
spent moving around the room, showing
the students the coins, asking for their
understanding of what the coins were
worth, and things of that kind. So that
preparation in fact was in place before the
enactment of this problem.
Japanese Participant In Japan, the
same problem is also handled in the first
year of senior high school. The focus is
how to solve this problem and how to find
those strategies. There needs to be time
for students to think about the problem.
Japanese Participant If students
understand the structure of the two-coin
problem, then they can utilize it in prob-
lem one (see the Homework handout in
Appendix G) and also in problem three. If
they understand the structure of problem
two, even if they (lo not write out every-
thing, they can use the results of problem
two for problem four. By drawing tables
they can grasp the structure of the
problem. If they learn that in elementary
school, and if they go up to the high
school, they can always think about the
meaning of the problem that is given.
U.S. Participant: What we have here is
what happens with many rich problems.
This problem is a different problem at the
third-grade level than it is at the sixth-
gra(le level, than it woul(1 be at the first
year of senior high school. You look for
(lifferent reactions from the students. The
only concern ~ had about this problem at
the thir(l-gra(le level was that the entire
time was spent in un(lerstan(ling the
problem. At the en(1 of the 30 minutes,
the students had no better way to solve
the problem than they had at the begin-
ning, which was trial and error. But
maybe at the third-grade level, that's all
we want. That is, perhaps it is enough to
simply introduce a problem of this kind, to
introduce the idea of trial and error, and to
solve it. At later grades of course, we look
PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
OCR for page 88
for systematic ways. And at the higher
grades, we look for the underlying math-
ematics of combinations and so on to
solve it.
Ball: The nature of these materials is
important to highlight. Because there is a
record of every day, the comments that
people are making right now in a profes-
sional development context would be
converted by the teacher educator into
questions for further inquiry by the
teacher e.g., Did students at the third-
grade level develop a more general
structure for understanding this class of
problems rather than simply discussing
whether children can or cannot do that?
When someone wonders about these
kinds of things, they would be invited to
turn back to the record, to look at the next
day's work, to look at students' written
work perhaps, or to read the teacher
journal. And this enables the kind of
developmental tracing of children's
learning not possible by simply visiting a
school or by reading one example. This is
a special feature of this kind of year-Ion"
record of practice. After all, teaching and
learning occur across a school year,
across time. One kind of work that this
kind of record permits is the opportunity
to look across days to see what happened
at the beginning of the next class, or what
sort of structure third graders ended up
developing? And how did the children
differ within the third-grade class?
U.S. Participant: ~ got the feeling that
after you looked past the mathematics,
one of the major purposes of this class
was creating a community of learners.
Especially from the way the teacher at the
end remarked about the way students
were listening to one another, giving each
other enough time, promoting the think-
ing of students without interfering others.
A teacher can get a lot of information from
the students about whether they were
using multiplicative methods or additive
methods in fin(ling out the amount of
money from the coins.
U.S. Participant: The lesson needs to be
seen in the context of an ongoing activity,
and the record we see is only a fragment
of a particular lesson. ~ think it did very
well in establishing certain rules of
behavior and operation. Listening to each
other, giving explanations, taking turns,
getting input from many students, all
these features were being established
very carefully. The full story is certainly
not here. But notice that Mick (li(1 come
up with nine solutions. In fact, he came
up with ten, which is also quite interest-
ing. So ~ thought as a recor(l, it woul(1 be
very interesting to see whether from the
recor(1 of the practice, the teacher was
able to observe what the students (li(l.
Japanese Participant As teachers make
a lesson plan for the lesson, it's very
important to think about the multiple
solutions that might come up and about
motivation as a very important factor.
Using the setting to naturally urge the
students to come up with the questions,
then come up with the solutions is impor-
tant. We (lo our best to urge the students
to come up with the solutions, to interest
them. Take the coins out of your pocket.
