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REFLECTIONS ON THE CONVOCATION
Edward Silver, Chair, Program Steering Committee for the Convocation,
Learning Research and Development Center, University of Pittsburgh

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Adaptedfrom the transcript of remarks by Edward Silver
Chair, Program Steering Committee for the Convocation
Learning Research and Development Center, University of Pittsburgh
Most of us came to this Convocation
because we have a serious commitment
to enhancing the quality and quantity of
mathematics learning in the middle
grades, such as the development of
important algebra and geometry con-
cepts. Most of us also came here because
we have a serious commitment to ad-
dressing other needs of young adoles-
cents such as their healthy social and
emotional development. The Convoca-
tion took as a premise that this dual
commitment to both the development of
mathematical ideas and the development
of children actually could, in many ways,
mask differences in perspective regard-
ing these two emphases; that is, individu-
als and groups might differ with respect
to the relative emphasis of these two
commitments in their work. For most of
us, although both are important, one
looms larger in our thinking than the
other. This (lifference in relative empha-
sis has been evident in the discussions.
A second premise was that the educa
tion of young a(lolescents would be
enhanced if we took off the mask,
expose(1 these (1ifferences in perspec-
tive, seriously examined them in order
to identify the convergences and diver-
gences, then trie(1 to crystallize some
issues, concerns, and questions that
would benefit either from some kind of
concerted action or from further serious
(leliberation. A third premise was that it
would be productive in the search for
these issues, concerns, anti questions,
to look for them along three different
(limensions curricular, pe(lagogical,
and contextual and we organized this
conference along these dimensions.
So now the question is: What have
we learne(l? Several (lifferent kin(ls of
learning might have occurred. We've
i(lentifie(1 some things we (lo agree on
anti some things we (lon't agree on.
We have gaine(1 an enriched un(ler-
standing of the issues and deeper
insights into questions. We now
understand some things a little deeper,

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a little differently than when we walked
in. And we have identified some poten-
tial areas for concerted action. Finally,
we have also identified a set of issues
about which we need to think much
harder and much longer.
In the remainder of my remarks, I'm
going to give you examples of a few
areas in which ~ think we made some
progress during the meeting. Let me
turn to the first one.
INTEREST AND RELEVANCE
We have broad agreement that the
study of mathematics is important for
young adolescents. There is also broad
agreement that we want mathematics to
be interesting to students. The question
is, then, how do we make it interesting.
We have had a set of examples pre-
sented to us that ~ think challenge a
view (which is very prevalent these
days) some of us might have held
walking into this Convocation that in
order for students to be interested in
mathematics, the mathematics has to be
relevant. That is, good mathematics for
middle grades students has to be tied to
some important application or some-
thing related in some important way to
students' lives.
Glenda Lappan told us on the first
night that we need to connect students
to things that are generally interesting
Ml DDLE GRADES CONVOCATION
to them, and Tom Dickinson described
students who were gathering (lata in
experiments about questions that were
genuinely of interest to them and then
displaying the data. Students were
([riving the investigations anti making
(recisions about how to (lisplay the (lata
in the best way to answer their ques-
tions. This is a good example of what it
might mean to have interesting math-
ematics for students.
But then, on the other hand, we saw
different examples in the videos. One
almost by inference in Linda Foreman's
case and the other more directly in the
video that Nannette Seago showed of
Cin(ly's teaching, in which students
were engage(1 in the investigation of
mathematical ideas in problems that
you could hardly call applied. They
were problems that (li(ln't come from a
meaningful context. They lacked the
connection to thematic or application
oriented settings that many of us might
take to be boun(1 up inextricably with
this notion of what's interesting to kills.
There is no question that in order for
young a(lolescents to learn mathemat-
ics, they're going to have to fin(1 it
interesting. The question is what is it
that makes the mathematics interest-
ing. The videos and the discussion
about the videos help to remind us that
students can find mathematics inher-
ently interesting. They can find "ap-
plied, real-world" tasks interesting, but

