Review for the Final Exam
Multicultural Mathematics
The final exam will be Wednesday, December 10th, 2:004:00pm
in Science Center 109. There will be a review session on Tuesday,
December 9th, 11am1:30pm, in Barbelin 221. I will also be in the
Learning Resource Center to tutor you on Tuesday from 3:00 to 5:00pm.
You may bring your calculator and two 3"x5" index cards with
your notes.
The final will consist of multiple choice and short answer questions
covering the mathematics of drumming, and essay questions covering the
whole course. You should review all the handouts from this section,
plus Homework #4 and the course packet p. 371380. Most of the handouts
may be printed from the
web siteif you are missing one please ask me for a copy.
Essay Questions: You can choose either two or three
of these essays, depending on how many multiple choice and partial credit
questions you do.

A 15th century writer called zero ``a sign which creates confusion
and difficulties'' (Menninger, 1957). More historical evidence
of difficulties with zero is detailed in p. 1914 in your couse packet,
and elsewhere. Explain, with examples, what confusions were caused
by zero. Given these difficulties, why was zero eventually adopted?

Read Liping Ma's article Exploring new knowledge: the relationship
between perimeter and area, on p. 287298 of your course packet.
On p. 297, she concludes that
In responding to the student's novel claim about the relationship
between perimeter and area, the U.S. teachers behaved more like laypeople,
which the Chinese teachers behaved more like mathematicians (Ma, 1999).
What is the essential difference, as Ma sees it, between a mathematician
and a layperson? Use specific examples, either from the article or
from elsewhere, to illustrate your answer. (Hint: you could
refer to our class discussion about the difference between a proof and
empirical observation.)

Reread Marcia Ascher's introduction to her book Ethnomathematics (it
was the last three pages of the first handout I gave youthe handout begins
with my article ``A Course in Multicultural Mathematics''). Read
Joseph's definition of ``modern mathematics'' (CP p. 18, under ``The Development
of Mathematical Knowledge''). What is ethnomathematics, as defined
by Ascher? Give specific examples of ethnomathematical topics
we discussed in class. What are the differences and similarities
between ethnomathematics and Joseph's definition of ``modern mathematics?''
Is ethnomathematics really mathematics? Give reasons for your opinion.

Read the regional studies from Africa Counts (CP p. 6195).
Choose either Southwest Nigeria or East Africa. What roles do mathematical
ideas play in this culture? For each role, can you find a parallel
in our own culture? Illustrate with specific examples.

Read the short story ``The Library of Babel'' by Jorge Luis Borges
(p1 p2 p3
p4). What mathematical ideas are the inspiration
for this story? How has Borges conveyed a sense of these mathematical
ideas? Be specific.
Topics to review

The mathematics of poetry

How Hemacandra counted the duration of a line using long syllables=2
and short syllables=1

How to write all the meters of a given duration

The Fibonacci numbers (what are they? what is the pattern?)

The correspondence between the Fibonacci numbers and the number of
poetic meters of a given duration

Frequency and period

The definitions of frequency and period

The relationship between frequency and period

The connection between tempo and frequency

Musical terms You should know the meaning of each term,
and be able to give an example of each

beat

note

drumhit

rest

rhythm pattern

duration

measure

time signature

tempo

rhythm cycle

complement

tihai

arithmetic series

polyrhythm

abcdrums notation

Understand the meanings of notation like dzz, (dzz)3, ((dzz)3
dd)2, z3, AABA, A4, (AABC)5

How to write out patterns in long form (for example, know that
(dzz)3 = dzzdzzdzz; or if A=dzd and B=dd, then AABA=dzddzddddzd)

How to compute the duration of a pattern, in number of beats (for example,
the duration of ((dzz)3 dd)2 is (3x3 + 2)x2 = 11x2 = 22; or if A=dzd
and B=dd, then the duration of AABA is 3+3+2+3=11)

Rhythm cycles

How to expand a rhythm cycle

How to write a rhythm cycle if the expanded form is given (for
example, dzzdzzdzz... = (dzz) )

How to recognize equivalent rhythm cycles

Modular arithmetic

How to recognize two numbers that are equivalent modulo some number

The relationship between modulo m and drumming with m
beats per measure

Modulo and the tihai ending

Arithmetic series

How to recognize in a composition

How to compute the duration of an arithmetic series (in music)

Summing a (mathematical) arithmetic series of the form 1+2+3+4+...+n
using the formula sum = (n+1)n/2.

Polyrhythms

How to recognize a polyrhythm of the form m against n
(for example, 3 against 4)

How to write a polyrhythm for two instruments in abcdrums notation

How to compute the number of beats needed to write out a polyrhythm

The least common multiple and its relationship to polyrhythms

True polyrhythms

Compositions

How to recognize the following in the context of a composition:
rhythm cycle, polyrhythm, arithmetic series, tihai ending

How to determine the total duration of the composition, and check that
all instruments end at the same time

How to determine the number of beats in the rhythm cycle, which we
defined to be the time signature