Transcript Helping All Children Become Proficient in Mathematics

Seeking Common
Ground
Jeremy Kilpatrick
University of Georgia
NCSSM Precalculus
“Teaching and Learning Cross-Country
Mathematics: A Story of Innovation in
Precalculus,” by J. Kilpatrick, L.
Hancock, D. S. Mewborn, & L. Stallings
In S. A. Raizen & E. D. Britton (Eds.),
Bold Ventures, Vol. 3: Case Studies of
U.S. Innovation in Mathematics
Education. Dordrecht, the Netherlands:
Kluwer, 1996
Outline
 What’s
the fuss about?
 Why seek common ground?
 What common ground?
 What complaints?
 What’s next?
 Where are the teachers?
What’s the fuss
about?
The New Math
Benjamin DeMott, “The Math
Wars.”
In Hells and Benefits: A
Report on American Minds,
Matters, and Possibilities.
New York: Basic Books,
1962.
Gurganus’s author’s note: A word to the
reader about historical accuracy
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1930s Federal Writers’ Project
found that many former slaves
recalled seeing Lincoln in the
South during the Civil War
Fanny Burdock (91): “We been
picking in the field when my
brother he point to the road and
then we seen Marse Abe
coming all dusty and on foot. . .
He so tall, black eyes so sad.
Didn’t say not one word, just
looked hard at us, every one us
crying. We give him nice cool
water from the dipper. . . .
Gurganus’s author’s note: A word to the
reader about historical accuracy
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“After, didn’t our owner or
nobody credit it, but me and all
my kin, we knowed. I still got
the dipper to prove it.”
In reality, Lincoln’s foot tour of
Georgia could not have
happened, but such scenes
were told by hundreds of slaves
“Such visitations remain, for me,
truer than fact”
The South is a realm where fact
and fable are both true
California Dreaming: Reforming
Mathematics Education
Suzanne Wilson
Why the new math reforms “failed”:
Weak mathematical knowledge of
leaders: “Not everyone was a
mathematician, and some of the
mathematicians . . . were not highly
respected” (p. 14)
 Misguided reforms: “The mathematics
was inappropriate . . . the wrong
mathematicians were involved” (p. 16)
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New Math Mathematicians
David Blackwell
 Robert Dilworth
 Mary Dolciani
 Andrew Gleason
 John Kelley
 Edwin Moise
 Peter Hilton
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Henry Pollak
 George Pólya
 Mina Rees
 Norman Steenrod
 Marshall Stone
 Albert Tucker
 Gail Young
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Andrew Gleason
Edwin Moise
Lipman Bers
George Pólya
Marshall Stone
Max Schiffer
Paul Rosenbloom
Henry Pollak
E. G. Begle
R. C. Buck
Robert Dilworth
*Participated in at least
one project
Lars Ahlfors
Garrett Birkhoff
Marston Morse
Richard Bellman
Mina Rees
New Math Reformer*
Morris Kline
André Weil
Critic*
*Signed “On the Mathematics Curriculum of the
High School,” Amer. Math. Monthly 69 (1962),
189-193: Math. Teacher 55 (1962), 191-195.
Math wars then and now
New math era
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Mathematicians push for
reform
Gulf between school and
university mathematics;
political and military
competitiveness
Opposed by teachers, parents,
and some mathematicians
Emphasis on content—abstract
structures—presented logically
and formally
Standards era
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Teachers (NCTM) push for
reform
Gulf between U.S. and
international performance;
economic and technological
competitiveness
Opposed by mathematicians,
parents, some teachers, and
policy makers
Emphasis on pedagogy—active
learning—with meaningful
content and investigations
Standards-Based Reform
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Termed “whole math,” like “whole
language”
Termed “new-new math,” like “new
math”
Groups of parents and
mathematicians formed
Lynne Chaney
June 1997
Kids are writing about “What We Can Do to Save the
Earth,” and inventing their own strategies for
multiplying. They’re learning that getting the right
answer to a math problem can be much less important
than having a good rationale for a wrong one.
Sometimes called “whole math” or “fuzzy math,” this
latest project of the nation’s colleges of education has
some formidable opponents. In California, where the
school system embraced whole math in 1992, parents
and dissident teachers have set up a World Wide Web
site called Mathematically Correct to point out the
follies of whole-math instruction.
http://ourworld.compuserve.com/
homepages/mathman/index.htm
Controversy
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New rhetoric: “Fuzzy math” “Parrot
math”
Stories of students not learning basic
facts
January 1998: Richard Riley,
U.S. Secretary of Education,
calls for a cease fire
in the “math wars”
Why seek common
ground?
