Transcript 40 Years of Teaching? What Has Happened?

40 Years of Teaching? What
Has Happened? Where Are We
Going?
Johnny W. Lott, Director
Center for Teaching Excellence
The University of Montana
Missoula, MT 59812
[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.
A Teacher’s Life Begins!
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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 Schedule
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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
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Written by Henry Van Engen, et al.
Reviewed in December 1965 Mathematics Teacher (R. M.
Shelton, Milliken University)
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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?
3
2
Other Memorable Texts
• 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
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
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),
“ 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,
“…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.”
Allendoerfer
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
“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 1963
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.
• Adopt the
scheme
of having
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arithmetic taught by arithmetic
specialists.
Put more emphasis on remedial
work in arithmetic in high schools
for those who need it.
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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
“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, selfsufficient, pure body of knowledge.
More 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.
Kline’s Proposal Principles
*Mathematics must be developed, not
deductively but constructively.
*Mathematics must be presented as
intuitively as possible.
*Mathematics is not an isolated, selfsufficient body of knowledge. It exists to
help man understand and master the
physical, economic and social worlds.
More Kline Principles
*Elementary math is not selfgenerating; 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 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.”
And 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.”
1970s—Years of Issues and Some
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 advertisements
May 4 - Kent State shootings
October 26 – Doonesbury debuts.
What is happening in the classroom?
Should mathematical logic be taught formally in
mathematics classes? (con-Peter Hilton)
Should performance contracting exist in math? (conTom Cooney and Larry Hatfield
Eliminate frills; taxpayers disenchanted.
Should math use behavioral objectives? (proAllendoerfer; con-Forbes)
Understanding, appreciating, etc. can only be
deduced from observed behavior.
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?
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.
1980s
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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.
1980s
Preparation for change culminating
with the publication of Curriculum
and Evaluation Standards for
School Mathematics by NCTM in
1989.
From 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)
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?
165 165 165 165 165
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7
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9
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660 1155 99014851320
99 ft
132 ft
1275 1275 10200 17850
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8 7650 17850
7650102001785035700 Answeris35700.
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.
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.
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 pennywise cost cutters.
4. Minimal-competency tests are indeed
minimizing competency.
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.
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
NSF Curricula Characterize Decade in
Mathematics
Curricula Overview:
Based on constructivism
Activity oriented
Context based
Integrated
Technology oriented
Example Test Item from SIMMS
Integrated Mathematics with Technology
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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.
2000-2005
NCTM
Publishes Principles and Standards for
School Mathematics
Discontinues Dialogues
Established On-Math, an electronic journal
Forms with MAA Joint Committee on
Mutual Concerns
Publishes Standards and Curriculum: A
View from the Nation.
Math Wars Ebb with Cooperation
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
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
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Use sense-making in testing in schools.
Move to integrated mathematics as the
mathematics program in the U.S.
Move to the metric system.
Consider research from other countries.
Learn how to use technology to prepare
students for the real world.
Where Are We Going? My Wishes—Not
My Crystal Ball
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Implement use of mathematics
specialists in elementary grades.
Change collegiate programs to
accommodate changes in high
school and before.
Harness the power of cell phone
technology for education.
Expect more taxpayer revolt as
baby boomers age and resist
paying for education.
Where Are We Going? My Wishes—Not
My Crystal Ball
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Decide how to use Computer
Algebra Systems (CAS).
New and different majors in
mathematics will be developed or
mathematics will decline as a
requirement for general education.
Calculus will not be the only entry
point for a mathematics major.
Statistics, or data analysis, will be
required in all high school classes.
References
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/