Syllabus

General Information

Title: MA4220: Number Theory
Time: Tuesday & Thursday at 12:30-1:45PM
Location: Hyde 349 (conference room)

Course Information and Policies

Corequisites

MA3110 or permission of instructor.

Course description

This course will be an introduction to number theory. We will study the set of integers, focusing on divisibility, congruences, arithmetic functions, sums of squares, quadratic residues and reciprocity, and elementary results on distributions of primes. As time permits we will also discuss applications in computer science and public-key cryptography. The course will take an axiomatic approach and focus on the construction of rigorous proofs from first principles.

Textbook

This semester we will be using the book Number Theory Through Inquiry by Marshall, Odell, and Starbird (ISBN: 978-0-88385-751-9). I expect you to be reading the textbook. I will not be covering every detail of the textbook and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the material in the textbook whenever necessary by asking questions in class or posting questions to the course forum.

In addition to working the problems in the textbook, I expect you to be reading the book. I will not be covering every detail of the book and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions in class or posting questions to the course forum.

This course will be different than any other math class that you have ever taken for two main reasons. First, you are used to being asked to do things like: "solve for $x$," "take the derivative of this function," "integrate this function," etc. Accomplishing tasks like these usually amounts to mimicking examples that you have seen in class or in your textbook. The steps you take to "solve" problems like these are always justified by mathematical facts (theorems), but rarely are you paying explicit attention to when you are actually using these facts. Furthermore, justifying (i.e., proving) the mathematical facts you use may have been omitted by the instructor. And, even if the instructor did prove a given theorem, you may not have taken the time or have been able to digest the content of the proof.

Unlike previous courses, this course is all about "proof." Mathematicians are in the business of proving theorems and this is exactly our endeavor. For the first time, you will be exposed to what "doing" mathematics is really all about. This will most likely be a shock to your system. Considering the number of math courses you have taken before you arrived here, one would think you have some idea what mathematics is all about. You must be prepared to modify your paradigm. The second reason why this course will be different for you is that the method by which the class will run and the expectations I have of you will be different. In a typical course, math or otherwise, you sit and listen to a lecture. These lectures are (hopefully!) polished and well-delivered. You may have often been lured into believing that the instructor has opened up your head and is pouring knowledge into it. I absolutely love lecturing and I do believe there is value in it, but I also believe that in reality most students do not learn by simply listening. You must be active in the learning you are doing. I'm sure each of you have said to yourselves, "Hmmm, I understood this concept when the professor was going over it, but now that I am alone, I am lost." In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called the Moore method (after R.L. Moore). Modifications of the Moore method are also referred to as inquiry-based learning (IBL) or discovery-based learning.

Much of the course will be devoted to students proving theorems on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word "produce" because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!

Furthermore, it is important to understand that proving theorems is difficult and takes time. You shouldn't expect to complete a single proof in 10 minutes. Sometimes, you might have to stare at the statement for an hour before even understanding how to get started. In fact, proving theorems can be a lot like the clip from the Big Bang Theory located here.

In this course, everyone will be required to

• write up quality proofs to assigned problems;
• present proofs on the board to the rest of the class;
• participate in discussions centered around a student's presented proof;
• call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar at first glance.

As the semester progresses, it should become clear to you what the expectations are. This will be new to many of you and there may be some growing pains associated with it.

Goals

(Adopted from Chapter Zero Instructor Resource Manual) Aside from the obvious goal of wanting you to learn how to write rigorous mathematical proofs, one of my principle ambitions is to make you independent of me. Nothing else that I teach you will be half so valuable or powerful as the ability to reach conclusions by reasoning logically from first principles and being able to justify those conclusions in clear, persuasive language (either oral or written). Furthermore, I want you to experience the unmistakable feeling that comes when one really understands something thoroughly. Much "classroom knowledge" is fairly superﬁcial, and students often find it hard to judge their own level of understanding. For many of us, the only way we know whether we are "getting it" comes from the grade we make on an exam. I want you to become less reliant on such externals. When you can distinguish between really knowing something and merely knowing about something, you will be on your way to becoming an independent learner. Lastly, it is my sincere hope that all of us (myself included) will improve our oral and written communications skills.

A little more propaganda

All of the secondary skills you will develop in this course are highly valued by society. Whether you become a teacher, a lawyer, an engineer, or an artist, what differentiates you from your competition is your ability to think critically at a high level, collaborate professionally, and communicate effectively.

Class presentations

(Adopted from Chapter Zero Instructor Resource Manual) Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken very seriously since they spearhead the work of the class. Here are some of my expectations:

• In order to make the presentations go smoothly, the presenter needs to have written out the proof in detail and gone over the major ideas and transitions, so that he or she can make clear the path of the proof to others.
• The purpose of class presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the proof clear to the other students.
• Presenters are to write in complete sentences, using proper English and mathematical grammar.
• Presenters should explain their reasoning as they go along, not simply write everything down and then turn to explain.
• Fellow students are allowed to ask questions at any point and it is the responsibility of the person making the presentation to answer those questions to the best of their ability.
• Since the presentation is directed at the students, the presenter should frequently make eye-contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.

Presentations will be graded using the rubric below.

4 Completely correct and clear proof or solution. Yay!
3 Proof has minor technical flaws, some unclear language, or lacking some details. Essentially correct.
2 A partial explanation or proof is provided but a significant gap still exists to reach a full solution or proof.
1 Minimal progress has been made that includes relevant information & could lead to a proof or solution.
0 You were completely unprepared.

