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.
I am also open to doing evening exams with a bit more time for you
to work if the class would prefer that and provided the scheduling
can be worked out satisfactorily.
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.
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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.