Department of Mathematics Mathematics 248, section D1 Fundamental Mathematics (Advanced Composition) 11:00 - 11:50 AM MWF 143 Henry Professor Daniel R. Grayson Email: Office: 357 Altgeld Hall Office phone: x3-6209 or 217-333-6209.

The course:

The official description is in the catalog. In this course we learn the fundamental ways to make rigorous proofs, we read our textbook, and we learn how to write proofs well in clear mathematical prose. We also learn how to use AMS-LaTeX for typing mathematical papers: LaTeX is the standard tool that almost every mathematician uses, and AMS-LaTeX is an augmented version produced by the American Mathematical Society.

This course fulfills the General Education Advanced Composition and Quantitative Reasoning II requirements. The student earns 4 hours of credit, 1 more than for Math 247. The extra hour gives us the extra needed to concentrate on learning writing skills.

The students:

The main responsibility of the student is to learn. Each student must speak up at the first opportunity when something is not clear, for the best time for an explanation of something confusing or too brief is never later and is always now. Each student will participate in the activities of the class, and contribute in meaningful ways to the joint enterprise. This includes reading the textbook.

An old saying is: "Give a man a fish and you feed him for a day, teach him how to fish and you feed him for a lifetime," attributed to Kuan-tzu, a Chinese Philosopher. A new saying might be: "Explain something to a student and you teach her for a day, teach her how to learn and you teach her for a lifetime." Thus I will not always explain everything to you -- instead I will try to nudge you into the right way of thinking about it so you can figure it out yourself. With enough nudges you'll learn to guide your thinking yourself.

Our grader is Jennifer Vandenbussche <jarobin1@math.uiuc.edu>. Feel free to email her with questions about how the homework was graded.

Problem solving:

In this course we will be solving some problems in small groups of three or four students. Hopefully the problems will be hard enough so that you won't all know how to attack them, but easy enough, so that they can eventually be solved after enough discussion and enough tiny hints from me.

Don't panic when confronted with a problem whose solution isn't apparent. Instead, think of ways to nibble away at the problem around the edges. Is there a related problem which might be easier to solve? If so, try solving it first.

No student should let the discussion of a problem stop until the solution is completely understandable. Nor should anyone ever belittle the ideas or questions of another --- we are trying to foster open and free discussion. Those of you who have good ideas have a responsibility to explain them carefully to the other members of your group.

Mathematical writing:

After each problem is solved by someone and the solution thoroughly discussed by the entire class, a paper containing the solution may be assigned. Aim the exposition at someone who hasn't participated in the class discussion. Occasionally this will mean that many delicate points will have to be explained. A high quality exposition will be both mathematically and grammatically correct, logical, compelling, and interesting.

Some of the writing assignments will be major ones, and will be accomplished as follows. Every student will write a first draft and bring enough copies to the next class for the professor and for the other members of the group. The group will discuss the first drafts, make critical but helpful comments about each other's writing, and decide how to merge the drafts into one excellent paper. One student will be chosen to merge them and produce the final draft to be turned in. Students should take turns being the producer of the final draft. Those students needing the most improvement in their writing skills should be the ones volunteering to be the producer; don't worry, the other students in the group will help you, because their grade depends on it, too.

Some of the writing assignments will be minor ones. For them, each student will write a paper independently of the other students. Do not look at other student papers or show them yours.

Double space and leave margins where the professor can write.

Some tips on good mathematical writing:

• Use complete sentences, correct grammar, and correct punctuation.
• Don't submit your first draft. Read it, get someone else to read it, and find out where it's unclear.
• Shorten your writing. The fewer words you make your reader read, the better, provided all the ideas are still presented clearly. After writing a sentence, examine it for phrases that have already occurred earlier in the text and try to eliminate them.
• Don't expect mathematical formulas and figures to speak for you. Refer explicitly to each one and tell the reader what you want the reader to get from it.
• Many mathematical adjectives and nouns have precise mathematical meanings, and an English synonym will not serve as a replacement. For example, "element" and "part" are not interchangeable when referring to an element of a set.
• Look at examples of writing proofs well in the book, and try to emulate the style.
• Global organization of a proof is important. Tell the reader what you are about to do, and then do it. Use paragraphs to delineate the parts of a proof.
• Define all symbols before using them.
• Start each sentence with a word, not a mathematical symbol.
• Two mathematical expressions or formulas in a sentence should be separated by more than just a space or by punctuation; use at least one word.
• When proving a statement of the form P=>Q don't mistakenly prove Q=>P by starting with Q and deducing something from it.
• Words have meanings: be aware of them. For example, an equation has an equal sign in it.
• Never say "it is easily verified that ..." or "it is easy to see that ...". The temptation to write such a phrase indicates you know some further justification is required, so provide something with content.
• Don't use abbreviations.

Links to tips on good mathematical writing:

The textbook:

You will be responsible for reading selected sections of the textbook and learning from it. See the hints below about mathematical reading, and ask me for help if you have any trouble.

We will also be doing problems and learning things that are not in the textbook.

The textbook is Mathematical Thinking: Problem-Solving and Proofs, by John P. D'Angelo and Douglas B. West, published by Prentice Hall, 2000, second edition, ISBN 0-13-014421-6. Look at the Errata.

• Scale : 100 A 90 B 80 C 70 D 60 F
• Writing assignments: 30%
• Homework: 10%
• Extra credit homework can add some points
• Hour exams (3): 30%
• Final exam: 30%

• A template for your papers written in TeX: template.tex
• A TeX file displaying many symbols: symbols.tex symbols.dvi
• The LaTeX project.
• The not so short guide to Latex, by Oetiker, Partl, Hyna, and Schlegl.
• Sources for TeX software
• Get AMS-LaTeX if your LaTeX doesn't include it. There is some documentation there, too.
• A sample paper written in AMS-LaTeX:
• MathSciNet, a searchable database of the mathematical literature
• GNU/Linux (the preferred operating system):
• You should be able to install tetex (a distribution of TeX), and emacs (the professional text editor), but the way to do that depends on your particular flavor of GNU/Linux.
• Mac OS X:
• XDarwin, an implementation of the X Window system.
• fink, a software distribution system. Use it to install tetex (a distribution of TeX), and emacs (the professional text editor).
• Microsoft Windows:
• cygwin, a complete distribution of GNU software. Includes XFree86 (the X Window system), and tetex (a distribution of TeX), and emacs (the professional text editor).
• MiKTeX, another implementation of TeX, simpler to install. Here are some usage hints from Christopher Lee and from Pam Reid.
• Various editors usable with TeX.
• Emacs for Windows, the professional text editor (not exactly the same as the one that comes with cygwin).

This web page: