HISTORY OF MATHEMATICS

Math 311. Spring 1998. Prof. Rickey

Textbook: A History of Mathematics. An Introduction, by Victor J. Katz, HarperCollins, 1993. Read it carefully before class so you are prepared to ask questions and to participate in discussions. Of course you should read it again later very carefully. We will cover most of the book, including some topics from the twentieth century.

Office Hours: I will be in my office, Room 406 MSC, Monday and Wednesday from 1:30 to 3:00. If my office hours change unexpectedly due to events beyond my control I will do my best to notify you by email. You can also talk to me after class. Other times are available by appointment. You are encouraged to stop in -- it is part of your education.

Aims of the Course:

  1. To give life to your knowledge of mathematics.
  2. To provide an overview of mathematics -- so you can see how your various courses fit together and to see where they come from.
  3. To teach you how to use the library and internet (important tools for life).
  4. To show you that mathematics is part of our culture.
  5. To indicate how you might use the history of mathematics in your future teaching.
  6. To improve your reading and analytic skills, especially in a technical situation.
  7. To improve your oral and written communication skills in a technical setting.

Remark: You should think about why you signed up for this course and what you hope to get out of it. If your goals are different than mine, let me know and perhaps I can accommodate you.

Stress of the Course: This course is designed as a survey of the history of mathematics. But far too much mathematics has been done in the past 4000 years to treat its entire history carefully, so we will concentrate on one theme: The development of calculus. To do this we will have to discuss the development of algebra, geometry, and trigonometry. Consequently we will discuss the history of most of the mathematics that is discussed in the high schools; this is intended to benefit the prospective teacher. We will not discuss many topics in modern mathematics.

Automathography via email: Each of you is required to get an email account so that I can communicate with you outside of class, and so that you can communicate with each other via a mail list called mathchat2. The first assignment involving email is to send me an automathography. You are to introduce your mathematical self to me. Tell me about the courses you have taken, what your favorites were, what you find hardest. Explain what your mathematical interests are, and what you plan to do after graduation. Reveal why you signed up for this course and what you expect to get out of it. If your aims for taking this course are different than those stated above, please let me know. If you have any anxieties about this course, or any special problems or needs, let me know. You are encouraged to be creative in your response; don't be pedantic and just answer the questions asked above; include whatever you wish. There are three purposes for this assignment: to make sure that you can use email, to introduce yourself to me, and for me to get some idea about your writing skills. Send your response by email by Friday January 16 (Yes, I realize this is not a class day).


Woops, forgot to include my email address. Here it is [well, my current email]: fred-rickey@usma.edu .

First Library Assignment: The purpose of these two assignments is to help you to learn your way around the library and the computer library catalogues. At first they may seem like busywork, but you will appreciate them when you are working on your paper.

For the first assignment you will be given the name of a mathematician and be asked to find out what you can about him or her. The questions asked on the sheet giving the name of your mathematician are designed to acquaint you with a few of the most useful reference works in the library dealing with the history of mathematics. Please answer the questions on this sheet and return it. [Each student was given a page with the name of Your Mathematician and questions to answer. They were also given a sample biography of Al-Khwarizmi.]

Some of the questions ask you to look up works by and about the mathematician. Your records on these can be turned in on 4 by 6 index slips (cards are too bulky; other sizes are unacceptable), but I would strongly encourage you to send them to me via email. Each should contain the author's name, title, date of publication (the publisher and place of publication are usually not of much interest), the library call number and any other information that is of interest to you. You should use the following format (for a book):

Klein, Felix (1849--1925)
The Evanston Colloquium. Lectures on Mathematics,
New York: Macmillan, 1894
Call Number: BGSU: SL 510.8 K63e

English translation of twelve expository lectures given at Northwestern University in conjunction with the Chicago World's Fair in 1893. One contains a new proof of the transcendence of pi, a result needed to show that one cannot square the circle with straightedge and compass.

You should also write a short two page biography of the individual. Include information about early education and background, contributions to mathematics as well as other fields, and whatever other information you have found that is of interest (e.g., anecdotes). You must supply references on your sources, as part of the intent of this exercise is for you to learn to deal with footnotes.

This biography should be posted on our class email list so that everyone in the class can read them. I encourage all students to read them carefully and make suggestions for improvement (this can be done off list, but do send me a copy). The purpose for posting all of these biographies is so that everyone can learn from them. After they have been posted for a week and people have had an opportunity to ask for clarification and to make suggestions, I will give you the opportunity to submit a revised copy for a grade. It is expected that many students will be asked to revise their paper.

You will be our class expert on this individual so be prepared to say something when the name comes up in class later. This assignment is due January 30. It is worth 50 points.

