History of Math Syllabus

Course Syllabus - Spring 2008

From the College Catalog : A history of the development of mathematical techniques and ideas from early civilization to the present, including the inter-relationships of mathematics and sciences.

Prerequisites: Math 1401 (Calculus I)

Texts: William Dunham, Journey through Genius: The Great Theorems of Mathematics, Penquin Books, 1990 and
Victor Katz, A History of Mathematics: An Introduction, 2^nd Ed., Addison-Wesley, 1998

For the sake of those who want to get right to the course requirements, I've placed my thinking about this course at the end of this document.

Readings:

The entire Dunham book will be read. For the first twelve weeks of the semester, a chapter a week is assigned. There will be no formal reading assignments from Katz, but it will be heavily used as an initial starting point for all projects and papers.

Assignments:

There will be two papers (a biographical and a topical paper) and three projects which will be evaluated to determine your grade. Graduate students (those enrolled in Math 5010) will have an additional fourth project, if there are at least three graduate students enrolled this fourth project will be a special team project otherwise it will be under the same rules as the other three projects. There are no exams or homework as such. Each student will be required to give at least one oral presentation to the class based on one of their projects/papers work. Written work will be prepared using a word processor and an electronic version will be required. In lieu of a written assignment, a web page may be constructed for the content (standards are the same for web pages and other written work).

Biographical Paper

This assignment will be due April 1. You are to select a mathematician from any period in history and write up a biography of that individual. No two students may choose the same mathematician, so as soon as you make your choice inform me and I will post on the website who you have chosen so that everyone else knows that that mathematician is no longer available. Selection is to be made by Feb. 12. To aid you in making this choice, I have posted a bibliography of biographical works. You are not limited to this list.There is also a list of choices made by my Spring '05 class which can be found in the Student Work section.

Look up your mathematician in the Dictionary of Scientific Biography, which is not available at Auraria Library (only the concise 1 volume version is on reserve there). Read the articles and take notes. The bibliographies in the DSB are often very useful.
Look up your mathematician in an encyclopedia. Record, for your bibliography, the name of the encyclopedia, date of publication, volumes, pages, and authors (signed articles are more reliable). Read the articles and take notes.
Note any discrepancies between the above sources. How do you decide which is more reliable?
Are there any books about your mathematician? Are there any books by your mathematician? Make sure those get listed in your bibliography.
Try to find information on your mathematician in the World Wide Web. For the bibliography, record the URLs; since the web changes so quickly, indicate also the date on which you accessed the page. Try to find out who is the author of the information on the page, and which institution, if any, sponsors it. Read the information, take notes, and compare with what you already know.
Look up your mathematician in Dauben's The history of mathematics from antiquity to the present : a selective bibliography (on reserve in the MathPhysics library at CU-Boulder). This might point to other relevant articles or books that you might want to consult or list. Another source of this kind is the Companion Encyclopedia to the History and Philosophy of Mathematics, ed. by I. Grattann-Guinness (also available in MathPhysics at CU_Boulder).
Have your mathematician's collected works been published? Does our library have a copy?
Write a biography of your mathematician. By this point, this should be an exercise in selecting and summarizing information. Your biography should include an outline of the person's life and a summary of his or her contributions to mathematics, science, and other fields. Indicate both who influenced your mathematician in significant ways, and who your mathematician influenced. Attach a bibliography.
Your writing should be concise and clear, but it must also be precise. Don't say things you don't mean. Be particularly careful with the description of your mathematician's scientific work - it's easy to say things that aren't correct.
Carefully document your biography. Don't quote without attribution. When you mention some fact that is not common knowledge, indicate its source. Your references should be specific enough that your readers can find the information for themselves (for example, when quoting from a book a page number is essential).

Topical Paper:

This assignment is due May 3. I have broken this down into phases with various due dates and credit amounts (total of 240 points).

Topic selection (10 points): The topic for your research project is to be selected by you, in consultation with me, by March 13. The topic should be a great theorem from mathematics history which can be developed into a chapter of Journey Through Genius. Note that Journey Through Genius chapters generally include historical information providing a context for the great theorem (much mathematics history and some general history), biographical information about the mathematicians involved, the great theorem itself together with a clear and lucid presentation of its proof, and an Epilogue describing subsequent ramifications and applications of the great theorem.

