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fir I. r~rm.r~_
The National Convocation on Middle Grades Mathematics was followed by three
separate Action Conferences, funded by support from the Department of Education
and additional support from the American Educational Research Association. The
following three papers synthesize the activities and discussion that occurred during
the three conferences.
ACTION CONFERENCE ON THE NATURE AND IMPACT OF ALGEBRA AT
THE MIDDLE GRADES
Organized by Hyman Bass, Columbia University
This action conference focused on providing school based decision makers with an
unclerstancling of the importance of bringing algebra into the micic3le gracles and the
issues involved in making this happen.
ACTION CONFERENCE ON RESEARCH IN THE TEACHING AND
LEARNING OF MATHEMATICS IN THE MIDDLE GRADES
Organized by Sandra Wilcox, Department of Teacher Education, Michigan
State University
The conference was clesignec3 around the question: What are the characteristics of
research that would be helpful and informative for teaching mathematics in the
micic3le gracles?
ACTION CONFERENCE ON THE PROFESSIONAL DEVELOPMENT OF
TEACHERS OF MATHEMATICS IN THE MIDDLE GRADES
Organized by Deborah Ball, University of Michigan.
The participants worked through a frame for considering the design and practice
of teacher development at the micic3le gracles, examining the ideas that drive
professional development in the light of what is known and unknown about teacher
learning.
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r.~.r~
Synthesis by Bradford Findell
Program Off cer, National Research Council
The Action Conference on the
Nature and Teaching of Algebra in the
Middle Grades brought together
mathematicians, mathematics educa
tors, middle school teachers, math
ematics supervisors, curriculum
developers, and others to discuss the
role of algebra in the middle grades.
The conference was provoked in part
by recent events and policy decisions
that have focused attention on algebra
in the middle grades.
Perhaps most prominent among these
events was the release in 1997 of results
in the Third International Mathematics
anti Science Stu(ly MESS, which
indicated, among other things, that the
U.S. curriculum at eighth grade is about
one year behind many other countries,
and that U.S. eighth graders perform, as
a group, below the international average
in mathematics achievement.
In the policy arena, several states
recently mandated that algebra be a
required course for high school gradua
tion, and the pressure has expanded
into the middle grades as well. In the
U.S. Department of E(lucation's White
Paper, Mathematics Equals Opportunity,
Secretary Riley recommends, on the
basis of a strong correlation with college
attendance, that students take algebra
or courses covering algebraic concepts
by the en(1 of eighth gra(le.
Despite the growing public attention
to algebra in the middle grades, there
has been little discussion (and perhaps
little public acknowle(lgment) of the fact
that there are more than a few possibili
ties for what algebra in the mi(l(lle
gra(les might look like. Thus, the
agenda for the Action Conference was
(lesigne(1 to bring some of these possi
bilities to light through presentations on
various views of algebra anti on re
search in the teaching and learning of
algebra, together with practical experi
ences of teachers anti (1istricts who have
been implementing some version of
algebra in the middle grades. Discus
OCR for page 139
sion was framed by six questions (See
Figure 1) that were presented by the
Action Conference Organizer, Hyman
Bass, a mathematician from Columbia
University and Chair of the National
Research Council's Mathematical
Sciences Education Board.
VIEWS OF ALGEBRA
The first presenter was km Fey, a
curriculum developer and mathematics
education researcher from the Univer
sity of Maryland. He began the after
noon by reminding the participants that
a big factor in the debate about the role
of algebra is the social and political
context. Algebra serves as a gateway to
postsecondary study and to scientific
and technical careers. And if algebra is
good in ninth grade, then it is better in
eighth grade, and some think even
better in seventh. But there has been
little attention to what the content of
algebra is or may be.
In the conventional view, algebra is
primarily about calculation with sym
bols Bob Davis called it a dance of
symbols. And algebra is used to solve
word problems. But students are not
very goo(1 at the wor(1 problems that we
teach them, never mind problems that
they haven't seen before.
The increase(1 availability of calcula
tors and computers provides new
ACTION CONFERENCE
demands and opportunities for the
teaching and learning of algebra. We
may concentrate on the design and use
of algorithms, on data modeling and
predictions, or on analyzing anti project
ing trends. Spreadsheets anti other
computing tools allow us to approach
such ideas graphically, numerically, anti
symbolically.
Fey acknowledged that there is a fair
amount of debate about what algebra is,
but suggeste(1 the question might not be
productive. Instead, he suggested that
what we want from algebra are some
concepts and techniques for reasoning
about quantitative conditions and rela
tionships. There are four major aspects
of such reasoning: representation,
calculation, justification, anti application.
Development of student understanding
and skill in these areas can begin in the
mi(l(lle gra(les. With such an approach,
he notes, it will not be sufficient to take
high school algebra and move it into the
middle grades.
