40 Years of Teaching? What Has Happened? Where Are We Going? Johnny W. Lott, Director Center for Teaching Excellence The University of Montana Missoula,

40 Years of Teaching?  What 40 Years of Teaching?  What Has Happened? Where Are Has Happened? Where Are

We Going?We Going?

Johnny W. Lott, DirectorCenter for Teaching Excellence

The University of MontanaMissoula, MT [email protected]

1965

August 11: Watts riots begin in Los Angeles, CA

August 13 – Jefferson Airplane debut in San Francisco August 18 – Vietnam War: Operation Starlite begins as 5500 US Marines destroy a Viet Cong stronghold in the first major American ground battle of the war. This in retaliation for a suspected attack on the US base at Chu Lai.

August 19 – 66 ex-SS personnel receive life sentences at the Auschwitz trial in Frankfort.

September – DeKalb County, Georgia, Johnny teaches first classes at Lakeside High School, grades 8-11.

1965

August 11: Watts riots begin in Los Angeles, CA

August 13 – Jefferson Airplane debut in San Francisco August 18 – Vietnam War: Operation Starlite begins as 5500 US Marines destroy a Viet Cong stronghold in the first major American ground battle of the war. This in retaliation for a suspected attack on the US base at Chu Lai.

August 19 – 66 ex-SS personnel receive life sentences at the Auschwitz trial in Frankfort.

September – DeKalb County, Georgia, Johnny teaches first classes at Lakeside High School, grades 8-11.

A Teacher’s Life Begins!A Teacher’s Life Begins!• New school, grades 8-11• Typical Day

– Bus Duty starts at 7:20 AM– Classes start at 8:00 AM– Classes end at 3:15 PM– Bus duty ends at 4:30 PM**Note that no teacher can clock out until 4:00.

• New school, grades 8-11• Typical Day

– Bus Duty starts at 7:20 AM– Classes start at 8:00 AM– Classes end at 3:15 PM– Bus duty ends at 4:30 PM**Note that no teacher can clock out until 4:00.

Teaching ScheduleTeaching Schedule• Period 1—Advanced Algebra and Trigonometry• Period 2—Algebra I• Period 3—Study Hall• Period 4—Geometry + Lunch + Study Hall• Period 5—Eighth Grade Mathematics• Period 6—Algebra I

• Period 1—Advanced Algebra and Trigonometry• Period 2—Algebra I• Period 3—Study Hall• Period 4—Geometry + Lunch + Study Hall• Period 5—Eighth Grade Mathematics• Period 6—Algebra I

Seeing Through Mathematics-Book 2, Scott Foresman

Seeing Through Mathematics-Book 2, Scott Foresman

– Written by Henry Van Engen, et al.

• Reviewed in December 1965 Mathematics Teacher (R. M. Shelton, Milliken University)

– Authors’ goal: Make sure that students learn all of the mathematics customarily in the first course in high school algebra

– Activities developed with concrete materials– General case reached through well-developed questions– Student develops major ideas of the topic for himself– Student has to do a considerable amount of thoughtful reading

and this should help to develop good habits for reading mathematics books at higher levels

– Authors use mathematical terms properly without being overly fussy

– Exercises are adequate– “I have not had an opportunity to use this book…”

– Written by Henry Van Engen, et al.

• Reviewed in December 1965 Mathematics Teacher (R. M. Shelton, Milliken University)

– Authors’ goal: Make sure that students learn all of the mathematics customarily in the first course in high school algebra

– Activities developed with concrete materials– General case reached through well-developed questions– Student develops major ideas of the topic for himself– Student has to do a considerable amount of thoughtful reading

and this should help to develop good habits for reading mathematics books at higher levels

– Authors use mathematical terms properly without being overly fussy

– Exercises are adequate– “I have not had an opportunity to use this book…”

Sample Question:

Consider the 2 and 3 pictured below.Which is the bigger numeral? Which is the bigger number?

23

Other Memorable TextsOther Memorable Texts

• School Mathematics Study Group (Allen, Frank, at al.). Geometry with Coordinates, Parts I and II. Yale University Press, 1962.

