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OCR for page 61

The sessions on teaching issues in the middle grades focused on the questions
.
What are the important characteristics of effective teaching in the middle grades?
Of effective teaching of mathematics in the middle grades?
· How can instruction in middle grades classrooms be organized to maximize
.
learning? How can we tell when learning is happening?
What tools and strategies wait make a difference in how middle grades students
learn mathematics?
USING VIDEO OF CLASSROOM PRACTICE AS A TOOL TO STUDY AND
IMPROVE TEACHING
Nanette Seago, Project Director, Video Cases for Mathematics Professional
Development, Renaissance Project.
PANEL REACTIONS TO THE "CINDY VIDEO"
TEACHING AND LEARNING MATHEMATICS IN THE MIDDLE GRADES:
STUDENT PRESPECTIVES
Linda Foreman, President, Teachers Development Group, West Linn,
Oregon.
PANEL RESPONSE TO FOREMAN STUDENT VIDEO
SUMMARY OF SMALL GROUP DISCUSSION ON TEACHING ISSUES IN
THE MIDDLE GRADES

OCR for page 61

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r]d
Her
[zag
far ~
Nanette M. Seago
Project Director, Video Cases for Mathematics Professional Development
A newly-funded NSF projects which
direct, is focused on developing video
cases as tools for use in mathematics
professional development. Currently, staff
are in the process of hypothesizing,
testing-out, analyzing, anti revising work-
ingtheories around the use of video as a
too} to promote teacher learning. There is
limited knowledge in the field about how
and what teachers' learn from professional
development experiences. It is mostly
uncharted territory. Not much is known
about what teachers learn from profes-
sional education (Ball, 1996), especially as
it pertains to the effect on teachers'
practice. Even less appears to be known
about how and why teachers develop and
use new un(lerstan(lings in their own
contexts. Conversations with colleagues,2
experiences of classroom teachers, and
experiences of professional developers
responsible for teacher learning are the
basis of our current knowledge.
Renecting on the practice of teachers
anti teachers of teachers, ~ have (level-
ope(1 some "working hypotheses" (Ball,
1996) about the use of video to promote
teacher learning. These frame the
design, development and formative
evaluations of my current work:
iNational Science Foundation; (ESI #9731339~; Host Institutions: San Diego State University Founda-
tion, West Ed
2I would like to acknowledge my colleagues Judy Mumme, Deborah Ball and her research group at
University of Michigan, Magdalene Lampert, Deidre LeFevre, Jim Stigler, Joan Akers, Judy Anderson,
Cathy Carroll, Gloria Moretti, and Carole Maples for their on-going help in thinking about these ideas.
3Ioan Akers is the formative evaluator of the VideoCases Project; Deidre LeFevre is a graduate student
of Magdalene Lampert and Deborah Ball at the University of Michigan and is using the developmental
process of our project to research the impact of different forms of video case facilitation on teacher
engagement and learning around issues of pedagogy, student learning, and mathematical content.

OCR for page 61

· teachers' mathematical and pedagogi-
cal learning is greatly enhanced by
connecting learning opportunities to
classroom practice to the actual
work of teaching.
· video cases of teaching can afford
teachers' opportunities to learn
mathematics as well as pedagogy;
· video cases of teaching can afford
teachers' opportunities to study and
learn about the complexities and
subtleties of teaching and gain ana-
lytic tools; and
· video case materials cannot stand
alone; the facilitation process is as
critical to what teachers learn as
teaching is to what students learn.
This paper is organized around some
questions to explore in the attempt to
understand how video can be used to
promote effective learning: What is the
work of teaching? What do teachers
need to know and be able to do in order
to do the work of teaching well? How
can video cases be used effectively to
develop teacher learning? Putting forth
working hypotheses and questions
invites others responsible for teacher
education to make explicit the teacher
learning theories or working hypoth-
eses that guide and frame their own
work so that the community can engage
in critical professional discourse and
inquiry into practice as professional
educators responsible for teacher
TEACHING ISSU
learning. Educational researchers can
use this work across multiple and varied
contexts to research why and how
teachers learn from professional educa-
tion experiences whether it is using
video or other practice-based materials
such as student curriculum, student
work, or written cases.
Teaching is thinking, intellectually
demanding work. Teaching is complex
and involves the interactive relation-
ship between content, students, and
teacher (Figure 11. Typically in profes-
sional development we tend to isolate
and separate this relationship which
can pull apart and oversimplify the
work of teaching. What gets left out
Figure 1. What Is the Work of Teaching?

