Syllabus	Math 135: First Steps into Advanced Mathematics	Spring 2008

About this Course

First Steps into Advanced Mathematics: This course will be probably be quite different from any Mathematics course you have ever taken. You may have heard that "First Steps is that proofs course." It probably is the first course you will have taken that focuses on proof, but it is also a bit more. First Steps is all about learning to read, write, speak, and communicate in the language of mathematics. To do this well will require a mastery of the tools and methods that mathematicians consider essential in the study of advanced mathematics. This course represents the "first steps" in acquiring this mastery, and it does so by forcing you to "do mathematics". As such, there will be little lecturing in this course as compared to other math courses you have taken. The class is run as a seminar and you will learn to use logic, deductive and inductive reasoning, set operations, and so on to explore mathematical topics and to communicate the results of these explorations precisely. The topics studied while developing these sophisticated skills are fundamental and permeate almost all branches of mathematics. The course could just as easily be called "Foundations of Mathematics" because most of the mathematics you study beyond this course will depend on the solid foundation you form here, and you will see many of these same topics in various forms, expanded on again and again in future courses.

Purpose: The primary purpose of this course is for you to become proficient, comfortable, and confident in using the language of mathematics to begin communicating with other mathematicians. In particular, you will learn how to make sense of abstract definitions and mathematical statements; recognize and construct rigorous proofs; pick out weak spots and fill in details in less rigorous proofs; and write, present, and defend proofs to a group of your peers in a clear, concise manner. You will also have fun and feel a certain pride as you begin considering yourself to be a talented and capable Mathematician studying with other Mathematicians.

Text and Resources: The main textbook for the course is Chapter Zero, Fundamental Notions of Abstract Mathematics by Carol Schumacher. Not only is this text very readable, it also expects your active participation while reading it. It is essential to your success in this course and as a future mathematician that you do read it, and do so in an interactive fashion. We will start at the beginning, and move systematically and thoroughly through Chapters 0-5. We will then apply the foundational skills learned to one or more of the remaining chapters which each represent a different topic in Mathematics. Along the way, there will be various other reading assignments of articles on the web, books on reserve in the library, handouts, and other resources which I think will be of interest to you and will assist you in becoming a mathematician.

Materials Required: You need a spiral notebook or composition notebook (not a 3-ring binder) for your journal work (see below). You may additionally want something separate to use for in-class notes and returned work. You need a good pencil and eraser (or several). All work that is submitted must be done in pencil.

Assessment Policies

Journal Assignments: Reading and uncollected homework will be assigned daily. All related work for these assignments will be kept in your journal. All work in your journal should be kept in order and completely separate from class notes and returned work. (I suggest getting a completely separate notebook for your class notes and returned work.) See the Journal Guidelines for more detailed information about format and content of the journal. The journal will be collected at each exam period and assigned a score based on a possible 40 points for each of the first two times, and a score based on a possible 20 points for the final time. (For a total of 100 points.) Note: it is OK to work on Journal assignments with other students, However, you must indicate who you worked with each time.

Presentations: Classroom presentations of various ideas, problems, and theorems will be required (usually from journal assignments). Each student must complete at least two presentations, one of which must be a formal proof. Required presentations are each worth 20 points and will be scored considering both mathematical accuracy and presentational quality. See the Presentation Guidelines for more detail about presentations. (Bonus presentation points may be earned - see below.)

Collected Homework: In addition to journal assignments, other problems and theorems will be regularly assigned. These will usually be collected at the next class period and will vary in value from 10 points to 40 points. Late homework will not be accepted. Occasionally, collaborative homework will be assigned. When this is the case, one write-up will be submitted, and each member of the collaborative group will receive the same score.

Quizzes: You should expect regular, short quizzes over material assigned for the journal. The purpose of these quizzes is to encourage and reward you for keeping up with the journal assignments. Each quiz will be worth 10 points. Quizzes may not be taken late.

Seminar Attendance: You are required to attend at least two mathematics/computer science seminar talks during the semester. Each of these two attendances will earn 10 points (you must stay for the entire talk). (Bonus attendance points may be earned - see below.)

Participation Score: All of the points awarded in the above categories will be added together and the percentage calculated will count as 50% of the course grade.

Exams: There will be two exams, each of which will include a take-home portion and an in-class portion. These exams together will count as 30% of the course grade. The in-class portions are scheduled as follows.

Final Exam: The final exam will be comprehensive in nature, and will include a take-home portion and an in-class portion. The final will count as 20% of the overall course grade. The in-class portion is scheduled as follows.

Attendance And Make-up Policies: You are expected to attend every class, to arrive at class on time, and to stay for the entire period. 3 or more unexcused absences will result in lowering your grade by a full letter grade. Each additional absence will lower it another letter grade. For purposes of grading, two tardies will be counted as an unexcused absence. A grade of zero will be assigned for any homework not submitted on time and for any quiz or exam not taken when administered to the class. The opportunity to "make-up" work missed will NOT be available except in instances required by The Colleges. In such cases, it is the student's responsibility to arrange for make-up work as soon as possible. (See HWS Catalogue, pp. 38-39.) In the rare event that I am required to provide "make-up" work, it will necessarily be significantly different from the work missed. This make-up policy will be strictly enforced. This make-up policy will be strictly enforced.

Bonus: Bonus is primarily offered to buffer any ill effects you may experience as a result of my strict make-up policy. You will earn 5 bonus points for each additional seminar talk you attend. You will earn up to 10 bonus points for each additional classroom presentation. Any bonus points earned are added to the overall points earned in the participation sections above (before dividing by total points possible in those sections). Bonus points available will be limited by the number of seminars occurring and by time available for classroom presentations. Required presentations take priority over bonus presentations.

Grading Scale: Scores will be weighted as indicated above and summarized below:

The initial course grade calculation will then be determined by the overall percentages:

The final course grade calculation will include consideration of factors such as attendance, conscientiousness, and level of participation. If your overall average falls in the upper 2% of a grade range, and you have 1 or fewer unexcused absences, have been conscientious about journal work and collected assignments, and have actively participated in class discussions, you will have a "+" appended to your grade. On the other hand, if your overall average falls in the lower 2% of a grade range, you may have a "-" appended to your grade if work has been missed or is incomplete, or if you have failed to participate at an acceptable level in class discussions.

Other

• Hobart and William Smith Colleges' Principle of Academic Integrity will be upheld. (See HWS Catalogue, p. 33. See also HWS Handbook of Community Standards.) Unless work is assigned collaboratively, you should not submit work for a grade that is not completely your own. Doing so will be grounds for receiving a failing grade in the course, and having a record of the event placed in your permanent file.

• All work submitted must be done in pencil and must be detailed and neat in order to receive full credit.

• A Note about the Center for Teaching and Learning (CTL): Hobart and William Smith Colleges encourage students to seek the academic collaboration and resources that will enable them to demonstrate their best work. Students who would like to enhance their learning and/or academic performance should contact the CTL. If you are a student with a disability for which you may need accommodations, you are required to register with the Coordinator of Disability Services at the CTL and provide documentation of the disability. Services and accommodations will not be provided until this process is complete. The web site for information pertaining to registration with the CTL and documenting disabilities is: http://www.hws.edu/studentlife/stuaffairs_disabilities.aspx.