Syllabus Math 251W: Foundations of Advanced Mathematics, Spring 2023

Required Text: Proofs and Fundamentals, A First Course in Abstract Mathematics, by Ethan Bloch

“In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigor. But at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired.” - Michael Atiyah

Course Objectives & Student Learning Outcomes

To paraphrase G. H. Hardy, a mathematician is an artist working in the medium of ideas. In this class you will transform from consumers of mathematics to creators; learning to use the tools of conjecture, abstraction, and logic to construct mathematical proofs. Proof writing is an integral part of all upper-level mathematics courses, and the method by which mathematicians extend the boundaries of what is known. It takes perseverance, creativity, clarity, logical thinking, and an ability to see connections and patterns - skills which will help you in whatever field you pursue. Through this course you will: learn to read, understand, and compose mathematical proofs; distinguish valid arguments from invalid ones; formulate, write, and present logical arguments; develop your aesthetic sense in mathematics; and strengthen your analytic reasoning skills.

Topics Covered

include, but are not limited to:

• Symbolic Logic

• Proof Techniques, including Induction Proofs

• Set Theory

• Equivalence Relations and Partitions

• Functions

• Modular Arithmetic and Modular Groups

Class Engagement: Students are expected to attend 2 of the math colloquium talks given throughout the semester. A schedule of talks can be found on the Math Department website, and linked to from the course WISE site. The goal of this attendance requirement is to introduce you to the diversity and vitality of current mathematics research, and to include you in the Math Department community. In addition, there will be in-class investigations, and out of class Perusall readings (including texts, podcasts, and videos) assigned to supplement class discussions. Your grade for this component of the course will be based on how meaningfully you engage in these activities and discussions, as based on your attendance and the quality of your in-class and Perusall contributions. Perusall assesses how integral your contributions are to the conversation – hence engaging with the Perusall readings early and often will increase your score.

Individual Exams: There will be one midterm and one cumulative final exam (celebrations? denouements? an assessment by any other name is just as exciting). More details about the structure of these exams will be given in class.

Group Exams: There will be two group exams. These exams are done in groups of two or three, with each group member having a unique problem to work on. Group members proof-read each other’s work and offer suggestions. Of the 50 points possible for each group exam, 40 will be based on your work and 10 will be based on your proof-reading of team member’s exams. Each team member is allowed one page, one-sided, of notes. For more information, see the handout on Group Exams https://sites.google.com/willamette.edu/groupexaminformationpage/home

Homework: Homework will consist of reading assignments, practice problems, and portfolio proofs (which must be generated using LaTeX). A list of useful LaTeX resources is available here: https://willamette.edu/~emcnicho/courses/LaTeX.html

Student Responsibility

I will do everything in my power to help you learn, but it is up to you to make the most of this experience by coming to class prepared and ready to engage in discussion. The following actions are all critical for your success in this class. If there are barriers to you doing any of the following, please let me know.

ASK QUESTIONS & THINK CRITICALLY. Throughout the course you should be asking yourself “Why are we doing this? Why does this method work? How is this related to other topics I’ve learned?” Ask questions in class, after class, during office hours, whenever! If you are confused or having problems with a certain section of the material talk to me about it. It is best to clear up confusions right away, and if you have a question, almost certainly other members of class do to. Be the brave one and speak up! The whole class will benefit from your questions.

DO THE ASSIGNMENTS. As you know, mathematics is not a spectator sport. At the same time, it can be a frustrating endeavor to engage in alone. Think of it as a team sport! I strongly encourage, and may even require, you to work with your fellow students on homework assignments.

SPEND TIME STUDYING. Willamette’s Credit Hour Policy holds that for every hour of class time there is an expectation of 3 hours of work outside of class. You should anticipate spending approximately 9 hours outside of class engaged in course-related activities. To master the content of this course you will need to invest time and effort. Set aside time for both homework and studying. A good approach would be to spend approximately 1-2 hours on reading the perusall assignments and taking notes on the assigned text sections, 3-6 hours doing the assigned problems, and 1-3 hours studying for tests. This course is completely manageable, and you can do well in it as long as you engage from the beginning.

How to Get Unstuck

Proof writing is a beautiful, but highly structured form of communication. The topics and skills covered in this class will be more abstract than those you have encountered in previous math courses. It is natural to feel a little lost or confused at times. What should you do? Discuss! Ask questions! You can start conversations on the course Discord channel, bring questions up in class, email me, go to the department tutoring, arrange to meet with me, arrange to meet with other students, attend office hours, etc. We now have the capability to meet in person or virtually, so there are even more ways to talk things through with me or your classmates.

Asking Questions in Class

Sometimes it's hard to know what exactly to ask, so here are some some ideas if you need inspiration:

• I'm a little confused on that last point, could you go over that again?

• How does this tie-into the overarching subject of this chapter?

• How did you know to take that step? I don't think I would have known to do that.

• Why is it important to <insert proof writing convention here>? Is that just the general practice of does it imply something specific?

• Can we quickly review the definition of <insert word here>? I've forgotten what that means.

