\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\usepackage{latexsym,amsmath,amssymb}
\usepackage[all]{xy}
\newenvironment{packed_enum}{
\begin{enumerate}
  \setlength{\itemsep}{1pt}
  \setlength{\parskip}{0pt}
  \setlength{\parsep}{0pt}
}{\end{enumerate}}
\newenvironment{packed_item}{
\begin{itemize}
  \setlength{\itemsep}{1pt}
  \setlength{\parskip}{0pt}
  \setlength{\parsep}{0pt}
}{\end{itemize}}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.0in}
\begin{document}
\centerline{Montserrat 104Q-- Mathematical Journeys:  From the Known to the Unknown}
\centerline{Syllabus -- Fall 2015}
\vskip 10pt
\leftline{Professor: John Little}
\leftline{Office: Swords Hall 331}
\leftline{Office Phone: (508) 793-2274}
\leftline{Office Hours: M 1-2pm, T 10am-12noon, W 1-3pm, R 2-4pm, F 11am-12noon,}
\leftline{$\qquad\qquad\qquad$ and by appointment}
\leftline{Course Homepage:  {\tt http://mathcs.holycross.edu/\~{}little/Mont201516/Fall.html}}
\vskip 10pt
\noindent
{\it 2015 Core Human Questions Cluster theme}
\vskip 10pt
How, then, shall we live when the journey may be as important as the origin 
or the destination?
\vskip 10pt
\noindent
{\it General description}
\vskip 10pt
This course is a part of the Core Human Questions cluster of the Montserrat 
program.  This means that at various times we will be addressing one 
or several of the following:
\vskip 10pt
\begin{itemize}
\item The general aims of Montserrat -- continued development of your
thinking, writing, and communication skills, and the connection of
living, learning and doing in college education,
\item The common readings and theme of this year's Core Human Questions cluster.
\item The specific subject matter of this seminar -- gaining an appreciation of
the intellectual journeys that people take within mathematics and 
the ways mathematicians create knowledge and communicate their understanding 
to others via proofs. 
\end{itemize}
\vskip 10pt
\noindent
{\it Subjects for this fall's seminar}
\vskip 10pt
The first full week of the semester will be devoted to discussing the 
first cluster common reading -- the {\it Odyssey} of Homer -- the story 
of one hero's long and adventure-filled journey home from the Trojan War.
First written down around 800 BCE as a record of a long tradition of
oral performance of poetry on this theme, the {\it Odyssey} has earned a place as 
the prototype of the epic journey in Western literature.  

After this, we will turn to the mathematical portion of the course.
Does it surprise you to learn that people are constantly adding to what we 
know about mathematics?  How does that happen, and how do new ideas come to 
be accepted? Ever since the time of the ancient Greeks (but significantly 
later in history than the time of Homer), {\it proofs} of mathematical results 
have played a key role in this. To relate this to the theme of our Montserrat 
cluster, the process of finding and writing down a proof can (very profitably) 
be seen as a sort of {\it intellectual journey}. The origin of this journey 
is what is known at the outset, the destination is the result to be proved. 
The journey itself is reflected in the sequence of logical steps that get 
us from the known to the (previously) unknown. Mathematicians would say 
these journeys into the unknown are often just as important as the destinations, 
mostly because the knowledge gained in finding a proof can also lead to other 
unexpected results and new understanding. 

But how does this relate 
to the more practical aspects of mathematics that you have studied before? 
For some mathematicians, in fact, the process 
of gaining understanding and finding proofs comes to be more important and 
interesting than any possible practical application of those ideas.
Our starting point in this part of the course will be the testimony of one 
particular British mathematician of the 20th century, G. H. Hardy, who left 
a very personal testament of what mathematics meant to him in his book 
{\it A Mathematician's Apology}. Hardy's ``take'' on the practical side of 
mathematics remains controversial, to say the least. But it should 
provide us with an opportunity to discuss the ethical implications of doing 
any scientific or mathematical work.   

