M326K PROJECT: HISTORY OF NUMBERS TIMELINE Spring 2004

**Final Project Due Date**: *Thursday, February 19*

**Progress Report Due Date**: *Thursday, February 5*

**Purpose**:

- To become familiar with the history of numbers and mathematical notation.
- To become familiar with resources in the history of mathematics.
- To become aware of how many aspects of mathematics that may have become second nature to you were important , significant discoveries or inventions at some point in history.

*Note*: Standard VI of the Texas State Board for Educator Certification
states: "The mathematics teacher understands the historical development
of mathematical ideas, the interrelationship between society and mathematics,
the structure of mathematics, and the evolving nature of mathematics and
mathematical knowledge." This standard is the basis for the UTeach Portfolio
requirement to "Illustrate knowledge of the history and cultural context
of mathematics; in particular, the evolution of mathematical concepts." This
project should give UTeach students a good start on this portfolio requirement.

**Overview**: This project is to be done* in groups of three students*.
(If the class size is not evenly divisibly by 3, then one or two groups
of two will be allowed.)

- All group members should participate in researching the information to go into the project
- All group members should participate in the creation of the final product.

Your assignment is to create a timeline of the development of number concepts and notations. The format of your timeline is up to you. It might be a paper, a scroll, an accordion fold, a notebook with some fold-out pages, a web site, or some combination of these. The only constraint on format is that it must be something I can readily carry to my office or access on the computer. In particular:

- Posters and oral presentations are
*not*acceptable. - Since PowerPoint is a medium for including visuals in an oral presentation,
it is
*not*a good choice for this project. - If you give your project to me on a computer disk, it must be either a CD or ZIP disk readable on a Macintosh

**Progress Report**: You will need to turn in a *progress report*
on *Thursday, February 5*. Each group member should have researched
at least one print source by then. Your progress report should include

- the names of all group members
- the tentative format
- a brief summary of your research so far and what still remains to
be done. This should include which resources
*each*group member has consulted so far.

**Minimum Requirements**:

1. In your timeline, you *must include* the first use (time and place)
in Western culture of *each* of the following. (You may not be able
to pinpoint all of these exactly, but should be able to give approximate
dates and places.) For our purposes, "Western culture" is defined as including
Mesopotamia, Egypt, Europe , and North America after European influence.
(This definition is intended to include the development of the mathematics
taught today in North America, Europe, and many other parts of the world.)

- The base 10 place value system. (Caution: Not every base ten system is a place value system. Not every place value system is a base ten system.)
- The symbol + for addition.
- The symbol - for subtraction.
- The symbol x for multiplication.
- The symbol for division.
- The concept of fraction
- Modern fraction notation (e.g., 3/4 or ).
- Decimal notation for numbers less than one. (i.e., use of the base ten place value system for numbers less than one.)
- The decimal point.
- The current standard algorithm for multiplication.
- The current standard algorithm for division. (Caution: There is something else that is called the "division algorithm. Be sure not to confuse these.)
- Zero.
- Negative numbers.
- The equal sign.
- Letters to stand for variables or unknowns.
- Exponents
- Irrational numbers.
- Complex numbers.
- Prime numbers.
- Pi.
- The number e (base of natural logarithms).
- Transcendental numbers other than _ [should be "pi" -- but might not appear correctly on your browser] and e.
- The idea of an infinite set.
- The Â [should be capital Greek Sigma -- but might not appear correctly on your browser] notation for sums.
- The symbol • [should be the standard infinity symbol -- but might not appear correctly on your browser] to mean infinity or infinite.

2. Since one purpose of this project is to become familiar with a variety of resources in the history of mathematics, you are required to use the following variety of resources in researching your project:

- At least three
- At least one
- At least two

**->** **See more information below on what references are or are
not acceptable.<-**

3. You must give a reference for each fact you mention.

**->** **See more information below on how to do this.<-**

4. Exact quotes must be placed in quotation marks. (A long quote
may be indented from surrounding text, with an introduction clearly indicating
that it is a quote.) An exact reference (including page numbers) must be
provided for each quote. *Copying material without providing proper indications
that it is a quote or without giving the source(including page number) is
plagiarism and is considered academic dishonesty.*

5. You must include *a list of the references you used*.

**->** **See more information below on the appropriate format
for references.<-**

6. You must also include in your project or hand in with it* a discussion
of the implications of what you have learned in your project for teaching
and learning mathematics.*

7. One project is to be turned in for the entire group. *In addition,
each group member must write and turn in individually the following items:*

- A discussion of what you found most interesting or worthwhile in working on the project.
- A discussion of what you encountered in working on the project that you would like to find out more about.
- A discussion of what each group member contributed to the project.

