Transcript 09-01-06-mini-DC - Fred Rickey's Home Page

Teaching a Course in the
History of Mathematics
Victor J. Katz
University of the District of Columbia
V. Frederick Rickey
U. S. Military Academy
Start
reading
now !
Seminar Rules Apply
• Ask any question at any time
• But, heed the schedule
• Email addresses are
[email protected]
[email protected]
Outline
I.
II.
III.
IV.
V.
How to Organize a Course
Approaches to Teaching History
Resources for the Historian
Student Assignments
How to Prepare Yourself
I. How to Organize a Course
1.
2.
3.
4.
5.
6.
7.
8.
9.
Who is your audience?
What are their needs?
What are the aims of your course?
Types of history courses
Textbooks for survey courses with comments
Textbooks for other types of courses
The design of your syllabus
Is a field trip feasible?
History of Math Courses on the Web
II. Approaches to Teaching History
1.
2.
3.
4.
5.
6.
Internal vs. External History
Whig History
The Role of Myths
Ideas from non-Western sources
Teaching ethnomathematics
Teaching 20th and 21st century
mathematics
III. Resources for the Historian
1. Books, journals, and encyclopedias
2. Web resources
3. Caveat emptor
IV. Student Assignments
1.
2.
3.
4.
5.
6.
Learning to use the library
What to do about problem sets?
Student projects
Possible student paper topics
Projects for prospective teachers
Exams
V. How to Prepare Yourself
1.
2.
3.
4.
5.
6.
7.
Start a reading program now!
Collect illustrations
Outline your course day by day
Get to know your library and librarians
Advertising your course
Counteract negative views
Record keeping
Are there other topics
you would like us to discuss?
• Note: We are not teaching history here
I. How to Organize a Course
1.
2.
3.
4.
5.
6.
7.
8.
9.
Who is your audience?
What are their needs?
What are the aims of your course?
Types of history courses
Textbooks for survey courses with comments
Textbooks for other types of courses
The design of your syllabus
Is a field trip feasible?
History of Math Courses on the Web
I.1. Who is your audience?
•
•
•
•
What level are your students?
How good are your students?
What type of school are you at?
How much mathematics or general history
do they know?
Answer: Not enough!
• Is the course for liberal arts students?
• What will they do after graduation?
I.2. What are their needs?
• If your students are prospective teachers,
what history will benefit them?
• Why are the students taking the course?
• How much “fact” do the students need to
know?
• Is this a capstone course for mathematics
majors that is intended to tie together what
they have learned in other course?
PROSPECTIVE
HIGH SCHOOL TEACHERS
• Teach more mathematics
• Make sure to deal with the history of topics
in the high school curriculum
• Discuss the use of history in teaching
secondary mathematics courses
• Stress the connections among various
parts of the curriculum
OTHER MATHEMATICS MAJORS
• History as a capstone course – helps to tie
together what they have learned
• Graduate school and academia
• Need to understand the development of
ideas and how to use these in future
teaching
• How and why abstraction became so
important in the nineteenth century
1.3. AIMS OF THE COURSE
•
•
•
•
•
To give life to your knowledge of mathematics.
To provide an overview of mathematics
To teach you how to use the library and
internet.
To indicate how you might use the history of
mathematics in your future teaching.
To improve your written communication skills
in a technical setting.
MORE AIMS
• To show that mathematics has been developed
in virtually every literate civilization in history, as
well as in some non-literate societies.
• To compare and contrast the approaches to
particular mathematical ideas among various
civilizations.
• To demonstrate that mathematics is a living field
of study and that new mathematics is constantly
being created.
I.4
•
•
•
•
•
•
Types of History Courses
Survey
Theme
Topics
Sources
Readings
Seminar
I.5 Survey Texts
•
•
•
•
•
•
•
•
•
Boyer
Burton
Calinger
Cooke
Eves
Grattan-Guinness
Katz
Hodgkin
Suzuki
1.6. Textbooks for other courses
• Dunham, Journey through genius: the great
theorems of mathematics
• Berlinghoff and Gouvêa, Math through the ages:
A gentle history for teachers and others
• Bunt, Jones, and Bedient, The historical roots of
elementary mathematics
• Joseph,The crest of the peacock: non-European
roots of mathematics
• Struik, A concise history of mathematics. New
York
Sourcebooks
• John Fauvel and Jeremy Gray, The
History of Mathematics: A Reader
• Ronald Calinger, Classics of Mathematics
• Jacqueline Stedall, Mathematics
Emerging: A Sourcebook 1540-1900
• Victor J. Katz, ed., The Mathematics of
Egypt, Mesopotamia, China, India, and
Islam: A Sourcebook
I.7. The design of your syllabus
•
•
•
•
•
•
•
Text
Aims
Outline
Readings
Assignments
Texts
Plagiarism
I. 8
Is a Field Trip Feasible?