Richard Schaar
Texas Instruments
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Managed TI calculator business since 1986, marketing
graphing calculators for mathematics education along with
the needed support programs for teachers
Frustrated over the lack of progress in K-12 mathematics
education
Worked with other Texans on an initiative under the
auspices of the Business Roundtable to help move the
states forward in improving mathematics education
Saw the math wars as a major stumbling block to progress
After talking with Jim Milgram (Stanford mathematician),
decided to convene a small group of people to find a middle
ground in the conflict
Got support from NSF and then MAA
“Peace commission”
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Richard Schaar, Texas Instruments, convener
Deborah Ball, University of Michigan
Joan Ferrini-Mundy, Michigan State University
Jeremy Kilpatrick, University of Georgia
James Milgram, Stanford University
Wilfried Schmid, Harvard University
What common
ground?
Article by Michael Pearson in
Aug./Sept. MAA FOCUS
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The MAA hopes to help encourage and facilitate
constructive discourse between mathematicians
and mathematics educators to seek common
ground in efforts to improve K-12 mathematics
teaching and learning
Success of two pilot meetings:
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At NSF in December 2004
At the MAA offices in June 2005
Document can serve as starting point for future
conversations
See http://www.maa.org/common-ground/
or Notices of the AMS, October 2005
Article by Michael Pearson in
Aug./Sept. FOCUS
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All students must have solid grounding in
mathematics to function effectively in today’s
world
Premises:
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Basic skills with numbers continue to be vitally
important for a variety of everyday uses
Mathematics requires careful reasoning about precisely
defined objects and concepts
Students must be able to formulate and solve problems
Areas of agreement: automatic recall of basic
facts, use of calculators in lower grades, learning
algorithms, fractions, teaching mathematics
in “real world” contexts, instructional
methods, teacher knowledge
Seeking Common Ground
A process:
 People working together
 Listening thoughtfully
 Valuing others’ opinions
 Taking time
 Agreeing on language
 Working hard toward a common
goal
What complaints?
K-12 Mathematics Education: How
Much Common Ground Is There?
Anthony Ralston
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A valuable exercise, with results
unexceptional to almost all FOCUS
readers, but fraught with difficulties:
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Blandness
Ambiguity
Disagreement in community—curriculum and
technology
Before attempt consensus, need a level of
respect in both communities
K-12 Mathematics Education: How
Much Common Ground Is There?
Anthony Ralston
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Ambiguity
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“Certain procedures and algorithms in
mathematics are so basic and have such wide
application that they should be practiced to the
point of automaticity”
“Calculators can have a useful role even in the
lower grades, but they must be used carefully,
so as not to impede the acquisition of fluency
with basic facts and computational procedures”
K-12 Mathematics Education: How
Much Common Ground Is There?
Anthony Ralston
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Disagreement in community
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“By the time they leave high school, a
majority of students should have studied
calculus”
“Students should be able to use the basic
algorithms of whole number arithmetic
fluently, and they should understand how and
why the algorithms work”
“The arithmetic of fractions is important as a
foundation for algebra”
“By the time they leave high school, a
majority of students should have
studied calculus”
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Although some should, and already do, take a full course in
calculus, most students should learn at least certain
fundamental ideas of calculus, such as rate of change, limit,
and derivative
Some 70% of the countries in TIMSS cover the topics of
elementary analysis (infinite processes and change) at
grade 12, and many address these topics in grades 9
through 11
A project involving incentives and district-wide commitment
in ten inner-city Dallas high schools has resulted in a ninefold increase to 330 out of 4161 graduates from 1995 to
2005 in the number of students receiving a score of three
or better on the AB Calculus Advanced Placement exam.
What’s next?
March meeting in Indianapolis
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Probability and data analysis in the elementary
curriculum
Algorithms in the curriculum
Technology in general
Calculus in high school
Algebra for all
Gap between policy (high standards) and teacher
beliefs and capacity
How can international studies and information be
used?
How should we weigh class size versus teacher
knowledge and capabilities?
Where are the
teachers?
Mathematics Teachers
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Are they in this fight?
What might they add to the
conversation?
Calculus as goal
 Role of definitions
 Applications as motivation
 Curriculum structure
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