Paper Outline (Preliminary Report) (60 points): Your preliminary report will be a detailed 2-3-page outline or description of what you intend for your paper to include. Your bibliography to date must be included with the preliminary report. Preliminary reports are due April 10 .

The paper itself (170 points): Your paper, of course, should be carefully prepared, organized and presented. It must include a title page and a bibliography, but otherwise should be organized as are the chapters in Journey Through Genius. The style also should be similar to that of Journey Through Genius---that is, it should be clear and lively. Number definitions, theorems, examples, and figures. You may find it helpful to number some or all of your equations as well. Be consistent in your numbering scheme and in your use of notation.

At least four (4) sources must be cited in the body of the paper. Cite these references by number (e.g. [3], where [3] refers to the third reference in your list, also labeled [3]) or author (e.g. (Smith, 1988)), rather than using footnotes. The paper must include a list of references, or bibliography. The list of sources you actually cite should appear first, followed (if you wish) by a list of additional sources consulted and/or recommended to the reader.

How much detail should you provide? Use Dunham's Journey Through Genius
as your model. More specifically, your goal should be to make the material you are presenting clear to a sophomore mathematics major who has not previously studied this material. Clear and correct explanations and arguments are essential; illustrative examples and diagrams are always helpful.

Criteria for evaluation include the accuracy of the history and the mathematics you present, the organization and clarity of your presentation, your use of appropriate and illustrative examples, your use of mathematical notation, and your use of the English language (this includes style and grace, as well as grammar and spelling).

The first draft is due April 29 (30 points); and the final draft is due May 8 (140 points).

The paper should be word processed and laser printed. An electronic version (for posting on the web site) should also be made. In order to save time later, I recommend that you type up sections of your paper as you work on them. In particular, be sure to compile your bibliography as you find and use references.

Suggested paper topics: Perfect numbers and the Euclid-Euler formula, Pythagorean triples, FLT n=4 case, sums of squares, Fibonacci sequence, properties of "Pascal" triangle, Stirling's formula, Wallis' formula, logarithm as an integral, logarithms of negative numbers, Konigsburg Bridge Problem and Euler circuits, Euler's formula, Four Color Theorem, Cauchy's "wrong" theorems and proofs, etc., etc.

Projects:

The projects are described in the separate projects page.

Due Dates and Syllabus:

These are given in the separate Semester Due Dates page.

Teams

It is entirely optional, but I will permit you to do at most two projects in teams of no more than three students. Team projects will be graded as teams, i.e., everyone on a team gets the same grade. However, be advised that the expectations for a team project are higher than for individual projects, and this will be reflected in the grading. The restrictions for time period and themes will hold for every member of a team. This means that teams will be easier to form near the beginning of the semester and difficult at the end.

Grading

As all material for grading will consist of papers (or webpages) I will use the standards that are set forth in Fernando Q. Gouvêa's Hints For Writing a Paper which will be handed out. I will also solicit input from your classmates before determining a grade when this is possible. Oral presentations will be evaluated by the class, but this will not be factored into a final grade.

The final grade will be determined as follows:

Projects: 3 @ 15% = 45%

Biographical Paper = 25%

Topical Paper = 30%

Posting Material

All written work is to also be submitted in electronic form. I will post these on this webpage so that everyone in class has an opportunity to read them (and possibly evaluate them).

Aids

As the coursework primarily involves doing library research, I have arranged a special class on Thursday, January 31. We will meet (at the usual time) in Room 245 in the Library. Librarian Diane Turner will help you to get the most out of the Auraria Library for your projects and papers. I strongly advise you not to miss this class, even if you think you know all there is to know about doing library research. A couple of good ideas about search strategies can save you hours of frustration.

If in your research you come across a mathematical topic which you need help with, let me know and I will try to develop a lecture for you. If the topic is of sufficiently general interest, I may give it to the class.

There are several hand outs that are posted on our webpage. Of particular interest is the self-check list for writing papers.