Then say, "Now ~ have three coins in my
han(l. How much it coul(1 it be?" This is
one way to check whether the chil(lren
un(lerstan(1 this problem. Some people
woul(1 say three cents or fifteen cents, just
haphazardly. But that's a good chance to
motivate them. Here you have a right
answer. ~ woul(1 probably repeat this two
or three times to interest the children.
Then maybe ~ woul(1 ask what is the
possible minimum amount of money?
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
OCR for page 89
What is the possible maximum amount of
money? And ~ would say, is it possible to
come up with four cents? Impossible.
What about six cents? What about
thirteen? Maybe it's possible. Here we
have three coins. Some amounts they can
make; some amounts they cannot. Okay,
now let us think about the cases, impos-
sible cases and possible cases, and let us
come up with all possible cases. The
question of how many solutions are there
is a mathematically interesting question.
But for children, is it very interesting?
Maybe they will wonder why is it impor-
tant to have knowledge about how many
solutions there could be? Maybe the
more important thing is four cents is
impossible, five cents is okay. What about
six, about seven? And do they have all of
the possible solutions? ~ think this kind of
approach is more important. If they can
make the complete table of those cases,
with three coins, they will list those
impossible cases. What about with four
coins? ~ think that could be the next step
Japanese Participant: The important
thing is to urge the children to think
about all of those possibilities and set up
the steps. Usually the Japanese teachers
try to think about how they can best set
up several stages of thinking for students.
But ~ know that this lesson is at the
beginning of the school year, so the
mathematical approach and the commu-
nity making are also the priorities. ~ know
that my comments should not be always
applied to that kind of situation, but ~
think the important thing is motivation.
Can we ask the proper questions so that
the children can follow the steps in a
mathematical way?
Bass: One important feature of the work
on this task that was not explicitly men-
tioned was beginning to teach the children
how to reason mathematically. So the
mathematical task provided a context for
that, but the very detailed, fastidious
explanations of why certain amounts
added up to certain amounts were not
only elementary exercises in addition but
they were the first steps in learning what
it means to give a reasoned, careful
explanation for a claim. And this, as one
woul(1 see in the later recor(ls of this
class, became a very important theme in
developing children's capacity to reason
mathematically.
CONCLUSION
Thank you very much for those very
interesting comments. ~ think we have
used up our time. ~ just want to make one
very brief remark. Some of the themes or
(Erections that you propose(1 coul(1 be
investigated in the actual record to see
what was either in the lessons before and
after, anti also in the teacher's journal. An
others of these, that are not necessarily
enacted in the lesson but simply math-
ematical elaborations of the task, can
become potential material for learning
mathematics in the context of practice and
using these for professional development.
PROFESSIONAL DEVELOPMENT THROUGH RECORDS OF INSTRUCTION
OCR for page 90
F~.le
I- -
~-~el
- ~ :~]
Aims
Margaret S. Smith, University of Pittsburgh
Over the past few years, my colleagues
en c! ~ at the University of Pittsburgh have
been exploring the potential of cases-
written accounts of teaching as sites for
investigating and analyzing mathematics
teaching and learning. The cases that we
have createc! are based on ciata that was
coDectec! Mom QUASAR, a national project
aimed at improving mathematics inshruc-
tion for students attending micicIle schools
in economically clisacivantagec! communi-
ties. Each case is based on actual events
that occurrec! as teachers enactec! reform-
orientec! instruction in urban micicIle school
classrooms. The cases are not meant to
represent best practice. Rather, they are
intenclec! to represent actual practice
what really happenec! when teachers set
about to teach mathematics in new ways.
The cases provide sites for teachers to
· · ~ · · · ~
engage in critique, inquiry, ant 1nvesUga-
tion into the practice of teaching.