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when this happens is the actual role of
the teacher in the thinking work of
teaching. For example: we spend time
doing mathematics because under-
standing mathematics is crucial to
teaching mathematics for understand-
ing and interpreting student thinking.
But learning mathematical content in
and of itself hasn't helped teachers deal
with figuring out the mathematical
validity in students' thinking, to recog-
nize the validity in partially-formed and
inadequately communicated student
thinking, or to analyze mathematical
misconceptions. But this kind of
mathematical analysis is a large part of
the mathematical content needed in
teaching for understanding.
Another example of professional
development is the examination of
student work and student thinking.
This is a critical part of the work of
teaching but not sufficient to help
teachers deal with the work of figuring
out what to do with student thinking
once they have it. Are there certain
strategies or solutions that the teacher
ought to highlight in whole-cIass discus-
sion? Does the order in which solutions
are shared matter in creating a cohesive
mathematical story line? How does one
use individual student thinking and
alternative approaches to further the
collective mathematical learning of the
whole class? Does a teacher stop and
pursue every chil(l's thinking, always, in
USING VIDEO OF CLASSROOM PRACTICE
every lesson? How does context factor
into the decisions?
Other efforts focus on teaching
strategies such as use of cooperative
groups, manipulatives, or writing in
mathematics separate from mathematical
learning goals and student learning.
Does one always use cooperative groups?
Are there advantages or disadvantages in
terms of student learning? For what
mathematical learning can manipulatives
be helpful? Do they ever hinder learn-
ing? Are some manipulatives more
helpful than others? If so, why? Does
how they get used make a difference?
These are just some of the recurring
dilemmas teachers face without suffi-
cient help in acquiring the skins and
knowle(lge necessary to make intellectu-
ally flexible decisions.
The work of teaching involves more
than "in-cIass time." It involves acts of
professional practice before, during, and
after the moments of face-to-face teach
ing. Some examples of the kind of work
teaching entails are:
Before
· Setting mathematical learning goals
· Choosing/analyzing tasks anti cur
riculum in relation to goals and
students
· Figuring out various student ap
proaches and possible misconceptions
· Learning about students
· Planning a mathematical story

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During
· Figuring out what kids are thinking/
understanding the mathematical
validity of student thinking on-the-spot
· Figuring out what to do with student
thinking once you have it seize or
not to seize? For what mathematical
learning goal?
· Balancing between individual student
and whole class mathematical think-
ing and learning analyzing and
deciding
· Building a mathematical story
· Budgeting time
· Learning about students
Abler
· Figuring out what to do tomorrow
based on today
· Analyzing student work
· Learning about students
PURPOSES FOR USING VIDEO
When most people think of the
purpose for using video with teachers,
they think of models, exemplars, or
illustrators of a point. The question this
poses is one worth researching. What is
being modeled or exemplified? What is
it that teachers learn and use in practice
from viewing models or exemplars?
Teachers approach new learning in the
way in which they were taught by
TEACHING ISSUES
following the procedure in an example.
Using this familiar "way of learning,"
they watch the video in search of proce-
dures to follow or features to copy. This
can create barriers to a deep level
examination of teaching, for it often
keeps the focus on surface and superfi-
cial features of classroom practice.
Video can be used with a different
frame, an analytic frame which focuses
on the analysis of teaching practice,
gaining the awareness necessary to
analyze and interpret the subtleties and
complexities involved in the relationship
between knowledge of content, stu-
dents, learning, and teaching. Analysis
involves studying the same video
episode through multiple lenses, as well
as the comparative study of multiple and
varied videos through the same lens. It
involves examining the details and
specifics of practice in this instance, in
this context, under these circumstances,
and perceiving the subtle particulars
(Schwab, 1978) in classrooms while
recognizing how these together form a
part of an underlying structure or
theory of teaching. Analysis involves
supporting and disputing assertions or
conjectures with evidence or reasoning.
Taking an analytic stance in framing
the use of video creates a set of teacher
learning issues for teacher educators/
professional developers to consider.
Gaining awareness of the subtleties and

