Introduction to Concepts of Mathematics

Textbook: An Infinite Descent into Pure Mathematics

Time

MWF, 9:30am-10:20am ET

Instructor

Dylan Wilson

Anonymous questions/feedback

Office Hours ( sign-up )

MW, 8am-9:20am ET

Introduction

Welcome! I'm so glad you found your way to this page. My intention for this course is that we will explore what it means to be a practicioner of mathematics, and what it is like to practice mathematics in community with one another.

The practice of mathematics involves an extremely well-developed collection of tools and skills for courageously facing down the unknown unfazed. With immense patience and perseverance, one returns again and again to that state of utter confusion in order to emerge on the other side, not necessarily less confused, but with a confusion that is clarified and precise. This capacity, to sit in the state of not-knowing without being paralyzed, might be one of the most important things to grow in order to succeed in math, and maybe even in life. Other companion skills and capacities include:

  • curiosity and wonder
  • beginning again when we get lost

The good news is that every one of these is trainable and learnable, and, indeed, this will be one of the pillars of the course.

Finally, mathematics is not done on an island. At the very least, we must be able to explain our ideas and arguments with others, but, even more than that, in order to not only succeed but thrive, we must learn how to be in community with others. This means we will cultivate:

  • an appreciation for diversity in views and styles
  • generosity with our ideas and our listening
  • joy for the successes of others
  • communication skills
Topics and Timeline

Ok, so probably you realized how long the intro was and that there wasn't any 'math' in it and you skipped here. No worries! Maybe this is more what you were looking for.

We will be using the following open-source textbook freely available here: Infinite Descent (version 0.4). Some parts of the textbook are under construction, and if needed I will supplement with notes or additional exercises/resources.

Below is a tentative schedule of topics and homework assignments; the further out they are, the more likely they are to change as we go along.

Week 1 (Homework 1)
Warm-up. Chapter 0.
Weeks 2-3 (Homework 2, Homework 3)
Logic. Chapters 1.1-1.3.
Weeks 4-5 (Homework 4, Homework 5)
Sets. Chapters 2.1-2.2.
Weeks 6-7 (Homework 6, Homework 7)
Functions. Chapters 3.1-3.2.
Weeks 8-9 (Homework 8, Homework 9)
Induction. Chapters 4.1-4.3 and Induction Excerpt.
Week 10-11
Review. Munkres excerpt
Week 12 (Homework 10)
Number theory. Chapters 6.1-6.2.
Week 13 (Homework 11)
More number theory.
Week 14-15 (Homework 12)
Modular arithmetic. Handout
Week 16 (Homework 13)
Gaussian arithmetic. Handout based on Keith Conrad's notes
Format and Structure
Weekly flow
On Mondays I will lecture on the material for the week. On Wednesdays I will go through examples, and you may do some exercises as well. On Fridays we will have a problem session and I will come around and answer questions. Homework is due Monday at the beginning of class and returned Friday. To get the most out of Monday's lecture, try reading or skimming the chapter before arriving in class.
First class
The very first class will be a sort of orientation. After giving an overview, we will have a chance to establish some community touchstones for communication and group-work, as well as go over a few additional suggestions for creating a more anti-oppressive space. before the first class. Update: we did this and the result is this document..
Start of class procedure
The eventual goal is to have everyone in their seats ready and accounted for in attendance by 9:30am, with only a notebook and pen or pencil at their desk (no backpacks, phones, tablets, computers, etc- those can be left at the side of the room). During the first week this won't happen on time because I need to learn your names: please come to the front and tell me your name so I can mark you as present. During the second week, please come to the front and I'll try to say your name. From the third week onwards, just give me a nod and make sure I see you as you walk in the door and I will mark you as present. See below for late/absence protocol.
Technology and resources
As a practice and a training for sitting with confusion, learning self-reliance, and not scattering our attention and energy, we will not be using any resources in this class other than the assigned textbook. No websites, forums, stackexchange, other textbooks, etc. The grading scheme is such that there is no incentive to use these other than avoiding the unpleasantness and discomfort of being confused. Please come to office hours or speak with me if this discomfort becomes overwhelming.
Assignments and Grading

The goal of this grading schema is for you to be able to spend as little time as possible worrying about grades, and as much time as possible learning. If you start to become worried about your grade, just talk to me about what's going on and we'll figure it out.

45%: Homework
Homework will be assigned at most weekly. It will be graded using the reasonable attempt metric.
15%: Projects
There will be two projects. You may work alone or in small groups, and explore a topic of your choosing, related to the course material, and a medium/format that works for you (article, video, oral presentation, etc.). We will meet to discuss your proposal and agree upon a rubric before you begin.
30%: In-Class Work/Attendance
Just show up and do some math and you will get this credit.
10%: Conversations
Depending on the size of the class and time constraints, I will meet with you and discuss the class (the material, the structure, your learning goals, etc.) All you need to do to receive credit is show up! I will send out more information about scheduling these conversations when the time comes.
Policies
I have attempted to devise policies that minimize the use of email. So please read carefully and try to see if one of the below protocols applies before sending emails.

Attendance

Please see "start of class procedure" for typical attendance protocol. If you arrive late please stay for a little after class to discuss anything missed. If you know you will be absent in the future, attend office hours ahead of time and get marked present. If you have an absence, attend office hours within a week of the missed class and get marked present. If you have unresolved absences or if there is some issue not covered here, sign up for a meeting through Calendly.

Collaboration

Please give credit where credit is due. If a member of your study group had a great idea and it appears in your homework, give them an acknowledgement! I'd like it if you didn't look up answers to homework questions online, and I hope the grading policy is such that you are not incentivized to do so.

Late Work

Show up at next office hours or stay after class to discuss.