40 Years of Teaching What Has Happened Where Are We Going

1
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

2
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.
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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.

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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

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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

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Sample Question Consider the 2 and 3 pictured
below. Which is the bigger numeral? Which is the
bigger number?
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Other Memorable Texts

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

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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.

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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
  ____.

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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.

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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.

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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.

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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

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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

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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

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• 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

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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.

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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.

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Klines 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.

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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.

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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.

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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.

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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.

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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.

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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?

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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.

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1980s

• 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.
• 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.

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1980s

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

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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)

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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
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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.

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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.

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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.

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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.

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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

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NSF Curricula Characterize Decade in Mathematics

• Curricula Overview
• Based on constructivism
• Activity oriented
• Context based
• Integrated
• Technology oriented

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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.

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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.

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Math Wars Ebb with Cooperation

• Dr. Richard Schaar, Texas Instruments, calls
  Peace Commission.
• Park City Mathematics Institute hosts NCTM and
  mathematicians for work together.

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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.

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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.

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

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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.

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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/