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 |
12% of the final grade |

Class
Participation |
8% of the final grade |

Written
Assignments |
15% of the final grade |

Take-home
Midterm |
15% of the final grade |

Two
In-Class Midterms |
30% of the final grade (15% each) |

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

In-class Midterm
#1 |
Wednesday 1 March, covers material through Chapter 3. |

Takehome Midterm |
12:00 noon Friday 24 March to 12:00 noon Tuesday 28 March |

In-class Midterm #2 |
Wednesday 19 April, covers
material from Chapters 4 and 5 |

Take-Home Final |
Due 11:30 a.m. on Wednesday 10
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. I will use a sophisticated randomization method to
select students to present in class and will try to ensure that
every student gets to present a roughly equal number of times.
If you are not prepared to present a problem, no problem! Just
take a pass, and we'll move on. As long as you don't pass more
than a couple of times in the semester, this will not adversely
affect your grade. But please **do not** go up to the board
without being fairly confident that you have a fairly complete
solution/proof. This will become apparent quickly and will not
benefit you or the rest of the class.

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 relatively 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 or solve a straightforward
problem.

**Takehome examinations:** One
midterm 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 this
document, the student handbook or consult with me.

*Students
who anticipate they may need accommodations in this course
because of the impact of a learning, physical, or
psychological disability are encouraged to meet with me
privately early in the semester to discuss their concerns.
In addition, students must contact Erin Salva,
Director of Student Accessibility and Support
Services **(740-427-5453** or salv ae@kenyon.edu),
as soon as possible, to verify their eligibility for
reasonable academic accommodations. Early contact will
help to avoid unnecessary inconvenience and delays.*