Representation is about expressing
complex relationships in efficient,
condensed, symbolic form. Tradition
ally, the typical algebra question has
been ' What is x?" But representations
such as data tables, graphs, symbolic
rules, and written expressions may be
used to record and describe numerical
patterns, formulas, patterns that change
over time, and cause and effect relation
ships.
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Figure ~ . Framing Questions
SOME IMPORTANT QUESTIONS
Attention to subject matter ~ attention to students
In teaching algebra in the middle Oracles, what are the tensions between attention to serious ancl
challenging mathematical content, on the one hancl, ancl, on the other hancl, sensitivity to clevelop
mental, social, ancl equity issues pertinent to adolescent chilclren?
Algebra as the language of mathematics
2. If one thinks of algebra as the language of mathematics, then does the learning of algebra entail some
of the same challenges encountered in the learning of reacling, ancl therefore call for a more cleliber
ate ancl focused attention to the task of teaching this formalization of mathematical expression ancl
communication?
Real world contexts ~ generalization and abstraction
3. Many have argued that in order to motivate student learning of mathematics, it should be presented
concretely in terms of real life problems ancl situations. This has been interpreted by some to require
that all mathematics learning be embeclclecl in complex empirical investigations ancl measurements.
Is this really warrantecl, both in terms of the presumptions about student motivation, ancl as effective
peclagogy? Does this shortchange the equally important mathematical processes of generalization
ancl abstraction, i.e., the distillation or clecontextualization of mathematical ideas from multiple
contexts?
Mathematics curriculum: covering mathematics ~ uncovering mathematics
A. TIMMS characterizes the U.S. curriculum in mathematics as a mile wide ancl an inch creep. One
manifestation of this is the pressure on teachers to race through an overloaded curriculum in both
standard ancl accelerated trackswith little time for student reflection ancl inquiry with new icleas, a
practice that flies in the face of what constructivist ideas tell us about the nature of learning. Is this
true? Ancl, if so, what can be clone to change this conclition?
Situating algebra in the mathematics curriculum?
5. How should algebra be situated in the curriculum? As a traditional focused algebra course, or
integrated with other subject areas, such as geometry? As a strand across many Oracles?
Mathematical curriculum: materials, design, selection criteria, . . .
6. What are the characteristics of currently available curriculum materials in terms of topic coverage,
pedagogical approaches, use of technology, support ancl guidance for teachers, etc. What kinds of
tradeoffs must be made in the adoption of one over another of these curricula? How can one measure
the impact of curricular choices on issues of equity, teacher preparation, community understanding,
program assessment, ancl articulation with elementary ancl high school programs?
THE NATURE AND IMPACT OF ALGEBRA
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Traditional approaches to algebra
have focused on symbolic calculation,
but calculation can be used to con
struct tables anti graphs of relation
ships, to solve equations, and to de
scribe rates of change and optimal
values; major ideas in calculus that are
often disregarded by calculus students
who rely only on symbolic calculation.
Calculation can aid in the construction
of equivalent representations for
quantitative relationships. The reason
the representations are equivalent is
that they both make sense for model
ing the same situation.
In algebra, justification, reasoning,
and proof are often considered in
conjunction with the properties of the
number system. But the properties of
the number system are the way they are
because the properties make sense. It
is not the case that 2 + 3 = 3 + 2 because
addition is commutative. Rather, addi
tion is commutative because verification
that they both equal 5 can be general
ized to all numbers.
Finally, applications can be pervasive
from start to finish, providing frequent
opportunities to move back and forth
between abstraction and the real worI(l.
Fey closed by pointing out that these
curricular goals should be implemente
with consideration of the ways students
learn. He suggested that the way
students encounter these i(leas must be
changed from the "demonstrate, imitate,
ACTION CONFERENCE
and practice" style that has dominated
algebra instruction.
Al Cuoco, Director of the Center for
Mathematics Education at EDC, pre
sented a view of algebra that was closer
to a traditional view, in that symbol
manipulation was more prominent anti
the problems were more often from the
world of mathematics. The emphasis
was not on the manipulation, however,
but rather on the ways of thinking that
emerge from consideration of the
historical (levelopment of the subject.
Many of his points were illustrated by
engaging mathematical problems.
Cuoco prefaced his presentation with
an acknowledgment that the points
made by Fey were important, and then
offered a list of possible answers to the
question, ' What is algebra?"
· Algebra is an area of mathematical
research.
· Algebra is the language of mathematics.
.
Algebra is a collection of skills.
· Algebra is generalize(1 arithmetic.
· Algebra is about "structure."
· Algebra is about functions.
· Algebra is about graphs.
· Algebra is about mo(leling.
· Algebra is a tool.
Historically, algebra grew out of a long
program of mathematical research that
looked for ways to solve equationsways
that (li(ln't (lepen(1 upon the particulars of
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the equations. Sometimes the methods
for solving these equations worked in
situations that had nothing to do with the
originalsituation. People started thinking
about properties of operations.