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complexities of practice does not hap-
pen easily. We exist in a strong culture
of quick fixes, definitive answers, and
oversimplification of the practice of
teaching. It wail take purposeful teach-
ing of analytic skills (Ball and Cohen,
1998) in order for teachers to use
analytic skills. This means it wait also
take the study of how teachers develop
and use those skills over time to under-
stand how to help teachers acquire
them.
Developing a culture where teachers
can learn to critically examine the
practice of others as well as themselves
wall be no easy task either, for there
currently exists a strong culture of
being nice. Teacher struggles and
challenges need to be seen as opportu-
nities to learn rather than secrets to
hide. Teachers need to be active mem
bers of an intellectual professional
community that values risk-taking,
diversity of opinion, critical debate, and
collaborative analysis. This has implica-
tions for the teachers of the teachers,
for they wall be responsible for develop-
ing this culture a culture foreign to
most U.S. teachers.
Video can be used for educative
purposes (Lampert an(1 Ball, 1998), as
opposed to a more open-ended "what
do you think?" approach. Open-ended
discussions can be useful in gathering
data on what teachers attend to, are
USING VIDEO OF CLASSROOM PRACTICE
aware or unaware of, or how they think
about teaching, learning and math-
ematics in practice, but for teaching
something such discussions are not
usually effective. Often the conversa-
tions become unfocused and scattered.
When viewing video, there is so much
data that sorting through it needs
focused guidance and structure. An
educative point of view means to plan
and focus on specific goals for teacher
learning. What can teachers learn
from this particular video and how can
they best learn it? The mathematical
an(1 pe(lagogical terrain of the vi(leo
needs to be mapped out in relationship
to what each episode offers teachers to
learn about mathematics, teaching, and
learning. An educative point of view
explores the question, how can we go
beyond this particular video or set of
videos to learn the big ideas of practice?
the big ideas of mathematics?
HOW DO WE USE VIDEO TO
PROMOTE TEACHER LEARNING?
What might it mean to plan anti
orchestrate professional education that
is guided by analytic and educative
purposes? What are the implications for
the work of teaching teachers? Just as
the practice of teaching is complex anti
involves the interactive relationship

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between teacher, students and content,
the practice of teaching teachers in-
volves the interactive relationship
between professional educators, teach-
ers, and teaching content (Figure 21.
While there is no one right way to
plan and facilitate a video session,
purposeful planning around three key
areas is helpful: (~) the content of the
video segments, (2) the learning goals
of the session and (3) audience (or
learners). A discussion of the planning
and facilitating these three areas using
the Cindy video as a context for analytic
and educative purposes in the work of
teaching teachers follows.
THE CONTENT OF THE VIDEO
The Cindy video clip lesson begins
with Cindy posing the problem, If you
lined up 100 equilateral triangles in a
row i_ (shared edges), what would
the perimeter be? This is the first part
of the larger mathematical problem of
the lesson: if you lined up 100 squares,
pentagons, hexagons in a row, what
would the perimeter be? Can you
generalize and find a rule for the perim-
eter of any number of regular polygons
lined up in a row? Figure 3 shows a
graph of Cindy's lesson with an arrow
marking the point in time that we drop
in with the video clip.
Figure 2. Teaching Teachers
Content of Video or
Other Professionc~l
Development Curriculum
TEACHING ISSU

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lust as teachers need to know the
content of tasks they pose to their
students, teacher educators need to
know the content of the material they
are using as tools for educating learn-
ers. Analyzing the material means to
analytically examine the terrain of the
video. What opportunity does it offer to
learn about mathematics? pedagogy?
students? learning? In what ways can it
be used to learn new professional habits
of mind? How can it be used to develop
skills in analysis and develop disposi-
tions of inquiry? The Cindy video has
been used to:
.
.
[earn mathematics. The math-
ematical content allows for teachers
to learn about multiple algebraic
representations of a geometric rela-
tionship in non-simplified forms. It
offers the opportunity to examine and
learn about recursive and relational
generalizations. The role of math-
ematical language and definitions are
also embedded opportunities for
working on mathematics within this
video.
[earn about student thinking/
reasoning. Focusing on student
conversation and responses (Nick,
Chris, and Lindsey) provides the
opportunity to learn what we can or
cannot tell about the student's appar-
ent understanding. Nick may have
USING VIDEO OF CLASSROOM PRACTICE
Figure 3. Cincly Lesson 1 . Overview of whole lesson (50
minutes)
10 min
8 min
10 min
1 5 min
7 min
Posing triangle problem
Individual/small group
Posing of additional
polygons problem
in small groups
the beginnings of making sense of the
geometric relationship by viewing the
whole of each in(livi(lual polygon's
perimeter multiplie(1 by the number
of total polygons, anti then subtract-
ing out the inside shared edges:
(n represents the number of polygons;
s represents the number of si(les of
each polygon).
We drop in