• School Mathematics Study Group (Allen, Frank, at al.). Geometry with Coordinates, Parts I and II. Yale University Press, 1962.

Geometry with Coordinates, School Mathematics Study Group, 1962

Geometry with Coordinates, School Mathematics Study Group, 1962The School Mathematics Study Group

(SMSG) began work in 1950s and had material on the land by 1960 in limited use.

This was a part of the revolution in mathematics following the launching of Sputnik by the USSR.

The School Mathematics Study Group (SMSG) began work in 1950s and had material on the land by 1960 in limited use.

This was a part of the revolution in mathematics following the launching of Sputnik by the USSR.

Geometry with Coordinates, Part I, School Mathematics Study Group

Geometry with Coordinates, Part I, School Mathematics Study GroupExamples:1. The absolute value of the difference between

the lengths of any two sides of a triangle is (greater than, equal to, less than) the length of the third side.

2. If the lengths of two sides of a triangle are 8 and 12, the length of the third side of the triangle must be greater than ___ and less than ____.

Examples:1. The absolute value of the difference between

the lengths of any two sides of a triangle is (greater than, equal to, less than) the length of the third side.

2. If the lengths of two sides of a triangle are 8 and 12, the length of the third side of the triangle must be greater than ___ and less than ____.

Renowned mathematician: Carl Allendoerfer wrote in the

Mathematics Teacher (December, 1965),

Renowned mathematician: Carl Allendoerfer wrote in the

Mathematics Teacher (December, 1965),

“ The first versions of the SMSG materials …were generally released in revised form for use in 1960. So now we have had five years of experience with them and can begin to form judgments of their effectiveness.”

“ The first versions of the SMSG materials …were generally released in revised form for use in 1960. So now we have had five years of experience with them and can begin to form judgments of their effectiveness.”

Allendoerfer continued,Allendoerfer continued,

“…history makes it clear that the SMSG materials were generally aimed at college-capable students. It was recognized that there were other students in the schools, and some people expressed pious hopes that with some simplification the SMSG materials could be made useful for them as well. I have seen no sign, however, that this was ever pressed very hard.”

“…history makes it clear that the SMSG materials were generally aimed at college-capable students. It was recognized that there were other students in the schools, and some people expressed pious hopes that with some simplification the SMSG materials could be made useful for them as well. I have seen no sign, however, that this was ever pressed very hard.”

AllendoerferAllendoerfer

In 1963, the College Board published data from 181 schools that normally sent students to take the College Board Mathematics Achievement Tests.

28% did not teach the structure of the number system;

34% did not teach set theory;56% did not teach logic;30% did not teach probability;30% gave no attention to calculus.

In 1963, the College Board published data from 181 schools that normally sent students to take the College Board Mathematics Achievement Tests.

28% did not teach the structure of the number system;

34% did not teach set theory;56% did not teach logic;30% did not teach probability;30% gave no attention to calculus.

Fundamentals of Freshman Mathematics by Allendoerfer

and Oakley

Fundamentals of Freshman Mathematics by Allendoerfer

and Oakley“When you study any new subject, your

first task is to learn the meanings of the special, technical terms which are introduced.” p. 2

“When you study any new subject, your first task is to learn the meanings of the special, technical terms which are introduced.” p. 2

The Cambridge Report of 1963The Cambridge Report of 1963The authors succumbed to euphoria.They thought that two years of calculus could be taught in

high school.The report shows no knowledge of the work of

psychologists [on how children learn].The report ignores the lower 7/8 of the ability group. These

children include the culturally deprived and those who are likely to be the poor of the next generation.

Allendoerfer

The authors succumbed to euphoria.They thought that two years of calculus could be taught in

high school.The report shows no knowledge of the work of

psychologists [on how children learn].The report ignores the lower 7/8 of the ability group. These

children include the culturally deprived and those who are likely to be the poor of the next generation.

Allendoerfer

Allendoerfer’s Proposed Interim Solution when the Cambridge

recommendations cannot be met.