Algebraic thinking involves reasoning
about calculations. When is the aver
age of two averages the average of the
whole lot? What is the sum, ~ + 2 + 4 +
~ + ... + 28? Which numbers can be
written as sums of a sequence of
consecutive whole numbers? In Euclid
ean division, why do the remainders
have to keep getting smaller?
Algebra involves reasoning about
operations. Consider the following two
problems:
1. Mary drives from Boston to
Washington, a trip of 500 miles. If she
travels at an average of 60 MPH on the
way clown and 50 MPH on the way back,
how many hours floes her trip take?
2. Mary drives from Boston to
Washington, and she travels at an
average of 60 MPH on the way clown and
50 MPH on the way back. If the total trip
takes 18 1/3 hours, how far is Boston
from Washington?
The first problem can be solved by a
sequence of direct computations. The
second requires reasoning about opera
tions. Students can solve it using a
"function machine" computer program
that allows reasoning about operations.
Triangular numbers have the recur
sion formula, a (n) = a (n  1) + n. While
THE NATURE AND IMPACT OF ALGEBRA
investigating these numbers, one
student saw that a (n) = n2  a (n  1) .
This is surprising enough. Cuoco wants
people to see that it is useful: Add the
two formulas to get the closed form.
Units digit arithmetic provides some
opportunities for reasoning about
calculations. What is the units digit of
24~3485 _ 75~332? Through questions
like this, students can very quickly
begin to reason about the system of
arithmetic module ten.
Some have argued that skins in symbol
manipulation are less important today, but
symbol manipulation is not just a skill.
Knowledge is not a copy of reality. To
know an object, to know an event, is not
simply to look at it and make a mental
copy or image of it. To know an object is
to act on it. To know is to mollify, to
transform the object, and to unclerstanc3
the process of this transformation, and as
a consequence to unclerstanc3 the way an
object is constructed. An operation is
thus the essence of knowledge; it is an
interiorizec3 action, which mollifies the
object of knowledge.
Cognitive development in children:
Development and learning (Piaget, 1964)
Symbol manipulation can also support
mathematical thinking. An(1 many
important mathematical ideas, such as
geometric series, the binomial theorem,
and the number theory behind
Pythagorean triples, require rather
sophisticated symbol manipulation.
OCR for page 139
During the question/answer period,
both Cuoco and Fey agreed on the
importance of algorithmic thinking an
reasoning about relationships. Fey
elaborated that the contexts provide a
series of hooks, but it is important to go
beyond the individual contexts to find the
commonalities among the representations
of various contexts. Hyman Bass sug
geste(1 that Fey's an(1 Cuoco's approaches
are not in opposition, but emphasize two
different aspects of the same thing.
RESEARCH ON THE TEACHING
AND LEARNING OF ALGEBRA
Grit Zaslavsky, a mathematics educa
tion researcher from Israel, began her
presentation by noting that in Israel
there is no debate about whether to
teach algebra. Because complete
coverage of the research was not pos
sible, she decided to share an example
of her own mathematical learning,
which developed into a collaborative
study of her graduate student, Hagit
Sela, and her colleague, Uri Leron.
Through this example she addressed
research issues associated with teach
ing and learning algebra.
She asked participants first to sketch
the graph off = x on axes without
scales, and then to sketch the graph of
fix) = x on axes where the scales were
different. For each case, she posed the
following questions:
1. What is the slope of the line you
sketched? How clic3 you determine it?
2. Does the line that you sketched
clivicle the angle between the two axes into
two congruent angles? How c30 you know?
3. Can you calculate tan fad, for the
angle a between the line you sketched
and the xaxis? If you were able to, how
clic3 you calculate it? If not, why not?
4. Describe your considerations,
reactions, dilemmas and other thoughts.
Typical graphs are shown in Figure 2.
For the first graph, most participants
Figure 2. Typical graphs
ACTION CONFERENCE
2
3
3
2
y
1 l
n
~3 2 1 ~ 1
~ 1
Ad/
1 Z
x
6
Y4
  1 ha

.
2
4
OCR for page 139
assumed that the scales were the same.
Zaslavsky pointed out that most people
think offal) = x as bisecting the right
angle of the coordinate system. None of
the four questions were problematic for
the first graph.
For the second graph, however, there
was disagreement, as some participants
focused on the scales of the axes and
others focused on the angle visually
made by the line in the first quadrant.
Zaslavsky pointed out that with the
advent of graphing technologies and the
possibilities for scale change, there was
a sense that everything was invariant
under scale changes. But she thought
that there were some things that were
varying the angle in particular.
This example raised many questions:
What is the slope of a linear function?
What is the relation between the slope
and angle? Is the slope a characteristic
of a Linear) function that is independent
of its representation? Or is it a charac
teristic of its graphical representation?
(Similar questions may be asked about
derivative.) Does it make sense to talk
about a function without reference to its
representation?
