Frequently Asked Questions.

No worries. First and foremost, refer to the syllabus. It'll outline course policies and you can check if and how your absence affects your grade. Then:

• Grab the notes from a friend.
• (If you are ill:) DO NOT SEND ME a doctor's note, your medical information, photo of your thermometer, rash, broken arm, bowel movement, … (You would be surprised. 🙃) The reason for this is because requiring a doctor's note is:
  1. Invasive: Your health is your business and you are the best judge of whether or not you are well enough to prioritize class. You should not be compelled to share your medical information with me. It's not part of my job to adjudicate your health issues as worthy of absence or not, and I would never want it to be.
  2. Unnecessary: Many medical issues may not require a doctor's visit but still necessitate missing class (mild fever, cold, migraines, dysmenorrhea, etc). Asking you to go out in public while moderately unwell just to get a piece of paper is dumb and, especially if you're contagious, antithetical to public health. And if you're not contagious, I would be irresponsible asking you to go unnecessarily to a doctor's office around potentially contagious people.
  3. Classist: Not everyone has access to affordable healthcare. Requiring a doctor's note amounts to charging students to miss class, which is abhorrent.
  4. Has perverse incentives: If I were to require doctor's notes and you were unable to provide a note for some reason (especially reasons (1.), (2.), or (3.)), then you may feel compelled to come to class while you are unwell. Never do this; the last several years have hopefully instilled in us the value of staying home when you are sick. Our class is not more important than your health, my health, and your classmates' health.
• If you missed a major assignment, like an exam, the syllabus should have a policy. If you need anything clarified, reach out ASAP, but first triple check your question isn't answered in the syllabus. Also, please do not ask me to handle your situation differently than the established policy. In the interest of equity I promise to apply all class procedures consistently. To do otherwise is to unfairly penalize students who respected the policies and acted under the expectation that they applied to everyone.
• If you have experienced an emergency and may be absent for a significant period of time, let me know. We'll discuss options; sometimes such circumstances may require a medical withdrawal, incomplete, or other administrative policy. The Dean of Students can also help you navigate this.

Notice that for minor, one-off absences, you don't have to email me. You can, and if you want to, please do — I appreciate being kept in the loop. But ultimately you're an adult and can decide for yourself if something else required your attention that day. I just require that you be responsible for the content you missed and accept any consequences for your absence as outlined by course policy. If you do email me, I don't typically want or need to know why you missed either. A simple "Hey Eric, I missed class today but I'm getting the notes from Alice; see you tomorrow." is perfectly fine by me. When you do email me, I will probably send you a link to this FAQ. 🙂

First: yes, you can (and should!) come to office hours! Everyone is always welcome. Office hours are a great resource that you need to take advantage of. I highly believe they're the #1 tool for success in college.

Typically you'll be given a handful of dedicated weekly times that your professor promises to be in their office, and you may drop in unannounced at those times. If you can't make any of my upcoming office hours, consult my work calendar to find a time that fits your schedule, and then email me to request it. Make sure to send your email well before your proposed day and time; that way I can reserve the time, or suggest another time if I'm not available. (I have more responsibilities than what are included on my work calendar.)

The primary use for office hours is asking questions. If you've been working the homework and get stuck, if there's a concept in lecture that was unclear, if you're reading ahead or came across a topic related to the course that you want to know deeper, these are all excellent opportunities to bring questions to office hours.

Here's some fantastic questions you could ask:

• "Hey Eric, I'm confused about how to solve A. I already know B, but when I try this problem, I keep getting C even though I think I'm supposed to get D. Where am I going wrong?"
• "When you passed back this assignment I missed question N. I looked up what I did wrong and reworked it on a new page; does it look correct now?"
• "Hi Eric, I was studying for next week's test and in my notes I wrote X. I'm not sure why it's important. Can you help me to understand the big picture?"
• "In class you mentioned topic Y but said it was outside the scope of the lecture. Can you say more about it now?" (As a bonus, when I do this, I usually explicitly invite students to ask me about it later in office hours! 🙂)
• "Hey Eric, I saw this really cool math thing, Z, the other day. Do you know about it? Can you tell me more? How does it connect to our class? Where can I look to learn more?"

Also, office hours are a much better place for questions than email. Email is not a great medium for discussing mathematics. You should come see me in person if you have questions. If you do email me a question that requires much more than a "yes/no" answer from me, I will probably reply that you should ask me in class or in office hours (and might point you to this FAQ).