Allendoerfer’s Proposed Interim Solution when the Cambridge

recommendations cannot be met.• Adopt the scheme of having arithmetic taught by arithmetic specialists.

• Put more emphasis on remedial work in arithmetic in high schools for those who need it.

– Overcome the mathematical blocks that exist in young people

– Teach them from scratch

• Adopt the scheme of having arithmetic taught by arithmetic specialists.

• Put more emphasis on remedial work in arithmetic in high schools for those who need it.

– Overcome the mathematical blocks that exist in young people

– Teach them from scratch

1966--June 2-Surveryor 1 lands on the moon becoming the first to soft land on another world

• June 5-Gene Cernan completes 2nd U.S. spacewalk on the Gemini 9 mission.

• June 6-James Meredith, civil right activist is shot while trying to march across Mississippi.

• June 13-The U.S. Supreme Court rules in Miranda vs. Arizona that police must inform suspects of their rights before questioning them.

• June 14-The Vatican announces the abolition of Index Liborum Prohibitum index of banned books.

• June 29-U.S. planes begin bombing Hanoi and Haiphong. • Johnny survived one year of teaching and taught

summer school for the one and only time in high schools.• August 1966-Johnny starts a masters program at Emory

University to learn all the math that he did not know in his textbooks in 1965-6

1966--June 2-Surveryor 1 lands on the moon becoming the first to soft land on another world

• June 5-Gene Cernan completes 2nd U.S. spacewalk on the Gemini 9 mission.

• June 6-James Meredith, civil right activist is shot while trying to march across Mississippi.

• June 13-The U.S. Supreme Court rules in Miranda vs. Arizona that police must inform suspects of their rights before questioning them.

• June 14-The Vatican announces the abolition of Index Liborum Prohibitum index of banned books.

• June 29-U.S. planes begin bombing Hanoi and Haiphong. • Johnny survived one year of teaching and taught

summer school for the one and only time in high schools.• August 1966-Johnny starts a masters program at Emory

University to learn all the math that he did not know in his textbooks in 1965-6

“New Math” in full swing“New Math” in full swing

Critics:

Morris Kline (April, 1966)Principles of modern mathematics curricula*Mathematics is or should be presented as a set of

deductive structures.

*Mathematics is to be presented rigorously.

*Mathematics is built up as an isolated, self-sufficient, pure body of knowledge.

Critics:

Morris Kline (April, 1966)Principles of modern mathematics curricula*Mathematics is or should be presented as a set of

deductive structures.

*Mathematics is to be presented rigorously.

*Mathematics is built up as an isolated, self-sufficient, pure body of knowledge.

More KlineMore Kline

*Mathematics is self-generating: axioms, concepts, and theorems come from purely mathematical sources.

*Students learn purely abstract concepts and if they learn these, the concrete will be automatically understood.

*The more terminology the better.

*Never use words where symbols can be substituted.

*Mathematics is self-generating: axioms, concepts, and theorems come from purely mathematical sources.

*Students learn purely abstract concepts and if they learn these, the concrete will be automatically understood.

*The more terminology the better.

*Never use words where symbols can be substituted.

Kline’s Proposal PrinciplesKline’s Proposal Principles

*Mathematics must be developed, not deductively but constructively.

*Mathematics must be presented as intuitively as possible.

*Mathematics is not an isolated, self-sufficient body of knowledge. It exists to help man understand and master the physical, economic and social worlds.

*Mathematics must be developed, not deductively but constructively.

*Mathematics must be presented as intuitively as possible.

*Mathematics is not an isolated, self-sufficient body of knowledge. It exists to help man understand and master the physical, economic and social worlds.

More Kline PrinciplesMore Kline Principles

*Elementary math is not self-generating; significant math grew out of real situations, needs, problems and phenomena.

*Math must be presented concretely.*Math must use as few terms as possible.

*Elementary math is not self-generating; significant math grew out of real situations, needs, problems and phenomena.

*Math must be presented concretely.*Math must use as few terms as possible.

