Class Schedule:  Tuesday, Friday 11:30 – 12:20, Wednesday 1 – 1:50, HH 211

Instructor:  B. Gold, HH247, 732-571-4451, bgold@monmouth.edu

Office Hours:  Monday 1-2, Tuesday 2:30 – 3:30, Wednesday 11:30-12:30, Friday 1-2; or by appointment or chance.

Required Text:  Number Theory Through Inquiry, by Marshall, Odell, and Starbird

Course Goals: Students will

Reasoned Oral Discourse outcomes:  The student will demonstrate the ability to:

• orally present ideas and information in a reasoned and effective manner with attention to elements of vocal and nonverbal quality;
• evaluate multiple sources of information and synthesize this material in a reasoned oral presentation;
• critically evaluate the style and substance of an oral presentation and pose appropriate questions; and
• respond to questions or challenges during a reasoned oral presentation in a group setting.

 Course methodology: The style of instruction in this course is called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Our text consists of a list of questions and theorem statements without proofs. The standing homework assignment is for each student to prove the next theorems or settle the next questions in the book, thus in a sense writing his or her own textbook. As students give presentations, members of the class are expected to ask questions when they don't understand something. I will also ask if other students have a different proof. Each student is responsible for determining whether the proof is correct. The correctness is not based on an authority figure (the instructor) asserting it is correct, but instead on the ability of the individual students to follow the argument. The instructor will only rarely present proofs. You will learn to think independently and to depend on your own minds to determine right from wrong. You will develop the central, important ideas of introductory number theory on your own. You will also learn to write mathematics, as this is a writing intensive course. 

I know that, by now, most of you are used to discussing problems with each other.  In this course, this is NOT allowed, except for the (computational) homework problems.  You are also NOT allowed to use other books, the Internet, or other sources to find proofs of the theorems or settle questions (except for those underlined and bolded in the schedule: for those, see the description two paragraphs later).  However, if you cannot do any of the theorems or examples assigned for the week, you should come see me, and I’ll try to help you.

After the week’s presentations are given, each student is to write up all presentations (s)he has given, and enough of other students’ presentations to fill a bit over one page (NO heading: you can’t use up space by typing your name on one line, the course name on the next, your e-mail address on a third, etc.), typed (using Word’s Insert Equation or Insert Symbol features as needed), 1 ½ spacing between lines at most, 12-point font.  (To be a Writing Intensive course, there needs to be at least 15 pages of writing that can be revised for content, style, and correct use of language (mathematical and English).)  I will grade these presentations on a 10 point scale, with 2 points for overall structure of the proofs, 4 points for correct details, 2 points for correct use of mathematical language, and 2 points for correct English.  If you lose ANY points for either English or mathematical language, you must resubmit the presentation within a week after it is graded, with these problems fixed to get any credit for the write-up.  You are also encouraged (but not required) to fix the mathematical content.  Your final grade for the write-up will be a weighted average of the original write-up and the revision.  I will choose the best-written version of each presentation for posting on the class e-campus page as the class version of the presentation.  These write-ups will be submitted via the eC@mpus dropbox for that week.

To satisfy several of the Reasoned Oral Discourse learning outcomes, twice during the semester each student will present a classical theorem by reading and synthesizing three different proofs of it.  These proofs are designated in boldface type and underlined in the schedule below.  To satisfy the outcome “the student will demonstrate the ability to evaluate multiple sources of information and synthesize this material in a reasoned oral presentation,” you will read three versions of the proof you are to present, and decide how to present the proof to the class.  Ideally you will synthesize several of the proofs, but at the least you will consider each and describe why you have chosen the proof you give.  Two of the versions must come from books (I have placed several on reserve for this course in the library; there are others in the mathematics department library, HH 205, and faculty members may have still others), and the third may be from a book, from the Internet, or you may develop it on your own.  All sources must be completely cited (including full URL for proofs from the Internet) for these presentations in your write-up, and you must, in your write-up for the week, discuss the range of proofs you read and why you chose the proof you gave.  At the end of your presentation, your fellow students will “critically evaluate the style and substance” of your presentation, considering both the content and “elements of vocal and nonverbal quality.”

Course Requirements: Daily class presentations, weekly computational homework (due Tuesdays) and written proof presentations (due Mondays at noon), two midterm examinations (Wednesday, February 27, and Wednesday, April 3), cumulative final examination.

Methods of Evaluation and Grading Policy: Midterms: 10% each; computational homework: 5%; in-class presentations: 30% (of which 5% is for asking good questions and careful evaluation of others’ presentations); write-ups of presentations: 30%; final examination: 15%.

On a scale of 0 to 100, grades of:

• A and A- will be assigned to scores of 90 and above
• B+, B and B- will be assigned to scores between 80 and 90
• C+, C and C- will be assigned to scores between 65 and 80
• D+, D and D- will be assigned to scores between 50 and 65
• F will be assigned to scores below 50.

If you are have to miss an examination, you must let me know prior to the exam or you receive an automatic 0. You must make arrangements with me some time the day of the exam about when you will take a make-up exam.

Attendance Requirement: Because each class will be devoted to student presentations of the problems, attendance is required. Any unexcused absences will result in grade reduction, and excused absences must be made up with replacement work.

Last date to Withdraw with automatic assignment of W grade: April 1, 2013.