A secondary use of office hours is for establishing professional connections and relationships. Office hours are a great place to talk about career options. If you're interested in academia, we can talk about your expectations and my experiences. If you're trying to build up your mathematical chops, we can discuss establishing a research project or a reading group together, maybe with a friend or two too. And if you're eventually looking to get a letter of recommendation from me, you should be doing some or all of this, sooner rather than later.

Here are a few things office hours are not:

• They're not a replacement for lectures. If you miss class, get the notes from a friend, and I'm happy to clarify specific questions, but I can't repeat a lecture for you.
• They're not personal tutoring. You should bring specific questions and not monopolize the time. Office hours will often have multiple students (sometimes by chance but also because I encourage that you bring your whole study group to office hours!), and I will divide my attention among you all. You're certainly welcome to listen to the answers to others' questions though!

No. I'm sorry. Here's why:

I strive to create a fair environment for everyone. Students should have confidence that the policies provide consistent expectations for everyone in the classroom, and that I am committed to applying the policies universally to ensure equal treatment. I will not deviate from them to make changes for individual students. To do so is to provide a "secret menu" that only students who ask can have access to, à la the squeaky wheel gets the grease. This is inequitable and unfair.

Please do not take the attitude that "it doesn't hurt to ask." It can and it does hurt to ask. It damages your reputation, shows you think yourself above the policies, and is unfair to your classmates who expect to be held to the established standards. It diminishes your chances of building and maintaining professional relationships. You will lose social capital among the department and will not be considered for opportunities outside of the classroom, like awards and scholarships, research projects, letters of recommendation, REUs or summer work, etc, which are some of the most important parts of undergrad. (It is access, not knowledge, which is perhaps the primary value of your four years at the university.)

If I do make any changes to our policies, I am obligated to do so for the entire class or not at all.

Here's some advice I wrote for a 5 week summer linear algebra class once. It's pretty universal though. When you read it, I think the only two things you should mentally tweak are that:

1. you'll have more than 36 contact hours in a regular semester, so adjust that calculation, and
2. if you're not in a proof-based class, focus more on calculations in the homework (but you still need to know the proofs, even if you won't be asked to produce your own on assessments). And grind tons of examples, until it's impossible to surprise you with a new kind of problem.

Because linear algebra is such an important class, it is worth your effort to master it. On top of that, this class is often a student's first exposure to upper level math — the only prerequisite for this course at our university is calculus 1 (not proof-based). So this course often requires an adjustment from the "plug-and-chug" experiences you've had in all your past math classes. Here are some general words of wisdom I wish I had heard (or, that I did hear, but wish I had taken to heart).

You'll have to invest a lot of work outside of lectures. That's especially true given the fast paced nature of a summer course, but it's true even for the typically paced 16 week series. Students often feel like they follow along in lecture, which is great! But please don't stop there complacently. The homework is incredibly important in this regard; math is not a spectator sport and only by doing do you figure out what you have and have not yet mastered.

The rule of thumb is that you should expect to spend about double to triple the time on material outside of class that you do inside it. We only meet for 36 hours all semester, so expect to spend over 100 hours throughout the summer on linear algebra. This need not all be spent on homework, and please note that that time should be productively spent. It's normal to struggle a bit at first (everyone does — no one comes out of the womb knowing this material!) but if you are staring at a question for 30 minutes without knowing where to start, you should readjust. Ask yourself:

• What information has this problem given me? Write it down.
• What am I being asked to do? What would a correct final answer look like? Write an example. When you are finished, compare your answer to what you wrote here.
• Do I know all the definitions of the words involved? Write them down.
• Are there any theorems which I can use that relate the inputs I am given to the outputs I am seeking? Write them down.
• Have I seen similar problems in the lecture notes or textbook? Pull them up and compare those problems to your current problem; depending on how similar they are, would the solutions also be similar? Write a solution to your problem which mirrors those solutions. Does it work? If not, where must they differ?

