Holy Cross Mathematics and Computer Science
Mathematics 243, section 1 -- Algebraic Structures
Syllabus, Fall 2006
Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred) or jlittle@holycross.edu
Office Hours: MWF 10-11, TR 1-3, and by appointment
Course homepage: http://mathcs.holycross.edu/~little/AS06/AS.html
Course Description
Algebraic Structures is the first half of a year-long second year
algebra sequence designed primarily for Mathematics majors. The
continuing course in the spring is called MATH 244, Linear Algebra,
and has this course as a prerequisite.
These second year algebra courses have two major goals:
- To introduce the foundations of the branch of mathematics
known as modern or abstract algebra, including
linear algebra, and
- To develop students' familiarity with, and skill in applying, the
basic strategies for developing and writing
mathematical proofs.
Within the Mathematics major, these courses (together with the Principles of
Analysis course, MATH 242) serve as bridges from the basic mathematics you
have seen in high school and in the calculus courses to the more
advanced courses you will normally take in your junior and senior years.
Because of this, Algebraic Structures and Linear Algebra can
perhaps best be described as part of a sort of
``boot camp for Math majors''. More advanced courses will
draw regularly on the concepts introduced here (algebraic structures such
as groups, fields, vector spaces, linear mappings and matrices, etc.) and in
those courses, the professors will assume that you are
familiar with the properties of those structures
that we will prove in this course. In addition, you will be expected
to be able to develop and clearly present logical proofs of your assertions
in those courses, so they will also assume the basic techniques and strategies
we will discuss here.
Although you will find that I am not usually a ``drill sergeant'' type,
there may be times
when the ``boot camp'' analogy will seem apt -- many Mathematics majors
find these courses to be among the most challenging they take at Holy Cross
because the whole way we work may well seem unfamiliar:
- Many of the questions on problem sets will not have a single number
or formula as an answer and more generally, learning ways to
perform calculations will rarely be the main focus of problems.
- Even the calculations we do will not always involve particular
explicit numbers, functions, etc.
- You will definitely not master this material by learning
rote procedures for solving a set collection of different types of problems
- A plausible reason for expecting something is true will not alone be
sufficient.
At times, you may ask why we are doing this. The ultimate answer is
that this ``abstract'' proof-oriented work is the way all mathematics
is communicated and in a sense it is what most of advanced
mathematics is really about, so you have to be prepared for it
if you decide to continue! The concept of mathematical proof is
the unique and distinctive feature of this branch of knowledge;
I think it is no exaggeration to say that it is one of the crowning
achievements of the human intellect. Even applied mathematicians (ones
who work on problems directly inspired by questions in the real world)
must develop new ideas to solve those problems, and then provide
convincing evidence (proofs) that what they claim is true so that others can
follow what they do.
To succeed in this course, it will help to realize
from the start that
- In many problems the goal will be first to develop, and
then to explain clearly, the complete logical argument that
establishes the truth of a general statement.
This means you will probably find you need to take more care
to develop, check, and record your thinking about problems than you
are probably used to. Suggestions: Never be in a hurry to
``get assignments over with''. Take your time, think things through
calmly and carefully, and always double- and triple-check your reasoning.
Start problem sets early so you are not pressed for
time at the due date. When things are not ``coming'' to you
put the work aside for a while, tend to your other classes,
take a walk or run, work out in the
wellness center, etc. If you have ``primed'' it sufficiently,
your unconscious mind will be working on
the problem even while you are outwardly doing other things! If after
your best effort, you can see some of the way to a solution but not
all the way, have the integrity to say what you are missing.
- You will have to think about the logical structure of the subject matter
and understand the definitions of concepts and the statements and proofs of
theorems to get the ``skeleton'' of the subject firmly in place in your mind.
This is also the reason for the weekly Definitions Quizzes (see below).
Suggestion: Read over your notes and the text
after every class and make
sure you understand what was done that day. Making a ``clean'' copy
of the class notes, with details filled in, original examples worked out,
added comments about things that you had to work to understand, etc.
is also highly recommended.
- You will need to commit a collection of key examples to memory and
be able to reason about their properties. Many problems on the problem sets
will deal with these key examples, and the point of doing the
problem is to help you add these new items to your ``permanent mental
furniture.''
Suggestion: Your involvement with problems and the
facts that are developed through them should not be over
when you turn the problem set in. Be sure you read through
comments on your work when the problem set is returned to see
where corrections or improvements are needed. Keep
your papers so you can refer back to them later.
You will have lots of chances
to develop and practice these new skills, and I will always be willing to
give you the benefit of my experience working with this kind of mathematics!
Even if you find this difficult at first, persistence and openness
to a different way of thinking will usually pay off in time.
Text
The text book for the course is Elements of Modern Algebra by
J. Gilbert and L. Gilbert, 6th edition. We will cover most of the
material in Chapters 1 - 4 and the Appendix on logic this semester.
Material We Will Study
During the semester we study the following topics
- Unit I: Logic, set theory and mappings (the language of mathematics) (about 13 days)
- Unit II: The integers and the integers modulo n (about 9 days)
- Unit III: First steps in the general theory of groups (about 14 days)
The other three days will be devoted to in-class examinations.