Each case has been conshructec! around
a cognitively challenging mathematical
task. Prior to discussing a case, we engage
teachers in an opening activity intenclec! to
give them an opportunity to explore the
mathematical icleas that are cenbral to the
case. This provides teachers with a
personal experience in working through
the mathematics on which to ciraw as they
interpret en c! analyze the work of the
teacher ant! her students cluring the class
porbrayec! in the case. The opening
activity also provides an opportunity to
explore the mathematics in the task in
more clepth.
The remainder of this discussion
focuses on a specific case entitlec! '~he
Pattern Trains: The Case of Catherine
Evans en c! David Young" (see Appendix
for a copy of the case). This is one of a
set of cases that was clevelopec! uncler the
auspices of COMET (Cases of Mathematics
Inshruction to Enhance Teaching), project
funclec! by the National Science Founcia-
tion that is creating materials for teacher
professional clevelopment in mathematics.
This case is one of four that explores
icleas relater! to algebra as the stucly of
patterns and functions.
The opening activity in the case of
Catherine and David, as shown in
Figure I, provides an opportunity for
teachers to look for the unclerlying
mathematical shructure of a pattern, to
use that shructure to continue the pattern,
en c! to clevelop a rule that can be used to
describe and build larger figures. The
task provides an interesting context for
discussing what algebra is en c! how
algebraic reasoning can be clevelopecI.
OCR for page 91
FIGURE ~ The opening activity from the pattern: The case of Catherine Evans and David Young.
; ~ if= ~
train 1 train 2 train 3
train 4
Solve
For the pattern shown, compute the perimeter for the first four trains, determine the perimeter for the tenth train without constructing
it, ancl then write a description that could be used to compute the perimeter of any train in the pattern. (Use the edge length of any
pattern block as your unit of measure.}
The first train in this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon is aclclecl. The first
four trains in the pattern are shown.
Consider
Fincl as many different ways as you can to compute (ancl justify} the perimeter.
The hexagon pattern task, featured in
this opening activity, can be solved in
several different ways. Consider, for
example, the responses produced by five
practicing middle school teachers who
participated in a workshop during the
summer of 1999. Linda's solution
(Figure 2) involves a recursive approach.
Linda recognized the general pattern of
adding 4 to find the perimeter of each
successive train, but her strategy required
knowing the perimeter of one train in
order to find the perimeter of the next train.
Barbara's solution (Figure 3) by con-
trast resulted in a generalization that can
be applied to any train. She determined
that each hexagon added four sides to the
perimeter of a train and that the first and
last hexagons also each contribute one
a(l(litional si(le.
Kevin's solution (Figure 4) also
involves adding four sides, but differs
slightly Tom the one proposed by Barbara.
PROFESSIONAL DEVELOPMENT THROUGH WRITTEN CASES
In this approach, Kevin explained that
each hexagon a(l(le(1 four si(les to the
perimeter of the train for each of the
hexagons in the middle of the train and
that each of the hexagons on the en(ls of
the train contribute(1 five si(les to the
perimeter.
Michael's approach (Figure 5) involved
first counting all six sides of each
hexagon. For each hexagon he then
subtracte(1 the vertical si(les (two per
hexagon), anti then a(l(le(1 on the two
vertical sides on the ends of the train.
Chris's solution (Figure 6) is a bit more
unusual. Chris thought about the hexagon
train as having a bottom anti a si(le anti a
top anti a si(le Marked by bol(1 lines).
She noticed that the perimeter of the
bottom anti a si(le was the same as the
perimeter of the top anti a si(le. Chris
note(1 that the bottom or the top was two
times the train number so that ~ nee(le(1 to
be added to the bottom and to the top.
OCR for page 92
FIGURE 2 Linda's solution to the hexagon pattern task.
~ / ~
train 1 train 2 train 3
6 10 14
+4 +4
~4
train 4
18
You just keep aclcling ~ each time. So the perimeter of the first train is 6, the perimeter of the second train is 10, the perimeter of the
third train is 1 A, and the perimeter of the fourth train is 18.
FIGURE 3 Barbara's solution to the hexagon pattern task.