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Nick's Response
Total Number of Sides - Shared
Sides: p = ns - 2tn - 1 )
ns is the number of polygons multiplied
by the number of sicles of each poly-
gon.
2 (n - 1) is the number of shared sicles,
i.e., sicles of the polygons not contribut-
ing to the perimeter. The perimeter is
the total perimeters of all polygons
minus the perimeter of the shared
sicles.
(Congruent
's)
hi\
hi,.\
clot\
got\
End side
Chris sees the relationship like Cindy
does tops + bottoms + 2 end sides,
where the tops + bottoms are the contri-
bution each polygon makes toward the
perimeter of the whole:
Chris's Response
Tops and Bottoms ~ 2 End Sides:
p = nils- 2) ~ 2
If top and bottom are consiclerec3
together, each polygon contributes the
same amount to the perimeter. Thus,
the perimeter equals the number of
polygons times the perimeter contrib-
utec3 by each polygon plus the two end
sicles.
TEACHING ISSU
Lindsey may be trying to make sense
of the relationship by taking out the en
polygons, counting the outsi(le e(lges of
each and dealing with the middle
polygons separately.
Lindsey's Response
Interior Polygons ~ End Polygons:
p = In - Hits - 2) ~ 2ts - 1 )
(n - 2) is the number of interior polygons
(s - 2) is the amount contributed to the
perimeter by each interior polygon
(s - 1) is the contribution of each end
polygon to the perimeter
The perimeter equals the number of
interior polygons times the amount
contributed to the perimeter by each
interior polygon, plus the contribution
toward the perimeter by the two end
polygons.
Examining students' thinking can
provi(le opportunities to work on math-
ematics. It also offers the opportunity
to examine the teacher-(lecisions around
each student's thinking. What appears
to be the teacher thinking anti reason-
ing for the (recisions around each
student's thinking? For what math-
ematical learning goals would she seize
in(livi(lual student thinking, when would
she not? When anti why might she slow
things ([own? or spee(1 things up?
Veronica introduces a recursive pattern
she sees from the table. Examining the

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table and being able to generalize the
pattern is an opportunity that can be
utilized.
.
[earn about teacher cIecision-
making. While it may seem obvi-
ous that teachers make constant
· 1 · .
conscious or unconscious cleclslons
while planning and orchestrating
discourse, this is not typically recog-
nized and critically examined by
teachers and administrators. In order
to analyze teacher decision-making in
light of content and students, one first
needs to become aware such deci-
sions even take place. You can't
analyze and interpret what you can't
see. It is the work of the facilitator to
"lift the veils so that one can see"
what isn't normally seen (Eisner,
19911. Using this video over multiple
sessions helps to move the learning
from awareness of decisions to the
level of analysis of decisions in light of
content and context. When focusing
on teacher decisions, the opportunity
exists to push at the current tendency
for teachers to be definitive in their
claims around teacher decisions.
Pushing for alternative possibilities,
alternative possible reasoning or
conjectures with supporting evidence
and arguments over time can create
the necessary dissonance for learning
new norms and practices.
USING VIDEO OF CLASSROOM PRACTICE
· Learn about the uncertainty in
teaching. These video segments
offer the opportunity for participants
to learn about the uncertainties
involved in a complex practice
(McDonald, 1992), especially the
recognized uncertainties involved in
the moments of not knowing if a
student's reasoning has mathematical
validity Lindsey and Nick). It can
highlight the recurring teacher
dilemma of figuring out what a stu-
dent is thinking and offer the opportu-
nity for teachers to gain multiple tools
for (leafing with these uncertainties
themselves. This learning can create
a tension for the facilitator between
not wanting teachers to leave with the
notion that all of teaching is uncertain
and on-the ny nor with the notion that
one can always predict what will
happen with certainty. The learning
focus is to gain understanding of the
importance of planning for possible
student approaches as well as gaining
tools for better (recisions in their own
moments of uncertainty.
GOALS FOR TEACHER LEARNING
It is important to (leci(le on goals for
teacher learning when using a video
case. Why are you using this case?
What outcomes (lo you seek from the
video case experience fs)? A case can be

