Embedded Files
Class Meetings
M W 1:10-2:40 pm, Ford 201
Final Exam Time: May 2 (F) 2-5pm
Additional course information, assignments, and due dates are available on the course Canvas site
Professor McNicholas, Ford 211
Office Hours: In addition to the office hours posted on the website below, I’m also available by appointment.
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
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
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
Graded Course Components*
Graded Course Components*
Your grade will be based on your level of achievement in each of the following graded course components. Exact grade cut-offs will be determined at the end of the semester, with a grade of 90% and above guaranteeing an A-, 80% and above a B-, and 70% and above a C-.
30% Problem assignments & group discussions
Homework may include Perusall assignments, proofs, discussions, and other assignments.
22% Weekly Quizzes
28% Midterms (celebrations? denouements?)
15% Cumulative Final Exam
5% In-class participation (3%), Mathematics Colloquium Attendance (2%)
*Recognizing the uncertainty inherent in our modern times, relative weights are subject to change
Problem Assignments & Group Discussions: Throughout the semester I will assign homework sets and readings that preview or reinforce class concepts, and provide the opportunity to explore and apply the theorems and techniques discussed in class.
Some of these assignments will be group homework sets and should be done collaboratively with your assigned homework group. You can either meet to do the problems together, or work on the problems individually and then meet to go over your solutions and select which solutions to turn in. For the collaborative assignments make sure everyone listed as an author contributes to the assignment, and understands all the elements of the solution. You will turn in a specified number of LaTeX-ed solutions and a 5-10 minute recording of your group conversation. I will grade both the written portion and the recorded conversation. We will discuss the rubric for grading recorded conversations in class.
While you are welcome to discuss individual homework assignments with your group members or other students in the class, your write-ups need to be uniquely your own. To avoid having write-ups that violate academic integrity policies, you should only scaffold individual proofs with your peers, waiting until you write them up on your own to use complete sentences and formal proof styles. Do not share latex files or PDFs of your individually written proofs.
A list of useful LaTeX resources is available here: https://willamette.edu/~emcnicho/courses/LaTeX.html. I will also assign Perusall readings that elaborate on class concepts or introduce you to related topics. Perusall assesses how integral your contributions are to the conversation – hence engaging with the Perusall readings early and often will increase your score.
Weekly Quizzes: There will be a short quiz most weeks consisting of one proof or problem (similar to something from that week's homework assignment) and possibly a definition or theorem statement that would have been used in that week's homework. For these quizzes, you will be allowed a notecard consisting of definitions and statements of theorems, but no worked-out proofs.
Individual Exams: There will be two midterms 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.
Engaging in the Mathematics Community: 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. 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 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 contributions.
Student Responsibility
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 the class do too. 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 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 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
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
Asking Questions in Class
Sometimes it's hard to know exactly what to ask, so here are 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 proofwriting convention here>? Is that just the general practice or 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
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.”
"I was a student in Multivariable Calculus and Foundations of Advanced Mathematics ... those were the two most difficult classes I have ever taken in my life- AND are the two that I got the most out of personally and intellectually at Willamette and even since finishing grad school. Though I never went on to do any formal mathematics, those two classes taught me so much about how to use logic and think critically, and especially write well (was in the top 98% in GRE writing, just saying...). Even though the content was difficult, I count my blessings every day that I registered for those classes."
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 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
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 preferred 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 for 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 that 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 are 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 the readings and homework assignments: You may, and are encouraged to discuss things with fellow students and get help from your professor, your notes, or similar resources. However, your submitted written work should be your own or a collaboration with your assignment group. Copy/pasting sections of another's assignment, reading and paraphrasing another source without proper citation, 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. While you are welcome to discuss individual homework assignments with your group members or other students in the class, your write-ups need to be uniquely your own. To avoid having write-ups that violate academic integrity policies, you should only scaffold individual proofs with your peers, waiting until you write them up on your own to use complete sentences and formal proof styles. Do not share latex files or PDFs of your individually written proofs.
on the use of Artificial Intelligence: The ease of use and availability of artificial intelligence is increasing at a rate that is hard to fathom. Open dialogue is crucial as we navigate this momentous shift in technology. For this reason, consult your professor before using AI on any course assignment, and clearly cite AI as a source whenever you do use it. Passing off the analysis, synthesis, or creative work of others as your own, whether surmised by an AI bot or not, is plagiarism. Reasons you might consider using AI in an inappropriate way and why you should instead speak with your professor:
feeling crunched for time and under pressure: We have all felt overwhelmed at times and unable to complete all the tasks before us. For that reason, there is a 24-hour grace period built into almost all assignments and I am often able to offer even greater flexibility in due dates if you reach out to me within 24 hours of the assignment due date.
do not see the benefits or purpose of the assignment: It is never my intent to waste your time or make you do menial tasks that could easily be done by an AI bot. If you do not see the educational value in an assignment, please ask me what the purpose is. It is possible that I need to restructure the assignment given the technological tools available, or better explain the intended student learning outcomes.
struggling to understand the material and synthesize new knowledge into an original product: When faced with an intellectual challenge, it is tempting to seek answers from the most readily available and anonymous source. However, it is only by engaging with challenges that we learn. If you need help processing a concept or solving a problem, turn to your classmates or me. You will gain a deeper understanding, a greater connection with your peers, and an improved ability to advocate for yourself by collaborating with others in the Willamette community as opposed to turning to an AI bot for an instantaneous answer. When you query an AI bot, you can not control the level of synthesis and analysis in the response. Even if your intent was merely to summarize a text, review a concept, or perform some other low-level task, the response you get may limit your ability to critically engage with the topic and shape your own opinions. For this reason, I encourage you to engage with AI in a cautious and thoughtful way. Given the impressive power of current AI bots and the speed at which they are improving, it is critical that you develop and strengthen skills that can not be easily supplanted by AI. If you are using AI to avoid intellectual challenges now, what will you offer an organization or employer that can't be assigned to an AI bot in the near future?
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, Emilio Solano, Assistant Provost for Institutional Equity and Community Engagement, is the contact person for all DACA/undocumented students. Emilio can provide those students with a number of external and internal resources that are available. His contact information is easolano@willamette.edu, Office: 302 UC, Phone: 503-370-6027.
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.
Land Acknowledgement: We are gathered on the land of the Kalapuya, who today are represented by the Confederated Tribes of the Grand Ronde and the Confederated Tribes of Siletz Indians, whose relationship with this land continues to this day. We offer gratitude for the land itself, for those who have stewarded it for generations, and for the opportunity to study, learn, work, and be in community on this land. We acknowledge that our University's history, like many others, is fundamentally tied to the first colonial developments in the Willamette Valley. Finally, we respectfully acknowledge and honor past, present, and future Indigenous students of Willamette.
Important Dates
Important Dates
1/20 (M) MLK Day - No Classes
1/27 (M) Add/Drop Deadline
3/17 (M) W Deadline
3/24-3/28 Spring Break
4/16 (W) SSRD - No Classes
4/29 (T) Last Day of Classes
5/2 (F) 2-5pm Foundations Final Exam
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