** **

**Grading**:

In most cases, all students in a group will receive the same grade. However, in cases where there is convincing evidence of very uneven participation, different students may be given different grades. Also, a student who omits their individual part described in item 7 , or whose individual report is below par, may receive a lower grade than the other students in the group

A "C" project will satisfy all of the minimum requirements (items 1 - 7 above) adequately, with at most 3 omissions or errors in the twenty-five required "first uses", and will be organized and written well enough to convey information clearly without distracting additions, errors or sloppiness.

An "A" project will meet all the requirements for a "C" grade and in addition
include at least two of the following features *in substantial amounts*:

- Discusses the difficulties involved in the development and acceptance of some of the concepts
- Includes cultures other than Western.
- Includes additional topics in numbers and operations.
- Shows how the development of the mathematical ideas interacted with other cultural events at the time.
- Very well organized, formatted, and written.

A "B" project will be between a C and an A project.

A "D" or "F" project will not meet the minimum requirements. In particular, a project involving plagiarism will receive a grade of D or F.

MORE ABOUT REFERENCES

I. *Books*

A. The following books have been put on reserve at the PMA library (fourth
level of RLM) for this class. They may be checked out overnight. Please
do not check out too many of these books at once, so others may use them.
*If you have difficulty checking out some books on this list, please let
me know, so I can change them to library use only*.

Bashmakova and Smirnova, *The Beginnings and Evolution of Algebra*

Berlinghoff and Gouvea, *Math through the Ages; A Gentle History for
Teachers and Others*

Cajori, *History of Elementary Mathematics*

Cajori, *History of Mathematical Notation*, Vol. 1

Cajori, *History of Mathematics*

Danzig, *Number: The Language of Science*

Eves, G*reat Moments in Mathematics *

Eves, *Introduction to the History of Mathematics *

Gazale, *Number:* *From Ahmes to Cantor*

Gillings, *Mathematics in the Time of the Pharoahs*

Gittlen, *History of Mathematics*

Joseph, *The Crest of the Peacock: Non-European Roots of Mathematics*

Katz, *A History of Mathematics*

Martzloff, *A History of Chinese Mathematics*

NCTM, *Historical Topics for the Mathematics Classroom*

Smith, *History of Mathematics, Vols. 1,2*

Swetz, Frank, *Capitalism and Arithmetic*

Swetz, Frank, *From Five Fingers to Infinity*

Wilder, *Evolution of Mathematical Concepts*

B. The following additional sources are also on reserve at the UT libraries indicated:

*Companion encyclopedia of the history and philosophy of the mathematical
sciences*.

(QA 21 C645 1994), on permanent reserve in both the PCL and PMA libraries is

Suzuki, Jeff, *A History of Mathematics*, on reserve at UGL for M
315C.

C. You do not need to restrict yourself to the books listed above. However,
the following books are **not** acceptable references, since they contain
inaccuracies:

Bell, *Men of Mathematics*.

Boyer, Carl, *A History of Mathematics
*

Ifrah, *From One to Zero*

Ifrah, *The Universal History of Numbers*

Menninger, *Number Words and Number Symbols*

II. *Web sites*

A. Here are some suggestions of web sites you **may** use:

David E. Joyce's History of Mathematics web site, http://aleph0.clarku.edu/~djoyce/mathhist/mathhist.html

The MacTutor History of Mathematics Archive,

http://www-history.mcs.st-and.ac.uk/~history/

Jeff Miller's Earliest Uses of Various Mathematical Symbols web site, http://members.aol.com/jeff570/mathsym.html

Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics web site

http://members.aol.com/jeff570/mathword.html

History of Negative Numbers web site, http://www.ma.utexas.edu/users/mks/326K/Negnos.html

B. Since web sites vary tremendously in quality*, if you want to use
a web site other than the ones above*, *you must check it out with
me before you use it.*

C*. *You may **not** use a web site written by a student as a
reference for your project.