Yet more Kline—on algebraYet more Kline—on algebra“The ninth year is the crucial year. It is

the student’s first real experience with mathematics… The content of ninth-grade algebra that I would recommend is the traditional one.”

“The ninth year is the crucial year. It is the student’s first real experience with mathematics… The content of ninth-grade algebra that I would recommend is the traditional one.”

And more Kline—on geometryAnd more Kline—on geometry

“…the essential new feature is the approach.”*Don’t start with a string of definitions.*Use no more than 10 axioms.*Add the topic of conic sections.*Teach how proofs come from physical facts

using location of buildings, astronomy problems, inclined plane, mass of the earth and “more homely applications.”

“…the essential new feature is the approach.”*Don’t start with a string of definitions.*Use no more than 10 axioms.*Add the topic of conic sections.*Teach how proofs come from physical facts

using location of buildings, astronomy problems, inclined plane, mass of the earth and “more homely applications.”

1970s—Years of Issues and 1970s—Years of Issues and Some DirectionSome Direction

January 5 - First episode of All My Children January 16 - Buckminster Fuller recognized by American Institute of Architects. March 17 - My Lai massacre charges April 1 - President Nixon signs the Public Health Cigarette Smoking Act into law banning cigarette television advertisementsMay 4 - Kent State shootings October 26 – Doonesbury debuts.

What is happening in the classroom?What is happening in the classroom?

Should mathematical logic be taught formally in mathematics classes? (con-Peter Hilton)

Should performance contracting exist in math? (con-Tom Cooney and Larry HatfieldEliminate frills; taxpayers disenchanted.

Should math use behavioral objectives? (pro-Allendoerfer; con-Forbes)Understanding, appreciating, etc. can only be deduced from observed behavior.

Should mathematical logic be taught formally in mathematics classes? (con-Peter Hilton)

Should performance contracting exist in math? (con-Tom Cooney and Larry HatfieldEliminate frills; taxpayers disenchanted.

Should math use behavioral objectives? (pro-Allendoerfer; con-Forbes)Understanding, appreciating, etc. can only be deduced from observed behavior.

What is happening in the classroom?What is happening in the classroom?

What should become of the high school geometry course? (Fehr, Eccles, and Meserve)

Has the time for accountability come? (Wells and Willoughby)

Should instruction be individualized?

What should become of the high school geometry course? (Fehr, Eccles, and Meserve)

Has the time for accountability come? (Wells and Willoughby)

Should instruction be individualized?

Motion Geometry?Motion Geometry?

UICSM (University of Illinois Committee on School Mathematics) introduced four paperback geometry books for middle school students in the 60s.

Only in the 1970s was the first high school transformational geometry book published.

UICSM (University of Illinois Committee on School Mathematics) introduced four paperback geometry books for middle school students in the 60s.

Only in the 1970s was the first high school transformational geometry book published.

1980s1980s

• Introduction of the IBM PC in 1981. • First commercial hand-held mobile phone – 1983. • Apple Macintosh first successfully commercially released in

1984.• Space Shuttle Challenger disaster in 1986.• Gorbachev introduces Glasnost and Perestroika in the Soviet

Union.• Soviet Union ends its military campaign in Afghanistan.• Ronald Reagan decides to invade Grenada in 1984.• Rubik’s cube, Cabbage Patch Kids and Trivial Pursuit entertain

us. • Popular artists include Michael Jackson, Bon Jovi, Duran Duran

Madonna, U2, and Menudo. • Steven Spielberg’s E.T. The Extra-Terrestrial opens in 1982.• AIDS epidemic is identified and named.

• Introduction of the IBM PC in 1981. • First commercial hand-held mobile phone – 1983. • Apple Macintosh first successfully commercially released in

1984.• Space Shuttle Challenger disaster in 1986.• Gorbachev introduces Glasnost and Perestroika in the Soviet

Union.• Soviet Union ends its military campaign in Afghanistan.• Ronald Reagan decides to invade Grenada in 1984.• Rubik’s cube, Cabbage Patch Kids and Trivial Pursuit entertain

us. • Popular artists include Michael Jackson, Bon Jovi, Duran Duran

Madonna, U2, and Menudo. • Steven Spielberg’s E.T. The Extra-Terrestrial opens in 1982.• AIDS epidemic is identified and named.