• Is it possible to quickly review why <thing 1> implies <thing 2>? It's not obvious to me.

• How would I recognize what kind of proof technique to use here? Are there particular clues to look out for?

Advice from Past Students to Future Students

On Writing Proofs:

“Start proofs as soon as they are available so you can ask your questions in class!”

“Using scaffolds to structure your proofs helps you solve them.”

“Just start the proof. Staring at it doesn’t help. Start writing and you’ll know more than you think.”

“Do not procrastinate. Let me say that again: DO NOT PROCRASTINATE! These things take time to work out and write up. Start early.”

“Takes time. It’s really detailed so be as specific as possible and give yourself time to LATEX it up.”

“Justify, justify, justify! Don’t assume everyone knows what you are talking about.”

“Eventually you’ll understand the proof outlines for each type of proof. However, the amount of time before it doesn’t suck varies. Some people get it right away, others take a few months.”

“It’s like a board game; it’s easy to play if you know the rules... Learn the rules.”

“Little details are important. Often understanding comes in whether you know to put the small stuff into your proof.”

“USE DEFINITIONS!!! That is how you do proofs.”

“Think systematically, don’t just wing it. Map out what you need to know first, then solve.”

“At least start at the beginning doing outlines. If you get it down, you can start just going for it. But start with the outlines.”

On Your Proof-Writing Abilities (once the class is over):

“Figuring out how to prove something was an awesome feeling! I really liked LATEXing and the logical reasoning behind the math."

“Proofs are an art. You will learn how to combine creativity and analysis.”

“They’re pretty good, I’m not entirely confident but my proof-writing abilities are extraordinarily better than they were at the beginning of the semester.”

“I feel so accomplished in my abilities now. I felt ashamed about my proof-writing at the beginning of the course. Now, I feel confident in it.”

“Proof-writing is the most academically challenging thing I’ve ever done; I’d say my proof-writing abilities are vastly improved.”

“They are vastly improved. I feel like I learned more about logic and stating arguments in structurally correct ways than I learned about math. Also, I learned a lot about writing everything precisely.”

“I feel like, after this class, I can do anything.”

On Class in general:

“Engage, ask questions. If something you say is wrong, you will remember why in the future.”

“Talk to people nearby. Math people are friendly and concerningly enthusiastic.”

“The transition from chapter1 to chapter 2 was really difficult and made me question my abilities, but it’s not impossible.”

“There is certainly an element of bonding over the mutual foundations struggle. You will make many foundations friends.”

“It’s very abstract, so sometimes I thought I understood but made more mistakes than I realized. It’s very notation heavy. Even when I understood concepts, sometimes the notation would trip me up.”

On Homework & LaTeX:

“Make sure your first draft is fairly complete so the preceptors can give you help and feedback.”

“It’s frustrating at first, but stick with it and you’ll figure it out. It actually doesn’t suck after a month or so.”

“LATEX is pretty much the coolest. You will learn how to code, but it will take some time. Exercise patience and care with LATEX.”

“It may seem new and mysterious at first, but you’ll be proud of the work you do once you finish the assignment.”

“Not as hard as you think. Once you get used to it, you’ll be happy you have it as a tool.”

On Group Exams:

“I liked the ability to work with classmates to improve our understanding.”

"Check in with group members every few minutes. They can give you tips/catch mistakes so you do not get stuck or get too far in a problem with an error.”

“They were fun-ish, all with the exam buzz.”

“Don’t blow them off! Prepare for a group exam just as you would for any other exam. The problems aren’t necessarily easier, but you have longer to do them; collaboration however can be difficult.”

On Colloquiums:

“I like having to go to these. They were interesting and it was neat to see what people are doing in math.”

“Go early! You’ll have more choice on going to ones that interest you.”

“Topics are not easy to get into, but widen my view of math.”

“I really liked these. They got me really excited about upper-level math. You get to taste some crazy mathematics.”

Course Policies

Inclusivity Statement: Willamette University values diversity and inclusion; we are committed to a climate of mutual respect and full participation. My goal is to create a learning environment that is effective, equitable, inclusive, and welcoming. If there are aspects of the instruction or design of this course that result in barriers to your inclusion, assessment of your work, or your achievement, please notify me as soon as possible. I will gladly honor your request to address you by your affirmed name or gender pronoun. Please advise me of this at any point in the semester so that I may make appropriate changes to my records. If I accidentally use an incorrect gender pronoun or mispronounce your name when addressing you, please let me know, in whatever manner makes you comfortable, so that I can make every effort to correct that error.

Students with disabilities are encouraged to contact the Accessible Education Services office in Smullin 155 at 503-370-6737 or Accessible-info@willamette.edu to discuss a range of options to removing barriers, including accommodations.

Missed Classes: Please make every effort to attend all classes and be an active participant in class activities and discussions. Much of your learning and understanding of new mathematical ideas will come from thinking and talking about them in class. If for some reason you are unable to attend class or turn in an assignment, please let me know as soon as possible, preferably before the missed class or assignment. I know how important athletic commitments, personal events, and religious traditions can be, and will do my best to accommodate your personal needs.