Statement on Academic Honesty: You are only allowed to work with others on the computational homework.  You must develop all presentations and write-ups entirely on your own, except for, as needed, consulting with me.  You should never be in physical or electronic possession of another students write-up for an assignment (except those I post on eC@mpus). Any violation of the academic policies (see the Student Handbook) will result in a grade of 0% for the assignment and may result in a failing grade for the course, a suspension, or an expulsion from Monmouth University.

Examination Rules: No student is permitted to have at his or her desk any books or papers that are not given out by the instructor. Possession of such material will be regarded as evidence of intent to use the information dishonestly. No communication between students during the examination is permitted. If there are questions, or if there is a need for additional material, the instructor should be asked. Details of calculations should be written on the pages of the exam. In accordance with the academic honesty policy of Monmouth University each exam will contain the following pledge:

I, ____________________________, certify that I have read the rules for examinations, and that I have abided by them. By signing, I affirm that I have neither given nor received aid during this examination, and I understand that violation of this affirmation may result in suspension or expulsion from Monmouth University.

Statement on Special Accommodations: Students with disabilities who need special accommodations for this class are encouraged to meet with me or the appropriate disability service provider on campus as soon as possible. In order to receive accommodations, students must be registered with the appropriate disability service provider on campus as set forth in the student handbook and must follow the University procedure for self-disclosure, which is stated in the University Guide to Services and Accommodations for Students with Disabilities. Students will not be afforded any special accommodations for academic work completed prior to the disclosure of the disability, nor will they be afforded any special accommodations prior to the completion of the documentation process with the appropriate disability office.

Outline of Course Content and Schedule:  All of these dates are very rough, because they depend on how fast the class gets through the material.  I will announce revisions in what is due the following week on Friday of each week, and will also post it in eC@mpus.  Numbers in decimal format are from the main part of the textbook (the number preceding the decimal is the chapter number), Ax is from the Appendix of our book.  Some homework problems are included with those from the book.

1/22 – 1/25: Presentations 1.1-1.6, A19-21; in your write-ups, everyone must include at least one induction proof.  Homework: 1.7, 1.8 

1/28 – 2/1: Presentations: A30, A31, 1.9 – 1.18; in your write-ups, everyone must include either A30 or A31. Homework: 1.19; you must use n > 7 as the mod in your examples.

2/4 – 2/8: Presentations: 1.20 – 1.23, 1.26, 1.27, 1.28 – 1.30; Homework: 1.25, 1.31.

2/11 – 2/15: Presentations: 1.32 – 1.34, 1.35, 1.38, 1.39 – 1.44, 1.45; Homework: 1.36, 1.37

2/18 – 2/22: Exam 1; Presentations: 1.48, 1.51, 1.53, 1.55 – 1.58, 2.1, 2.3;

Homework: 1.50, 1.54, 2.2, 2.4, 2.5, 2.6 and the following two problems:

1. Find gcd(4144,7696), using the Euclidean algorithm.  Use this to find integers x and y such that  4144x + 7696y = gcd(4144,7696).  Then, find a formula which gives all integer pairs (x,y) such that 4144x + 7696y = 7gcd(4144,7696).

2. Find gcd(530418,100083), using the Euclidean algorithm.  Use this to find integers x and y such that  530418x + 100083y = gcd(530418,100083).  Then, find a formula which gives all integer pairs (x,y) such that 530418x + 100083y = 5gcd(530418,100083).

2/25 – 3/1: Presentations: 2.7, 2.9, 2.12, 2.13, 2.19 – 2.24; Homework: 2.10, 2.11, 2.14, 2.15, 2.16, 2.17 

3/4 – 3/8: Presentations: 2.25 – 2.31 (must give new proofs, using the FTA), 2.32 – 2.34, 2.35, 2.37, 2.38; Homework: Find five primes congruent to 3 modulo 4; and 3.1.

3/11 – 3/15: Presentations:2.41 – 2.43, 3.2 – 3.4, 3.14, 3.16, 3.17; Homework: 2.44 (the first 3 of each), 3.5, 3.7, 3.15, 3.18 (parts 1-3, and look at part 4)

3/18 – 3/22: Spring break 

3/25 – 3/29: Presentations: 3.19, 3.20, 3.24, 3.27, 3.28, 3.29; Homework: 3.21, 3.22, 3.25, 3.26, 4.1, and the following:

Find all simultaneous solutions of the following system of congruences: 

2x ≡ 2 (mod 4)

3x ≡ 4 (mod 5)

5x ≡ 3 (mod 9)

(Be careful!!! The first equation has MORE than one solution!  But the numbers are all small and this doesn't get too ugly.)

4/1 – 4/5: Exam 2; Presentations: 4.2 – 4.6, 4.8 – 4.11, 4.13, 4.14, 4.15, 4.16, 4.17; Homework: 4.7 (for some n > 7), 4.12 (compute a^p-1 (mod p) for at least TWO of the primes 5, 7, 11, and at least TWO a's greater than 1 and less than p-1 (and not those done in class)), 4.19. 4.20

4/8 – 4/12: Presentations: 4.18, 4.21, 4.28 – 4.31, 4.32, 4.33; Homework: 4.22, 4.23, 4.27, 4.34, 4.35

4/15 – 4/19: Presentations: 4.36, 4.38, 4.40, 4.41, 4.42; Homework: 4.37, 4.39

4/22 – 4/26: Presentations: 5.1 – 5.6, 5.8; Homework: 5.7

4/29 – 5/3: Presentations: 6.1 – 6.6, 6.8, 6.11 – 6.15, 6.17; Homework: 6.7, 6.9, 6.10, 6.16 (for n = 10)