However, you should not let the rubric deter you from presenting if you have an idea about a proof that you'd like to present, but you are worried that your proof is incomplete or you are not confident your proof is correct. You will be rewarded for being courageous and sharing your creative ideas! Yet, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute.

I will always ask for volunteers to present proofs, but when no volunteers come forward, I will call on someone to present their proof. Each student is expected to be engaged in this process. The problems chosen for presentation will come from the daily assignments. After a student has presented a proof that the class agrees is sufficient, I will often call upon another student in the audience to come to the board to recap what happened in the proof and to emphasize the salient points.

Homework

Daily Homework: Homework will be assigned each class meeting, and students are expected to complete (or try their best to complete) each assignment before walking into the next class period. All assignments should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation and spelling. You will almost always write a draft of a given solution before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment.

The Daily Homework will generally consist of proving theorems from the textbook. On the day that a Daily Homework is due, the majority of the class period will be devoted to students presenting some subset (maybe all) of the proofs of the theorems that are due that day. At the end of each class session, students should submit their write-ups for all of the proofs that were due that day. Daily Homework will be graded on a $\checkmark$-system.

Students are allowed to modify their written proofs in light of presentations made in class; however, you are required to use the felt-tip pens provided in class.

Weekly Write-ups: In addition to the Daily Homework, you will also be required to submit two formally written proofs each week. You may choose any two theorems that were turned in during a given week to submit the following Wednesday by 5PM. For example, you may choose any two theorems that were turned in during week 2. These problems are due by 5PM on Wednesday in week 3. Beginning with the second Weekly Write-up, you will be required to type your submission. You can either email me your Weekly Write-ups (preferably a PDF), share them with me via PSU Google Docs, or submit them in person.

The Weekly Write-ups are subject to the following rubric:

4 This is correct and well-written mathematics!
3 This is a good piece of work, yet there are some mathematical errors or some writing errors that need addressing.
2 There is some good intuition here, but there is at least one serious flaw.
1 I don't understand this, but I see that you have worked on it; come see me!
0 I believe that you have not worked on this problem enough or you didn't submit any work.

Please understand that the purpose of the written assignments is to teach you to prove theorems. It is not expected that you started the class with this skill; hence, some low grades are to be expected. However, I expect that everyone will improve dramatically. Improvement over the course of the semester will be taken into consideration when assigning grades.

Any Weekly Write-up problems that you received a score of 1, 2, or 3 on can be resubmitted up until one week after the corresponding problem was returned to the class. The final grade on the problem will be the average of the original grade and the grade on the resubmission. Please write "Resubmission" on top of any problem that you are resubmitting and keep separate from any other problems that you are turning in.

Unlike a traditional Moore method course, you are allowed and encouraged to work together on homework. However, each student is expected to turn in their own work. In general, late homework will not be accepted. However, you are allowed to turn in up to 5 homework assignments (daily or weekly) late with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. Your overall homework grade will be worth 25% of your final grade.

You can find the list of homework assignments here. I reserve the right to modify the homework assignments as I see necessary.

Exams

There will be 2 midterm exams, which are tentatively scheduled for the following Thursdays: March 10 and April 21. There will also be a cumulative final exam, which will be on Tuesday, May 17 at 11:00am–1:30pm. Each exam will consist of an in-class portion and a take-home portion. Each exam will be worth 15% of your overall grade. Make-up exams will only be given under extreme circumstances, as judged by me. In general, it will be best to communicate conflicts ahead of time.

Rules of the Game

You should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course. On the other hand, you may use each other, the textbook, me, and your own intuition.

Basis for Evaluation

Category Weight Notes
Homework 25% a mix of Daily Homework and Weekly Write-ups
Presentations/Participation 30% each student must present at least twice prior to each exam
Exam 1 15% in-class portion on Thursday, March 10
Exam 2 15% in-class portion on Thursday, April 21
Final Exam 15% Tuesday, May 17 at 11:00am–1:30pm

Getting Help

There are many resources available to get help. First, I recommend that you work on homework in groups as much as possible, and to come see me whenever necessary. Also, you are strongly encouraged to ask questions in the course forum on our Moodle page, as I will post comments there for all to benefit from. I am always happy to help you. If my office hours don't work for you, then we can probably find another time to meet. It is your responsibility to be aware of how well you understand the material. Don't wait until it is too late if you need help. Ask questions! Lastly, you can always .

To effectively post to the course forum, you will need to learn the basics of LaTeX, the standard language for typesetting in the mathematics community. See the Quick LaTeX guide for help with $\LaTeX$. If you need additional help with $\LaTeX$, post a question in the course forum on our Moodle page.

Math Activity Center

You can also visit the Math Activity Center, which is located in Hyde 351. This student-run organization provides peer tutoring services for most 1000 and 2000 level math courses and some 3000 level courses. Tutors are typically math majors interested in teaching math and practicing their instructional skills. You can drop in anytime during open hours.

Student Handbook

The PSU Student Handbook addresses policies pertaining to students with disabilities, religious observation, honor code, general conduct, etc. The Handbook can be found here.

ACT for Growth

All teacher education majors are subject to the Areas of Concern/Targets for Growth policy, which is located here.