As part of a research project,
Zaslavsky and her colleagues were
investigating approaches to this set of
tasks in various populations. Students,
mathematics teachers, mathematics
educators, and mathematicians all
shared a sense of confusion and to
THE NATURE AND IMPACT OF ALGEBRA
some extent, inconsistency or disequi
librium. All felt a need to rethink, re
define, or reconstruct meaning for
what they hall thought of as fun(lamen
tal and rather elementary concepts:
slope, scale, anti angle.
In their research (as in the Action
Conference) there were several qualita
tively different approaches to tackling
the problem. Individual people brought
their own perspectives. For some, slope
is a geometric concept. When the scale
is 1:3, the line does not bisect the angle.
For others, slope is an analytic concept,
and the "fact" that the line bisects the
angle is a clear result of the analytic
calculation. Still others questione(1 the
meaning of y = x if the units of x are
ifferent from the units of y.
What (toes all this have to (lo with
learning algebra in the middle grades?
Learning is about constructing mean
ing. This meaning can change over
time, across learners (even experts) and
across contexts. The teacher shoul
provide a rich context for building
different perspectives and meanings.
In algebra, even the notion of variable
has several meanings: unknown, vary
ing, generalization, etc. Meaningful
learning takes place when the learner
(leals with a "real problem" in the sense
that the problem is real to him or her.
Research supports a contextual ap
proach in which students engage with
problems to which they can relate.
OCR for page 139
of trajectories for teachers' professional
growth (what is appropriate to learn at
different points in a teacher's career),
how to support communities of inquiry
for continuing professional growth.
5. We need to better understand
what the system needs to be like, in
order to take the best prepared novice
and support her/him.
6. We need to better understand what
makes rich teaming environments for
teachers (including standardsbased
curriculum materials), what they learn in
these environments, and how to scale up
this activity. This includes understanding
productive connections between re
search and professional development.
7. We need to develop research
tools and methodologies for examining
and describing complex settings like
classrooms.
8. We nee(1 to enlist teachers as
colleagues in developing knowledge
about these issues.
Student learning
1. We know a lot about student
thinking in some domains of the middle
grades curriculum. We need to expand
this knowledge to other content areas as
well as across content domains
2. Much of what we know about
student thinking has unstated premises
(e.g., social anti instructional conditions
are assumed) and we need to be ready
to question what we know.
ACTION CONFERENCE
3. We need to broaden the "we" in the
research community. We may learn that
there is more shared knowledge than
there is share(1 theoretical commitment.
4. We nee(1 to combine mo(le} buil(l
ing with mode} testing.
5. We need to better understand
whether and in what ways reform
curricula support the development of
mathematical power for all students,
particularly poor students, students of
color, and students with special needs.
6. We need to look at good teaching
as a context for student learning.
7. We nee(1 to learn more about
distributed learning communities, and
about the relation of individual thinking
to shared social practice.
Communicating with various
constituencies
Here we raised a number of issues.
1. Who are we doing research for?
2. What does it mean to do system
atic research? What is the role of
collaboration between teachers and
researchers?
3. Does our research address problems
that are widely recognized as significant?
Do we know the concerns that parents
have and how concerns change?
4. How do we package what we
believe is useful and compelling for others
(e.g., school boards, parents, commu
nity)? How do we present date thatis
credible and acceptable to the public?
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REFERENCES
Fennema, E., Sowder, J., & Carpenter, T. (in
press). Creating classrooms that promote
understanding. In T. Romberg and E.
Fennema, (Eds.), Teaching and learning
mathematics with understanding Hillsdale, NI:
Erlbaum.
Huntley, M.A., Rasmussen, C.L, Villarubi, R.S.,
Sangtong, J., & Fey, J.T. (in press). Effects of
Standardsbased mathematics education: A
study of the CorePlus Mathematics Project
Algebra/Functions strand. {oavrnalfor
Research in Mathematics Education.
Gravemeijer, K (1998~. Developmental research
as a research method. In A. Sierpinska and I.
Kilpatrick (Eds.), Mathematics education as a
research domain: A search for identity (An ICMI
Study Publication, Book 2 (pp. 277295~.
Dordrecht, The Netherlands: Kluwer Aca
demic Publishers.
Greenwald, A., Pratkanis, A., Lieppe, M., &
Baumgardner,M. (1986~. Under what
conditions does theory obstruct research
progress? Psychological Review, 93, 21~229.
Henningsen, M., & Stein, M.K (1997~. Math
ematical tasks and student cognition: Class
roombased factors that support and inhibit
highlevel mathematical thinking and reason
ing. {oavrnalfor Research in Mathematics
Education, 28~5), 524549.
Hiebert, I. (1998~. Aiming research toward
understanding: lessons we can learn from
children. In A. Sierpinska & I. Kilpatrick
(Eds.), Mathematics education as a research
domain: A search for identity. Dordrecht, The
Netherlands: Kluwer Academic Publishers.
Hiebert, I. (1999~. Relationships between
research and the NCTM Standards. {oavrnal
for Research in Mathematics Education.
Hoover, M., Zawojewski, J., & Ridgway, J. (1997~.