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JOEL RECORDING IMPORTANT
IDEAS
As we work on different problems,
we come across "big ideas" that
seem to keep coming up, even in
seemingly unrelated topics.... This is
where our journals come in. They
allow us to record our thoughts and
processes so that we may look back
later and work through our thought
processes again.
Each student's journal is different, as
a journal is a place for records of per
sonal struggles, discoveries, and in-
sights that help illustrate what we have
Figure 1. Journal excerpts ~ and 2
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TEACHING ISSUES
been working on in class.... I have
chosen a few excerpts from my journal
to help demonstrate how we record
important ideas and look back later if
we are stuck.
It is by coming across examples such
as this that we have learned that it is
very important to record the important
mathematics ideas we come across, and
to try applying methods we came up
with earlier, even if at first look it seems
as though the methods have nothing to
do with the idea we're examining. In
this way, new i(leas make sense and are
easier to un(lerstan(l....
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OCR for page 61

Figure 2. Journal excerpt 3
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KYLE MATHEMATICAL TRUST
Learning mathematics is a journey.
... Our teacher trusts us as capable
· Answer a question without letting the
stu(lents get involve(l,
· Not allow students to invent their own
mathematical thinkers who can find our procedures,
own way.... That is, she believes there is
a mathematician w~thin each of us.
Therefore, she does not lead, show, or
· · ~
gu~de us ~n our Journey....
A teacher that does NOT have math-
ematical trust in her students may:
· Show a best way to solve a problem,
· Ask a leading question that looks for
an expected answer,
S T U D E N T P E R S P E C T I V E S
· Not allow students to fee} disequilibrium.
The above actions discourage the use
of new ideas and different approaches.
They also take away the stu(lent's
opportunity to fee} the joy of learning
and doing mathematics. ~ have noticed
that when someone telIs me how to
solve a problem, my thinking stops. On
the other han(l, when someone allows

OCR for page 61

me to wrestle with an idea, ~ find myself
inventing more strategies and ~ get a
stronger grasp of the idea. Solving a
problem myself helps me clarify my
understanding of the mathematics and
leads to important conjectures.
Last year our class invented the
strategy "completing the square" to
solve quadratic equations. This fall,
when the idea came up again, we felt
certain we could solve any quadratic
equation. So, our teacher sai(l, "Just for
fun, if ax2 + be + c = 0, what is x?" After
Figure 3. Journal entries
- " ~ 1 It ~ ~ 1 "=9 ~ ~ ~ ., , . .. , at
~-~ For rat . : ~
the first day of exploration our teacher
asked us if we wanted clues. We pro
tested and ended that day and the next
(lay of class in (lisequilibrium. Finally,
after three (lays of buil(ling mo(lels,
cutting and rearranging pieces, debat-
ing and discussing, and no clues, we
found the value of x.
The journal entry on the left below
shows how ~ solved a specific quadratic
equation. The journal entry on the right
shows how ~ use(1 that i(lea to generalize
about any quadratic equation.
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TEACHING ISSUES

