MATH 347: Fundamental Mathematics

Answer 1: A rather dry description can be found in the course catalog at

http://courses.uiuc.edu/cis/catalog/urbana/2008/Fall/MATH/347.html

Answer 2: This course is intended to introduce you to what might, with some pretension, be called real mathematics, as studied by those who devote their life to the subject. We will emphasize the following three aspects of mathematics:

Mathematics as a rigorous discipline: Central to mathematics is the notion of proof -- giving convincing arguments to support your mathematical claims. For you to participate in this walk of life, you will need to understand what a mathematical statement is, what premises you are ``allowed'' to start from in reasoning mathematically, and what constitutes valid rules of reasoning.

Mathematics as a broad discipline: You all know from calculus (at least) that mathematics is much more than simple arithmetic. But your previous experience might not have convinced you that it is all that much more; maybe it's arithmetic plus derivatives and integrals! We will shatter this pernicious notion once and for all, by exposing you to techniques from across the spectrum of mathematics, including aspects of combinatorics, number theory and real analysis (this last may be viewed, again somewhat pretentiously, as calculus done right).

Mathematics as a social discipline: Mathematicians, despite often being averse to normal social interaction, are required by the realities of the academic life to publish their work. This involves explaining to their colleagues what they have been studying, how they have been studying it, what their results are, and (perhaps most importantly) how they arrive at their results. This is only possible if there is a standard grammar to mathematical language that governs how mathematics is to be written and read. Fluency in this language is one of the most important skills you will take away from this course if you plan to continue in mathematics.

It is often said (but deserves to be repeated, in boldface) that mathematics is not a spectator sport. Your attendance and participation is expected. We will often discuss problems collectively as a class, and I expect you to share your ideas, and not merely to (like a parasite) take from others. We will deal throughout the term with some genuinely difficult concepts. You should not feel ashamed if you have difficulty wrapping your mind around something during this term; please interrupt the lecture to ask for clarification when this happens! In return for your participation, my promise to you is that I will do everything in my power to make our classroom a question-safe space.

Textbook (required)

Content-wise, I hope to cover Chapters 1-4 (elementary concepts), Chapters 5-8 (introduction to combinatorics and number theory), Chapters 10 and 12 (more on combinatorics), and Chapters 13-15 (introductory real analysis). However, these goals are flexible, and depending on how we are progressing I will not hesitate to omit nonessential topics if necessary.

An up-to-date list of reading assignments and corresponding homework can be found at

http://www.math.uiuc.edu/~pppollac/347/schedule.html

You are responsible for checking that page on a regular basis to keep up to date with reading assignments and homework.

Instructors and lectures

Lectures are MWF, from 12:00 PM - 12:50 PM in Altgeld 343. (You can remember this because 343=7^3.)

The instructor for this course is:

Paul Pollack E-mail: pppollac at illinois dot edu

Office: 301 Altgeld

Office hours (tentative): Monday 3 PM-4 PM, Wednesday 1 PM-2 PM, Thursday 11 AM-12 PM and 3PM-4PM and by appointment

Please put your tuition money to good use and attend my office hours if you have questions. This time is already set aside for you; you are not interrupting! I prefer that you stop in physically if you have substantial questions about the homework or the lecture; basic questions (such as clarification on definitions, or questions about the structure of the course) can often be handled by e-mail, which I check rather frequently. Please be reasonable however -- e.g., if you e-mail me at 1 AM on Saturday night/Sunday morning, do not expect me to see your message until Monday afternoon.

Homework

Homework is assigned weekly. Written homework assignments should be typewritten or carefully handwritten and should be stapled. Moreover, you are expected to write in coherent, complete, and correct sentences, and to provide justification when requested. Since we are emphasizing the social aspect of mathematics in this course, the quality of your exposition (i.e., how well you communicate your solutions and and your proofs) will be a factor in your grades, both on the homework and on the exams. Most problems will be from the text of D'Angelo and West, although I may sometimes supplement these problems with my own exercises; in this case the assignment will be physically handed out in class as well as posted electronically at the course website.

Late homework will not be accepted without prior permission of the instructor. Such permission, which will only be granted under extraordinary circumstances, should be discussed with the instructor at least a week in advance.

Exams

There will be three in-class midterm exams as well as a final exam.

The bulk of the questions on the exam (as well as the bulk of the points) will come from short answer questions. These will be comparable in difficulty to assigned homework problems.

Midterm exam 1: Friday, September 26
exam 2: Wednesday, October 22
exam 3: Friday, November 21

Final exam: Wednesday, December 17, 1:30-4:30 PM, our classroom

Determination of the final course grade

The three midterm examinations will be worth 15 percent of your grade.
The final exam is worth 25 percent.
Homework accounts for the remaining 30 percent. Your lowest score on a homework assignment will be dropped.

As usual, an A is a grade in the closed interval [90,100], a B is grade in the half-open interval [80,90), a C is a grade in [70,80), etc. I do not give partial grades (such as A+, B-, etc.) Individual exams and homework scores are not curved. However, a curve may be applied at the end: this will only happen if I decide that one or more of the exams was written unfairly and a poor test of your skills.

Statement on academic integrity

On exams no help can be either given or received.

That being said, I would strongly encourage you to work with others on the homework, and to discuss freely your ideas with your classmates. As iron sharpens iron, so one man (or woman!) sharpens another. Your journey to mathematical maturity need not be a lonely one! However, while you may collaborate with others in finding the solution to a problem, the final write-up must be your own. That is, it must reflect your understanding of the material (someone cannot dictate to you what to write), it must be physically written (or typewritten) by you, and what you write must not be a copy of someone else's completed assignment.

If the statement of a problem is unclear or you are not familiar with all the definitions, you should feel free to consult outside references, including outside textbooks, online references, and mathematically inclined friends. (Even easier, if I am checking e-mail at the time, is to just ask me!) However, you may not look up solutions to the exercises online. Furthermore, you may not collaborate on problems with those outside your section of Math 347.

Further clarification on what is acceptable can be found in §1-401 of the student code, which is available electronically:

http://www.admin.uiuc.edu/policy/code/article_1/a1_1-401.html

Please read that document and consult me if you have any more questions. The University and I take academic integrity seriously, and consequences for a breach may be rather unpleasant (see §1-403 of the linked document for some possibilities).

Special accommodations

Students with disabilities who may require special accommodations should talk to me as soon as possible. Appropriate documentation concerning disabilities may be required. For further information, please consult

http://www.disability.uiuc.edu/resourceguide/

Also, students who have schedule conflicts with the course due to religious observances (or other matters of equally significant weight) should come talk to me. In either case, please apprise me of the situation within the first two weeks of class (by September 5, 2008).