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 5 components:
includes written assignments,
class participation, and
|50% of the final grade|
|Take-home Midterm||20% of the final grade|
|In-Class Midterm||5% of the final grade|
|Take-home Final||20% of the final grade|
|In-Class Final||5% of the final grade|
|In-class Midterm||Monday 25 March, covers material through Section 4.2.|
|Takehome Midterm||12:00 noon Friday 22 March to 12:00 noon Tuesday 26 March|
|Takehome Final||12:00 noon Friday 3 May to 6:30 p.m. Thursday 9 May|
|In-class Final||6:30 p.m. on Thursday 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 (email@example.com), 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.