OCR for page 61

Because our teacher trusted us and
we trusted ourselves, we invented an
algorithm for solving any quadratic
equation dater we found out that other
mathematicians had also invented that
aIgorithm).
Learning is a journey. Mathematical
trust keeps us going and allows us to
travel in new directions without worry-
ing about getting lost or taking the
routes that others do.
CONCLUSIONS
Working with these students for four
years has stirred my thinking about
learning and teaching and enabled my
growth as a writer. More importantly
perhaps, it has left me with food for
thought about teaching mathematics.
Following are a few ideas on my mind,
prompted by the students' comments
and work and by the Convocation
panelists and participants. Perhaps they
wait provide thought or discussion
starters for you, an(l/or perhaps you
have other ideas to add to the list.
· It is possible to form a remarkable
mathematics community and cIass-
room culture when youngsters and
their teacher stay together over time.
How can this mode} be adapted to
work in the mainstream mi(l(lle
school setting?
S T U D E N T P E R S P E C T I V E S
We teachers and curriculum develop-
ers can impose artificial limits on
students by the questions we don't
invite or pose and by our own concep-
tions of learning, teaching, anti
mathematics. How do we learn to
recognize this in our actions and
work?
· While it is the case that I~in(lsay, Joel,
and Kyle each went on to explore
applications of the area of a trapezoid,
center of rotation, and quadratic
formula, their papers suggest it was
the mathematical ideas themselves
that were engaging. What motivates
students to engage in thinking about
mathematical ideas? What makes a
problem "real" for students? What is
meaningful context?
If one agrees that these students
provide evidence that it is possible to
cultivate interest in serious math-
ematical content, what are the instruc-
tional practices that are most influen-
tial in cultivating such interest?
What is important and relevant
mathematics for mi(l(lle gra(les? What
is worthwhile mathematical activity?
Note: (leriving the quadratic formula
was not a part of my original lesson
plan; however, as Kyle pointe(1 out,
the class spent 3 (lays wrestling with
the challenge. What may have been
gained or lost by taking this math-
ematical excursion? It seems to fit
Julie's criteria for worthwhile activi

OCR for page 61

ties. What are other criteria to add to
Julie's list?
· Many teachers were educated in a
system that promoted the notion that
only certain people can do mathemat-
ics. Can students come to recognize
and nurture their "inner mathemati-
cians" if the teacher does not believe
every student has a capable math-
ematical mind? What professional
development experiences are neces-
sary for teachers to develop a sense of
"mathematical trust" in their students?
· Contrary to many common (lescrip-
tors, doing mathematics is an emo-
tional experience, and those emotions
can be positive ones, e.g., empower-
ment and passion for the subject
matter, pride in discovery and inven-
tion, respect for disequilibrium, and
joy over solving challenging prob-
lems. What teacher actions best
facilitate the development of such
feelings about mathematics?
· When students are called upon to
communicate mathematically, they
learn the language of mathematics as
they learn to speak any language-
simultaneously using invented lan-
guage (e.g., smooching) and formal
language (e.g., disequilibrium, trans-
formations). It can be uncomfortable
for teachers as they strike a balance
between accepting students' invented
language anti teaching formal math-
ematical language.
TEACHING ISSU
· As Erica pointed out, teaching that
focuses on how students think about
mathematics has a powerful influence
on students' learning as well as their
views of themselves as mathemati
cians. It also provides the teacher
rich information about the extent to
which students un(lerstan(1 anti are
able to integrate mathematical ideas
and processes into their own way of
thinking. For example, in Lindsay's
and Toel's explanations of their
thinking, their use of transformations
provides evidence of their sense of
geometry as a process; anti we see
evidence of I~in(lsay's anti Kyle's
sense about the integrated nature of
algebra and geometry by their use of
algebraic symbols to represent the
geometric and algebraic relationships
they could "see" in models. Because
teaching that centers on how stu-
dents' think is so different from the
mathematics instruction most teach-
ers and parents have experienced, it
is particularly challenging to shift
away from practice that emphasizes
telling students what to think. Long
term professional (levelopment anti a
curriculum that emphasizes students'
mathematical thinking are essential
for teachers to make this shift.
The examples given in this paper
provide a glimpse of what is possible
when students are immerse(1 over time

OCR for page 61

in a Standards-based curriculum;
however, it is important to keep in
mind the fact that this project lasted for
four years. lust as implementing
reform is a challenge and requires
long-term support for teachers, the
benefits of reform-based teaching are
not immediately apparent in students.
Had these students written their papers
even a year earlier, some students
would have expressed doubts about
how and what they were learning (they
kept close tabs on activity in their
peers' more traditional cIassrooms
S T U D E N T P E R S P E C T I V E S
the media, and even some mathematics
teachers, told them they would never
learn what they needed to know); there
would have been more disequilibrium
about certain mathematical i(leas that
are described with confidence here;
their parents may have expressed
doubt due to the uproar about math-
ematics reform in the local news; and
their teacher would have been a little
less secure in her conviction to main
tain high expectations, trusting
everyone's (lisequilibrium was a sign of
new learning about to occur.