There is a more detailed day-by-day schedule posted on the
course homepage.
As always, it may become necessary to add, subtract, or rearrange topics.
I will announce any changes in class and on the course homepage.
Course Format
In order for a student to get as much as possible out of this or any
course, regular active participation and engagement with the ideas
we discuss are necessary.
To get you more directly involved in the subject matter of this course,
regularly throughout the semester the class will break down into groups
of 3 or 4 students for one or more days, and each group will work individually
for (a portion of those) class periods on a group discussion exercise.
I will be responsible for designing and preparing these exercises, and
I will be available for questions
and other help during these periods. Each group will keep a written record
of their observations, results, questions, etc. which will be handed in.
The other meetings of the class will be structured as
lecture/discussions.
Grading Policy
Grading for the course will be based on
- Three in-class tests, together worth
40% of the course grade. Tentative dates:
- Friday, September 29.
- Wednesday, November 1.
- Friday, December 1.
- A three-hour final exam, worth 25% of the course grade. The
final examination will be given at 8:30 a.m. on Wednesday, December 13.
- Weekly individual homework assignments, given out in class. The homework
will count as 20% of your course grade. The individual assignments
are a very important part of this course and keeping up to date will
be necessary to succeed with this material. Note: Because of the
enrolment in this class, in order to get graded work back to you in
a reasonable amount of time, it may become necessary for me to grade
only selected problems on some of the problem sets. If I need to take this
option, I will not announce which problems will be graded,
and you will be expected to do and hand in all of the announced
problems in any case.
No credit will be given for late homework,
except in the case of an excused absence.
- Written reports from small group discussions -- one report
from each group. Information regarding the expected format
will be given out with the assignment. Together, worth 10% of the course grade.
- Weekly "Definitions Quizzes" (given the first 10 minutes of class
on Fridays when there is no exam, starting Friday, September 8).
The average of the best 5 out of 9 will form the other 5% of the course grade.
If you ever have a question about
the grading policy or your standing in the course, don't hesitate to ask me.
Departmental Statement on Academic Integrity
Why is academic integrity important?
All education is a cooperative enterprise between teachers and
students. This cooperation works well only when there is trust and
mutual respect between everyone involved.
One of our main aims as a department is to help students become
knowledgeable and sophisticated learners, able to think and work
both independently and in concert with their peers. Representing another
person's work as your own in any form (plagiarism or ``cheating''),
and providing or receiving unauthorized assistance on assignments (collusion)
are lapses of academic integrity because they subvert the learning process
and show a fundamental lack of respect for the educational enterprise.
How does this apply to our courses?
You will encounter a variety of types of assignments and examination
formats in mathematics and computer science courses. For instance,
many problem sets in mathematics classes and laboratory assignments
in computer science courses are individual assignments.
While some faculty members
may allow or even encourage discussion among
students during work on problem sets, it is the expectation that the
solutions submitted by each student will be that student's own work,
written up in that student's own words. When consultation with other
students or sources other than the textbook occurs, students should
identify their co-workers, and/or cite their sources as they would for
other writing assignments. Some courses also make use of collaborative
assignments; part of the evaluation in that case may be a rating of each
individual's contribution to the group effort.
Some advanced classes may use take-home
examinations, in which case the ground rules will usually allow no
collaboration or consultation.
In many computer science classes, programming projects are
strictly individual assignments; the ground rules
do not allow any collaboration or consultation here either.
What are the responsibilities of faculty?
It is the responsibility of faculty in the department to
lay out the guidelines to be followed for specific assignments in
their classes as clearly and fully as possible, and to
offer clarification and advice concerning those guidelines
as needed as students work on those assignments.
The Department of Mathematics and Computer Science upholds the
College's policy on academic honesty.
We advise all students taking mathematics or computer science courses
to read the statement in the current College catalog carefully and
to familiarize themselves with the procedures which may be
applied when infractions are determined to have occurred.
What are the responsibilities of students?
A student's main responsibility is to follow the guidelines laid down
by the instructor of the course. If there is some point about the
expectations for an assignment that is not clear, the student is responsible
for seeking clarification. If such clarification is not immediately available,
students should err on the side of caution and follow the strictest possible
interpretation of the guidelines they have been given.
It is also a student's responsibility to protect his/her
own work to prevent unauthorized use of exam papers, problem solutions,
computer accounts and files, scratch paper, and any other materials used in
carrying out an assignment. We expect students to have the integrity to say
``no'' to requests for assistance from other students when offering that
assistance would violate the guidelines for an assignment.
Specific Guidelines for this Course
In this course, all examinations will be closed-book and given in-class.
No sharing of information with other students in any form will
be permitted during exams. On group discussion write-ups, close collaboration
is expected. On the problem sets, discussion of the
questions with other students in the class, and with me during office
hours is allowed, even encouraged.
If you do take advantage of any of these
options, you will be required to state that fact in a "footnote"
accompanying the problem solution. Failure to follow this rule
will be treated as a violation of the College's Academic
Integrity policy.