Effects of the Connected Mathematics Project
on student attainment. www.mth.msu.edu/
cmp/effect.html.
Lambdin, D., & Lappan, G. (1997, April).
Dilemmas and issues in curriculum reform:
Reflections from the Connected Mathematics
project. Paper presented at the Annual
Meeting of the American Educational Re
search Association in Chicago, IL.
National Center for Educational Statistics. (1996~.
P?`rsaving excellence: A study of U.S. eighth
grade mathematics and science achievement in
international context. Washington, DC: Author.
National Center for Educational Statistics. (1997~.
P?`rsaving excellence: A Study of U.S. foavrth
grade mathematics and science achievement in
international context. Washington, DC: Author.
Reese, C.M., Miller, KE., Mazzeo, I., & Dossey,
I.A. (1997) NAEP 1996 mathematics report
cardfor the nation and the states: Findingsirom
the NationalAssessment of Educational
Progress. Washington, DC: National Center for
Education Statistics.
Sowder, I.T., Philipp, RA., Armstrong, B.E., &
Schappelle, B. (1998~. Middle grades teachers'
mathematical knowledge and its relationship to
instruction. Albany,NY: SUNY Press.
Sowder, J.T., & Philipp, R (in press) The role of
the teacher in promoting learning in middle
school mathematics. In T. Romberg & E.
Fennema (Eds.), Teaching and learning
mathematics with understanding Hillsdale, NI:
Erlbaum.
Stein, M.K, Grover, B.W., & Henningsen, M.
(1996~. Building student capacity for math
ematical thinking and reasoning: An analysis
of mathematical tasks used in reform class
rooms. American Educational Research
{oavrnal, 33~2), 455488.
Stein, M.K, & Lane, S. (1996~. Instructional
tasks and the development of student capacity
to think and reason: An analysis of the
relationship between teaching and learning in
a reform mathematics project. Educational
Research and Evaluation, 2~1), 5080.
Stigler, I.W., & Hiebert, I. (1999~. The teaching
gap. New York: Free Press.
Thompson, P. W. (1995~. Notation, convention,
and quantity in elementary mathematics. In
Sowder, J.T., & Schappelle, B.P. (Eds.~.
Providing a foundation for teaching mathemat
ics in the middle grades (pp. 199219~. Albany,
NY: SUNY Press.
RESEARCH IN THE TEACHING AND LEARNING OF MATHEMATICS
OCR for page 139
Synthesis by Megan Loef Franke, University of California, Los Angeles and
Deborah Ball, University of Michigan
National attention to teaching, higher
student achievement, and the need for
more and betterqualified teachers is on
the rise. Professional development and
teacher education are of increasing
interest and concern.
All the concern for professional
development is occurring at a time
when views on teacher learning con
tinue to evolve. The change that teach
ers are being asked to make in order to
enact standardsbased reforms are
ambitious and complex (Little, 1993;
Cohen and Hill, 19981. Little points out
that the current reforms require teach
ers "to (liscover anti (1evelop practices
that embody central values and prin
ciples." Here teachers are seen as
learners, "teaching and learning are
inter(lepen(lent, not separate functions...
[teachers] are problem posers and
problemsolvers; they are researchers,
and they are intellectuals engaged in
unraveling the process both for them
seives anti for [their stu(lents]"
(Lieberman & Miller, 19901.
The Action Conference on Profes
sional Development was (lesigne(1 to
afford an opportunity to examine
promising approaches to professional
levelopment.
The premise was that extant knowI
edge about professional development is
underdeveloped. Ideology and belief all
too often dominate practice and policy.
The Conference was intended to create
an analytic and practical conversation
about the sorts of opportunities in
professional development most likely to
lea(1 to teachers' learning anti improve
ments in their practice. With a focus on
mathematics at the mi(l(lle gra(les, the
structure of the Action Conference was
groun(le(1 in analysis of the practice of
teaching middle grades mathematics,
considering the major tasks teachers
OCR for page 139
face and what knowledge and skill it
takes to perform those tasks. This
analysis of teaching practice was used to
take a fresh look at the kinds of opportu
nities for learning that teachers need.
What does teaching cIemanc! of
teachers arc! what cIoes this
imply for teacher learning?
Deborah Ball, as conference leader,
framed the workshop discussion with
the diagram (Figure 11. Ball pointed out
that the focus of the discussion through
out the workshop would be from the
vantage point of teacher educators, and
would consider sites through which
teachers might most profitably learn
mathematics content needed in teach
ing, based on tasks which teachers
regularly do as part of their teaching.
Participants observed that the diagram
helps make clear that the content to
learn becomes more complex at each
level. Schoolchildren are learning
mathematics. Teachers are learning
about mathematics, but also about
children's learning of that mathematics,
and about the teaching of that math
ematics. The diagram does not incorpo
rate all elements of what would be
needed in a national infrastructure for
systemic change. It depicts relations
among learning mathematics, learning
to teach mathematics, learning to teach
about the teaching of mathematics.