OCR for page 61

=.1.=
1~01~1
~. rig
Three panelists were invited to
comment: Hyman Bass, a university
mathematician, Sam Chattin, a career
middle grades teacher, and Deborah
Ball, a researcher on teaching and
teacher educator. Sam Chattin, who
does not teach math, reacted to the
video thinking about middle grades
students in the context of social groups.
He observed that the students in the
video had formed an effective social
group and seemed to have made a
conscious choice to stay in the group,
probably because of the affirmation they
received about their ability to do math-
ematics and be successful. The teacher
had clearly done some modeling about
learning mathematics and about group
behaviors. He noted that a strong
commitment to the social group was
evident in the students' willingness to
raise money to travel to Washington,
DC. The body language of the students
making their presentations indicated
they felt very secure, and it was clear
that a(lults hall ma(le them that way,
free to make mistakes with no censure.
They use(1 their own wor(ls (e.g.,
"smooched"), an indication they felt free
to translate what they knew to their own
world. Probably the most significant
observation for the audience was his
remark that the audience laughed when
the students were the most serious.
Chattin pointed out that it is hard for
adults to recognize just how serious
middle grades students are.
Hyman Bass described himself as a
university mathematician "infected" with
observing elementary teaching and
thinking about how students learn
mathematics. He shared Sam's impres-
sions anti saw in the students a renec-
tionoftheteacher'spractice: attitude
towards content, classroom culture,
philosophy, anti principles of teaching.
Bass felt the students saw themselves as
mathematicians with the pri(le of (liscov-
ery when they realize(1 they were part
of history. The mathematical topics
covere(1 algebra anti geometry, which in
his view have been excessively sepa-
rate(1 in the stan(lar(1 curriculum.
Particularly nice was the use of transfor-
mational geometry, cutting anti pasting
to find the area of the trapezoid (which
preserves area), and in one of the
presentations the use of rotations and

OCR for page 61

the center of rotations as a way to
approach the problem. He observed
that some of the methods of teaching
were obvious in the student presenta-
tions, such as keeping a journal of
mathematical ideas. When one of the
students was given a problem about
using perpendicular bisectors of chords
to find the center of a circle, although he
had the answer, he returned to his
records in the journal and made the
connections with the mathematics he
had recorded to learn why his answer
worked. According to Bass, the tape
revealed products of enlightened teach-
ing where the students were able to
communicate about mathematics and
had a passion for the subject.
Deborah Ball noted the challenge of
commenting on teaching when teaching
was not visible on the video segment.
However, she said, several key observa-
tions were possible about what the
teacher must have done for students to
be able to do the things displayed on the
video. She framed her remarks around
the conjecture that this teacher had
PANEL RESONSE TO FOREMAN STUDENT VIDEO
actively held and communicated high
expectations for students. She said that
teachers can move children if they hold
expectations that students can learn and
(lo not take refuge in "Most (or my) kills
can't (lo this." Ball's first point was that
the teacher in this case hall to teach her
students how to use her questions as a
way to learn; they had to learn to make
sense of the way she teaches. Second,
the teacher had to cultivate interest in
the mathematics. Thinking students are
not interested is the static view; the
teacher had done something to make
these students interested and involved.
A thir(1 point was that the teacher hall to
cultivate a language within which the
class could work. She had taught them
some formal wor(ls that were not part of
their vocabulary (e.g., "(lisequilibrium")
but she also accepted their words
("smooshed"~. And finally, she must
have created some incentives for stu-
dents to learn to work this way. The
students hall been given high incentives
to engage in mathematically soun
work.

