Seeking Common Seeking Common GroundGround

Jeremy KilpatrickJeremy KilpatrickUniversity of GeorgiaUniversity of Georgia

NCSSM PrecalculusNCSSM 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

OutlineOutline

What’s the fuss about?What’s the fuss about? Why seek common ground?Why seek common ground? What common ground?What common ground? What complaints?What complaints? What’s next?What’s next? Where are the teachers?Where are the teachers?

What’s the fuss What’s the fuss about?about?

The New MathThe New MathBenjamin 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 Gurganus’s author’s note: A word to the reader about historical to the reader about historical

accuracyaccuracy 1930s Federal Writers’ Project 1930s Federal Writers’ Project

found that many former slaves found that many former slaves recalled seeing Lincoln in the recalled seeing Lincoln in the South during the Civil WarSouth during the Civil War

Fanny Burdock (91): “We been Fanny Burdock (91): “We been picking in the field when my picking in the field when my brother he point to the road brother he point to the road and then we seen Marse Abe and then we seen Marse Abe coming all dusty and on foot. . coming all dusty and on foot. . . .

He so tall, black eyes so sad. He so tall, black eyes so sad. Didn’t say not one word, just Didn’t say not one word, just looked hard at us, every one looked hard at us, every one us crying. We give him nice us crying. We give him nice cool water from the cool water from the dipper. . . . dipper. . . .

Gurganus’s author’s note: A word Gurganus’s author’s note: A word to the reader about historical to the reader about historical

accuracyaccuracy ““After, didn’t our owner or After, didn’t our owner or

nobody credit it, but me and nobody credit it, but me and all my kin, we knowed. I still all my kin, we knowed. I still got the dipper to prove it.”got the dipper to prove it.”

In reality, Lincoln’s foot tour In reality, Lincoln’s foot tour of Georgia could not have of Georgia could not have happened, but such scenes happened, but such scenes were told by hundreds of were told by hundreds of slavesslaves

““Such visitations remain, for Such visitations remain, for me, truer than fact”me, truer than fact”

The South is a realm where The South is a realm where fact and fable are both truefact and fable are both true

California Dreaming: Reforming California Dreaming: Reforming Mathematics EducationMathematics Education

Suzanne WilsonSuzanne Wilson

Why the new math reforms “failed”:Why the new math reforms “failed”: Weak mathematical knowledge of Weak mathematical knowledge of

leaders: “Not everyone was a leaders: “Not everyone was a mathematician, and some of the mathematician, and some of the mathematicians . . . were not highly mathematicians . . . were not highly respected” (p. 14)respected” (p. 14)

Misguided reforms: “The mathematics Misguided reforms: “The mathematics was inappropriate . . . the wrong was inappropriate . . . the wrong mathematicians were involved” (p. 16)mathematicians were involved” (p. 16)

New Math New Math MathematiciansMathematicians

David BlackwellDavid Blackwell Robert DilworthRobert Dilworth Mary DolcianiMary Dolciani Andrew GleasonAndrew Gleason John KelleyJohn Kelley Edwin MoiseEdwin Moise Peter HiltonPeter Hilton

Henry PollakHenry Pollak George PólyaGeorge Pólya Mina ReesMina Rees Norman SteenrodNorman Steenrod Marshall StoneMarshall Stone Albert TuckerAlbert Tucker Gail YoungGail Young

New Math New Math Reformer*Reformer*

*Participated in at *Participated in at least one projectleast one project

Critic*Critic*

*Signed “On the Mathematics Curriculum *Signed “On the Mathematics Curriculum of the High School,” of the High School,” Amer. Math. Monthly Amer. Math. Monthly 6969 (1962), 189-193: (1962), 189-193: Math. Teacher 55Math. Teacher 55 (1962), 191-195.(1962), 191-195.

George George PólyaPólya

Henry PollakHenry Pollak

Lipman BersLipman BersAndrew GleasonAndrew Gleason

Edwin MoiseEdwin Moise

Robert Robert DilworthDilworth

Max Max SchifferSchiffer

Lars Lars AhlforsAhlfors

Mina ReesMina Rees André WeilAndré Weil

Morris KlineMorris Kline

Marshall StoneMarshall Stone

R. C. BuckR. C. BuckRichard Richard BellmanBellman

E. G. BegleE. G. Begle

Garrett Garrett BirkhoffBirkhoff

Paul Paul RosenbloomRosenbloom

Marston Marston MorseMorse

Math wars then and nowMath wars then and now

New math eraNew math era Mathematicians push for Mathematicians push for

reformreform Gulf between school and Gulf between school and

university mathematics; university mathematics; political and military political and military competitivenesscompetitiveness

Opposed by teachers, Opposed by teachers, parents, and some parents, and some mathematiciansmathematicians

Emphasis on content—Emphasis on content—abstract structures—abstract structures—presented logically and presented logically and formallyformally

Standards eraStandards era Teachers (NCTM) push for Teachers (NCTM) push for

reformreform Gulf between U.S. and Gulf between U.S. and

international performance; international performance; economic and technological economic and technological competitivenesscompetitiveness