Next, to get a feeling for the powerful attraction this sort mathematical
work can have on some (maybe even some of you!), we will take a close look
at Book I of Euclid's {\it Elements}.  (This work is also the 
source of several particular proofs that Hardy discusses, but his examples
come from later sections.)  Much as 
the {\it Odyssey} has served as a prototype for stories about journeys, the 
{\it Elements} has served as a prototype for textbooks of mathematics, right
down to the present day.  As we will see, Book I is a very tightly-organized
sequence of theorems and proofs leading to a masterful proof of a familiar
result from plane geometry that all of you have seen (although you may not
know a proof for it).  

Following the October vacation, we will take a short break from our mathematical journey 
for the second cluster common reading, {\it Huckleberry Finn} by Mark Twain.  
Twain's novel has been seen as the source of most of distinctively American 
literature and we will see how Huck and Jim's trip down the Mississippi
River in pre-Civil War America is another sort of epic journey.

The second mathematical portion of the course will be devoted mainly to 
the question: How do we {\it find proofs}?  We will have seen by this point
in the course that this step in the process is not addressed at all in Euclid's 
work. Many, many generations of students have found this frustrating, to say the 
least.  And indeed, having only the ultimate destination or {\it finished 
product} in the form of a polished proof, {\it \`a la Euclid}, is arguably not the 
most productive way to learn to make use of mathematical ideas, although teachers
of mathematics have different opinions about what might replace it(!)  In this part 
of the course, we will look at the ``messier'' cycle of conjectures (informed 
guessing), attempted proofs, counterexamples, and refutations by which actual
proofs come into being through the interaction of groups of real people.     

Our final cluster common reading, Cheryl Strayed's memoir {\it Wild} will
round out this first semester.   