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or
Using the videos and pane! observa
tions as a backdrop, the discussion groups
addressed the following questions: What
are the important characteristics of
effective teaching in the middle grades?
Of effective teaching of mathematics in
the middle grades? How can instruction
in middle graces classrooms be organized
to maximize learning? How can we tell
when learning is happening? And, what
tools and strategies will make a difference
in how middle grades students learn
mathematics? What are the important
characteristics of effective teaching in the
middle graces? Ofeffective teaching of
mathematics in the middle grades?
The answers to the first question
clearly reflected the middle school
philosophy, with an emphasis on a safe
learning environment where students
work in a social caring classroom and
learning mathematics is treated as a
social activity. There was strong sup-
port for student-centered classrooms
and reinforcement of student ideas
and work. Some caution was voiced
about using praise to reward less-than
I ~Cw.2
Ark
adequate performance. As the groups
struggled to identify effective math-
ematics teaching, many mentioned that
"quiet discomfort" signals new learning
and that a sense of disequilibrium is
essential to learning, resecting the
message from one of the videos. A
clearly identified characteristic of
effective middle grades mathematics
teaching was the need for strong con-
tent knowledge on the part of the
teacher. This was particularly signifi-
cant when mathematics was viewe(1
from the perspective of a challenging
mi(l(lle gra(les curriculum that goes
beyond computational skills. The
groups i(lentifie(1 the following charac-
teristics of effective mi(l(lle gra(les
mathematics teachers. Effective mi(l(lle
gra(les mathematics teachers:
· have high expectations for their
students
· have students who are involve(1 in
active learning situations anti en-
gage(1 in communicating mathematics
· (resign their lessons with well (lefine(1

OCR for page 61

goals and a coherent message making
connections within the course and the
curriculum
.
are able to build understanding from
concrete models, knowing how and
when to bring the ideas to closure
· understand the importance of asking
the right questions in ways that
promote thinking and allowing
sufficient time for students to respond
· listen to their students' responses so
they know where those students are
in their mathematical understanding
and use this knowledge to develop
student learning
· are flexible yet have created a familiar
structure and routine for their classes
· provide models for student learning
by the way they teach.
How can instruction in middle grades
classrooms be organized to maximize
learning? The groups stressed the
critical role of the school administration
in creating a structure and support for
student learning, from setting the
school environment to ensuring that
class disruptions were minimized. The
groups also consistently mentioned the
need for time for teachers to work
together (leveloping lessons anti think-
ing through the curriculum, for flexible
blocks of time (not necessarily block
scheduling), for manageable class sizes,
and for clear articulation between
gra(les. Over a fourth of the groups
SUMMARY OF SMALL GROUP DISCUSSION
supported "looping" having a teacher
remain with a class of students over
several years. There was strong sup-
port for using teams as a way to create a
community of teachers.
How can teachers tell when learning
is taking place? Students provided such
evidence when they were engaged and
able to explain the mathematics they
were learning to others. Students who
un(lerstan(1 can apply mathematics to
solve problems and have the ability to
revise their thinking based on their
investigations.
What tools anti strategies will make a
difference in how middle grades
students learn mathematics?
Manipulatives, calculators including
graphing calculators, and computers all
were referenced by the discussants as
important tools to help students learn
mathematics. Assessment as a too} to
enhance learning was suggested, as
well as engaging students in writing
and projects. Strategies for helping
students learn mathematics inclu(le
creating a warm and open environ
ment, where there were clear and
consistent policies among the team
members. Parents should be informed
anti involve(l. Mentoring anti buil(ling
a community of teachers were reoccur-
ring themes. Teachers should be
working with other teachers on les-
sons, visiting classes, anti (1esigning
professional (levelopment activities

OCR for page 61

around the context of the content
teachers were teaching. Teachers
should be engaged in posing questions
and debating answers to stimulate
student thinking. Attention should be
paid to the developmental levels of
students, although questions were
raised about what the term develop
mentally appropriate meant and how
teachers would understand this in
terms of their students.
The comment was made that ' five are
not taught to teach, only about teach-
ing." Video as a way to initiate a discus-
sion of teaching was perceived as both
positive and negative. The initial ten
TEACHING ISSU
dency to criticize can overtake the
discussion. There was concern that
viewers might not recognize good
teaching. The risk of going public as a
teacher and standing for inspection was
too great to make this a useful medium.
Those who found viewing actual in-
stances of teaching useful, appreciated
the different thinking that can result
from talking about actual practice. A
video allows a situation to be viewed
repeatedly from different lenses. The
groups did agree, however, that the
viewer should have a well (lefine(1 focus
in order to make the viewing and
ensuing discussion useful.