Ball in(licate(1 that mathematicians,
mathematics education researchers and
teacher educators have all created lists
specifying knowle(lge for teachers.
Although these lists enable profession
als in the field to discuss content in
professional development, these lists
are not grounded in an analysis of the
work of teaching. Such lists tend not to
consider questions about the math
ematical content knowledge that is
necessary in the context of teaching or
the knowledge of mathematics required
when teaching extends beyond how to
a(l(1 fractions or identify geometric
patterns. It includes being able to frame
a mathematically strategic question,
come up with the right example, con
struct an equivalent problem, or under
stan(1 a chil(l's nonstan(lar(1 solution.
The kind of mathematical knowledge it
Figure 1. Teaching, Learning, and Learning to Teach
Mathematics
PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATH EMATICS
\
OCR for page 139
takes to teach a problem goes beyond
knowing the content that the students
are learning, and includes the capacity
to know and use mathematics in the
course of teaching.
Participants considered the implica
tions of this perspective on mathemati
calknowledge. In the practice of
teaching, the teacher's mathematical
knowledge is called upon in many
different ways in figuring out what a
student means, in listening to what
students say, in choosing and adapting
mathematical tasks, and in knowing
whether the students are understand
ing. How do teachers use their knowI
edge of the content, their knowledge of
pedagogy anti other knowle(lge in an
interaction with their students? How do
they decide which students' ideas to
pursue? How do they decide which
paths will further the particular content
in which they are engaged? How do
teachers decide what a student means
by a question or an answer, or judge
whether students actually understand
the concept in question? It is in these
interactions that knowledge of the
content is critical.
Ball, together with Joan Ferrini
Mun(ly, Center for Science, Mathemat
ics and Engineering Education at the
National Research Council, led an
activity designed to engage participants
in considering mathematics knowledge
as it is used in one task of teaching. Ball
ACTION CONFERENCE
presented a mathematics problem and
asked participants to solve the problem
and to describe the mathematical
territories into which it might head.
I have some pennies, nickels, and climes
in my pocket. I put three of the coins in
my hand. How much money conic I have?
(NCI~M, 1989)
Having analyze(1 the mathematical
territory of the problem, participants
were then asked to make a "downsized
or unsized version" that was mathemati
cally similar in structure. The partici
pants considered what students might
say or do. Was making the problem
more complicated numerically a means
of "upsizing"? What (li(1 it mean to make
a "similar" problem, anti what sorts of
mathematical knowle(lge anti reasoning
did it take to do this? Participants
discussed different versions of the
problem and considered the mathemat
ics they used to create and evaluate
them.
What cIo we know about
professional cIeve~opment,
teacher learning, arc! the
improvement of practice?
Although there is a lot of professional
development in the U.S., much of it is
ineffective. Approaches to professional
development in the U.S. are often
fragmented and incoherent, with little
basis in the curriculum teachers will
OCR for page 139
have to teach. Much professional
development centers around features
such as manipulatives or cooperative
groups rather than on the substance of
improving students' mathematics
learning. Generally, evidence about the
effects of professional development on
student achievement is scattered and
thin. Too often, professional develop
ment is defined by belief and propa
gated by enthusiasm. Some profes
sional development works, but there is a
large gap even between effective profes
sional development and changes in
practice.
Often teachers view the focus of
mathematics professional development
as engaging in process: learning how to
be a mathematics teacher or figuring
out what the process of teaching should
look like. Yet as teachers identify
process goals, the substance of the
process is left unchallenged. Teachers
need opportunities to think about how
the processes in which they engage
serve the students' [earning as well as
their own, how the processes relate to
the mathematical ideas, student think
ing, and mathematical discourse.
As one example, Ball cited the Cohen
and Hill (1998) study in California that
found that professional development
that was grounded in the curriculum
that teachers had to teach that is,
provided teachers with opportunities to
learn about the content, to learn how
children think about that same content,
and to learn ways to represent that
content in teaching made a significant
difference in student scores. Their
findings indicated that professional
development that made a difference
provided opportunities to learn that
· grounded content in the student
curriculum;
· were about students' thinking about
that content;
were about ways to connect students
and content;
· and were situated in context, materi
als, and sites of practice.
.