Opposed by mathematicians, Opposed by mathematicians, parents, some teachers, and parents, some teachers, and policy makerspolicy makers

Emphasis on pedagogy—Emphasis on pedagogy—active learning—with active learning—with meaningful content and meaningful content and investigationsinvestigations

Standards-Based ReformStandards-Based Reform

Termed “whole math,” like “whole Termed “whole math,” like “whole language”language”

Termed “new-new math,” like Termed “new-new math,” like “new math”“new math”

Groups of parents and Groups of parents and mathematicians formedmathematicians formed

Lynne Chaney Lynne Chaney June 1997June 1997

Kids are writing about “What We Can Do to Save the Kids are writing about “What We Can Do to Save the Earth,” and inventing their own strategies for Earth,” and inventing their own strategies for multiplying. They’re learning that getting the right multiplying. They’re learning that getting the right answer to a math problem can be much less important answer to a math problem can be much less important than having a good rationale for a wrong one. than having a good rationale for a wrong one. Sometimes called “whole math” or “fuzzy math,” this Sometimes called “whole math” or “fuzzy math,” this latest project of the nation’s colleges of education has latest project of the nation’s colleges of education has some formidable opponents. In California, where the some formidable opponents. In California, where the school system embraced whole math in 1992, parents school system embraced whole math in 1992, parents and dissident teachers have set up a World Wide Web and dissident teachers have set up a World Wide Web site called Mathematically Correct to point out the follies site called Mathematically Correct to point out the follies of whole-math instruction.of whole-math instruction.

http://ourworld.compuserve.com/ http://ourworld.compuserve.com/ homepages/mathman/index.htmhomepages/mathman/index.htm

ControversyControversy

New rhetoric: “Fuzzy math” “Parrot New rhetoric: “Fuzzy math” “Parrot math”math”

Stories of students not learning basic Stories of students not learning basic factsfacts

January 1998:January 1998: Richard Riley, Richard Riley, U.S. Secretary of Education, U.S. Secretary of Education, calls for a cease fire calls for a cease fire in the “math wars” in the “math wars”

Why seek common Why seek common ground?ground?

Richard SchaarRichard SchaarTexas InstrumentsTexas Instruments

Managed TI calculator business since 1986, marketing Managed TI calculator business since 1986, marketing graphing calculators for mathematics education along with graphing calculators for mathematics education along with the needed support programs for teachersthe needed support programs for teachers

Frustrated over the lack of progress in K-12 mathematics Frustrated over the lack of progress in K-12 mathematics educationeducation

Worked with other Texans on an initiative under the Worked with other Texans on an initiative under the auspices of the Business Roundtable to help move the auspices of the Business Roundtable to help move the states forward in improving mathematics educationstates forward in improving mathematics education

Saw the math wars as a major stumbling block to progressSaw the math wars as a major stumbling block to progress After talking with Jim Milgram (Stanford mathematician), After talking with Jim Milgram (Stanford mathematician),

decided to convene a small group of people to find a middle decided to convene a small group of people to find a middle ground in the conflictground in the conflict

Got support from NSF and then MAAGot support from NSF and then MAA

““Peace commission”Peace commission” Richard Schaar, Texas Instruments, convenerRichard Schaar, Texas Instruments, convener Deborah Ball, University of MichiganDeborah Ball, University of Michigan Joan Ferrini-Mundy, Michigan State UniversityJoan Ferrini-Mundy, Michigan State University Jeremy Kilpatrick, University of GeorgiaJeremy Kilpatrick, University of Georgia James Milgram, Stanford UniversityJames Milgram, Stanford University Wilfried Schmid, Harvard UniversityWilfried Schmid, Harvard University

What common What common ground?ground?

Article by Michael Pearson in Article by Michael Pearson in Aug./Sept. MAA FOCUSAug./Sept. MAA FOCUS

The MAA hopes to help encourage and facilitate The MAA hopes to help encourage and facilitate constructive discourse between mathematicians constructive discourse between mathematicians and mathematics educators to seek common and mathematics educators to seek common ground in efforts to improve K-12 mathematics ground in efforts to improve K-12 mathematics teaching and learningteaching and learning

Success of two pilot meetings:Success of two pilot meetings: At NSF in December 2004At NSF in December 2004 At the MAA offices in June 2005At the MAA offices in June 2005

Document can serve as starting point for future Document can serve as starting point for future conversationsconversations

See See http://www.maa.org/common-ground/http://www.maa.org/common-ground/ or or Notices of the AMS,Notices of the AMS, October 2005 October 2005

Article by Michael Pearson in Article by Michael Pearson in Aug./Sept. FOCUSAug./Sept. FOCUS

All students must have solid grounding in All students must have solid grounding in mathematics to function effectively in today’s mathematics to function effectively in today’s worldworld