/\~ ~ ~
train 1 train 2 train 3
There are four sides for each hexagon plus two on the ends.
So fortrain 4: 4 ~ 4 + 2 = 18
P = 4x + 2
train 4
FIGURE 4 Kevin's solution to the hexagon pattern task.
train 1 train 2 train 3 train 4
Each hexagon on the inside of the train acids four sides to the perimeter. The first ancl last hexagons in the train acid five sides each
to the perimeter.
~ · 2+ 10
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
OCR for page 93
FIGURE 5 Michael's solution to the hexagon pattern task.
train 1 train 2 train 3 train 4
You find the total number of sides for all the hexagons and then subtract two sides for each hexagon for the insicles. Then you need
to acid two back on for the encls.
So for train A: 6 · ~ - 2 · ~ ~ 2 = 18
P= 6 x - 2x +2
FIGURE 6 Chris's solution to the hexagon pattern task.
~1~1
train 1 train 2 train 3
train 4
For each train, the perimeter of one side and the bottom is the same as the perimeter of one side and the top. So the perimeter of
one side and top or bottom is 2x ~ 1, so you have 2 of these.
So for train A: 2~2 · ~ ~ 1 } or 2~9} = 1 8
P= 22x+1}
As you can see from these sample
solutions, teachers have many different
yet interesting ways of connecting the
diagram with a symbolic representation.
After teachers have solved the task, we
have found it helpful to explore the
mathematics in more depth before moving
on to a discussion of the case. One
possibility would be to make a list of all
the symbolic representations generated
by the teachers, and ask them if the
PROFESSIONAL DEVELOPMENT THROUGH WRITTEN CASES
representations are equivalent and to
explain the rationale for their (recision.
Alternatively, you may want to explore the
mathematical content and processes
embedded in the hexagon pattern task.
This can lead to a (1iscussion of math-
ematical ideas such as generalization, the
or(ler of operations, the (listributive
property, equivalence, and perimeter.
There are many other questions that
could be asked depending on your goals
OCR for page 94
for teacher learning, the context within
which you and the teachers are working,
and the teachers' prior knowledge and
experience.
Once teachers have had a mathematics
experience related to the case, they are
ready for the case discussion. Since the
case is not self-enacting, you must create a
professional learning task for teachers
which serves to focus their investigation
and analysis of the case. One task that my
colleagues and ~ have found helpful in
analyzing the case of Catherine and David
is as follows: Indicate the ways in which
you think Catherine's and David's classes
are the same and the ways in which you
think they are different. Be sure to cite
line numbers from the case to support
your claims. [At this point participants
are given time to work in small groups to
generate charts that made salient the
similarities and differences between
-
Catherine and David's classes.] This
small group work was followed by a whole
group discussion of similarities and
(lifferences. Table ~ contains a record of
the responses pro(luce(1 by workshop
participants during the group discussion.
This task of finding the similarities and
differences between Catherine and
David's classes requires comparing an
event that occurred in one class with an
event that occurred in the other class,
analyzing the two events in order to
determine whether they have anything in
common, and noting what is the same or
what is (lifferent about the events. This
activity generally brings to light many key
issues related to mathematics teaching
and learning that can be further explore(l.
Another example of a similarities/(liffer-
ences list generated by practicing mi(l(lle
school teachers is shown in Box ~ at the
end of the (locument.
TABLE 1 Chart Generatec! by Practicing Mic~clIe School Teachers in Response
to the Similarities anc! Differences Task
Similarities
Differences
.
Willingness to change
· Same task
· Positive attitudes
· Encouraged student involvement
Commitment to new program
Same school and same Oracle level
Both teachers were part of a community
· Catherine focused on doing proceclures; Davicl
focused on understanding relationships (between
number of blocks and perimeter).
· Catherine was more concerned with success and
directed student thinking so they would be more
successful; David was more concerned with
student unclerstancling.
