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40 Years of Teaching What Has Happened Where Are We Going


June 5-Gene Cernan completes 2nd U.S. spacewalk on the Gemini 9 mission. ... Never use words where symbols can be substituted. Kline's Proposal Principles ... – PowerPoint PPT presentation

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Title: 40 Years of Teaching What Has Happened Where Are We Going

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
  • jlott_at_mso.umt.edu

1965August 11 Watts riots begin in
Los Angeles, CAAugust 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 Teachers Life Begins!
  • New school, grades 8-11
  • Typical Day
  • Bus Duty starts at 720 AM
  • Classes start at 800 AM
  • Classes end at 315 PM
  • Bus duty ends at 430 PM
  • Note that no teacher can clock out until 400.

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

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
  • Student develops major ideas of the topic for
  • 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?
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
  • 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.
  • 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.

  • 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
  • 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

Allendoerfers 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

  • 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
  • Johnny survived one year of teaching and taught
    summer school for the one and only time in high
  • 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,
    self-sufficient, 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

Klines Proposal Principles
  • Mathematics must be developed, not deductively
    but constructively.
  • Mathematics must be presented as intuitively as
  • 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 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.

Yet more Klineon algebra
  • The ninth year is the crucial year. It is the
    students first real experience with mathematics
    The content of ninth-grade algebra that I would
    recommend is the traditional one.

And more Klineon geometry
  • the essential new feature is the approach.
  • Dont 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.

1970sYears 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?
    (con-Tom Cooney and Larry Hatfield
  • Eliminate 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 should become of the high school geometry
    course? (Fehr, Eccles, and Meserve)
  • Has the time for accountability come? (Wells and
  • 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.

  • 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
  • Ronald Reagan decides to invade Grenada in 1984.
  • Rubiks cube, Cabbage Patch Kids and Trivial
    Pursuit entertain us.
  • Popular artists include Michael Jackson, Bon
    Jovi, Duran Duran Madonna, U2, and Menudo.
  • Steven Spielbergs E.T. The Extra-Terrestrial
    opens in 1982.
  • AIDS epidemic is identified and named.

  • 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?
99 ft
132 ft
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
  • 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
    penny-wise cost cutters.
  • 4. Minimal-competency tests are indeed minimizing

1990s characterized by conflict and change in
  • NCTM
  • publishes Assessment Standards and Professional
    Teaching Standards.
  • creates Teaching Children Mathematics,
    Mathematics Teaching in the Middle School,

1990s characterized by conflict and change in
  • Math Wars Erupt over Standards and NSF
  • 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
  • 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.

  • NCTM
  • Publishes Principles and Standards for School
  • Discontinues Dialogues
  • Established On-Math, an electronic journal
  • Forms with MAA Joint Committee on Mutual
  • 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
  • No Child Left Behind instituted.
  • Schools sink under testing.
  • Vouchers outlawed in Florida.
  • Schools and states sue to halt unfunded
  • NSF funding static DoED funding increased.

Where Are We Going? My WishesNot My Crystal Ball
  • 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 WishesNot My Crystal Ball
  • 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
  • Expect more taxpayer revolt as baby boomers age
    and resist paying for education.

Where Are We Going? My WishesNot My Crystal Ball
  • 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.

  • 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
  • The Forum Accountability, MT 65 (November
  • 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/
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