OPPORTUNITY TO LEARN
SITUATED IN SITES OF PRACTICE
The Action Conference explored the
promise and potential problems of
situating opportunities for teachers'
learning in practice. Much teacher
professional development offers teachers
the opportunity to learn a new idea or
activity and transfer that idea to their
classroom. Professional development
focused on helping teachers develop
expertise within the context of their
practice emphasizes the interrelationship
of ideas and practice. Some professional
developers engage teachers with cases of
teachers engaged in teaching mathemat
ics and discuss what the teacher did and
PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATH EMATICS
OCR for page 139
why. Some professional developers
engage teachers in the curriculum that
they are going to teach and use that as
the practice context, while others engage
teachers with video examples of cIass
room mathematics practice, or some with
student work. Each of these situates
opportunities for teacher learning in
practice. In each case the teacher is
pushed to consider how the mathemati
cal ideas play out in the context of
teaching mathematics, and they are
competed to consider the details related
to the students' thinking, the nature of
the mathematics and its relation to their
instruction. Each of these approaches
affords opportunities forIearning. Each
holds pitfalls that might impede such
learning. Critical questions then are:
How might the potential affordances be
exploited? How might the potential
pitfalls be managed? To learn more about
what people are trying and how it is
working, anti to sharpen the ability to
examine their own designs and practices
as professional developers, participants
engage(1 in several examples of profes
sional development projects.
WRITTEN CASES
Margaret Smith, Pennsylvania State
University, engage(1 the participants in a
case study of student work The Case of
Ed Taylor; Smith, Henningsen, Silver, &
ACTION CONFERENCE
Stein, 1998) and worked through the S
Pattern Task (Figure 21. Analyzing
student responses le(1 the participants to
discuss the importance of the mathemati
cal knowledge of the teacher in analyzing
the mathematics students do. The partici
pants offered the foLow~ng potential
affordances for professional developers
using cases as a medium for learning.
Cases offer the possibilities of
metacognition about the mathematics,
provi(le an in(lepth look at the mathemat
ics because the student thinking in the
case is visible; provide the opportunity to
study the particular mathematics involved
in the case in context, provide an opportu
nity to read mathematics, can be used in
many ways with different groups including
administrators, and create opportunities to
learn mathematics in teaching. Potential
pitfalls included the challenge of asking
questions that focus on the mathematics,
focusing more on pedagogy than the
mathematics, making the best use of lime,
hard to read mathematics, and the fact the
teachers may not know the mathematics.
C U RRIC U LUMMATE RIALS
Karen Economopoulos, TERC, en
gaged participants in an experience
designed to show how curriculum
materials might serve as a site of practice
where teachers might learn mathematics
in their work. Economopoulos posed two
OCR for page 139
Figure 2. The S Paltern Task
·~
·~.·~.
·~. ·~.·~.
·. ·. ·~.·~.
· ·. ·~.·~.
·. ·~. ·~.·~.
a. Sketch the next two figures in the S Pattern.
b. Make observations about the pattern that help describe larger figure numbers in the pattern.
c. Sketch ancl describe two figure numbers in the pattern that were larger than 20.
cl. Describe a method for finding the total number of tiles in the larger figure numbers.
e. Write a generalized clescription/formula to find the total number of tiles in any figure in the pattern.
questions for rejection and discussion:
How might curriculum materials such as
these offer professional development
opportunities for leachers? How might
these materials influence or support a
teacher's daily decisions? Potential
affordances to exploit suggested by the
participants included: The material
speaks directly to the teacher; the
mathematics is very near the surface,
new materials generate thinking and
learning. It is possible for administrators
to see more depth in the mathematics.
Materials can provide the opportunity to
see a broader view of mathematics.
Materials can move the mathematics
preferred by the teacher forward. Conti
nuity is built into the lessons; the com
mon material promotes communication
among the teachers. The pitfalls to be
managed included the forgoing: It is
easy to turn the search for the mathemat
ics into a makean (ltake situ ation (levoi
of thinking. If teachers (lo not un(ler
stand the content, using the materials to
focus on the mathematics might increase
frustrationIevels. Relearning familiar
materials might result in resistance. In
some tasks, it might be easy to avoid the
mathematics.
VIDEOTAPES OF TEACHING AND
LEARNING
Nanette Seago, Mathematics Renais
sance Project, engage(1 participants in
examining a videotape of a professional
(levelopment workshop, "Facilitating the
Cindy Lesson Tape." The teachers had
previously viewe(1 a videotape of an
eighth gra(le math lesson. The tape
PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATH EMATICS
OCR for page 139
provided an opportunity to consider the
challenges of using videotape as an
instructional medium in professional
development. Watching this video
focused on the task of "facilitation" of a
teacher group discussion and raised
explicit issues about the challenges of
keeping the professional development
work centered on specific learning
goals teachers' learning of mathemat
ics, for example. There was considerable
interest expressed in the potential of
videotape as a medium and discussion
about its role in helping teachers think
about their practice. Seago encouraged
the viewers to think back to the framing
diagram of the conference and the use of
video in the context of teachers as
learners. Affordances cited included the
opportunity to look at a shared instance
of teaching, to see mathematics in use by
teachers and students. Some of the
pitfalls included the problem of keeping
teachers focused on particular content,
questions about the quality of the teach
ing portrayed, anti the (difficulty of
concentrating on the mathematics in the
midst of the many other things to watch.