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they can also find tasks that aren't
applied in the real world interesting-
because these are tasks that arise
within a classroom community of
students who are investigating math-
ematical ideas about something that
they value.
Yesterday, Deborah Ball talked about
the notion that students' interest can be
cultivated. That is, students bring their
interests to the classroom, but teachers
have the capacity to cultivate new areas
of interest, as well. In many of the
things that I've been involved with in the
Quasar Project, and in the work that
we're doing related to the new NCTM
stan(lar(ls, we've been trying to grapple
with this idea of how you cultivate
students' interest and thoughtful en-
gagement in classrooms. It is very clear
that we have examples, some of which
you saw yesterday an(1 many of which
you can see in other locations, demon-
strating that students can be challenged
an(1 supporte(1 in engaging with math-
ematical ideas and find them quite
interesting in a variety of ways.
What we should take from this is not
that students should never see context
nor that everything has to be embedded
in context. Rather, our thinking about
this issue is enriched when we resect
back on the set of examples that we saw
anti the way in which they represent
mathematical possibilities. They show
us that students are engaged when they
CLOSING REMARKS
have interesting things to think about,
and mathematics is filled with interest-
ing things to think about. We need to
give students the chance to see math-
ematics as being interesting and some-
thing to think about.
As ~ reflected on this matter of rel-
evance and engagement, ~ recalled
something from my first year of teach-
ing. ~ taught seventh grade in the South
Bronxin New YorkCity. Oneofthe
students in my class was named Jeffrey.
~ want to tell you about Jeffrey. He was
very pleasant, and he had learned that
the way you get through school is to
smile at the teacher and be polite.
Jeffrey was a won(lerfu} little boy, but
academic school was not a priority for
him. Nevertheless, during the year,
Jeffrey, for some reason, became very,
very interested in palindromes. For
those of you who (lon't know about
palin(lromes, a number like i,331 is a
palin(lrome because if you write the
number forwar(1 or backwards, it's the
same number. Jeffrey got very inter-
este(1 in palin(lromes not because he
could apply them to his every (lay life,
but because they struck his curiosity.
Jeffrey spent most of the seventh gra(le
in an in(lepen(lent exploration of palin-
dromes. And it turns out that you can
learn a lot of algebra by exploring
palindromes and looking at the struc-
tures of these numbers and what hap
pens if you multiply them by certain

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numbers, what happens if you combine
them in certain ways, how many possi-
bilities there are for certain forms, and
so on. Jeffrey had his own independent
study course going on because that's
really what engaged his interest.
To bring closure about what makes it
interesting for kids to grapple with
mathematics, the examples we've seen
at this Convocation suggest that the
"interest" can be in the tasks themselves
that we give students, or it can be in
challenges raised by the tasks and in the
process of grappling. Students can find
it very interesting and can learn a lot
from struggling with challenging tasks.
They can derive a tremendous amount
of well-deserved self-esteem from this.
Mathematics is not easy. It is not always
fun. It is something that's worth strug-
gling with and worth doing well. This
struggle can be a very rewarding
experience, and it can be meaningful.
Now ~ want to draw attention to a
second area that ~ heard as a popular
topic at this Convocation algebra.
ALGEBRA
There were many mentions of algebra
in the plenary sessions, and from
looking at the records from the discus-
sion groups, it is clear that algebra came
up quite frequently there. It is quite
possible that some of us came to this
Ml DDLE GRADES CONVOCATION
meeting with a view of algebra in the
middle grades as a course very much
like the first year course in high school.
That is, algebra for middle school
students would mean that students in
some gra(le before high school would
take this course, whether it's eighth
grade or seventh grade or sixth grade.
This conception of algebra as the only
notion of algebra in the middle grades
was called into question by much of the
discussion and many of the examples
that we saw. That is not to say that one
cannot have a one-year course in aIge-
bra. But even a one-year course that
focuses on algebra can be (lifferent than
what we might expect. If you think
about the video involving Cindy, she was
teaching an algebra course, but the way
she was teaching that algebra course
strikes me as somewhat (lifferent than
our caricature of the way in which the
first year of high school algebra is
typically taught.
If we think about the set of i(leas that
Glen(la talked about on the first night,
the set of ideas that you might have read
about in the first discussion session on
algebra in the mi(l(lle gra(les section
(lrawn from the Principles and Standards
for School Mathematics: Discussion
Draft, or in other materials for this
Convocation, you get a (lifferent view.
This view suggests it might be possible
to think about algebra and the develop-
ment of algebraic ideas over grades six

