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{\bf\centerline{Math 223: Introduction to Mathematical Reasoning}}
{\bf\centerline{Spring, 2004}}
\medskip {\bf Meeting Time:} MWF 1:55 p.m.--2:45 p.m.
\smallskip {\bf Meeting Place:} Hayes-Healey  215
\smallskip {\bf Instructor:} Matthew Dyer
\smallskip\hskip 1cm {\bf Office:} Hayes-Healey 140
\smallskip\hskip 1cm{\bf Office hours:} Tu 12:00-1:00 p.m, F 12:00--1:00 p.m.
  or by appointment
\smallskip\hskip 1cm {\bf Phone:} (574) 631-6082
\smallskip\hskip 1cm {\bf Email:} dyer.1@nd.edu
\smallskip{\bf Textbook:} Mathematical Thinking (Problem Solving and 
Proofs)     2nd Edition,
by John D'Angelo and Douglas West.\smallskip
  {\bf  Why this course:}  Up till now, most of your mathematics
courses have likely emphasized examples, computation, and intuitive
understanding of mathematics.  This course will emphasize careful mathematical
arguments.  By addressing questions about familiar things like numbers
(Are there finite or infinitely many prime numbers?  Do all rational numbers
have rational square roots?) and sets (What does it mean for a set to
have ``infinitely many'' elements?  Do all sets with infinitely many
elements have the same size?), we will see how it is that one justifies
statements in mathematics.  In a nutshell, the subject of this course is
numbers, and its goal is to help you understand, invent, and present proofs.

  \smallskip {\bf    What we'll cover:}     The material covered falls into
roughly three categories (with a fair amount of overlap).

(i)  Basic material concerning sets, logic, and methods of proof.  Chapters
1--4 of the textbook.

(ii)   Algebra and Number Theory (i.e. things relating to integers).
Chapters 5--8.

(iii)   Analysis (i.e. things relating to real numbers).  Chapters 13--15.

As the course unfolds, we might deviate a bit from this plan, but I expect
we'll mostly hold to it.

  \smallskip{\bf  How you will be evaluated:}

(i)   Homework:    Worth 30\% of your final grade.

(ii)   Two Midterm Exams:    Given in class, each worth 20\% of your
final grade.


  (iii)  Final Exam:     Worth
30\% of your final grade.







Grade cutoffs for the major grades (A,B,C,D,F) will be assigned for each exam
so you may judge your exam performance.

\smallskip{\bf Exams:}


\hskip1cm Exam 1: Friday February 20

\hskip1cm Exam 2: Friday April 2.

\hskip1cm  Final Exam:     Monday, May 3 at 8 AM. Room TBA.

The in-class exams are intended to examine material
covered in class after the preceeding test, but knowledge of earlier 
material may be required also.
The final exam will cover material from the whole course.

Under extreme circumstances, I will consider giving a makeup exam, but
you'll need an excuse from whoever it is at ND that officially doles these
things out.  Also, I reserve the right to shift weights in the final grade
so that a makeup exam is less heavily weighted than its regularly
scheduled counterpart.

\smallskip{\bf Homework}
  Homework will be assigned and collected (more or less)
every Wednesday. I encourage you to
collaborate with each other on homework problems.  Discussing homework
  with one another
and with me  is a good way both to learn the material and gain facility in
communicating it to others.  In fact for each
assignment, you may join forces with  {\sl    one or two  }   other
students to turn in a single, jointly written set of solutions
(with all collaborating students names listed on the top).

Late homework will not be accepted.  Regardless of how much you have left
to do or how good your reason is, I will only grade the work that you
give me at the beginning of class the day it's due.  If you think
you've got a good reason (e.g. medical emergency or something equally dire)
for not finishing some assignment, we can talk about {\sl discounting }
it, but there will be no late grading.
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\smallskip{\bf Honor Code} This course will be conducted under the Honor Code.
Copying another student's work in an exam is of course a violation of the code.
During exams, where I expect you to work alone, you must
direct all questions and comments concerning the exam directly to me.
Any reliance on other students or other exams is out of bounds.

Discussion of homework problems with other class members is permitted,
  as long as
your  written homework is prepared by those (one to three) students whose name
is listed on the assignment  and reflects their
final understanding of the homework problems. However, direct copying of
  another student's (or group of student's)
  written work is a violation of the Honor Code.

    \smallskip
{\bf General advice for the course}

(i)  One of the most effective ways of
learning mathematics is by solving a  range of problems involving the
subject matter and skills you are learning. By doing this, you consolidate your
understanding of the material and skills, begin to build a useful mathematical
  intuition
for them, and gain experience in problem solving. In this course,
  with its emphasis on understanding, inventing and  presenting proofs,
  this is especially important.
For these reasons (as well as its contribution to your assessment) the homework
should be regarded as an integral part of the course.

  Additionally,
your mathematical understanding is of little use if it cannot be imparted to
others, and carefuly writing homework solutions will develop your skills in
mathematical communication. You should make every effort to make your written
homework solutions as clear as possible. Some points to consider in doing this
  are that your solutions should be written in complete sentences, with major
portions separated into paragraphs, all symbols you use should be defined
  and you should provide justifications for all major steps you use.

(ii)  Please ask me questions, in or out of class. That helps you
(you can correct misunderstandings or uncertainties before we move on
to new material, which often  may build on the foundation you're unsure about),
  helps the other class members (many of whom  may have wanted to
  ask a similar question) and helps me (by giving me more a more explicit
  idea about
  what points  may be causing difficulty).
The exception to
this concerns last minute questions about homework.  I give at least a week
to complete each homework assignment, and since I expect you to use the
full week, I'll be loathe to deal with basic questions about an assignment
(like {\sl "how do I start problem 3?"}) the day before it's due.

(iii) I'd be very happy to see you for office hours.
  If you can't get to  the office hours I've set, don't hesitate to contact me
to set up an appointment.

(iv)
Keep up with the reading in the class.  Reading math is a skill that
takes time to learn, but the payoff is very high.  By learning to read
you free yourself to learn on your own terms, independent of professors and
courses and schedules (and high tuition!).

(v) (For those who took the last item seriously.)
The main thing about reading math books is that you must read
slowly and {\sl actively}.
In order to make sense of even well-written mathematics, you must often stop to
think about what a particular phrase or sentence means, go back to remind
yourself of a previous definition or theorem, consider how a statement
could and could not be rewritten, or do a pencil and paper computation
to convince yourself that some equation is true.  You'll find that this
approach to reading is infectious.  Once you get used to it, you'll tend
to approach lectures and newspapers and novels the same way and that
all these things become much more fun and edifying once you make yourself
an integral part of them.





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