This is likely your first exposure to proofs, which can be intimidating (but very rewarding too! They inspire my love of math). In general, following a logical argument can be complicated, but some advice below may make it easier. When you read a theorem, ask yourself:

• What are the hypotheses?
• What is the conclusion?
• What are all the definitions?
• What is an example of an object that meets the hypotheses? What does the conclusion say about the object? Can you work that out by hand to confirm? An example does not a proof make, but are there aspects about your example that feel universal — like they would work for any example? If you're unsure, repeat this bullet point with an example that's as different as you can manage while still meeting the hypotheses.
• Before reading the proof, attempt a proof yourself. After your attempt, compare to the notes or textbook. There are often many ways to prove something, but were your approaches similar? Does the proof do something or address a feature that your attempt did not anticipate?
• When you read the proof, can you follow the order of steps? Ask yourself: if I currently know X, then we should do Y, and therefore Z …
• After reading the proof, take an example of an object that meets the hypotheses, and apply every step of the proof to that example. Confirm that after you're done, your example satisfies the conclusion.
• There will never be an extraneous hypothesis in our class (to the best of my knowledge). Where is each hypothesis used in the proof?
• What happens if you remove a hypothesis? What part of the argument fails? Can you construct a specific counterexample without that hypothesis where the conclusion fails?

In addition to the homework, it's very productive to read the textbook, both in advance and after the fact. Before a lecture, read through the corresponding section. At this stage, pay particular attention to definitions and examples. Namely:

• For each definition, can you write down an example without looking at the book?
• Double check the textbook's examples. Confirm that everything they say is an X actually is and that all the arithmetic they do is correct. (Often, the textbook chooses to skip some detail checking for expedience or to defer to the exercises — do this detail checking!)
• When you get to theorems, take the advice above, but be aware that it's okay if something doesn't click the first time. Focus mostly on the theorem statement and examples. You should attempt a proof and read the textbook's, but it may take time to fully internalize.
• Flag any concepts you're uncomfortable with, and pay particular attention to those during lecture.

After class, read the textbook again with an eye for self-reflection and detail-scrubbing. In particular:

• Did you make any mistakes when you first read?
• Were there nuances pointed out in lecture that didn't come across when you first read?
• Returning to the theorems, how does the lecture's proof compare to the book's, and how do both compare to your first attempt?
• Do you now confidently understand the concepts you flagged? If not, follow up.
• Can you explain the section? Could you produce a lecture to your peers? How about to people unfamiliar with the material? A "yes" here is strong evidence of your mastery (but be honest with yourself!). Put it to the test and lecture to your roommate, pet, rubber duck, …

Recommendation letters are a really important part of my job and of your professional trajectory. I'm honored to write letters for students that I highly recommend. People that review your applications often give the letters a lot of weight (among all applicants, academic success is often a given, so letters are one of the only ways to be competitive). Letters of recommendation are a way for me to stake my reputation and goodwill on your behalf, so I want your letters to be the best that they can be, for your benefit and mine, so that my recommendations continue to be well-regarded. I've gotten to where I am in my career thanks to the people that wrote letters for me, and part of making mathematics and academia more equitable is carrying that torch forward to the students that have earned it. I don't take these facts lightly. Letters are indispensable and not "just another box to check" on your application. If you'd like me to write you a letter, here are things you need to do (and why):

• Have recently taken my class and earned an A. Academic excellence is a necessary (but not sufficient!) condition for a recommendation. "Recent" isn't a concrete thing, but if it's been a while you might need to remind me who you are, which doesn't beget a strong letter.
• Have made an impression and sought a connection with me. Letters are a chance to humanize you and discuss your soft skills, in addition to your technical abilities. A strong letter uses specific examples and anecdotes about our relationship to demonstrate why you're well-qualified for whatever you're applying to. In order of decreasing impact:
  1. if we never did a project or reading group together,
  2. if you never came to office hours and had an academic conversation with me,
  3. if you did not actively participate regularly (i.e., almost every day) in class,
  then all my letter can say is your grade in the class, and that's something your transcript already does. I also only want to recommend students who are friendly, polite, and a pleasure to be around.
• (For certain applications:) Be a mathematics major. Especially if your application is math or STEM focused, letters are better received when they come from professors in your major/field. At the very least, even if you're not a mathematician, it'd be good if you've taken more than one class with me.
• Be in good academic standing. On principle I will not recommend a student who has been found in violation of academic integrity. If you have ever "grade-grubbed" or asked for class policies not to apply to you, I refuse to recommend you. You need to have passed all your classes.
• Not apply to the following kinds of organizations.
  1. The military industrial complex.
  2. Intelligence agencies.
  3. Policing and the prison industrial complex.
  4. Any organization that relies heavily on insecure employment, poor labor conditions, and/or union-busting.
  5. Any organization whose existence requires destructive environmental practices.
  6. Any organization which uses machine learning systems trained on data sets unwillingly or unwittingly taken from the works of artists, writers, designers, or other creatives.
  I am happy to have an in-person discussion about my reasons for this stance. I've adopted this list nearly verbatim from Matthew Gentry Durham's, itself adapted from the Just Mathematics Collective's.

