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Foundations: 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.
Inquiry-Based Learning: The classroom approach used in this class is called Inquiry-Based Learning, or IBL. IBL is a pedagogical strategy in which students are led to develop mathematical concepts and discover mathematical connections for themselves. The faculty member serves as mentor and moderator. The time in class is structured as a collaborative learning experience in which everyone works together to deeply understand the mathematical ideas. There is a lot of evidence that, when students actively discuss mathematical ideas with other students on a regular basis, they learn more deeply and the knowledge stays with them longer than when they work in isolation. Thus I encourge you to work together in small groups outside of class as well as in the class.
Your brain needs time to process ideas and made connections: When working out new mathematical ideas for yourself, thinking time is essential. You should not expect to get every problem instantly. This means you need to arrange your out of class work time in shorter, more frequent sessions rather than planning a massive number of hours the night before an assignment is due. Schedule an hour to an hour and a half most days of the week to work on Foundations. You will find that your brain keeps processing between times and the ideas come to you when you least expect them. But only if you have primed your brain by doing some preliminary thinking beforehand.
Working with your fellow students: 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. This not only provides a more profound learning experience, I think it also makes the class more fun.
Office hours: Most students need to consult with me as they work through the problems for this class. Thus frequent work in office hours is an expected part of the course. I encourage you to plan Foundations work time around my office hours so that we can chat at times when you feel stuck or confused.
Text: Chapter Zero: Fundamental Notions of Abstract Mathematics, 2e by Carol Schumacher. Addison-Wesley, 2001.
Grading: The grade will be calculated based on 5 components:
| Class preparation, class participation, and in-class presentations | 30 % of the final grade |
| Written Assignments | 20 % of the final grade |
| In-Class Midterm #1 | 5% of the final grade |
| Take-home Midterm | 15 % of the final grade |
| In-Class Midterm #2 | 15% of the final grade |
| Final Examination | 15% of the final grade |
| In-class Midterm #1 | Friday, February 12, 2016 |
| Takehome-Midterm | Any consecutive 60 hour period between 11 a.m. on Friday, March 25 and 5 p.m. on Tuesday, March 29. |
| In-class Midterm #2 | Monday, April 18, 2016. |
| Final Examination | Takehome portion distributed Friday, May 6 (last day of classes) Tuesday, May 10, 2016 at 10:30 a.m. (90%). + Roundtable discussion from 10:30-11:30 a.m.(10%) |
Class preparation and in-class work: Foundations is 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. Much of the time in class is spent in discussing ideas that students have already been working on outside of class. Thus for most class periods you will have problems to work on in preparation for class. That work will serve as a springboard for class discussion that deepens the insights gained before class with careful class discussion that teases out subtleties and sharpens understanding.
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. The general rule of thumb for college courses is 2-3 hours of work outside of class for every hour in class. This class is certainly on the high end of that scale. Thus class work is the most substantial portion of the grade. The grade for classwork has several components: class preparation, 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 math 222 is primarily a language course, you will be expected to learn to clearly and precisely 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. In addition to considering mathematical content, the grade 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. The problems will be graded on a scale of 1 to 5. (I reserve the right to assign 6 points to an exceptionally well written or elegant proof!) You should not think of the grade as representing a percentage but, rather, as delivering a message:
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. At least initially, if you are in doubt about whether or not to say something that you feel is pertinent, always do so!
Cooperative learning for written assignments: I encourage students to discuss problems with me and with each other. This is true for written work as well as for class assignments. If you work with other students on an assignment, you should note this on your paper. (There is no penalty, this is just a matter of academic honesty.) However, write-ups are individual. Be sure that the group discussions stop before the writing begins. Discuss the problems in a group, then go off by yourself to write up the solutions for submission. It is not OK for a group to work out a problem together, copy it down, and turn in identical write-ups----even if all members of the group contributed equally to its production. Individual digestion of the ideas and individual writing must be the end of the process. Note that if you all write something together on a whiteboard, take a picture with your phone, and then separately copy down what is on the phone you have not accomplished the "individual digetion and writing" that I expect. You can use joint work as a guide, but the final write-up should come out of your head. (See Academic Integrity, below.)
You are the authority: Many of the proofs you will do in class are "standard" in the sense that you would have no problem finding full solutions in books or online. But the idea here is for you to struggle with these ideas and produce solutions of your own. (Often with help from Prof. Schumacher or in consultation with other students, as noted above.) Thus when you work on class problemm, you may not consult any books except the textbook. You are, of course, free to use any class notes, any previously proved theorems, and anything that is distributed in class.You may not consult outside sources, including on-line sources and human sources not associated with the class. In particular, it is fine to consult with students in your class or with Professor Schumacher, but you should not ask students who took the course previously to help you or to provide you with copies of their work. (If students in the other section are working on the same problems as you are at the same time, you may work with them as you would with a student in your class.)
LaTeX: During the semester you will be introduced to a powerful mathematical typsetting program called LaTeX. After that, you will be expected to LaTeX on problem on each written homework assignment and all of your mid-term and final takehome exams.
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, but sometimes presenters will be chosen by me or randomly, by the roll of a die. Asking for volunteer presenters, 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 midterms: The purpose of these exams will be to encourage everyone to gain a command of the basic mathematical facts that are discussed in class. Typical questions will ask you to define important terms, answer true/false and short answer questions, give examples and counterexamples on the basic material and perhaps state an important theorem or two. You may be asked to give proofs of a theorems that have already been presented and discussed in the class.The second midterm is worth more than the first and will thus be commensurately more difficult. This is natural because by the time of the second midterm we will have much more sophisticated skills and will have studied deeper ideas.
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 Prof. Schumacher until all exams have been turned in. You may not consult any books except the textbook. You may not consult outside sources, including on-line sources. However, you are free to use any class notes, any previously proved theorems, and anything that is distributed in class. In addition, all guidelines for written assignments also apply to take-home exams. You will be asked to turn in LaTeX'd solutions to all of your takehome exams.
Academic Integrity: You are encouraged to work with other students in our class on everything except exams. It is, however, understood that all written work that you turn in must finally be your own expression. In working alone or with other students on problems, it is understood that your only sources of information will be the book, any notes you took in or for class, and brainpower. (And, of course, Prof. Schumacher, who is happy to answer questions at any time.)
For further information see the student handbook or consult Prof. Schumacher.
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 Student Accessibility and Support Services (SASS) at 740-427-5453. The Coordinator of SASS, Erin Salva (salvae@kenyon.edu), will review your concerns and determine, with you, what accommodations are appropriate. 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. Except in extraordinary circumstances, accommodations must be arranged with Ms. Salva and with me at least one week before they are to take effect.
Title IX; responsible employee: As a member of the Kenyon College faculty, I am concerned about the well-being and development of my students and am available to discuss any concerns you may have. However, I need for you to know that as a Kenyon faculty member I am obligated by federal law to share certain information with the college's Title IX coordinator. This is to ensure our student's safety and welfare are being addressed, consistent with the requirements of that law. These disclosures include but are not limited to reports of sexual assault, relational/domestic violence, and stalking.