• Visit a rare book room
• Visit a museum
• Visit a book store
I.9. HM Courses on the Web
Many individuals have placed information
about their courses on the web.
See the url on p. 1 of the handout, which
will take you to Rickey’s pages on this
minicourse.
Note especially the sources course of
Gary Stoudt, whose url is on p. 6 of the
handout.
II. Approaches to Teaching History
1.
2.
3.
4.
5.
6.
Internal vs. External History
Whig History
The Role of Myths
Ideas from non-Western sources
Teaching ethnomathematics
Teaching 20th and 21st century
mathematics
II.1. Internal History
•
•
•
•
Development of ideas
Mathematics is discovered (Platonism)
History written by mathematicians
Mathematics is the same, whether created
in Babylon, Greece, or France; i.e.,
mathematics is “universal”
II.1. vs. External History
•
•
•
•
Cultural background
Mathematics is invented
History written by historians
Mathematics influenced by ambient culture
(Story of Maclaurin)
• Biographies
II.2. Whig History
It pictures mathematics as progressively and
inexorably unfolding, brilliantly impelled along its
course by a few major characters, becoming the
massive edifice of our present inheritance.
Does history then only include ideas that were
transmitted somehow to the present or had
influence later on?
Or do we try to understand mathematical ideas in
context?
Examples of ideas that were
probably not transmitted
Indian development of power series
Babylonian solution of “quadratic equations”
Islamic work on sums of integral powers
Chinese solution of simultaneous
congruences
Gauss’s notebooks
Examples of ideas that probably
were transmitted
• Basic ideas of equation solving
• Trigonometry, both plane and spherical
• Basic concepts of combinatorics
II.3. The Role of Myths
• What myths do we tell?
• What myths do we want future teachers to
tell their students?
• Do we tell the truth and nothing but the
truth? (We cannot tell the “whole truth”.)
II.4. Ideas from
non-Western sources
Why non-Western Mathematics?
• Not all mathematics developed in Europe
• Some mathematical ideas moved to
Europe from other civilizations
• Relevance of Islam, China, India today
• Mathematics important in every literate
culture
• Compare solutions of similar problems
• Diversity of your students and your
students’ prospective students
Chinese Remainder Theorem
• Why is it called the Chinese Remainder
Theorem?
• The first mention of Chinese mathematics
in a European language was in 1852 by
Alexander Wylie: “Jottings on the Science
of the Chinese: Arithmetic”
• Among the topics discussed is the earliest
appearance of what is now called the
Chinese Remainder problem and how it
was initially solved in fourth century China,
in Master Sun’s Mathematical Manual.
Chinese Remainder Theorem
• We have things of which we do not know the number; if
we count them by threes, the remainder is 2; if we count
them by fives, the remainder is 3; if we count them by
sevens, the remainder is 2. How many things are there?
• If you count by threes and have the remainder 2, put
140. If you count by fives and have the remainder 3, put
63. If you count by sevens and have the remainder 2,
put 30. Add these numbers and you get 233. From this
subtract 210 and you get 23.
• For each unity as remainder when counting by threes,
put 70. For each unity as remainder when counting by
fives, put 21. For each unity as remainder when
counting by sevens, put 15. If the sum is 106 or more,
subtract 105 from this and you get the result.
Indian proof of sum of squares
A sixth part of the triple
product of the [term-count
n] plus one, [that sum]
plus the term-count, and
the term-count, in order,
is the total of the series of
squares. Being that this
is demonstrated if there is
equality of the total of the
series of squares
multiplied by six and the
product of the three
quantities, their equality is
to be shown.
Teaching a Course in the
History of Mathematics
Victor J. Katz
University of the District of Columbia
V. Frederick Rickey
U. S. Military Academy
Islamic Proof
• This example is taken from the Book on the
Geometrical Constructions Necessary to the
Artisan by Abu al-Wafā’al-Būzjānī (940-997).