REFLECTIONS ON THE
IMPROVEMENT OF
PROFESSIONAL DEVELOPMENT
A panel, moderated by Mark Saul, from
BronxviDe Public Schools, consisting of
ACTION CONFERENCE
Iris Weiss, Horizon Research; Stephanie
Williamson, Louisiana Systemic Initiative;
and John Moyer, from Marquette Univer
sity, reacted to the issues raised at the
Conference through the lens of their
experiences. Weiss, from the perspective
of large scale research about implementa
tion, argued for the need to help teachers
develop some way to filter and make
decisions. She also raised a concern
about the issue of scale; how to move
from a teacher and a classroom to the
system, and in the process, the need for a
central vision an(1 coordination an(1 for
some form of quality control. Weiss
in(licate(1 her belief that the issue is a
design problem. Professional develop
ment mo(lels (lesigne(1 for a best case or
for small numbers (lo not work when
scaled up. Moyer, resecting on profes
sional development done with urban,
large city mi(l(lle gra(les teachers, in(li
cated that scaling up had to be done
creatively and in small steps. Leaders
who had been developed from earlier
small projects became facilitators in the
larger one but also remained as part of the
ca(lre of learners. He observe(1 that one
of the most successful efforts was to
observe teachers as they taught, with the
observer writing down what the teacher
sai(l. When the observer later aske(1 why
the teacher ha(1 use(1 those wor(ls, the
teachers began the process of resection
on their practice that le(1 to some lasting
changes. Williamson (lescribe(1 the work
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done in the state of Louisiana through the
National Science Foundation funded
ment leaders to engage in the study of
professional development. As a
LASIP systemic initiative and the focus on consequence, professional developers
leacher consent knowledge. Williamson
noted that the guidelines for the program
require collaboration among school
systems and universities and stimulated
partnerships among K12 and higher
education. An outgrowth of the programs
has been the development of a "Core of
Essential Mathematics for Grades K~"
that focuses on growth of important ideas
across grade levels and is now guiding
decisions for both professional develop
ment and preservice programs about
what teachers need to know and be able
to do to teach these concepts in depth.
At the close of the panel, Nora Sabelli
from the National Science Foundation
discussed the new opportunities for
research on middle grades mathematics
in the Department of Education/
National Science Foundation Research
Initiative. She indicated that the focus
of the initiative will be on longterm
agendas, strategic plans for implementa
tion, and ensuring that there is the
human, methodological and institutional
capacity for converting schools into
learning communities.
SUMMARY
Participants discussed the lack of
must use their work with teachers as
sites for ongoing learning and re
search as they create opportunities for
teachers to learn working to ensure
that those opportunities impact teach
ers' knowledge and practice as well as
student achievement. Professional
developers are expected to work with
large numbers of teachers and do so
quickly. Under these conditions,
professional developers have the
opportunities to see themselves as
learners. The Action Conference did,
however, take seriously professional
development as a field and attempted
to create a frame for thinking about
theoretical, research, and practice
based learning.
Most professional development
providers are convince(1 that the
approaches they take enable teachers
to learn and students to benefit; other
wise, they would not pursue the ap
proaches. However, little is known
about what various approaches afford
or do not afford, especially in relation
to classroom practice and student
achievement. Little is known about the
details of the various approaches to
professional (levelopment. The mes
sage as the field considers the issues
raised at the Conference is to reflect on
opportunities for professional develop the circle of learners and on their
PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATH EMATICS
OCR for page 139
relation to middle grades mathematics
and to each other, to design research
studies around professional develop
ment approaches, and to think deeply
about the mathematics middle grades
teachers need to know to teach well
and how they can come to know that
mathematics for themselves.
REFERENCES
Cohen, D.K, & Hill, H. (1998, January). State
policy and classroom performance: Mathemat
ics reform in California. CPRE Policy Briefs.
Philadelphia, PA: University of Pennsylvania.
Lieberman, A, & Miller, L. (1990~. Teacher
development in professional practice schools.
Teachers College Record, 92~1), 105122.
Little, I.W. (1993~. Teachers' professional
development in a climate of educational
reform. Educational Evaluation and Policy
Analysis, 1562}, 129151.
ACTION CONFERENCE
National Council of Teachers of Mathematics.
(1989~. Cavrric?`l?`m and evaluation standards
for school mathematics. Reston, VA: Author.
Smith, M., Henningsen, M., Silver, E.A, & Stein,
M.K (1998) The case of Ed Taylor. Unpub
lished document, University of Pittsburgh.
PARTICIPANT BACKGROUND
MATERIAL
Ball, D.L (1997~. Developing mathematics
reform: What don't we know about teacher
learning but would make good working
hypotheses. In Friel, S., & Bright, G. (Eds.),
Reflecting on our work. NSF Teacher Enhance
ment in K6 Mathematics (pp. 79111~.
Lanham, MD: University Press of America.
Good Teaching Matters (1998~. Thinking K16.
Vol 3, Issue 2. Education Trust.
Rhine, S. (1998, JuneJuly). The role of research
and teachers' knowledge base in professional
development. Research News and Comment
(pp. 2731~.