Some thoughts about this course

There are several ways to organize a History of Mathematics course. Three of the most popular are:

Historical Survey: Starting with pre-Greek mathematics this approach follows the historical development of mathematics through the ages. In a semester course, this approach rarely gets beyond the development of Calculus in the 17^th Century. This is probably the most common type of course.
Famous Mathematicians: Concentrating on a handful of the most influential mathematicians, the subject is developed by extending out from their work to those that were influenced by it and what they added to it. This is the approach taken by the popular text of Calinger, A Contextual History of Mathematics.
Famous Theorems: In this history of ideas approach, a handful of significant theorems are examined in detail. Included in this examination are the social context in which the results are embedded and the mathematicians who contributed to the result. This is the organizational principle of Dunham's Journey through Genius.

Each of these approaches has advantages and disadvantages. There can be no single organizational principle that works best; the subject is too complex to yield to such simplifications. In designing this course I have tried to look at the objectives of the course with fresh eyes and not be blinded with what has been done previously. I realized, fairly early, that I did not wish to give a survey course. I have two objections to this type of course. The landscape of mathematics is vast, and any survey that attempts to be comprehensive will have to be superficial and just gloss over the interesting history and mathematics. On the other hand, if the material is dealt with in depth, not much of the subject will be covered, in particular, the roots of modern mathematics will never be examined. If you reject, as I have, the survey approach, then you must ask yourself what should a student walk away with from a history of mathematics course? It is hard to answer this question in terms of content – what is valuable information for one student may be a useless factoid for another. I would rather phrase my answer in terms of skills and concepts. Thus, I would like for each student:

to gain an appreciation of the complex interplay between mathematics and the social milieu in which it is developed,
to develop some skill at critical analysis of historical data,
to expand and refine the ability to find information on any topic, and
to improve the ability to explain technical material in written form as well as orally.

While there are many other goals I could add, I've kept the list short so that we may concentrate on them given the time constraints of the course.

Mathematical Content

I have examined a number of History of Math courses that have web pages. A statement to the effect that this is a math course and therefore there will be a considerable emphasis on the mathematics is very common. In designing this course I have asked myself why this must be so. It seems to me that the real reason lies in the fact that the course is taught by mathematicians and not historians. There are very few mathematical historians (there is only one department of Math History in the US (Brown University) and only a handful of Ph.D. programs) so the vast majority of these courses must be taught by mathematicians who have not specialized in math history. It seems to me to be apparent that in order to keep the comfort level of the instructors relatively high, a large dose of mathematics would be necessary.

I too am a research mathematician with no training in math history and as such I am also attracted by this lure. However, I make a distinction between the history of mathematics and the appreciation of mathematics. One can use the study of the history of mathematics to increase the appreciation of mathematics, but this is not a two-way street and it is a bit of a stretch to try to make this go the other way. On the other hand, there is a need to have an upper level course which develops an appreciation of mathematics; especially for those whose careers include the teaching of mathematics. As there is no course other than this one in which a broad appreciation can be developed, it is important to include this as part of the course. Therefore, I am really going to present a course which ought to be titled “The History and Appreciation of Mathematics”. In the “appreciation” part of the course, we will be concerned with mathematical content. To this end, I am using the Dunham book, Journey through Genius, as the basis of the lectures I present in class. As a class we will read the entire book and one of the papers that is assigned is to essentially add a new chapter to that book.

And here is the obligatory CLAS statement

Spring 2008 CLAS Academic Policies

The following policies pertain to all students and are strictly adhered to by the College of Liberal Arts and Sciences (CLAS).