He had noticed that artisans made use of
geometric constructions in their work. But, “A
number of geometers and artisans have erred in
the matter of these squares and their
assembling. The geometers [have erred]
because they have little practice in constructing,
and the artisans [have erred] because they lack
knowledge of proofs.”
Islamic Proof
I was present at some meetings
in which a group of geometers
and artisans participated. They
were asked about the
construction of a square from
three squares. A geometer
easily constructed a line such
that the square of it is equal to
the three squares, but none of
the artisans was satisfied with
what he had done.
Islamic Proof
Abu al-Wafa then presented one
of the methods of the artisans, in
order that “the correct ones may
he distinguished from the false
ones and someone who looks
into this subject will not make a
mistake by accepting a false
method, God willing. But this
figure which he constructed is
fanciful, and someone who has
no experience in the art or in
geometry may consider it
correct, but if he is informed
about it he knows that it is false.”
Islamic Proof
Why Was Modern Mathematics
Developed in the West?
• Compare mathematics in China, India, the
Islamic world, and Europe around 1300
• Europe was certainly “behind” the other three
• Ideas of calculus were evident in both India and
Islam
• But in next 200 years, development of
mathematics virtually ceased in China, India,
and Islam, but exploded in Europe
• Why?
II.5. Teaching Ethnomathematics
• Mathematical Ideas of “traditional peoples”
• What is a “mathematical idea”?
Idea having to do with number, logic, and
spatial configuration and especially in the
combination or organization of those into
systems and structures.
• Can these mathematical ideas of
traditional peoples be related to Western
mathematical ideas?
Examples of Ethnomathematics
• Mayan arithmetic and calendrical
calculations
• Inca quipus
• Tracing graphs among the Bushoong and
Tshokwe peoples of Angola and Zaire
• Symmetries of strip decorations
• Logic of divination in Madagascar
• Models and maps in the Marshall Islands
Books on Ethnomathematics
• Marcia Ascher, Ethnomathematics: A
Multicultural View of Mathematical Ideas
(1991)
• Marcia Ascher, Mathematics Elsewhere:
An Exploration of Ideas Across Cultures
(2002)
II.6. TEACHING 20TH AND 21ST
CENTURY MATHEMATICS
New concepts:
•
•
•
•
Set Theory and Its Paradoxes
Axiomatization
The Statistical Revolution
Computers and Computer Science
II.6. TEACHING 20TH AND 21ST
CENTURY MATHEMATICS
Recently resolved problems:
•
•
•
•
Four Color Problem
Classification of Finite Simple Groups
Fermat’s Last Theorem
Poincaré Conjecture
II.6. TEACHING 20TH AND 21ST
CENTURY MATHEMATICS
Unresolved problems:
• Hilbert’s 1900 list of Problems
–
Which Problems Are Still Unresolved?
See Ben Yandell, The Honors Class (2002)
• Clay Millennium Prize Problems
–
–
Riemann Hypothesis
Birch and Swinnerton-Dyer Conjecture
See K. Devlin, The Millennium Problems (2002)
III. Resources for the Historian
1. Books, journals, and encyclopedias
2. Web resources
3. Caveat emptor
Twenty Scholarly Books
• Jens Høyrup, Lengths, Widths, Surfaces: A
Portrait of Old Babylonian Algebra and Its Kin
(2002)
• Eleanor Robson – Mathematics in Ancient Iraq:
A Social History (2008)
• Kim Plofker – Mathematics in India: 500 BCE –
1800 CE (2009)
• Jean-Claude Martzloff, A History of Chinese
Mathematics, translated by Stephen S. Wilson
(1997)
Books
• Victor J. Katz, ed., The Mathematics of Egypt,
Mesopotamia, India, China, and Islam: A
Sourcebook (2007)
• D. H. Fowler, The Mathematics of Plato's
Academy: A New Reconstruction (1987, 1999)
• S. Cuomo, Ancient Mathematics (2001)
• Reviel Netz, The Transformation of Mathematics
in the Early Mediterranean World: From
Problems to Equations (2004)
Books
• J. Lennart Berggren, Episodes in the
Mathematics of Medieval Islam (1986)
• Glen Van Brummelen, The Mathematics of the
Heavens and the Earth: The Early History of
Trigonometry (2009)
• Jeremy Gray, Worlds Out of Nothing: A Course
in the History of Geometry in the 19th Century
(2007)
• Hans Wussing, The Genesis of the Abstract
Group Concept (1984)
Books
• Ivor Grattan-Guinness, ed. From the
Calculus to Set Theory, 1630-1910: An
Introductory History (1980)
• C. H. Edwards, The Historical
development of the calculus (1979)
• Judith V. Grabiner, The Origins of
Cauchy's Rigorous Calculus (1981)
• Stephen M. Stigler, The History of
Statistics (1986)
Books
• Gerd Gigerenzer et al, The Empire of Chance:
How Probability Changed Science and Everyday
Life (1989)
• Ed Sandifer, The Early Mathematics of Leonhard
Euler (2007)
• Robert Bradley and C. Edward Sandifer, eds.,
Leonhard Euler: Life, Work and Legacy (2007)
• Ivor Grattan-Guinness, ed., Landmark Writings
in Western Mathematics, 1640-1940 (2005)
Ten Popular Books
• Derbyshire, John. Prime Obsession: Bernhard
Riemann and the Greatest Unsolved Problem in
Mathematics, 2003.