· Types of questions: Catherine's had one right
answer; Davicl's required explanation.
· Catherine made tasks easier; Davicl Respect
students solve the original tasks.
· Davicl's students form generalizations
"approaching symbolic"; Catherine's could apply
rule to large numbers but not any number.
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
OCR for page 95
The COMET project shares Shulman's
(1986) view that
the strength of cases is that they can be
used to straddle the space between gener-
alizations and particularities, between the
kinds of abstract, formal, codified knowl-
edge that can be taught in the absence of
context and the kinds of knowledge that are
experientially derived, often informal, and
perhaps lacking in precision.
Hence a case allows you the ability to go
from the very particular things that
happen within a specific classroom to
seeing those instances as examples of
some larger class of phenomena that we
consider to be important in teaching and
learning. For example, the case of
Catherine and David was designed to
provide specific instantiations of key
mathematical ideas such as generaliza-
tions identifying patterns, perimeter,
intuitive notions of variable, and connec-
tions among representations. Within the
case, there are opportunities to look at
particular examples of each of these ideas.
In addition, if you look across the set of
cases related to algebra as the study of
pattern and functions, you would see the
same set of ideas woven throughout the
cases. This provides an opportunity to
explore ideas in more than one context
and from more than one perspective.
Each of the cases is also designed to
make salient"pedagogical moves" that
support or inhibit student learning.
Moves that support student learning
include teachers pressing for explanation
and meaning, modeling high-level perfor-
mance, allowing students sufficient time
to explore and think, drawing conceptual
connections, and building on prior knowI-
edge. Pedagogical moves that inhibit
student learning include shifting the focus
to following rehearsed procedures;
removing problematic aspects of the class,
and allowing insufficient time for students
to explore and think. So again, in each case
you can see specific events that connect to
these more general ideas about math-
ematics teaching and learning. These
ideas can be explored over a set of cases
as well as in a teacher's own practice.
In facilitating the discussion of the case,
we generally start by making a record of
the similarities and differences that
teachers identified, resulting in the
creation of a chart similar to the one
shown in Table 1. Our goal is to then
move from the specific things that happen
in the case (as represented in the chart)
to more general ideas. So the chart
becomes not an end in itself but rather a
starting point for additional discussion.
This discussion might begin by focusing
on a specific difference that was noticed.
For example, in further discussing the
types of questions pose(1 by Catherine anti
David (see Table I, third bullet in the
differences column), the facilitator might
want to press teachers to analyze the
learning opportunities that were or were
not affor(le(1 by each teachers' question-
ing strategies, thereby explicitly connect-
ing teaching and learning. The facilitator
might also want teachers to look at the list
of differences and to begin to look for
commonalties across events in the list in
order to see specific instances as a subset
of a larger class of phenomena. For
example, fostering students' thinking
might serve as a bigger idea around
which to organize a number of different
classroom events such as helping students
un(lerstan(1 relationships anti requiring
students to provide explanations.
The ultimate goal of a case (liscussion is
to create generalities that teachers will be
able to (lraw upon in situations outside the
case. The point is for teachers to take
something away from the analysis of the
case that can be use(1 to think about their
own teaching. For example, a (liscussion
PROFESSIONAL DEVELOPMENT THROUGH WRITTEN CASES
OCR for page 96
Similarities Between Catherine ant! Davic!
· Both teachers were struggling to change their practice.
· The classes were working on the same task. Each student in the classes just presents
one idea. (In a Japanese class, each individual student is encouraged to present
multiple ideas.)
Both teachers asked students to share their thinking and asked for different solutions.
· Both tried to capitalize on student solutions. The problem about the 1 00th train is
from the students (lines 490-4951.
· Both Jo things that they hadn't planned on and so monitor and adjust their teaching
as they are going. Both of them give homework directly from the class.
· Both seemed to realize it was important to work toward generalizations of what they
were teaching. Both were going to get to big mathematical ideas in the end.