Premises:Premises: Basic skills with numbers continue to be vitally Basic skills with numbers continue to be vitally

important for a variety of everyday usesimportant for a variety of everyday uses Mathematics requires careful reasoning about precisely Mathematics requires careful reasoning about precisely

defined objects and conceptsdefined objects and concepts Students must be able to formulate and solve problemsStudents must be able to formulate and solve problems

Areas of agreement: automatic recall of basic Areas of agreement: automatic recall of basic facts, use of calculators in lower grades, learning facts, use of calculators in lower grades, learning algorithms, fractions, teaching mathematics algorithms, fractions, teaching mathematics in “real world” contexts, instructional in “real world” contexts, instructional methods, teacher knowledgemethods, teacher knowledge

Seeking Common GroundSeeking Common Ground

A process:A process: People working together People working together Listening thoughtfullyListening thoughtfully Valuing others’ opinionsValuing others’ opinions Taking timeTaking time Agreeing on languageAgreeing on language Working hard toward a common Working hard toward a common

goalgoal

What complaints?What complaints?

K-12 Mathematics Education: How K-12 Mathematics Education: How Much Common Ground Is There?Much Common Ground Is There?

Anthony RalstonAnthony Ralston

A valuable exercise, with results A valuable exercise, with results unexceptional to almost all FOCUS unexceptional to almost all FOCUS readers, but fraught with difficulties:readers, but fraught with difficulties: BlandnessBlandness AmbiguityAmbiguity Disagreement in community—curriculum and Disagreement in community—curriculum and

technologytechnology Before attempt consensus, need a level of Before attempt consensus, need a level of

respect in both communitiesrespect in both communities

K-12 Mathematics Education: How K-12 Mathematics Education: How Much Common Ground Is There?Much Common Ground Is There?

Anthony RalstonAnthony Ralston

AmbiguityAmbiguity ““Certain procedures and algorithms in mathematics are so Certain procedures and algorithms in mathematics are so

basic and have such wide application that they should be basic and have such wide application that they should be practiced to the point of automaticity”practiced to the point of automaticity”

““Calculators can have a useful role even in the lower Calculators can have a useful role even in the lower grades, but they must be used carefully, so as not to grades, but they must be used carefully, so as not to impede the acquisition of fluency with basic facts and impede the acquisition of fluency with basic facts and computational procedures”computational procedures”

K-12 Mathematics Education: How K-12 Mathematics Education: How Much Common Ground Is There?Much Common Ground Is There?

Anthony RalstonAnthony Ralston

Disagreement in communityDisagreement in community ““By the time they leave high school, a By the time they leave high school, a

majority of students should have studied majority of students should have studied calculus”calculus”

““Students should be able to use the basic Students should be able to use the basic algorithms of whole number arithmetic algorithms of whole number arithmetic fluently, and they should understand how and fluently, and they should understand how and why the algorithms work”why the algorithms work”

““The arithmetic of fractions is important as a The arithmetic of fractions is important as a foundation for algebra”foundation for algebra”

““By the time they leave high school, a By the time they leave high school, a majority of students should have majority of students should have studied calculus”studied calculus”

Although some should, and already do, take a full course in Although some should, and already do, take a full course in calculus, most students should learn at least certain calculus, most students should learn at least certain fundamental ideas of calculus, such as rate of change, limit, fundamental ideas of calculus, such as rate of change, limit, and derivativeand derivative

Some 70% of the countries in TIMSS cover the topics of Some 70% of the countries in TIMSS cover the topics of elementary analysis (infinite processes and change) at elementary analysis (infinite processes and change) at grade 12, and many address these topics in grades 9 grade 12, and many address these topics in grades 9 through 11through 11

A project involving incentives and district-wide commitment A project involving incentives and district-wide commitment in ten inner-city Dallas high schools has resulted in a nine-in ten inner-city Dallas high schools has resulted in a nine-fold increase to 330 out of 4161 graduates from 1995 to fold increase to 330 out of 4161 graduates from 1995 to 2005 in the number of students receiving a score of three 2005 in the number of students receiving a score of three or better on the AB Calculus Advanced Placement exam.or better on the AB Calculus Advanced Placement exam.

What’s next?What’s next?

March meeting in IndianapolisMarch meeting in Indianapolis

Probability and data analysis in the elementary Probability and data analysis in the elementary curriculum curriculum

Algorithms in the curriculum Algorithms in the curriculum Technology in general Technology in general Calculus in high school Calculus in high school Algebra for allAlgebra for all Gap between policy (high standards) and teacher Gap between policy (high standards) and teacher

beliefs and capacity beliefs and capacity How can international studies and information be How can international studies and information be

used?used? How should we weigh class size versus teacher How should we weigh class size versus teacher

knowledge and capabilities?knowledge and capabilities?

Where are the Where are the teachers?teachers?

Mathematics TeachersMathematics Teachers

Are they in this fight?Are they in this fight? What might they add to the What might they add to the

conversation?conversation? Calculus as goalCalculus as goal Role of definitionsRole of definitions Applications as motivationApplications as motivation Curriculum structureCurriculum structure