• Dunham, William. The Calculus Gallery:
Masterpieces from Newton to Lebesgue, 2005
• Havil, Julian. Gamma: Exploring Euler’s
Constant, 2003.
• Maor, Eli Trigonometric Delights, 1998.
• Netz, Reviel and William Noel. The Archimedes
Codex, 2007
Popular Books
• Nahin, Paul J. An Imaginary Tale: The Story of –1,
1998.
• __________ When Least Is Best: How Mathematicians
Discovered Many Clever Ways to Make Things as Small
(or as Large) as Possible, 2003.
• Salsburg, David. The Lady Tasting Tea: How Statistics
Revolutionized Science in the Twentieth Century, 2001.
• Sobel, Dava. Longitude: The True Story of a Lone
Genius Who Solved the Greatest Scientific Problem of
His Time, 1995.
• Wilson, Robin. Four Colours Suffice: How the Map
Problem Was Solved, 2002.
III.1.a. Collection
Table of Contents
• Archimedes
• Combinatorics
• Lengths, Areas, and
Volumes
• Linear Equations
• Exponentials and
Logarithms
• Negative Numbers
• Functions
• Polynomials
• Geometric Proof
• Statistics
• Trigonometry
III.1.b. History Journals
•
•
•
•
•
•
Historia Mathematica
Isis
Archive for History of Exact Sciences
The British Journal for the History of Science
Annals of Science
History of Science
III.1.b. Popular Journals
•
•
•
•
•
•
The MAA journals
Mathematical Intelligencer
Physics Teacher
Scientific American
Mathematical Gazette
Bulletin of the British Society of the History
of Mathematics
III.1.c. Encyclopedias
• Companion encyclopedia of the history
and philosophy of the mathematical
sciences, edited by I. Grattan-Guinness
• Dictionary of Scientific Biography
• The Encyclopaedia Britannica, 11th ed
• The Encyclopedia of Philosophy
• Dictionary of American Biography,
III.2. WEB RESOURCES
These are so abundant and the
search engines are so good, that
it seems futile to attempt anything
comprehensive.
Here are a few especially useful
ones.
Rome Reborn
Earliest Extant Euclid, 888
www.ibiblio.org/expo/vatican.exhibit/Vatican.exhibit.html
The Euler Archive
http://www.math.dartmouth.edu/~euler/
E53 – Solutio problematis
ad geometriam situs pertinentis
http://www.math.ubc.ca/people/
faculty/cass/Euclid/byrne.html
http://www.lib.cam.ac.uk/
RareBooks/PascalTraite/
David Joyce’s History of
Mathematics Homepage
• http://aleph9.clarku.
edu/~djoyce/
mathhist/
IV. Student Assignments
1.
2.
3.
4.
5.
6.
Learning to use the library
What to do about problem sets?
Student projects
Possible student paper topics
Projects for prospective teachers
Exams
IV.1. Learning to use the library
• Tour of library
• Your mathematician
• Prize nomination
• Vita
IV.2. Problem Sets
• Solve a problem as it was solved in a
particular time period
• Complete the development of a particular
idea or procedure
• Solve an “old” problem using modern tools
and compare methods
• Generalize an “old” problem solving
procedure
Discussion Problems
• Compare and contrast methods
• Develop a lesson for the classroom based
on a particular historical development
• Discuss the pedagogy of an old textbook
IV.3. Projects
Written projects and/or oral reports?
