Syllabus -- Mathematical Structures, section 2, Fall 2017

Holy Cross Mathematics and Computer Science

Mathematics 243, section 2 -- Mathematical Structures

Syllabus, Fall 2017

Professor: John Little
Office: Swords 331
Office Phone: 793-2274
email: jlittle@holycross.edu
Office Hours: M 2-4pm, T 9-11am, W 10-11am, R 1-3pm, F 10-11am, and by appointment
Course Homepage: http://mathcs.holycross.edu/~little/MATH243-2017/Math243Home.html

Course Description

Mathematical Structures is a new course offered for the first time in Fall 2017. It is part of a recent restructuring of the requirements for the Mathematics major at Holy Cross for students in the class of 2020 and later classes. This new course combines some topics from the previous course Algebraic Structures (old MATH 243) and others from Principles of Analysis (old MATH 242, which will no longer be offered). The main ``agenda'' for the new MATH 243 is not entirely the mathematical subject matter we will cover (although that is fundamental for later courses). Rather, the important purposes of this new course also include:

introducing the language and ways of thinking associated with modern, abstract mathematics, and
developing students' familiarity with, and skill in applying, the basic strategies for developing and writing mathematical proofs.

Within the Mathematics major, this course will serve as a bridge from the basic mathematics you have seen in high school and in the Calculus sequence (including MATH 241 -- Multivariable Calculus) to the more advanced courses (especially the new required courses in Modern Algebra and Real Analysis) you will take in your junior and senior years. Because of this, we are most interested in your reactions and comments as we proceed and I will be soliciting your opinion about how things are going even before the CEF's given at the end of the semester.

Mathematical Structures (and Linear Algebra -- MATH 244) 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 and in Linear Algebra and 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 courses at this level to be more challenging than the other Mathematics courses they take later at Holy Cross because the whole way we work may well seem unfamiliar. (Those of you thinking of pursuing secondary teaching as a career might find it interesting to know that this is truer now than it was in the past. Traditional high school geometry classes, in particular, were much better at preparing students for this kind of work than typical high school mathematics classes taught today!) Some of the unfamiliar aspects:

Learning ways to perform calculations will rarely be the main focus of what we do in class or ask you to do on assignments.
Even the calculations we do will not always involve particular explicit numbers, functions, etc. although there will be some of this type of work.
You will definitely not master this material by learning rote procedures for solving a set collection of different types of problems.
You will also not master this material by looking things up online constantly. We are not using a standard text book and you will not find solutions for any of the problems posted anywhere before the assignments are due. (I will post solutions after the due dates so you can see models to work from for the future.) The goal is for YOU to work out solutions to problems by THINKING about them!
What we will be asking you to do is often to find, and then clearly present, a complete logical argument for showing that some statement is true under clearly articulated conditions. A plausible reason for expecting something is true will not alone be sufficient.

At times, you may find yourself asking ``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 very real sense it is what most of advanced mathematics is really about, so you have to be prepared for it if you decide to continue in the Mathematics major! 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 (those who work on problems directly inspired by questions about the real world) must sometimes 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 have done.

To succeed in this course, it will help to realize from the start that:

You will want to start problem sets early so you are not pressed for time at the due date.
You will probably find you need to take more care to develop, check, and record your thinking about problems than you are 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.
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 and don't be afraid to ask for help(!)
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 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.''
Finally, 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 the photocopied bound course pack Mathematical Structures by Prof. Hwang (my name is on the title page too, but he did almost all of the writing!) This is available (only) through the HC bookstore.

It is expected that Holy Cross students will have textbooks and other required class materials in order to achieve academic success. If you are unable to purchase course materials, please go to the Financial Aid office where a staff member will be happy to provide you with information and assistance. For the book for this course, you may also consult with Prof. Hwang or with me and we can provide an electronic version free of charge.

Material We Will Study

During the semester we study the following topics

Unit I: Basic logic, set theory, and proofs (about 8 days)
Unit II: Discrete structures: The natural numbers, the integers and the integers modulo n (about 18 days)
Unit III: Continuous structures: The real and complex numbers, completeness, sequences, series (about 10 days)

The other three days will be devoted to in-class examinations or to review, etc. 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 midterm exams, together worth 50% of the course grade. Tentative dates (if done in class):
- Friday, September 29.
- Friday, November 3.
- Wednesday, December 6.

Because everyone can have a bad day, I will count the lowest exam only half as much as the other two (that is, weights of 20%,20%,10% for the exams, in decreasing order by the numerical score.)

A two-hour final exam, worth 25% of the course grade. The final examination will be given at the time established by the Registrar for MWF 11-11:50 classes during the regular final exam period; watch for announcements.
Weekly individual homework assignments, posted on the course homepage. The homework will count as 10% of your course grade. The individual assignments are an 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 5% of the course grade.
Weekly 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 10% 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 a 78 course average would usually convert to a C+ letter grade. It would never convert to a letter grade of C or below, but it might convert to a B- or above depending on the distribution of scores in the class as a whole.)

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 (or possibly at an evening time). No sharing of information with other students or consultation of any online or other sources 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.

Other Policies and Information

College policy on excused absences from class (this is a link available in the online version).
Any student who feels the need for accommodation based on the impact of a disability should contact the Office of Disability Services to discuss support services available. Once the office receives documentation supporting the request for accommodation, the student would meet privately with Disability Services to discuss reasonable and appropriate accommodations. The office can be reached by calling 508-793-3693 or by visiting Hogan Campus Center, room 215A.
It is my intent that students from all diverse backgrounds and perspectives be well-served by this course, that students' learning needs be addressed both in and out of class, and that the diversity that the students bring to this class be viewed as a resource, strength and benefit. It is my intent to present materials and activities that are respectful of diversity: gender identity, sexuality, disability, age, socioeconomic status, ethnicity, race, nationality, religion, and culture. Your suggestions are encouraged and appreciated. Please let me know ways to improve the effectiveness of the course for you personally, or for other students or student groups.