No mathematical background beyond high school geometry and algebra is required
for this course, but 
students should be open to new points of view and willing to think deeply.
\vskip 10pt
\noindent
{\it Some groundrules}
\vskip 10pt
Most class meetings will be devoted to discussions, oral presentations,
or work in smaller groups, so your active participation will be important
for the success of what we do. 
\begin{itemize}
\item Unless specifically directed otherwise, please \emph{turn off}
all cell phones, tablet or laptop computers and other similar
electronic devices for the duration of each class meeting -- your full 
attention and participation will be necessary.
\item No distracting or ``provocative'' clothing, headgear, or other 
personal items in class, please.
\item In on-campus events, off-campus trips, etc. you are representing
this seminar, the Montserrat program, and, by extension, Holy Cross in a wider 
community.  Take responsibility and regulate your behavior accordingly. 
\item In discussions, there may be times that you disagree with Prof. Little
or with a classmate.  Please feel free to express that disagreement 
and be prepared to say why you disagree and back up your ideas with evidence.  
But please keep the conversation civil and respectful.
\end{itemize}
\vskip 10pt
\noindent
{\it Course objectives}
\vskip 10pt
\noindent
The major objectives of the seminar will be:
\vskip 10pt
\begin{packed_enum}
\item To acquaint you with the role of proofs in mathematics, to 
learn some famous and influential examples of proofs from Euclid's
{\it Elements}.
\item To acquaint you with some of the process by which proofs 
are developed by mathematicians and to give you the experience
of finding proofs of your own.
\item {\it Individually}, to further your development as speakers and writers. 
\item {\it As a group}, to create an atmosphere where civil and 
constructive conversations can take place.  
Many of the topics we discuss will have controversial aspects and reasonable 
people can have very different viewpoints on them.  Recognizing that, everyone 
(including Prof. Little, of course) should listen carefully and seek to 
understand where others are coming from, especially when your first
inclination might be to disagree.   Good academic {\it writing} can also 
be seen as a conversation between the writer and others who have thought 
about similar questions.  So this way of doing things will carry over into 
the way we approach writing assignments as well.
\item To participate in and contribute to the common activities of the 
Core Human Questions Cluster of the Montserrat program.
\vskip 10pt
\end{packed_enum}
\noindent
{\it Texts}
\vskip 10pt
\noindent
The texts for the course are:
\begin{packed_enum}
\item[1)]  Homer, \emph{The Essential Odyssey}, translated and edited by Stanley 
Lombardo, Hackett Publishing.
\item[2)]  Twain, Mark, \emph{Adventures of Huckleberry Finn}.  Any unabridged and unexpurgated
edition is OK; an inexpensive Dover republication will be available in the HC bookstore.
\item[3)]  Strayed, Cheryl, \emph{Wild: From lost to found on the Pacific Crest Trail}, Vintage Paperback.
\item[4)]  Hardy, G.H. \emph{A Mathematician's Apology}, with foreword by C.P. Snow, (Canto Classics) Cambridge University Press.
\item[5)]  Euclid, \emph{Elements}, Book I, online at:
\vskip 10pt
\centerline{\tt http://aleph0.clarku.edu/\~{}djoyce/java/elements/elements.html}
\vskip 10pt
\item[6)]  Lakatos, I. \emph{Proofs and Refutations}, Cambridge University Press.
\item[7)]  Davis, P. and Hersh, R. \emph{The Mathematical Experience} Study Edition, Birkhauser, 2012.
\end{packed_enum}
 See course schedule for more information
about when will be looking at each one.
\vskip 10pt
\noindent
{\it Course schedule}
\vskip 10pt
A detailed day-by-day course schedule and listing of required and suggested outside events
will be maintained on the course homepage (and will be accessible through the 
Moodle course management system).  That listing is a tentative, evolving schedule,
so it may change and you will probably want to refer to it frequently.  
Any important changes will also be announced in class well in advance.   
\vskip 10pt
\noindent
{\it Assignments and grading}
\vskip 10pt
\begin{packed_enum}
\item[1)]  {\it Midterm exam} (15 \% of course grade) -- tentative date: Friday, October 30.
\item[2)]  {\it Final exam} (20 \% of course grade) -- the final will be given at the 
regular time for MWF 9 am classes, when that is determined.
\item[3)]  {\it Writeups from group project days and problem sets} (about 5 assignments -- 15 \% of course grade)
\item[4)]  {\it Three roughly 3-5 page papers}, and other shorter writing assignments.  
(Information and guidelines to be distributed later) (30 \% of course grade)   
\item[5)]  {\it Presentation}:  During our reading of the {\it Elements} of Euclid, 
each student, working with a classmate, will prepare an oral 
presentation, approximately 15-minutes in length, on one of the Propositions 
in Book I, as assigned by Professor Little (10 \% of course grade)
\item[6)]  Class participation (10 \% of course grade)
\end{packed_enum}
\vskip 10pt
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):
\vskip 10pt
\begin{packed_item}
\item A -- 94 and above
\item A- -- 90 - 93
\item B+ -- 87 - 89
\item B -- 84 - 86
\item B- -- 80 - 83
\item C+ -- 77 - 79
\item C -- 74 - 76
\item C- -- 70 - 73
\item D+ -- 67 -- 69
\item D -- 60 - 66
\item F -- 59 and below.
\end{packed_item}
\vskip 10pt
\noindent
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, although it might be a B+ in some circumstances.) If you ever have a question about the grading policy or your standing in the course, don't hesitate to ask me.
\vskip 10pt
\noindent
{\it Advice on how to succeed in this class}
\vskip 10pt
\noindent
{\bf A good ``work ethic" is key.}  As you should be able to tell
from the course description above, you do not need to be a 
``math genius'' to do well in this course.  But you will need 
to put in a consistent effort and keep up with the reading and 
assignments.

\vskip 10pt
\noindent
{\bf Come to class.}  Unless you are deathly ill, have a genuine family 
emergency, are away at a game or meet of a college athletic team, etc. 
plan on showing up here at 9:00 am every Monday, Wednesday, 
and Friday this semester.  Many of the class meetings will be 
structured around discussions or student presentations.  Your participation 
is expected and needed for the success of the course!

\vskip 10pt
\noindent
{\bf Take notes and use them.} This may seem obvious, but it is worth 
saying!  Used intelligently, your notes can be a valuable resource as 
you work on problem sets and prepare for the exams.