Joint or individual projects?
IV.4. Possible Student Paper Topics
•
•
•
•
•
•
•
Bourbaki
Julia Robinson and Hilbert's tenth problem
Alan Turing
Dürer's Polyhedra
The Four Color Problem
Holbein's Ambassadors
Daniel Bernoulli & the spread of smallpox
IV.5. Projects for Prospective
Teachers
• Compare the Babylonian, Mayan, and Hindu-Arabic
place-value systems in their historical development and
their ease of use. Devise a lesson on this.
• Analyze the history of the limit concept from Eudoxus to
the mid-eighteenth century, including Berkeley's
criticisms and Maclaurin's response. Create a lesson.
• Discuss how the history of the solution of cubic
equations from the Islamic period through the work of
Lagrange can be used in algebra classes.
• Compare the teaching of algebra (or geometry) in the
eighteenth century and the twentieth by studying
textbooks.
IV. 6. Exams
•
•
•
•
Mathematical Problems
Short Answer Questions
Multiple Choice Questions
Essay Questions
Mathematical Problems
1. Translate a Babylonian problem and solution
into modern terms
2. Solve a quadratic problem of Abu Kamil by first
converting it into one of the six types of
quadratic equations and then using the method
for that type. For example, suppose 10 is divided
into two parts, each one of which is divided by
the other, and the sum of the quotients is 4 ¼.
Find the two parts.
3. Use Fermat’s method to find the maximum of
bx – x3
Mathematical Problems
4.
Find the relationship of the fluxions of x and y
on the curve x2 + xy + y3 = 7 using one of
Newton’s procedures.
5. Derive the quotient rule of calculus by an
argument using differentials.
6. Give a geometric argument using differentials
and similar triangles to show that
d(sin x) = cos x dx.
7. Show that one can solve the cubic equation
x3 + d = bx2 by intersecting the hyperbola
xy = d and the parabola y2 + dx – db = 0.
Short Answer Questions
1. Order the following mathematical discoveries by
their approximate date, beginning with the
earliest:
a. Solution of the system of equations which we express
as xy = a, x + y = b.
b. Earliest explicit expression of the multiplicative rule
for combinations.
c. Development of the base 60 place value system.
d. Development of the base 10 place value system.
e. First statement of the parallel postulate.
f. First extant rigorous proof of the rule for combinations
expressed in b.
Short Answer Questions
2.State one mathematical contribution of
each of the following: Cardano, Bombelli,
Viete, Harriot.
3.What is Cardano’s Ars Magna and why is
it important?
4. Trigonometry was originally developed to
_____________________.
Essay Questions
1. Outline the major contributions to trigonometry of the
civilizations of Greece, India, and Islam.
2. Today, mathematics is often thought of as the intellectual
exercise of proving theorems using logical reasoning and
beginning with explicitly stated definitions and axioms.
Were the Babylonians and the Egyptians, then, “doing
mathematics”? Explain.
3. Describe the proof method called today the method of
mathematical induction. Was the proof by Levi ben
Gerson giving the formula for the number of
permutations of a set of n objects a proof by
mathematical induction? Why?
Essay Questions
4. Is mathematics invented or discovered? Discuss with
reference to at least four mathematical concepts
discussed this semester.
5. Compare and contrast Newton’s and Leibniz’s versions
of the calculus. In your answer, include ideas in
differentiation, integration, solving differential equations,
and applications to physical problems.
6. Compare and contrast the use of axioms by Euclid with
the use of axioms around the turn of the twentieth
century. For the latter period, you may pick one or two
axiom sets to provide a focus for your discussion.
7. Why is “rigor” in analysis so important? After all,
Newton and Leibniz worked out the basics of the
calculus without it. Give examples to support your
argument.
V. How to Prepare Yourself
1.
2.
3.
4.
5.
6.
7.
Start a reading program now!
Collect illustrations
Outline your course day by day
Get to know your library and librarians
Advertising your course
Counteract negative views
Record keeping
Start
reading
now !
V.1. Start a reading program now!
• Read a survey text
• Read journal articles
• Read deeply in the history of a
mathematical field you know well
How to Learn More History
• Go to talks at meetings
• Join the Canadian Society for the History
and Philosophy of Mathematics and the
British Society for the History of
Mathematics
• Join/start a seminar in the history of
mathematics
V.2. Collect illustrations
•
•
•
•
•
Pictures of mathematicians.
