The central purpose of this course is to introduce you to careful use of language in the context of mathematical reasoning and proof. The course is meant to make you think about mathematics in a completely new way, in a more mature way. It should set you on the path to becoming a mathematical producer rather than mathematical consumer.

As part of this venture, we will discuss the basic principles of logic and various proof techniques, applying them in the context of the essential building blocks of mathematical structures: sets, relations (including orderings and equivalence relations), functions, etc. While the class will introduce you to some new mathematics, the emphasis of the course is on process rather than content. I rarely lecture; you and your fellow students will prove virtually all the theorems yourselves and present them to each other in a seminar setting. Thus I hope that the most important lines of communication will be between students rather than instructor to student as is the case in many classes.

**Grading:** The grade will be
calculated based on 7 components:

In-class Presentations |
20% of the final grade |

Class
Participation |
8% of the final grade |

Written
Assignments |
16% of the final grade |

Take-home
Midterm |
20% of the final grade |

In-Class
Midterm |
8% of the final grade |

Take-home
Final |
20% of the final grade |

In-Class
Final |
8% of the final grade |

In-class Midterm |
Monday 24 March, covers material through Section 4.2. |

Takehome Midterm |
2:00 pm Friday 21 March to 2:00 pm Tuesday 25 March |

Takehome Final |
2:00 pm Friday 2 May to 1:30 p.m. Friday 9 May |

In-class Final |
1:30 p.m. on Friday 9 May |

**Class work:**
Foundations is very likely somewhat different from other math
courses you have had. Because the purpose of the class is
to change the way that you think and reason about mathematics,
it is essential that you become immersed in the work of the
course. It is not enough to respond to what an instructor
does or tells you. You and your fellow students are the
ones that make things happen in class. Without your active
participation, nothing will happen. Perhaps more than in
any class you take, you will get benefit out of the course in
direct proportion to how much effort you put in. Thus
class work is the most substantial portion of the grade.
It has several components: written assignments, class
presentations and class participation generally. (This
last includes contributing to class discussions, asking good
questions, and active participation when another student is
presenting work at the board.) And, I should add,
attendance. If you don’t attend you can’t
participate. You are expected to be in class; if you
aren’t your grade will be adversely affected.

**Written Assignments:**
Since Foundations is primarily a language course, you will be
expected to learn clearly and precisely to express mathematical
ideas in writing. Several times during the semester you will be
asked to write up and turn in the proof of some theorem.
Each proof will be assigned two grades---one grade for content
and one grade for form. The content grade will reflect the
extent to which the appropriate ideas are expressed in your
write-up. That is, whether you understood the mathematical
ideas required for the proof. The grade for form will take
into consideration clarity of expression, completeness, proper
usage of both English and mathematical grammar, and whether you
really said what you meant to say.

Of course, form and content cannot be entirely divorced. If your writing is sufficiently muddled that the reader cannot tell what you meant to say, both grades will suffer. Likewise, if you really don’t understand what it is you are trying to say, the writing will be fuzzy and unclear. However, it is not impossible to distinguish the factors; the grades will be separated so you can see where improvement is needed.

When you write up an assignment, you are expected to include sufficiently many details to enlighten someone who does not already know what you are trying to say. This may require that you restate a definition or previous theorem and say how it is used in your proof. Do not be afraid to include too many details. If you are in doubt about whether or not to say something that you feel is pertinent, always do so!

**Class Presentations:** I have
said that most of the class will consist of students presenting
work to each other. You will be expected to do your share
in this. Most of the time I rely on volunteers to make
presentations. This makes it possible for students to
present the work about which they feel most confident. But
the fact that so much of the grade depends on this participation
means that all students must volunteer on something like a
regular basis. Don’t assume that because others volunteer,
you (or your grade) are off the hook. The good news is
that you probably won’t end up having to get every problem
assigned during the semester. If you don’t get it, someone
else will, and you will get to see the fruits of their labors.

The person who is presenting his or her work at the board is not the only person with responsibilities in a presentation. The students sitting at their desks have as central a role to play. Students presenting their work are not meant to replace a seasoned polished lecture that would be given by an experienced instructor. Nor should they be made to. They are counting on their fellow students to help them by making clarifying suggestions and asking questions. I will feel free to ask questions of persons who are sitting down.

**In-class exams:** The purpose of
these exams will be to encourage everyone to gain a command of
the basic mathematical facts that are discussed in class.
The questions will be straightforward for anyone who has been
digesting the material along the way. Typical questions
will ask you to define important terms, answer true/false and
short answer questions on the basic material and perhaps state
an important theorem or two. You may be asked to give a
simple proof of a fact that has already been presented and
discussed in the class.

**Takehome examinations:** Both
midterm examinations and a portion of the final examination will
be take-home exams. You will be required to construct
proofs for theorems that you have not seen before. You are
on your honor not to discuss take-home exams with anyone but me
until all exams have been turned in. You may not consult
any books except the textbook, but you are free to use any class
notes, any previously proved theorems, and anything that is
distributed in class. All guidelines for written
assignments also apply to take-home exams.

**Academic Honesty: **You are encouraged to work with other
students on everything except exams. (It has been my experience
that most students who thrive in this course are part of a small
group of 2-4 students who work together regularly outside of
class. I think this also makes the class more fun.)
It is, however, understood that all written work that you turn
in must finally be your own expression. For further
information see the student handbook or consult with me.

**Disabilities:** If you have a physical,
psychological, or learning disability that may impact your
ability to carry out assigned course work, feel free to discuss
your concerns in private with me, but you should also consult
the Office of Disability Services at 5453. The Coordinator of
Disability Services, Erin Salva (salvae@kenyon.edu), will review
your concerns and determine, with you, what accommodations are
appropriate. (All information and documentation of disability is
confidential.) It is Ms. Salva that has the authority and the
expertise to decide on the accommodations that are proper for
your disability. Though I am happy to help you in any way I can,
I cannot make any special accommodations without proper
authorization from Ms. Salva.