Syllabus -- Algebraic Structures, section 3, Fall 2012

Holy Cross Mathematics and Computer Science

Mathematics 243, section 3 -- Algebraic Structures

Syllabus, Fall 2012

Professor: John Little
Office: Swords 331
Office Phone: 793-2274
email: little@mathcs.holycross.edu or jlittle@holycross.edu
Office Hours: MF 10am - 12noon, TWR 2pm - 4pm, and by appointment
Course homepage: http://mathcs.holycross.edu/~little/AS12/AS12Home.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 as you study for exams and prepare for other work.

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, 7th 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. Many of the class meetings will be structured as lecture/discussions -- but with an emphasis on the ``discussion'' part. I will regularly ask questions and seek responses from the class. To get you even more directly involved in the subject matter of this course, through the semester the class will break down into groups of 3 or 4 students for one or more days, and each group will work separately 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.

Grading Policy

Grading for the course will be based on

Three in-class tests, together worth 45% of the course grade. Tentative dates:
- Friday, September 28.
- Friday, November 2.
- Wednesday, December 5.
A two and one half hour final exam, worth 25% of the course grade. The final examination will be given at the regular time for MWF 8:00am classes. Watch for announcements of this and please do not make plans to leave campus for the semester break prematurely.
Weekly individual homework assignments, posted on the course homepage. The homework will count as 15% 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. See the course homepage for some guidelines on preparing and submitting your problem sets. No credit will be given for late homework, except in the case of an excused absence.
Weekly "Definitions Quizzes" (given the first 10 minutes of class on Fridays when there is no exam, starting Friday, September 7). The average of the best 5 out of 9 will form 10% of the course grade.
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 5% of the course grade.

I will be keeping your course average in numerical form throughout the semester, and only converting to a letter for the final course grade. The course grade will be assigned according to the following conversion table (also see Note below):

A -- 94 and above
A- -- 90 - 93
B+ -- 87 - 89
B -- 84 - 86
B- -- 80 - 83
C+ -- 77 - 79
C -- 74 - 76
C- -- 70 - 73
D+ -- 67 -- 69
D -- 60 - 66
F -- 59 and below.

Note: Depending on how the class as a whole is doing, some downward adjustments of the above letter grade boundaries may be made. No upward adjustments will be made, however. (This means, for instance, that an 85 course average would never convert to a letter grade of B- or below. But a 79 course average might convert to a letter grade of B- depending on the distribution of averages across the whole class.)

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 others 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. In addition, you must follow these groundrules concerning the problem sets:

Do not divide up the problems on an assignment among members of a group and then compile the solutions for the group members to submit by copying work supplied by your group-mates (i.e. even with proper citation of who is in your working group). (Reason: This may seem like a good labor-saving device. But in practice, it means that if you have not really worked through some of the problems yourself, then you may not have really understood what is going on in them. You may also be turning in work that contains errors you have not caught because you have not thought and worked through the solution individually!)
Do include notes giving credit for office hours help you have received directly (individually or with others).
Do not delegate members from your group to come to office hours to ask about specific questions, and then compile help I gave to those individuals into solutions that are submitted by all members of the group. Do not credit office hours or incorporate ideas I shared there in your solutions if you have not been there yourself. In other words, collaboration on a problem must cease when some members of a group have been to office hours asking about a problem and others have not been there. (Reason: I try to make office hours as interactive as possible, so that I am pointing you in the right direction to solving a problem rather than showing you the whole solution myself. I try to get you to supply many or most of the key ideas and many technical details. If you copy work that relies on the interaction with me and other students without having been there, you have not gone through that process yourself and you don't have any right to present that work as your own.)
If anyone asks you to do something that violates these groundrules, don't comply with that request. If you have a question about whether something you want to do is OK or violates these groundrules, ask Prof. Little.

Failure to follow these groundrules will be treated as a violation of the College's Academic Integrity policy.