If you don't meet all of the above conditions but believe you have a compelling reason why I should write you a letter anyways, let me know! I'm happy to hear you out and I've written letters before for strong students who may not meet every condition above. In your email, please indicate that you've read everything here and explain why you deserve an exception. I am uncompromising on my ethical concerns but typically reasonably flexible otherwise. Knowing you well trumps almost everything else.

If you do meet every condition listed above, please email me to ask for a letter. Here's what I'll need from you:

• Your CV. CV stands for curriculum vitae (literally, the course of your life). In academia, unlike most other disciplines, we use CV to mean a document that displays your complete academic and professional history, as opposed to a résumé, which is a summarized version, focused more on your professional history than your academic, that you might use to apply to a (non-academic) job.
• Your transcript. Unofficial version is fine. Please don't spend money on my behalf.
• Your performance in my classes. Remind me which classes they were, what semesters they were held, and what grades you earned (or just point them out when you send your transcript). If there were any projects or major assignments in our courses, resend them to me.
• Information about the application. What are you applying to? Do they have a program announcement? How do I submit my letter? Are there multiple different programs (e.g., applying to several grad schools)?
• Why you're applying. What are your career and life goals? How does this application further those?
• Your parts of the application. This may include things like your statement of purpose, your research summary, etc; I want everything. I need these to be PDF. (I don't know what the application requires, but they probably want PDF too.)
• The names and details of your other letter writers. Why did you pick them to write letters? How can I complement their perspectives?
• Information that you suggest I put in my letter. I'll ultimately decide what I write, but are there particular accomplishments or qualities about you that you'd like me to discuss? In particular, please share with me at least two or three personal anecdotes about our relationship (even if you don't specifically want them in the letter); this helps jog my memory and highlights to me the things you've valued about our relationship and why you'd like a letter from me in particular. (Note that to do this well, you need to have made an impression and sought a connection with me, like I mentioned above!)
• My due date. It needs to be a long way away, in order to give me ample time to write your letter well. I'm not going to set a time frame in stone, but at least over a month is really preferred. If it's too soon I'll say no. It's also YOUR responsibility to keep reminding me about the deadline. You have my permission to pester me incessantly when it gets close.

Don't worry; most of this stuff you'll likely need for the application itself anyways. Go ahead and send all this info at the same time as when you ask for my letter. I'll likely need to see it before I can commit to "yes" or "no," this avoids too much back-and-forth, and even if I end up unable to write your letter, it benefits you to have all this information together in one place. Your other letter writers will probably appreciate that you've gathered it too.

You will also need to waive your right to view my letter. This is a cultural norm that may seem odd at first, so let me explain. It is standard, especially in academia, for applicants to waive the right to view their letters. You are not the target audience for your letter, and often a candid letter assesses you honestly by making direct comparisons to your peers; for example, see this sample letter by Evan Chen. This is an unequivocally positive endorsement, but it would not be appropriate for you to read such a letter about yourself. Another significant reason is that when you don't waive your right, the people who review your application will sometimes (fairly or not) view your letters with skepticism. I do not want to risk your application being viewed less favorably, the reviewers finding your actions gauche, or my letter coming across as insincere or not candid. Thus I will not write a letter if you can view it. Please also note that waiving your right does not mean your letter writers will now talk poorly about you or sabotage your application! I only agree to write letters at all if I can write a strong one. If you don't trust a professor to have your best interests at heart when you waive your right to view, you probably don't have a relationship that warrants a letter anyways.

After you hear back about your application, please let me know how it went! I love celebrating my students' successes with them. 🙂

This might all seem like a lot, but it's pretty par for the course. You can see similar advice from Tarik Aougab, Adam Boocher, Evan Chen, Keith Conrad, Taylor Dupuy, Matthew Gentry Durham, Megumi Harada, Leo Herr, Chris Kapulkin, Gizem Karaali, Kiko Kawamura, Kiran Kedlaya, Michael Orrison, Rob Pollack, Bjorn Poonen, Dhruv Ranganathan, Joe Silverman, Mark Tomforde, Ravi Vakil, David Zureick-Brown, and I'm sure 1000 other people whose resources guided me through the norms of academia.