Math 300, section 1 - Fall 2009
 Fundamental concepts of mathematics
  MWF 10:10-11:00, LGRT 113
Instructor: Tom Braden   TA: Peter Krafft
Office hours: Tues 2-3, Thurs 3-4, and by appointment
Mon 11:30-12:30, Wed 2:30-3:30
email: braden at math dot umass dot edu
pkrafft at student dot umass dot edu
Office: LGRT 1240
LGRT 1117
Phone: 545-1732


Announcements:

The final exam will be Wednesday, December 16 from 1:30-3:30 in LGRT 321.

Here is a handout that we will use when we study group theory.
Here's a second handout continuing where the first one left off.

Solutions to the last homework are up.

Here are solutions to the second exam.
Here are solutions to the first exam.

Homework page 

Tentative schedule of lectures:
Week 1 (9/9, 11): Introduction to proofs (1.1)
Week 2 (9/14, 16, 18): Logic, sets, proof strategies (1.2 through 1.6)
Week 3 (9/21, 23, 25): Divisibility of integers (2.1 through 2.3, 2.5)
Week 4 (9/28, 9/30, 10/2): Divisibility continued.
Week 5 (10/5, 7, 9): Congruences, modular arithmetic (3.1 through 3.5)
Week 6 (10/13, 14, 16): Review for Exam #1, begin proof by induction
Week 7 (10/19, 21, 23): Proof by induction (cont'd), binomial theorem (chapter 4)
Week 8 (10/26, 28, 30):  Functions (6.1 through 6.5)
Week 9 (11/2, 4, 6): Cardinality (6.6) 
Week 10 (11/9, 11/13): Permutations (6.9) and beginning group theory (handout)
Week 11 (11/16, 18, 20): More group theory, Review for Exam #2
Week 12 (11/23, 25):  still more group theory (handout)
Week 13 (11/20, 12/2, 12/4): group theory, Complex numbers (chapter 8)
Week 14 (12/7, 9, 11): Complex numbers cont'd, review for Final Exam


Math 300 is an introduction to rigorous abstract mathematics.  In lower-level classes like calculus, the emphasis is on applying formulas and theorems to specific problems.  In this class, we will be more concerned with how the ideas and formulas arise, and why they are true.  We will learn what a proof is, how to read, create, and present proofs, and how to tell a correct proof from an incorrect one.   It is like learning a new language: we need to learn the grammar (logical deduction) and the vocabulary (sets, functions, and other basic structures), but it also helps to have something to say, so we will also study some beautiful mathematics along the way.  Most of the actual mathematics we will study will concern the behavior of different classes of numbers: natural numbers,integers, rational numbers, real numbers, and complex numbers, as well as generalizations such as modular arithmetic and polynomials.  At the end we will also explore some basic ideas of symmetry and group theory.

Text:  Gilbert and Vanstone, An introduction to mathematical thinking, supplemented by occasional handouts.

Topics: Logic, sets and proofs (chapter 1), beginning number theory (chapters 2 and 3), induction, recursion, and the binomial theorem (chapter 4), rational and real numbers (chapter 5), functions and cardinality (chapter 6), complex numbers (chapter 8), polynomials (chapter 9), groups (handout).

Co-seminar:  All students need to register in Spire for the 1-credit co-seminar, listed as Math 391A.  This will meet in small groups at various times during the week; students will be assigned to a seminar time during the first two class periods.  These sections are very important for doing well in this course; if you are unable to attend your section one week, you should attend another one.  There will be occasional short quizzes during section, consisting of one or two basic questions.  They may or may not be announced in advance.

Grading:  your grade will be based on:
There will also be occasional extra-credit challenge problems.

Attendance: Attendance is required both during lectures and at the discussion sections.  Attendance AND participation are important ingredients for your success in the course. Frequent absences will be reflected in your grade.

General remarks: For many of you this will be your first mathematics class where concepts and proofs are more important than algorithmic, formula-based computation.  Do not make the mistake of treating this like a calculus class!  You will need to be much more active in your learning.  In order to really understand the concepts, you should ask lots of questions like: What happens if I change this definition in some way? What goes wrong if I leave out an assumption from this theorem? Are there any other examples that work like this?  Is this like something I've already seen?  You should do this in class, while reading the book, and while working on homework problems.

Our textbook is fairly accessible, but it still has many more ideas per page than your average calculus book.  Don't be discouraged if you must read slowly, or read many passages more than once: that is what is expected.  I strongly encourage you to read the material for a lecture before you come to class.  You should read actively -- when the book gives examples to illustrate an abstract definition, spend some time to convince yourself that they do in fact work, try to think of other examples of the same type, etc.

Homework Rules and Guidelines:  Homework will be due on Fridays at the start of lecture, unless otherwise stated. Late homework will not be accepted, but I will drop your lowest homework grade.

When you get stuck on a problem (and you will get stuck from time to time), I encourage you to seek help from me, your TA, your classmates or other students.  Many times talking with someone else will help you see a new way to approach a problem which you hadn't seen before.   I especially recommend that you work with your fellow students in groups.  But, if you are stuck on a problem and seek help from someone else, you owe it to yourself to try to understand the concepts and ideas that come up in the discussion -- don't just memorize the series of steps leading to the solution. Then, go home and reconstruct the argument for yourself in the privacy of your own brain, to make sure you are not merely reproducing mindlessly something you have not thought through. Remember that during tests and quizzes, you will have to rely on your own understanding of the material.

Here are the rules for collaborating on homework problems:

I. You must list the names of all people with whom you discussed each specific problem.
II. You MUST write your solutions completely independently.

Part of what you will be learning in this class is how to communicate mathematics to other people, so your homework will be graded on understandability as well as correctness.  Doing your homework will usually be (at least) a two-step process, where you first work out how to do it, and then rewrite your solution, getting rid of any false starts or unnecessary steps.