Every student MUST check and verify their schedule prior to the published drop/add deadlines. Failure to verify a schedule is not sufficient reason to justify a late add or drop later in the semester. It is the student’s responsibility to make sure that their schedule is correct prior to the appropriate deadlines.
CLAS students must use their email.cudenver.edu email address. Email is the official method of communication for all University of Colorado at Denver and Health Sciences Center business. All email correspondence will take place using your UCDHSC email address. Go to http://www.cudenver.edu/registrar to update and/or change your email address.
Students are NOT automatically added to a course off a wait list after wait lists are dropped. If a student is told by a faculty member that they will be added off the wait list, it is the responsibility of the student to complete the proper paperwork to add a course. Students are NOT automatically added to a course off the wait list after the 5^th day of the semester when wait lists are dropped.
Students are not automatically notified if they are added to a class from a wait-list. Again, it is the responsibility of the student to verify their schedule prior to any official dates to drop or add courses.
Students must complete and submit a drop/add form to make any schedule changes. Students are not automatically dropped from a class if they never attended, stopped attending or do not make tuition payments.
Late adds will be approved only when circumstances surrounding the late add are beyond the student’s control and can be documented independently. This will require a petition and documentation from the student. Late adds will only be approved if the student has not taken any exams, quizzes, or has not completed any other graded assignments. Independent verification of this from the professor of record will be required. Please note that the signature of a faculty member on an add form does not guarantee that a late add petition will be approved. Petitions are available in NC 2024.
Late drops will be approved only when circumstances surrounding the late drop have arisen after the published drop deadlines, are beyond the student’s control, and can be documented independently. This will require a petition and documentation from the student. Pre-existing circumstances (circumstances that existed prior to the published drop deadlines) regarding illness, work, family, or other confounding issues will not be considered adequate reason to drop or withdraw from courses after the published University and/or College drop deadlines. Please note that the signature of a faculty member does not guarantee that a late drop petition will be approved. Petitions are available in NC 2024.
Students wishing to graduate in spring of 2008 must meet with their academic advisor by the end of the drop/add period to obtain a graduation application. This application must be completed and submitted by 5 PM on February 6, 2008. You can obtain an application ONLY after meeting with your academic advisor. There are no exceptions to this policy or date.
Students are responsible for completing financial arrangements with financial aid, family, scholarships, etc. to pay their tuition. Students will be responsible for all tuition and fees for courses they do not officially drop using proper drop/add procedures and forms.
Students who drop after the published drop/add period will not be eligible for a refund of the COF hours or tuition.

Important Dates

January 22, 2008; First day of Class
January 27, 2008 ; Last day to be added to a wait list using the SMART system.
January 27, 2008: Last day to add a course using the SMART system.
January 22-January 28, 2008; Students are responsible for verifying an accurate spring 2008 course schedule via the SMART registration system. Students are NOT notified of their wait-list status by the university. All students must check their scheduled prior to January 28, 2008 for accuracy.
January 28, 2008: LAST DAY TO DROP WITHOUT DROP CHARGE.
January 28, 2008: Wait Lists are dropped. Any student who was not added to a course automatically from the wait list by this date and time MUST complete a drop/add form to be added to the class. Students are NOT automatically added to the class from the wait list after this date and time. If your name is not on the official student roster, you are not registered for the course.
January 29, 2008: First day instructor may approve request to add a student to a full course with a Schedule Adjustment Form.
January 29 – February 6, 2008: Drops allowed using SMART system. Drop charge applies. Full term courses may be added using Schedule Adjustment Form.
February 6, 2008 at 5 PM; Last day to add structured courses without a written petition for a late add. This is an absolute deadline and is treated as such. This deadline does not apply to independent study, internships, and late-starting modular courses.
February 6, 2008 at 5 PM; Last day to drop a spring 2008 course with a tuition refund minus the drop charge and no transcript notation. Drops after this date will appear on your transcript. This is an absolute deadline and is treated as such.
February 6, 2008 at 5 PM; Last day to completely withdraw from all spring 2008 courses with a tuition refund and no transcript notation. Drop charge applies. Drops after this date will appear on your transcript. This is an absolute deadline and is treated as such.
February 6, 2008 at 5 PM; Last day to request pass/fail option for a course.
February 6, 2008 at 5 PM: Last day to request a no credit option for a course.
February 6, 2008 at 5 PM: Last day to register for a Candidate for Degree.
February 6, 2008 at 5 PM: Last day to petition for a reduction in thesis or dissertation hours.
February 6, 2008 at 5 PM: Last day to apply for spring 2008 graduation. You must make an appointment and see your academic advisor to apply for graduation.
After February 6, 2008 all schedule changes require a petition. Petitions are available in NC 2024.
February 27, 2007: Early Alert System OPEN for faculty.
March 10, 2007: Early Alert System CLOSES for faculty.
April 7, 2008 at 5 PM; Last day for non CLAS students to drop or withdraw from all classes without a petition and special approval from the student’s academic Dean. This is treated as an absolute deadline.
April 18, 2008 at 5 PM; Last day for CLAS students to drop or withdraw from all classes without a petition and special approval from the student’s academic Dean. This is treated as an absolute deadline.
No schedule changes will be granted once finals week has started. There are NO exceptions to this policy.

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