If there is sufficient interest we can convert these biographies into web pages. Yes, this would be for extra credit. Say up to 25 points, depending on the value added. Of course your name would go on the page.

Note: You are strongly encouraged to start a permanent file of library slips. This is a lifelong habit that is well worth developing. This is especially true of those of you who intend to become teachers, as you will have many occasions to want to have more information about a topic that you once read about. There are also several software packages available for keeping track of references (I use EndNote Plus), but I know of no freeware.

Second Library Assignment: The second library assignment is designed to acquaint you with the periodical literature. You are to turn in short synopses of ten papers on the history of mathematics (both words are crucial) that you have looked up and read. You should post this information to our class email list. The primary purpose of doing this is that it will provide everyone with lots of ideas for their major paper. Each is to contain complete bibliographic details: author, title (in quotes), periodical, volume, date (in parentheses following the volume number), and pages. The slip must also contain a short summary of the paper. The most interesting of these will be edited and posted on the web, so write your synopses with an eye towards encouraging your peers to read the paper. But be honest, if the paper is uninteresting, boring or not well written, say so.

A typical slip (for a journal article) is as follows:

Hogan, Edward
1971 "Robert Adrain: American mathematician,"
Historia Mathematica, vol. 4 (1971), pp. 157-172.

A very good, interesting biographical sketch concentrating on Adrain's publication of two journals, his teaching, and his mathematics. Very easy to read.

There are three ways to do this assignment and I encourage you to try them all:

  1. Pick up one of the journals listed in the bibliography (to be distributed) and browse until you find a historical paper that interests you.
  2. Pick a topic that interests you, look it up in Kenneth O. May's Bibliography and Research Manual of the History of Mathematics and then go find the paper.
  3. Use one of the databases available through the library computer. I will bring a computer to class one day and discuss how you can do this.
The last two techniques will be the one you will have to use when writing your research paper, so I encourage you to do some of this. Another possibility is to follow up references that you encounter in your reading. Be sure to look in some old journals as they can be lots of fun.

The ten articles are to come from at least four different periodicals (books are not permitted), and deal with at least four different mathematical topics. Once an article has been posted it should not be duplicated; to prevent problems in this regard, you should post your work shortly after you have done it; don't wait till you get your synopses of all 10 articles completed.

This assignment is due February 6. It is worth 50 points.

You may find it easier to work on these two library assignments simultaneously. At the same time you should begin thinking about a topic for your research paper.

While doing these assignments you are strongly encouraged to browse in the library. Whenever you pick up a volume of a journal to look up one thing see what else it has of interest. Be sure to make slips on anything you find that interests you. Don't rely on your memory; I guarantee it will fail you.

Internet Exercise: The quality of the information on the internet is highly variable and judging its validity is a highly valuable skill. You will be given a sequence of web sights to look up and to comment upon. Some of these have been selected because they are very poorly done. Others have been chosen because they are well done. You will be asked to decide which and to explain the reasons for your decisions.

These exercise will be posted on the web and I will notify you of the URL when that happens. Your comments on the sights should be posted on our email list so that all can benefit from what you learn.

This exercise is due February 13. It is worth 50 points.

Exams: The midterm exam will be given March 27; the final exam on Friday May 8 from 3:30 to 5:30 P.M. A sample exam will be distributed prior to each exam. There will be true-false questions, matching, fill-in-the-blanks, multiple choice questions, short answer questions, and essay questions. Some questions will be closely related to the exercises at the ends of the chapters. The exams will include all of the material in the book, as well as that discussed in class.

Research Paper: You are to write a paper on a topic of your choice. This is meant to be an interesting and enjoyable assignment, not a chore. So choose a topic with care. The only restriction that I impose is that it cannot be on the mathematician that your first library assignment dealt with.

You should think about the choice of a topic for your paper while you are doing the two library assignments. The exercises at the ends of the chapters in Katz suggest many possible topics. Some students prefer to write about a mathematician, others prefer the history of some mathematical topic. You are encouraged to talk to me (during office hours or via email) about possible topics. As soon as you have an idea, please let me know so that I can suggest possible references or make comments about the reasonableness of your choice of topic. On February 25 I would like each of you to give me your topic via email; only rarely will I veto a topic as unreasonable. Here are some paper topics that students have chosen in the past.

