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  [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
books
• At least one
web site.
• At least two
articles in appropriate professional journals such as The Mathematics Teacher.

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

 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.

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:

1. A discussion of what you found most interesting or worthwhile in working on the project.
2. A discussion of what you encountered in working on the project that you would like to find out more about.
3. 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.

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, Great 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:

Book with More than One Author:

Book with Editor:

Book with More than One Editor:

Article from a Journal:

Article from a Book:

Website:

 Book Also Available at a Website

(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: