Mathematics 115: Mathematics through its History

Main Topics†††

 

Interim 2002

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 Euclid's Elements (axiomatic system, constructions, 2-column proofs)

Circles and approximations of π in various civilizations

Geometric algebra of Greeks and Arabs (including completing the square)

Women in mathematics†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †††††††††††

Topology:Platonic solids & Eulerís formula, map coloring, Moebius strips

 

Interim 2000

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 Euclid's Elements (axiomatic system, constructions, 2-column proofs)

Approximations of π in various civilizations

Geometric algebra of Greeks and Arabs (including completing the square)

Perspective drawing

Topology:Platonic solids & Eulerís formula, networks & Euler paths, map coloring,

Women in mathematics†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Moebius strips

Pascal's Triangle

 

Interim 1999

Number systems from various cultures

Using the abacus

Pythagorean Theorem

Alternative algorithms for arithmetic

Egyptian mathematics

Geometry in Euclid's Elements

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

††††††††††††††††††††††††††††† Tablas para los ninos (first arithmetic printed in California)

††††††††††††††††††††††††††††† 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 New World)

††††††††††††††††††††††††††††† 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:†††† Euclid's Elements:geometry, including basic definitions; constructing

†††††††††††††††††††††††††††††††† 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 Samos - Brian

††††††††††††††††††††††††††††† Archimedes' volume formulas (method of the lever) - Brian

††††††††††††††††††††††††††††† Hypatia's astrolabe - Wendy

Video:††††††† The Tunnel of Samos

 

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 Africa and the South Pacific, Konigsberg Bridge Problem

†††††††††††††††††††††††††††††††† - 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 U.S. census:counting vs. statistics

 

Voting methods:Condorcet, de Borda

 

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