1980s1980s

Preparation for change culminating with the publication of Curriculum and Evaluation Standards for School Mathematics by NCTM in 1989.

Preparation for change culminating with the publication of Curriculum and Evaluation Standards for School Mathematics by NCTM in 1989.

From the Arithmetic TeacherFrom the Arithmetic Teacher“The fundamental problem with

arithmetic is to find effective computational techniques for coping with the usual array of arithmetic situations that are encountered in real life.” Madell, AT, (October 1987)

“The fundamental problem with arithmetic is to find effective computational techniques for coping with the usual array of arithmetic situations that are encountered in real life.” Madell, AT, (October 1987)

Jack drew a sketch of a lot as shown. To fence the lot, the cost is $12.75 a rod. How much will the

fence cost if 16.5 ft = 1 rod?

Jack drew a sketch of a lot as shown. To fence the lot, the cost is $12.75 a rod. How much will the

fence cost if 16.5 ft = 1 rod?

132 ft

99 ft

165 165 165 165 165

4 7 6 9 8

660 1155 990 1485 1320

1275 1275 10200 17850

6 8 7650 17850

7650 10200 17850 35700 Answer is35700.

John Newton writes,John Newton writes,

Pattern blocks are a natural medium for stimulating spatial and geometric thinking...With Logo, those same ideas can appear in bright colors on a computer screen or printout.

Pattern blocks are a natural medium for stimulating spatial and geometric thinking...With Logo, those same ideas can appear in bright colors on a computer screen or printout.

Of the 1980s, Stephen Willoughby writes,

Of the 1980s, Stephen Willoughby writes,

Many circumstances inhibit meaningful change:

1. Lack of teacher time to evaluate texts and prepare to teach new materials.

2. Tradition of adopting K-8 series prevents upper-grade texts from being more advanced.

Many circumstances inhibit meaningful change:

1. Lack of teacher time to evaluate texts and prepare to teach new materials.

2. Tradition of adopting K-8 series prevents upper-grade texts from being more advanced.

Of the 1980s, Stephen Willoughby writes,

Of the 1980s, Stephen Willoughby writes,

3. Adopting “best sellers” instead of “carefully developed, innovative, field-tested programs that have been proven effective and that integrate all important strands, including problem solving, thinking, and communication” is a bad thing by penny-wise cost cutters.

4. Minimal-competency tests are indeed minimizing competency.

3. Adopting “best sellers” instead of “carefully developed, innovative, field-tested programs that have been proven effective and that integrate all important strands, including problem solving, thinking, and communication” is a bad thing by penny-wise cost cutters.

4. Minimal-competency tests are indeed minimizing competency.

1990s characterized by conflict and change in mathematics.

1990s characterized by conflict and change in mathematics.

NCTM *publishes Assessment Standards and Professional Teaching Standards.*creates Teaching Children Mathematics, Mathematics Teaching in the Middle School, Dialogues.

NCTM *publishes Assessment Standards and Professional Teaching Standards.*creates Teaching Children Mathematics, Mathematics Teaching in the Middle School, Dialogues.

1990s characterized by conflict and change in mathematics.

1990s characterized by conflict and change in mathematics.

*Math Wars Erupt over Standards and NSF Curricula!

Leaders: Side 1: Frank Allen, James Milgram,

Richard Askey, Hung-Hsi Wu

Side 2: NCTM

*Math Wars Erupt over Standards and NSF Curricula!

Leaders: Side 1: Frank Allen, James Milgram,

Richard Askey, Hung-Hsi Wu

Side 2: NCTM

NSF Curricula Characterize Decade in Mathematics

NSF Curricula Characterize Decade in Mathematics

Curricula Overview:Based on constructivismActivity orientedContext basedIntegratedTechnology oriented

Curricula Overview:Based on constructivismActivity orientedContext basedIntegratedTechnology oriented

Example Test Item from SIMMS Integrated Mathematics with

Technology

Example Test Item from SIMMS Integrated Mathematics with

Technology

• According to a newspaper report, the trees in a certain land area are being cut at a rate of 15% a year. The lumber company claims that it replaces 2000 trees every year in the area. Discuss the future tree production of this land area if the plan continues.

