Mathematics 115: Mathematics
through its History
Main Topics
Number systems from various cultures
Using
the abacus
Alternative
algorithms for arithmetic
Pythagorean
Theorem
Pascal's
Triangle
Egyptian
mathematics (doubling, unit fractions, false position)
Pythagorean
pebble arithmetic (triangular, oblong, and square numbers)
Geometry in
Circles and approximations
of π in various civilizations
Geometric
algebra of Greeks and Arabs (including completing the square)
Women in
mathematics
Topology: Platonic solids & Euler’s
formula, map coloring, Moebius strips
Number systems from various cultures
Using
the abacus
Alternative
algorithms for arithmetic
Pythagorean
Theorem
Egyptian
mathematics (doubling, unit fractions, false position)
Pythagorean
pebble arithmetic (triangular, oblong, and square numbers)
Geometry in
Approximations
of π in various civilizations
Geometric
algebra of Greeks and Arabs (including completing the square)
Perspective
drawing
Topology: Platonic solids & Euler’s
formula, networks & Euler paths, map coloring,
Women in
mathematics Moebius strips
Pascal's
Triangle
Number systems from various cultures
Using
the abacus
Pythagorean
Theorem
Alternative
algorithms for arithmetic
Egyptian
mathematics
Geometry in
Approximations
of π
Geometric
algebra
Pascal's
Triangle
Early North
American arithmetic texts
Perspective
drawing
Division
algorithm and ID numbers
Knot theory
Graph theory
and map coloring
Women
in mathematics
Topics and
More Topics
Throughout the semester (at least
weekly):
Mathematical
games (include magic squares here) - Bree (Fridays)
Special
problems such as the river crossing, weighted coin, or cannibal problems
Women
mathematicians - Michaelene (References: Math
Equals, Teri Perl, more)
Topics/activities: Earliest evidences of counting: Ishango bone, etc.
Begin number
systems and their development (see tomorrow's topic)
Topics/activities: Number systems and their development (Babylonian/Mesopotamian, Egyptian, Greek, Maya, Chinese, Indo-Arabic, Inca quipus) - Nicole and Bree
Topics/activities: Abacus and counting board arithmetic (with
candy!) - Nicole and
Michaelene
Addition and
subtraction: alternate algorithms for subtraction, checking
by casting out 9's
Robert Recorde and the = sign (1500s)
Topics/activities: Multiplication and division: Egyptian doubling and halving - Bree
Other algorithms for
multiplication and division - Bree
Sumario Compendioso
(first math book published in
Mathematics
education throughout history
Topics/activities: Egyptian unit fractions and their use in
practical problems - Bree
Decimals, Simon Stevin (1500s) - Alissa
Practical
mathematics: Egyptian formulas for areas
and volumes,
including volume of a frustum of a pyramid - Michaelene and Rachel
What constitutes /
qualifies as a mathematical formula?
Topics/activities: Figurate numbers (Pythagorean pebble
arithmetic--with candy, of
course), perfect and amicable numbers, primes, Sieve of
Eratosthenes,
some additional Greek number theory - Michaelene
and Wendy
Do humans invent
or discover mathematics? Do animals do
math?
Topics/activities:
triangles, squares, pentagons, and hexagons; quadrature -
Wendy
What constitutes a
mathematical proof?
Further
applications of Greek geometry, including Eratosthenes'
measurement of the circumference of the Earth - Wendy
Tunnel of
Archimedes' volume
formulas (method of the lever) - Brian
Hypatia's
astrolabe - Wendy
Video: The Tunnel of
Topics/activities: Estimates of π by Eudoxus, Archimedes,
and others - Michaelene and
Rachel
Video: The Story of Pi
Topics/activities: Origins of the Pythagorean Theorem,
including picture proofs and
applications by early Chinese mathematicians; also Mesopotamian,
Egyptian,
Indian, and Greek versions - Michaelene and Rachel
Pythagorean
triples and Fermat's Last Theorem, including Sophie
Germain's
contributions to FLT
Videos: The Theorem of Pythagoras, The Proof
Topics/activities: Moebius strip and cutting one hole in a
sheet of notebook paper to form
a ring through which Brian can fit his body - Brian
(obviously!)
Building 3D models
of the Platonic solids (regular polyhedra) and
discovering Euler's formula - Rachel
Four-Color Theorem
- Rachel
Topics/activities: Graphs from
- Brian
Knot theory -
Nicole and Jen
Video: Life by the Numbers 3
Topics/activities: Solving linear equations - method of false
position, samples of problems
from different cultures and societies, negative numbers,
irrational
(incommensurable) numbers
Do you believe in
negative numbers? irrational numbers? imaginary nos.?
History of
arithmetic (again) and algebraic notation: how more succinct
notation made mathematics easier (and life harder?)
Topics/activities: Methods for finding square roots, including
Babylonian algorithm
- Rachel
Greek geometric
algebra (again), then geometric algebra in the Arab
world: al-Khwarizmi
and completing the square, quadratic formula
(give idea of solution of the cubic equation by 16th Century
Italian
algebraists by providing picture of completing the cube) -
Wendy,
Mike
Are pictures
better than words?
Topics/activities: Infinite series (including Zeno's
paradoxes, geometric series, Oresme's
series, geometric progressions, Gauss sum, harmonic series)
- Brian
Fibonacci numbers
- Bree
Topics/activities: "Pascal's" Triangle; Indian and
other combinatorics, esp. Bhaskara
Probability -
Native American dish game; gambling and Pascal - Brian
Topics/activities: Perspective in art and drawing, visualizing
the fourth dimension
- Nicole and
Rachel
And you thought
arithmetic was a boon to capitalism!
Video: Life by the Numbers 1
Topics to be
included in 4-unit version of course:
Modular
arithmetic (congruence), including bar codes and cryptography; begin with Maya
and other calendars
- Mike
Greek logic
and Lewis Carroll logic games - Alissa
Divisibility: Division Algorithm, Euclidian Algorithm
The parallel postulate and
non-Euclidian geometry
Symmetries and
tiling in various cultures
From tracing
out curves (e.g. witch of Agnesi) to coordinate geometry (Agnesi, Descartes)
Florence
Nightingale and graphical statistics (to aid in the prevention of disease)
The
Voting
methods: Condorcet, de Borda