III. *Professional Journals*

Here are some examples of professional journals that you might use as
references. *If you wish to use other periodicals as sources, be sure to
check them out with me for suitability first.*

*The Mathematics Teacher*. This is the professional journal for secondary
math teachers.

- You can find a list of
*Mathematics Teacher*articles on math history online at http://my.nctm.org/eresources/MT/index/subjects/history.asp - Back issues of this journal are at the PCL library.
- Some current issues are in the UTeach Resource Center in Painter Hall.
- Some articles from recent years can be accessed online. To find these, look up "Mathematics Teacher" in UTNetCat and click on the appropriate link.
- I have back issues through 1989 and might be able to loan some out for short periods.

*Mathematics Teaching in the Middle Grades*. This is the professional
journal for middle grades math teachers.

- You can find a list of
*Mathematics Teaching in the Middle Grades*articles on math history online at http://my.nctm.org/eresources/MTMS/index/subjects/history.asp - Back issues of this journal are at the PCL library.
- Some articles from recent years can be accessed online. To find these, look up "Mathematics Teaching in the Middle Grades" in UTNetCat and click on the appropriate link.

*The College Mathematics Journal.* This is a journal aimed at students
and their teachers in the first two years of college.

- You can find a list of
*College Mathematics Journal*articles in recent years and a link to a searchable database online at http://www.maa.org/pubs/cmj.html - Issues since 1984 are in the PMA library
- Older issues are in the PCL library.

*Mathematics Magazine* publishes articles related to college mathematics.

- Contents of recent articles and a link to a searchable database are online at http://www.maa.org/pubs/mathmag.html
- Available in the PMA library
- Some older volumes available online through UTCat.

*The American Mathematical Monthly*. This contains a wide range of
expository articles on mathematics

- Contents and article summaries from recent years are online at http://www.maa.org/pubs/monthly.html
- The journal is available in the PMA library.

*The Mathematical Intelligencer* publishes expository articles about
mathematics, mathematicians, and the history and culture of mathematics.

- Issues through 2000 are in the PMA library
- Some articles are available online through UTCat

IV. **How to List References . **Use the format for references that
is used in the publications of the National Council of Teachers of Mathematics.
Here are some examples:

*Book with Single Author*:

Boyer, Carl, *A History of Mathematics*.

*Book with More than One Author*:

Boyer, Carl B. and Utta C. Merzbach. *A History of Mathematics*.
2^{nd} ed. New York: John Wiley & Sons, 1989.

*Book with Editor*:

Calinger, Ronald, ed., *Classics of Mathematics*. Englewood Cliffs,
N.J.; Prentice Hall, 1995.

*Book with More than One Editor*:

Swetz, Frank, John Fauvel, and Otto Bekken, eds. *Learn From the Masters!*
Washington, D.C.: Mathematical Association of America, 1995.

*Article from a Journal*:

Espenshade, Pamela H. "A Text on Trigonometry by Levi Ben Gershon 1288
- 1344." *Mathematics Teacher* 60 (October 1967): 628 - 37.

*Article from a Book*:

Bowsher, Lester E. "History of the Terms 'Ellipse,' 'Hyperbola,' and
'Parabola.' " in *Historical Topics for the Mathematics Classroom, *pp.
222 - 24. Reston, Va.: National Council of Teachers of Mathematics, 1989

*Website*:

"The MacTutor History of Mathematics Archive." www-history.mcs.st-and.ac.uk/~history/

* Book Also Available at a Website*

National Commission on Excellence in Education. *A Nation at Risk:
The Imperative for Educational Reform*. National Commission on Excellence
in Education, 1983. Available at www.ed.gov/pubs/NatAtRisk. World
Wide Web.

(You can find more examples at http://my.nctm.org/eresources/mt/reference_mt.asp
or by looking in recent copies of the *Mathematics Teacher*.)

V. **How to give references for facts.** Use the format used in the
publications of the National Council of Teachers of Mathematics. Example:

*If you are stating that the first use of the word xirxium was in Austin,
Texas on December 16, 1982, and your source for this information was p.
13 of the book by Jones published in 1984 that is listed in your list of
references, you would write:*

The first use of the word xirxium was in Austin, Texas on December 16, 1982 (Jones, 1984, p. 13).

If there are two books or articles listed in your list of reference that are both by Jones and both published in 1984, you need to list their dates in your list of references as 1984a and 1984b; then you refer to these books as (Jones, 1984a) and (Jones, 1984b)