Each paper must meet the following requirements:

  1. The papers are to be on the history of mathematics. They can be neither all history nor all mathematics. Each should contain a reasonably non-trivial piece of mathematics as well as the history and background of that mathematics.
  2. Enough expository material should be included so as to make the paper self-contained. If you have doubts, ask a friend to read it. Having someone else read your paper critically is the best way to improve the exposition.
  3. You should use a variety of research materials and must give careful references to your sources. You will want to use books and encyclopedias, but I especially encourage you to use the journals (a necessity for B work). Your paper should include a bibliography listing your sources and they should be cited in the body of your paper when appropriate. The best sources to use are original sources, but, admittedly, that is hard to do. Their use is, however, required for A work.
  4. The paper must be prepared using a wordprocessor (you may write in symbols if the wordprocessor you are using does not handle them); if you don't know how to use one, now is the time to learn. Other issues such as the length, format, etc., are up to you. Since you will be startled by this last comment, let me point out that papers have a natural length. You are telling a story which needs certain background, exposition, and detail. When that is successfully done, stop; you have finished. You should turn in two copies of your paper as I intend to keep one copy.
The grading of your paper will be based on a number of factors, including: the historical and mathematical content; the significance, interest, accuracy, and completeness of the material; the accuracy, scope and significance of your references, and the sensitivity with which they are used and cited; and finally, the style in which it is written (poorly written papers will not be accepted). As in Olympic figure skating, your score will be a combination of technical performance and artistic merit. The grade of A will be given only for truly excellent work which uses original sources; B for good solid work that makes use of high quality journal articles; C for average work; D and F for unsatisfactory work. All grades are possible.

Here are some suggestions for writing your research papers.

To guarantee that you devote sufficient thought to your paper a topic is due February 25 and a preliminary report and outline is due March 18. The latter should include (a) your topic, (b) a few words about what you intend to do, (c) an outline, (d) your preliminary bibliography, and (e) any questions you have about your paper. This is best done via email as it makes it easier for me to incorporate comments. The more details you include, the more feedback you will get from me. Some students choose to turn in a first draft a little later. Feel free to ask questions and to indicate the problems you are having. The intent of this preliminary report is twofold: It should aid you in writing your paper and it allows me to make suggestions. Remember, the secret of good writing is rewriting.

The final version of your paper is due April 22.

Problems: There are many problems in the textbook and they contain a great deal of information about the history of mathematics. You should read all of them (as well as the references and footnotes for they give you some idea of the wealth of information available). Pick one problem from each chapter and do it (I encourage you not to pick the most routine ones). I will also suggest problems that you should pick from. You should do these problems in groups of cardinality three (this is called cooperative learning). Get together and work the problem, then write up a full solution with explanations. Have others in your group check and amend the write up before it is turned in. Put all three names on the paper. When the problem is turned in, it will be given to another group of three for grading. The second group will check if the problem has been done correctly, and will make comments on how good the explanation was, and will make suggestions for improvement. I will then look at the paper and make comments to both groups. Each problem is worth 15 points for doing it, and 5 for correcting it. Groups doing especially challenging problems or turning in exceptional solutions will receive bonus points. Groups can do additional problems for extra credit. You can turn in these problems when you finish with them, but must do so within 3 class days of the discussion of the chapter in class (this is to prevent everything from being turned in at the end of the semester when all of us will be too busy to deal with them effectively).

Each week I will call for volunteers (or, if needed, designate people) to put problems on the board.

Class Attendance and Participation : Your are expected to attend class. Roll will be taken at the beginning of the semester so that I can learn your names. If you miss class you are expected to find out what happened in class from a classmate and to learn the material on your own. You are also required to present a written note (on nice paper) to me explaining your absence and apologizing. I will randomly call on people to discuss what they learned in the readings, and to suggest those topics that they are having trouble with.

Plagiarism: According to the Random House College Dictionary plagiarism is "the appropriation or imitation of the language, ideas, and thoughts of another author, and representation of them as one's original work." Scrupulous care must be taken to avoid this in your writing. Naturally the source of a direct quotation must be cited. But also when you take the ideas of another and rephrase them you must cite your source. In historical work everything except the common and readily available facts needs a reference to the work where you learned this information. Mutilation of library materials is a crime, both literally and figuratively. Xerox is cheap and readily available, so there is no excuse for defacing library holdings in any way.

Cheating of any form will not be tolerated and will be treated with the utmost severity. See your student handbook for details.

SUMMARY OF DATES:

Date Assignment Points
January 16 Automathography 25
January 30 First library assignment 50
February 6 Second library assignment 50
February 13 Internet assignment 50
February 25 Topic for paper
March 27 Midterm Exam 100
March 18 Outline of paper 25
Problems 200
April 22 Paper 150
Classroom performance 50
May 8, 3:30--5:30 Final exam 150

You are expected to turn in the assignments on the dates indicated above. There will be a 10% penalty for each class that your assignment is late.


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If you have comments, send email to V. Frederick Rickey at fred-rickey@usma.edu .