Review for the Final Exam
Multicultural Mathematics

The final exam will be Wednesday, December 10th, 2:00-4:00pm in Science Center 109.  There will be a review session on Tuesday, December 9th, 11am-1:30pm, 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. 371-380.  Most of the handouts may be printed from the web site-if 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.

  1. 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. 191-4 in your couse packet, and elsewhere.  Explain, with examples, what confusions were caused by zero.  Given these difficulties, why was zero eventually adopted?
  2. Read Liping Ma's article Exploring new knowledge:  the relationship between perimeter and area, on p. 287-298 of your course packet.  On p. 297, she concludes that

  3. 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.)

  4. Reread Marcia Ascher's introduction to her book Ethnomathematics (it was the last three pages of the first handout I gave you--the 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.
  5. Read the regional studies from Africa Counts (CP p. 61-95).  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.
  6. 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

  1. The mathematics of poetry
    1. How Hemacandra counted the duration of a line using long syllables=2 and short syllables=1
    2. How to write all the meters of a given duration
    3. The Fibonacci numbers  (what are they? what is the pattern?)
    4. The correspondence between the Fibonacci numbers and the number of poetic meters of a given duration
  2. Frequency and period
    1. The definitions of frequency and period
    2. The relationship between frequency and period
    3. The connection between tempo and frequency
  3. Musical terms  You should know the meaning of each term, and be able to give an example of each
    1. beat
    2. note
    3. drumhit
    4. rest
    5. rhythm pattern
    6. duration
    7. measure
    8. time signature
    9. tempo
    10. rhythm cycle
    11. complement
    12. tihai
    13. arithmetic series
    14. polyrhythm
  4. abcdrums notation
    1. Understand the meanings of notation like  dzz, (dzz)3, ((dzz)3 dd)2, z3, AABA, A4, (AABC)5
    2. 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)
    3. 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)
  5. Rhythm cycles
    1. How to expand a rhythm cycle
    2. How to write a rhythm cycle if the expanded form is given  (for example, dzzdzzdzz... = (dzz) )
    3. How to recognize equivalent rhythm cycles
  6. Modular arithmetic
    1. How to recognize two numbers that are equivalent modulo some number
    2. The relationship between modulo m and drumming with m beats per measure
    3. Modulo and the tihai ending
  7. Arithmetic series
    1. How to recognize in a composition
    2. How to compute the duration of an arithmetic series (in music)
    3. Summing a (mathematical) arithmetic series of the form 1+2+3+4+...+n  using the formula sum = (n+1)n/2.
  8. Polyrhythms
    1. How to recognize a polyrhythm of the form m against n  (for example, 3 against 4)
    2. How to write a polyrhythm for two instruments in abcdrums notation
    3. How to compute the number of beats needed to write out a polyrhythm
    4. The least common multiple and its relationship to polyrhythms
    5. True polyrhythms
  9. Compositions
    1. How to recognize the following in the context of a composition:  rhythm cycle, polyrhythm, arithmetic series, tihai ending
    2. How to determine the total duration of the composition, and check that all instruments end at the same time
    3. How to determine the number of beats in the rhythm cycle, which we defined to be the time signature