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through eight in some way that doesn't
require a separate course for the year
we decide to teach algebra. This kind of
algebra instruction would be integrated
algebra, integrated with the stu(ly of
numbers, with the study of geometry,
and so on. This is, in fact, the approach
that is represented in most of the
curriculum materials that have been
produced and released in recent years.
Many of those materials differ in the
way that they go about doing this, but
they all share a commitment to trying to
develop algebraic knowledge or fluency
in a more integrated way throughout the
middle grades rather than concentrated
in a single year. This view of algebra is
really quite different than the view that
some of us might have had coming into
this Convocation. And it is a view that
challenges us to think harder about
what it would mean to learn algebra in
the middle grades.
Now ~ want to connect that to a last
point which ~ think came through very
clearly in the pane} session, much more
strongly than it had in earlier sessions.
Good curriculum and good intentions and
good practices and pedagogy may not be
enough. There is a range of policy and
political matters that need to be consid-
ered. As we heard this morning, other
kin(ls of support from parents, a(lministra-
tors, and organizational context matter,
and they matter a great deal.
There were a number of people who
CLOSING REMARKS
talked about the politics that surround
reform ideas, whether they were middle
school reform ideas or mathematics
reform ideas. And many kinds of
politics have been mentioned in this
Convocation community politics,
district politics, school politics, personal
politics, professional politics, and so on
~ remember from my first year of
teaching when we were trying to create
"open classrooms," which was then the
avant-garde reform idea. But we didn't
have the kind of physical space needed
for open classrooms. We had a very old
building with lots of walls. In response,
we rearrange(1 space anti use(1 the
hallways. We arranged students in
groups rather than having them sitting
in straight rows of desks. But we didn't
have tables in fact, we didn't have any
of the things that now are standard
practice. Instead, we had individual
student desks, and so the desks were
organized into small groups to allow
students to work together. Some
students would work in different loca-
tions in the room, some out in the
hallway, and so on. Every night Tom,
the janitor, would come to my room,
take all of the (leeks, whether they were
in the hallway or in (lifferent corners of
the room, wherever they were anti
arrange them into straight rows. Tom
had (lone this for 22 years in this school
An(1 the fact that a new teacher thought
that the furniture was going to be

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arranged in some other way had no
impact whatsoever on Tom.
Every night Tom would come and
move the desks. And it quickly became
a joke. The first thing the students
would do in the morning was rearrange
the furniture. No intervention by the
principal, no (liscussion with Tom,
nothing would have an impact on this.
We did this all year long, and mine
wasn't a unique case in the building.
There were other teachers doing the
same thing. So there is politics even at
that level, let alone dealing with commu-
nities and parents and so on.
How does this relate to algebra in the
middle grades? At this Convocation,
we've heard that there is a pretty w~de-
spread view of algebra, what it means
and how it looks. Essentially, this is the
view ~ described as the one that many of
us might have had walking into this
Convocation algebra that looks just
like the first year of a high school
course, all happening in one year. An(1 if
that is, in fact, the common perception
of what algebra is, and if you're trying to
implement some other way of teaching
algebra, there's likely to be a problem
because people won't understand it,
whether it's Tom the janitor or whether
it's Tom's great grandson who is now in
your class or his grandson who's a
parent in the community. There is a
serious set of issues that have to be
addressed in terms of unpacking for
Ml DDLE GRADES CONVOCATION
ourselves and for the whole community
what it means to say that students are
learning algebra, and what it would
mean for all students to learn algebra.
Can we design programs so that
students succeed in learning algebra in
the middle grades? Bob FeIner's com-
ment about "no acceptable casualties" is
a very important one. We don't have a
very good conception of what this
means. Our programs in mathematics
have not always been designed so that
everybody could be successful with
them. Mathematics education has
generally been organize(1 to fin(1 the few
students who could be successful, so
they could get on to the next course.
Some folks are working har(1 to change
this way of thinking, anti it is now a goal
for many that all students should be
more successful in mathematics. But
we need to recognize that there is a
huge education and political job to be
done in "unpacking" what it means for
"all students" to learn algebra, if we
want something that's (lifferent from
just taking that high school course one
or two or three years early. We need to
(levelop a broa(1 un(lerstan(ling of this
notion of algebra with others, including
administrators anti parents anti other
members of the public. We nee(1 to
systematically examine different in-
structional anti curricular arrangements
that are (lesigne(1 to have all students
learn algebra. We have a lot of hunches