Title pages of famous works.
Significant pages from important works.
Maps.
Quotations from famous mathematicians.
Emilie du Chatelet
Pacioli’s Summa
Reisch’s Margarita Philosophica
V.3. Outline Your Course Day by Day
• Decide on nature of course (Chronological,
Themed, Combination)
• Pick out “key” general concepts and order them
• For each key concept, pick out specific topics to
cover
• Choose a topic or topics for each available day
• Pick materials related to each chosen topic
• Give yourself flexibility, for undoubtedly you will
have planned too much
Descartes
Outline for day xx
Descartes on Analytic Geometry
• Biographical information worth mentioning:
– Attended a good school. Recent work of Galileo and
his telescope was discussed.
– A sickly lad. Lay in bed.
• Fly and analytic geometry. True?
– Importance of contact with the Dutch. Latin.
– Queen Christina of Sweden. Death.
Scientific work of Descartes
•
•
•
•
•
•
•
•
•
•
Philosophia mathematica. Newton used this title.
Method. Cogito ergo sum. Tell Ari Katz joke.
Geometry is an appendix. You can read it. Translations.
Optics: Snell's law, rainbow.
Started analytic geometry. Oblique axes. xyz for variables. Exponent
notation
Curves: geometric vs. mechanical. Examples.
He went from geometry to algebra, not v.v.
Had a method to solve any problem (and Newton believed him!).
Folium of Descartes. Fermat has a better method for tangents.
Says you can't do arc length. Set up for van Heuraet and Newton.
Powerpoint to Prepare
•
•
•
•
Portraits:
• Seated, Schooten, stamps, Vic Norton's cartoon on aliasing.
Title pages:
• Geometrie: 1637, 1649, 1659. Newton read the second Latin edition.
• Translations: Smith-Latham, Olscamp.
Quotations:
• Rules of problem solving. For prospective teachers especially.
• Newton on reading Descartes.
• This method will solve all problems.
Selected Pages:
• Solution of quadratic equations.
• Conchoid is geometric.
• Finding tangents by the two circle method.
• Heuraet on arc length from second Latin edition (1659).
• Folium of Descartes. 1638 definition. Graphs of Newton and
L'Hospital.
• Things to take to class to pass around:
– Smith-Latham translation of Geometry.
– Olscamp translation of the whole Method.
– Polya's How to Solve It.
• Things to read before class:
– Sections of the text the students are to read.
– The DSB article on Descartes.
– Look at J. F. Scott's, The scientific work of
René Descartes, 1952
– Read section on Descartes in GrattanGuinness’s Landmarks
V.4. Get to know your
library and librarians
•
•
•
•
•
•
Look at every book
Find the specialized librarians
Tell them your interests
Ask them to help your students
Be determined to find answers
Visit a rare book room
V.5. Advertising your course.
•
•
•
•
Talk to former students
Send email to majors
Post flyers
Talk to colleagues
V.6. Counteract Negative Views
• Among some mathematicians the history of mathematics
is not regarded as a serious pursuit.
• It is worth your while to spend some time talking to your
colleagues about your course. Point out to them that you
are doing significant amounts of mathematics in your
course (give some illustrations). Point out that it is not a
course in anecdotes.
• Students must master a great deal of material and they
are required to write about mathematics in a way that
shows that they have mastered the details.
V.7 Record keeping
• Immediately record full reference for items
you photocopy
• Record what references you use for each
class
• Record how you could improve the class
(and what not to do again)
Start
reading
now !
PowerPoint and additional information
available at
http://www.dean.usma.edu/
departments/math/people/
rickey/hm/mini/default.html
Advertisement
Using Historical Topics of Mathematics
Effectively in the High School Curriculum
David Kullman (Miami University of Ohio) will
host a workshop on June 23-25, 2009 at Bluffton
University. The cost is $125 which includes
lunches.
Contact Duane Bollenbacher at
[email protected] or 419-358-3296
Further information will appear on the webpage
of the Ohio Section of the MAA.
Advertisement
History of Undergraduate Mathematics
in America (HUMA II)
Meeting is planned for the summer of 2010
at West Point.
Your research contributions are welcome.
Contact Fred Rickey.