\vskip 10pt
\noindent
{\bf Use the texts and class notes actively.} Reading about mathematics
is not like reading a novel.  You will probably need to read and think over things 
more than once.  You may want to work through examples to understand
some of the topics that we do.  

\vskip 10pt
\noindent
{\bf Set up a regular study schedule and work at a steady pace.} It's not easy 
to play catch-up in a mathematics course (even when the course is part of a 
first-year program with additional goals beyond the mathematics).  You should 
expect to budget at least 6 hours in a typical week for work outside of class. 
The best way to use your time is to do a few problems, some reading from the 
books, and reviewing of class notes every day.

\vskip 10pt
\noindent
{\bf Most importantly, if you are having difficulty learning something, 
get help as soon as possible.}  You can do this by asking questions during 
class (any time something isn't clear), or seeing me during office hours.
\vskip 10pt
\noindent
{\it Statement on academic integrity}
\vskip 10pt
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.  To be become an engaged and advanced learner, you must be able to think and work both independently and in concert with your peers.  
 
The College academic honesty policy states: 
``As an institution devoted to teaching, learning, and intellectual inquiry, Holy Cross expects all members of the College community to abide by the highest standards of academic integrity. Any violation of academic honesty undermines the student-teacher relationship, thereby wounding the whole community. The principal violations of academic honesty are plagiarism, cheating, and collusion.

Plagiarism is the act of taking the words, ideas, data, illustrative material, or someone else's statements, and presenting them as one's own without full and proper acknowledgment.

Cheating is the use of improper means or subterfuge to gain credit or advantage. Forms of cheating include the use, attempted use, or improper possession of unauthorized aids in any examination or other academic exercise submitted for evaluation; the fabrication or falsification of data; misrepresentation of academic or extracurricular credentials; and deceitful performance on placement examinations. It is also cheating to submit the same work for credit in more than one course, except as authorized in advance by the course instructors.

Collusion is assisting or attempting to assist another student in an act of academic dishonesty.”
The full statement on Academic Honesty in the College Catalog is available at
\vskip 10pt
\centerline{\tt http://www.holycross.edu/catalog/acad\_program.pdf}
\vskip 10pt
\noindent
If you do not know how to correctly cite reference materials, consult with your professor, the campus Writer’s Workshop, or visit one of the links below:

\noindent
Holy Cross Department of History:
\vskip 10pt
\centerline{\tt http://academics.holycross.edu/history/academichonesty.htm}
\vskip 10pt
\noindent 
Wadsworth Cengage, ``Plagiarism Prevention Zone'' 
\vskip 10pt
\centerline{\tt http://college.cengage.com/english/plagiarism\_prevention.html}
\vskip 10pt
\noindent
NOTE:  If in doubt about what you plan to do or write violates academic honesty, PLEASE ASK!
The temptation to engage in an act of academic dishonesty may arise, 
but the chance possibly to enhance a single grade is not worth the loss 
of your personal integrity. 
\vskip 10pt
\noindent
{\it Specific guidelines for this course}
\vskip 10pt
In this course, all examinations will be closed-book. No consultation of outside sources of information or
sharing of information with other students in any form will be permitted during exams. In group discussions,
close collaboration with the other members of your group is expected. 
For the papers, please refer to the specific guidelines for each assignment regarding what
is acceptable and how to cite sources other than the course texts if that is allowed.  
In general, some discussion of ideas for your paper with classmates and with Prof. Little
will be acceptable as you prepare to write.  But your final written work should be your own 
and any ideas you use from other sources should be acknowledged.  
On individual problem sets, discussion of the questions with other students in the class
and with me during office hours is allowed, {\it even encouraged}. 
However, your final problem solutions should be prepared individually and 
the wording and organization of your final problem solutions should be entirely 
your own work.  Moreover, if you do take advantage of any of the above options for 
discussion of problems with others, you will be required to state that fact in a 
footnote accompanying the problem solution. Failure to follow these rules will be 
treated as a violation of the College's Academic Integrity policy.  
\end{document}