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and a lot of opinions about which way is
best or which way will work. Many
people believe that if we just did it the
old way, it would be fine. But, if we
could be more precise about what it is
that we're trying to get students to be
able to do, what it is that we want them
to know, and how it is that we would like
them to be able to perform, we could
then ask whether or not taking that high
school course one or two or three years
early really meets these goals. We
would also have to ask ourselves
whether putting students through a
well-taught version of an innovative
middle school mathematics program
does this. We have to push ourselves to
ask this evidence question. What is the
evidence that we can, in fact, produce
the kinds of competence we want in our
students through these different in-
structional approaches?
So for me, this issue has gotten no
less complex. But it strikes me that this
is a place where we have an opportunity
to begin to work together, because ~
think the middle school community and
the mathematics education community
are both very interested in finding ways
to increase the competence and confi-
dence of students with respect to
mathematics. We want all students to
have the opportunities that mathemati-
cal competency and mathematical
proficiency affords them. Some people
call it "mathematical power." Some
CLOSING REMARKS
people don't like that term, but that's
what it's about. It's about having math-
ematics, owning it, having it be your
own and being able to open the doors
that mathematics can open. We want
this to happen for students, and we want
this to happen in ways that are sensitive
to their needs. This Convocation has
crystallized some of this for us, sharp-
ened some of the issues, and left us with
a number of other issues about which
we have to continue the conversation.
The issue of"mathematics for all stu-
dents" is one in which we might be
really to begin to act on.
OTHER ISSUES
There are also a few other issues of
note that were raised in this morning's
conversation an(1 in the (1iscussion
groups. The issue of teacher preparation
was not (liscusse(1 explicitly in any of the
sessions but was certainly a running
theme along with teacher professional
(levelopment. An(1 those two coalesce
around questions about teacher short-
ages and turnover. Occasionally, there
was mention of the special needs of
students anti teachers in high poverty
communities. This is a very important
issue that is (lifferent in rural communi-
ties and in urban communities. This
plays out in the mi(l(lle gra(les in quite
(lifferent ways. The organization of

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schools is often different in those com-
munities, and the ways of thinking about
specialist teachers and generalists is
quite different also. More conversation
is needed about these issues of diversity.
Racial and ethnic diversity, linguistic
diversity, special needs students, and
mainstreaming all need to be considered
as we move forward.
Technology was strangely absent in
most of the conversations, although it
came up in this morning's discussions
about the home-school interface in
reference to students who have access
to technology of a certain kind in the
home, but not in school. Technology is
very important for students of this age,
but technology also has an impact on
what's taught, how it's taught, and what
the possibilities are for teacher profes-
sional development and teacher assis-
tance. Consider, for example, the
amount of help that could be provided to
teachers to do a better job through the
uses of technology. Some people ~ know
in this audience are thinking very hard
about that. And we need to be looking
at that.
And then lastly, ~ want to mention the
notion of identity because Mary Kay
Stein talked a little bit about it this
morning, and it came up very strongly
in the discussion group sessions.
Maybe not everybody would attach the
word identity to this notion, but some
participants are asking questions about
how a teacher should balance attention
Ml DDLE GRADES CONVOCATION
to competing demands for a group of
students. How do you balance your
attention to the student with your
attention to the discipline or the subject
matter? How do we deal with the
generalist/specialist potion? How do
you balance affiliation with fellow
teachers of mathematics versus mem
bership on a cross-disciplinary team of
teachers for a group of students? There
are ways to frame the question that set
up false dichotomies as if it has to be
one or the other. Those of you who live
in classrooms every (lay know it isn't
that simple. But it is clear that how you
think of yourself has an influence on
what happens in classrooms. When you
think of yourself as a mathematics
teacher, you have a particular set of
resources and colleagues as well as a
set of constraints on what you do.
When you think of yourself primarily as
a middle school teacher, then you have a
different set of resources and colleagues
anti so on. We have to think about ways
of forming a community that has a joint
identity and that helps to move the
agenda of this Convocation forward.
And ~ just want to close by reiterating
something that Steve Gibson ma(le a
point of saying this morning that we
nee(1 to keep in min(l. Engaging in
(liscussions anti (1ialogues such as we
have experience(1 at this Convocation is
the way in which we're likely to make
progress. Thank you for being part of
this very productive first step.