Cell Phone/Screen Policy: Unless specifically stated for class use, no phones, laptops, iPads, or other devices which take your eyes off your fellow classmates and the class discussion are allowed. Electronic devices must be turned off or silenced during class meetings. If your phone goes off, or it is clear that you are using one of these devices during class, you will be responsible for bringing treats for the entire class at the next class meeting.

Health: As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and reduce your ability to participate in daily activities. Willamette services are available and treatment does work. If you think you need help, please contact Bishop Health as soon as possible at http://willamette.edu/offices/counseling/. Crisis counseling is available 24/7 at WUTalk: 503-375-5353 and Campus Safety is available at 503-370-6911. Emergency resources are also available from the Psychiatric Crisis Center at 503-585-4949 and the National Suicide Prevention Lifeline at 1-800-273-8255.

Willamette is a community committed to fostering safe, productive learning environments, and we value ethical sexual behaviors and standards. Title IX and our school policy prohibit discrimination on the basis of sex, which regards sexual misconduct including discrimination, harassment, domestic and dating violence, sexual assault, and stalking. We understand that sexual violence can undermine students academic success, and we encourage affected students to talk to someone about their experiences and get the support they need. Please be aware that as a mandatory reporter I am required to report any instances you disclose to Willamette’s Title IX Coordinator. If you would rather share information with a confidential employee who does not have this responsibility, please contact our confidential advocate at confidential-advocate@willamette.edu. Confidential support also can 4 be found with SARAs and at the GRAC (503-851-4245); and at WUTalk - a 24-hour telephone crisis counseling support line (503-375-5353). If you are in immediate danger, please call campus safety at 503-370-6911.

Academic Integrity: Students of Willamette University are members of a community that values excellence and integrity in every aspect of life. As such, we expect all community members to live up to the highest standards of personal, ethical, and moral conduct. Students are expected not to engage in any type of academic or intellectually dishonest practice and encouraged to display honesty, trust, fairness, respect, and responsibility in all they do. Plagiarism and cheating are especially offensive to the integrity of courses in which they occur and against the College community as a whole. These acts involve intellectual dishonesty, deception, and fraud, which inhibit the honest exchange of ideas. Ignorance of what constitutes plagiarism shall not be considered a valid defense. Plagiarism and cheating may be grounds for failure in the course and/or dismissal from the College. For more information, see http://willamette.edu/cla/catalog/policies/plagiarism-cheating.php. If you are unsure of what constitutes cheating, please ask me. These are the practices I expect you to follow in each of the components of the course:

on collaboration: You may, and are encouraged to discuss things with fellow students, and get help from your professor, textbook, notes, or calculator. However, your submitted written work should be your own or a collaboration with your assignment group. Copy/pasting sections of another's assignment or code, reading and paraphrasing another source, or providing your assignment to be copied by others, are violations of university policy. When working on a group assignment, all members of the group should contribute to producing all components of the finished product. Writing your name on work you did not contribute to in a substantive manner is a violation of university policy.

on individual tests: Throughout the course there will be individual tests and/or quizzes. Your work on these assessments should be your own, and you may not receive help from any source other than your professor (unless directed otherwise).

Intellectual Property & Privacy: Class materials and discussions including recorded lectures are for the sole purpose of educating the students enrolled in the course. The release of such information (including but not limited to directly sharing, screen capturing, or recording content) is strictly prohibited, unless the instructor states otherwise. Doing so without the permission of the instructor will be considered an Honor Code violation and may also be a violation of other state and federal laws, such as the Copyright Act. 5

DACA/Undocumented Student Advocate: Willamette is committed to supporting our DACA/Undocumented students in a variety of ways. This year, Olivia Muñoz is the contact person for all DACA/undocumented students can provide those students with a number of external and internal resources that are available. Her contact information is email:omunoz@willamette.edu, Office: 3rd Floor UC, Phone: 503-370-6447.

SOAR Center Offerings (Food, Clothing, and School Materials): The Students Organizing for Access to Resources (SOAR) Center strives to create equitable access to food, professional clothing, commencement regalia, and scholarly resources for WU and Willamette Academy students. The SOAR Center is located in the Putnam University Center’s third floor, across from the Harrison Conference Room. The space houses the Bearcat Pantry, Clothing Share, and First-Generation Book Drive and is maintained by committed students and advisers. Please check www.willamette.edu/go/soar for current hours of operation and email soar-center@willamette.edu for any questions or concerns.

Important Dates

Week 1: 1/17 (Tue) First day of classes

Week 3: 1/30 (M) A/D Deadline

Week 4: 2/6 (M) C/NC Deadline

Week 6: 2/20-21 (M-Tue) Mid-semester break - No Classes

Week 10: 3/22 (W) Withdraw Deadline

Week 11: 3/27-31 (M-F) Spring Break - No Classes

Week 13: 4/11-14 Registration for Fall

Week 14: 4/19 (W) SSRD - No Classes

Week 16: 5/3 (W) Last Day of Classes

Week 17: 5/9 (Tue) Final Exam 2-5pm