There was no activity to create formulas for expressions in either of these classes.
There is no evidence that the teachers were selecting students with a sequence of
ideas in mind, or that the teachers knew what the students had Jone in small groups
or individually. There was no evidence that they were calling on students in a particu-
lar order.
Differences Between Catherine ant! Davic!
· One difference is that Catherine is in the first year of teaching this curriculum, and
David is in his second year, which points out that teachers themselves need to learn
how to Jeal with reform and what it might look like.
· Catherine had a set amount of time in mind for the lesson and when that time was
over, she said, "We have to move on. It's time for another topic." Whereas, David
felt more comfortable continuing his lesson into another day when the content was not
covered to his satisfaction.
· Catherine used a square to measure perimeter versus David who used a segment to
mark off and measure perimeter. This might have implications for students' unJer-
stanJing about what perimeter is.
· There was a difference in the level of support. Catherine was going through a
change of practice with colleagues who were at the same place in learning how to do
this. David was coming into an established community that had gone through this
change and was trying to catch up. It wasn't clear whether he had the same opportu-
nities to look at videos of his class and discuss it with colleagues. He Jid not seem to
have the same opportunities to reflect as Catherine JiJ.
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
OCR for page 97
· Catherine seems to get a little bit more impatient, and when a student doesn't seem to
get an idea right away, she is there helping. For example, she seems to be literally
moving the hands of the boy who was showing the perimeter on the overhead.
David tends to be more willing to take time, ask more provoking questions, and wait
for the students to make sense of things.
David seems to push more for the multiple strategies, having one student explain, and
then ask whether anybody else had Jone it a different way. He gave five different
explanations for the formula, whereas Catherine got one from a student and pre-
sented another.
They introduce the topic in a different way. Catherine starts the lesson by asking the
students to make generalizations about the patterns and only brings up the word
"perimeter" when a student mentions it. David's initial introduction to the lesson is to
find the perimeter for the four trains.
· Catherine seemed to be narrow in how she asked questions, with answers that she
wanted from the students, rather than being open to the answers that the students
gave.
· David had questions for example, about noticing a relationship, giving a bit of
direction to the student in terms of what are the kinds of things you might look for.
Catherine asks questions such as, "How many on the enJ2" (line 164) How many will
there be altogether (line 1661. They are very specific one-answer questions.
David says "How are these two numbers related2/' (line 569) in his effort to help
students find a connection between the train number and the perimeter. He is giving
questions with several possible answers.
· It seems as though Catherine was validating the students' answers, which would
introduce something that the students would then seek, versus David who was encour-
aging open discussions and not necessarily commenting on correctness.
The relationship between questions and evidence of student learning or understand-
ing possibly came from the relationship between the questions Catherine was asking
and what it was she thought she was getting (lines 205 and 2451.
Catherine seems to be focused on asking questions with a numerical answer. What's
another perimeters Whereas David seemed to be assessing student understanding
based on their ability to explain how they got their answer and communicate an
understanding that way.
.
.
PROFESSIONAL DEVELOPMENT THROUGH WRITTEN CASES
OCR for page 98
regarding Catherine's concern about
student success may provide teachers
with a new lens for considering what it
means for students to be successful and
for considering whether "imitation"
indicates understanding of mathematics.
The hope is that this "lens" would sensi-
tize teachers to similar decision points in
their own practice.
What can be gained by using materials
like this? My colleagues and ~ contend:
In order to grab hold of classroom events,
to learn from examples, and to transfer
what has been learned in one event to
learning in similar events, teachers must
learn to recognize events as instances of
something larger and more generalizable.
Only then can knowledge accumulate. Only
then will lessons learned in one setting
suggest appropriate avenues of action in
another (Stein et al., 2000, p. 341.
PROFESSIONAL DEVELOPMENT THROUGH THE USE OF RECORDS OF PRACTICE
Representative terms from entire chapter:
using records