• According to a newspaper report, the trees in a certain land area are being cut at a rate of 15% a year. The lumber company claims that it replaces 2000 trees every year in the area. Discuss the future tree production of this land area if the plan continues.

2000-20052000-2005

NCTMPublishes Principles and Standards for School MathematicsDiscontinues DialoguesEstablished On-Math, an electronic journalForms with MAA Joint Committee on Mutual ConcernsPublishes Standards and Curriculum: A View from the Nation.

NCTMPublishes Principles and Standards for School MathematicsDiscontinues DialoguesEstablished On-Math, an electronic journalForms with MAA Joint Committee on Mutual ConcernsPublishes Standards and Curriculum: A View from the Nation.

Math Wars Ebb with CooperationMath Wars Ebb with Cooperation

Dr. Richard Schaar, Texas Instruments, calls Peace Commission.

Park City Mathematics Institute hosts NCTM and mathematicians for work together.

Dr. Richard Schaar, Texas Instruments, calls Peace Commission.

Park City Mathematics Institute hosts NCTM and mathematicians for work together.

Federal Government Proliferates Education Policy Arena

Federal Government Proliferates Education Policy Arena

No Child Left Behind instituted.Schools sink under testing.Vouchers outlawed in Florida.Schools and states sue to halt “unfunded

mandates.”NSF funding static; DoED funding

increased.

No Child Left Behind instituted.Schools sink under testing.Vouchers outlawed in Florida.Schools and states sue to halt “unfunded

mandates.”NSF funding static; DoED funding

increased.

Where Are We Going? My Wishes—Not My Crystal Ball

Where Are We Going? My Wishes—Not My Crystal Ball

• Use sense-making in testing in schools.Use sense-making in testing in schools.• Move to integrated mathematics as the Move to integrated mathematics as the

mathematics program in the U.S.mathematics program in the U.S.• Move to the metric system.Move to the metric system.• Consider research from other countries.Consider research from other countries.• Learn how to use technology to prepare Learn how to use technology to prepare

students for the real world.students for the real world.

• Use sense-making in testing in schools.Use sense-making in testing in schools.• Move to integrated mathematics as the Move to integrated mathematics as the

mathematics program in the U.S.mathematics program in the U.S.• Move to the metric system.Move to the metric system.• Consider research from other countries.Consider research from other countries.• Learn how to use technology to prepare Learn how to use technology to prepare

students for the real world.students for the real world.

Where Are We Going? My Wishes—Not My Crystal Ball

Where Are We Going? My Wishes—Not My Crystal Ball

• Implement use of mathematics Implement use of mathematics specialists in elementary grades.specialists in elementary grades.

• Change collegiate programs to Change collegiate programs to accommodate changes in high school accommodate changes in high school and before.and before.

• Harness the power of cell phone Harness the power of cell phone technology for education.technology for education.

• Expect more taxpayer revolt as baby Expect more taxpayer revolt as baby boomers age and resist paying for boomers age and resist paying for education.education.

• Implement use of mathematics Implement use of mathematics specialists in elementary grades.specialists in elementary grades.

• Change collegiate programs to Change collegiate programs to accommodate changes in high school accommodate changes in high school and before.and before.

• Harness the power of cell phone Harness the power of cell phone technology for education.technology for education.

• Expect more taxpayer revolt as baby Expect more taxpayer revolt as baby boomers age and resist paying for boomers age and resist paying for education.education.

Where Are We Going? My Wishes—Not My Crystal Ball

Where Are We Going? My Wishes—Not My Crystal Ball

• Decide how to use Computer Algebra Decide how to use Computer Algebra Systems (CAS).Systems (CAS).

• New and different majors in New and different majors in mathematics will be developed or mathematics will be developed or mathematics will decline as a mathematics will decline as a requirement for general education.requirement for general education.

• Calculus will not be the only entry Calculus will not be the only entry point for a mathematics major.point for a mathematics major.

• Statistics, or data analysis, will be Statistics, or data analysis, will be required in all high school classes.required in all high school classes.

• Decide how to use Computer Algebra Decide how to use Computer Algebra Systems (CAS).Systems (CAS).

• New and different majors in New and different majors in mathematics will be developed or mathematics will be developed or mathematics will decline as a mathematics will decline as a requirement for general education.requirement for general education.

• Calculus will not be the only entry Calculus will not be the only entry point for a mathematics major.point for a mathematics major.

• Statistics, or data analysis, will be Statistics, or data analysis, will be required in all high school classes.required in all high school classes.

ReferencesReferences

Allendoerfer, Carl B. “The Second Revolution in Mathematics,” The Mathematics Teacher 58 (December 1965): 690-695.

http://math.berkeley.edu/~wu/Kline, Morris. “A Proposal for the High School Mathematics Curriculum,” The Mathematics

Teacher 59 (April 1966): 322-330.Maddell, Robert. “One Point of View: The Fundamental Problem of Arithmetic,” Arithmetic

Teacher 35(October 1987): 2.Mathematics Teacher. “The Forum: Individualization of Instruction,” MT 65 (May 1972).… “The Forum: Accountability,” MT 65 (November 1972).… “The Forum: What Should Become of the High School Geometry Course?” MT 65 (February

1972).… “The Forum: The Utility of Behavioral Objectives,” MT (December 1971).… “The Forum: Some Pros and Cons of Performance Contracting in Mathematics,” MT (October

1971).… “The Forum: Should Mathematical Logic Be Taught Formally in Mathematics Classes?” MT

(May 1971).National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School

Mathematics. Reston, VA: NCTM, 1989. Newton, John E. “From Pattern-blocks Play to Logo Programming,” Arithmetic Teacher 35 (May

1988): 6-9. School Mathematics Study Group (Frank Allen, et al.). Geometry with Coordinates, Parts I and

II. New Haven: Yale University Press, 1962.Van Engen, Henry, et al. Seeing Through Mathematics, Books 1-3. Chicago: Scott, Foresman &

Co., 1964.Willoughby, Stephen. “One Point of View: Accomplishments of the 1980s,” Arithmetic Teacher

36 ( September 1988): 11.http://math.berkeley.edu/~wu/

Allendoerfer, Carl B. “The Second Revolution in Mathematics,” The Mathematics Teacher 58 (December 1965): 690-695.

http://math.berkeley.edu/~wu/Kline, Morris. “A Proposal for the High School Mathematics Curriculum,” The Mathematics

Teacher 59 (April 1966): 322-330.Maddell, Robert. “One Point of View: The Fundamental Problem of Arithmetic,” Arithmetic

Teacher 35(October 1987): 2.Mathematics Teacher. “The Forum: Individualization of Instruction,” MT 65 (May 1972).… “The Forum: Accountability,” MT 65 (November 1972).… “The Forum: What Should Become of the High School Geometry Course?” MT 65 (February

1972).… “The Forum: The Utility of Behavioral Objectives,” MT (December 1971).… “The Forum: Some Pros and Cons of Performance Contracting in Mathematics,” MT (October

1971).… “The Forum: Should Mathematical Logic Be Taught Formally in Mathematics Classes?” MT

(May 1971).National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School

Mathematics. Reston, VA: NCTM, 1989. Newton, John E. “From Pattern-blocks Play to Logo Programming,” Arithmetic Teacher 35 (May

1988): 6-9. School Mathematics Study Group (Frank Allen, et al.). Geometry with Coordinates, Parts I and

II. New Haven: Yale University Press, 1962.Van Engen, Henry, et al. Seeing Through Mathematics, Books 1-3. Chicago: Scott, Foresman &

Co., 1964.Willoughby, Stephen. “One Point of View: Accomplishments of the 1980s,” Arithmetic Teacher

36 ( September 1988): 11.http://math.berkeley.edu/~wu/