The "service work" imbalance

Imagine you're sitting at a computer with a button you can click. You're in a room full of other people who are also participating in this experiment. You've been randomly placed in a group with two other people, though you don't know which two.

If, at the end of a two minute timer, none of you have clicked your button, all three of you earn $1. But if someone clicks the button, the round immediately ends, that person earns $1.25, and the other two group members earn $2.

There are 10 rounds. Nobody knows who volunteered in which round.

What would you do?

When researchers ran this experimental game of chicken, they found that in mixed-gender cohorts, women volunteered 3.4 out of 10 times on average, while men volunteered only 2.3 out of 10 times. In contrast, when the group of people in the room was single-gender, everyone volunteered about 2.7 times out of 10.¹

The researchers concluded that in mixed-gender groups, women are more willing to volunteer to do tasks that incur a cost to themselves, and men are more likely to hope someone else (perhaps a woman) will do it. This wasn't because the women were better at the task or enjoyed it more, nor was it because women were nicer and more altruistic: in single-gender groups, women and men volunteered at the same rates! So what's going on?

This experiment was conducted by the authors of The No Club: Putting a Stop to Women’s Dead-End Work. These four successful women started a "No Club" because they kept finding themselves agreeing to take on work tasks that someone needed to do but that did not benefit their careers. Together, they slowly got better at saying "no" to these non-promotable tasks (NPTs). This made their workloads more manageable and gave them more time for promotable (i.e., career-benefitting) work. But then they noticed that the NPTs they turned down were just being taken on by other female colleagues instead.

When they began to conduct studies and interviews on the question, they found that women were doing more than their fair share of NPTs across industries ranging from academia to law firms to the TSA to bartending. Women were likelier to volunteer or to be asked to do these tasks, and once asked, women were likelier than men to say yes.²

One study of a consulting firm's records on their employees' billable hours found that women spent about 200 hours more per year than men on non-promotable work! For women in more junior roles, this time ate into their time spent on promotable work; for senior women, it was on top of their promotable work. The senior women were simply doing more work in total than senior men—perhaps they realized they had to in order to get where they are!

One of The No Club’s authors, economics professor Linda Babcock, gives a personal example, comparing her schedule on a non-teaching day to that of a male colleague. Hers looked like:

8:30-10:30 IRB meeting

10:30-12 curriculum review meeting

12-1:30 student presentation

1:30-2 interview with reporter

2-3 executive education meeting

3-4 research

4-5 prepare talk for womens' group

5-6 faculty meeting

His looked like:

8:30-12 research

12-1:30 student presentation

1:30-5 research

5-6 faculty meeting

He had time for 6 more hours of research. This is an extreme example, but it illustrates the point.  

In the fall semester of last year, I remember apologetically telling a fellow graduate student I couldn't co-run LGBTQ math socials with her. I was doing too much of this sort of thing already: Math Includes, Real Representations, Girls' Angle, DRP, WiSTEM, and being a GSC rep. When I listed them all, I realized it might be wise to cut down on these things in the spring semester. In retrospect, I mostly failed to do this.

Last month, I was asked for the first time to review a paper for a TCS³ conference. I agreed, of course: reviewing papers is a service to the community, and for me it's also a new skill to develop. Coincidentally, less than a week later, I was asked again by someone else to review a different paper. Now I was torn. I felt bad saying no, but I was busy and I knew it would take away from my research time. Eventually, I clicked on the paper to glean some hint of how time-consuming it would be to review. I promptly saw that my PhD advisor was one of the authors, upon which I breathed a sigh of relief and informed the requester that I had to say no due to a conflict of interest.

The replication crisis has taught me to take all social science studies with a grain of salt. But I can certainly say this stuff comes up in my own life. There's a name for NPTs in academia, and it’s "service work," and I do a lot of it. And I wouldn't be surprised in the least if this is correlated with my gender.

So what do we do about this?

Naturally, we should change the current norms and expectations so that we no longer expect women to take on service work at higher rates than men. Individual women can volunteer less and be more willing to say "no" when asked to take on an excess of service work. And everyone—especially people in positions of authority—can consciously strive to be gender-balanced in who they ask to do service work, to fight back against subconscious biases toward asking women.

There is however a complication: sometimes you do specifically need women⁴ for a particular role. For example, on a faculty committee focusing on diversity and inclusivity, you obviously want representation of perspectives from underrepresented groups. For a math mentorship program for women, you want female mentors to serve as role models. And so on.

I think the trick is to pay attention to when a woman is actually needed for the role, and when a man would do. Don't do what a university referenced in The No Club did—they bragged that their faculty committees were all 50% women, despite only a quarter of faculty being women! Although this practice was well-intentioned, it meant female faculty had to spend a lot more of their time on committee-related service work, while men were comparatively off the hook. They would have done well to think carefully about which committees needed gender parity most.

Finally, departments can make service work promotable, or at least less non-promotable, by counting mentorship and service as an important factor when choosing candidates to hire or promote. This was on my mind last year when I was asked to serve on the grad student interview committee for new faculty candidates (another NPT, I now realize, ha!). We grad students were there to judge whether the candidates would be good teachers and PhD advisors, but I made sure to ask each candidate a question about what they were doing to support diversity in math. I'm not sure if my initiative-taking actually affected anything—optimistically, one hopes that these questions about service are included elsewhere in the interview process. But I'm not privy to the rest of the process, so who knows.

Reader, how about you? What is the gender balance of service work in your department or office? And what can you do about it?

Thanks to Rebecca Giblon, Holden Lee, and Dennis Yi for feedback on a draft of this post.

1 Multiplying this by three, you can see that there were some rounds in which no one volunteered. ^↩

2 The authors acknowledged as a limitation of their research that their studies focused only on men and women; more research is needed to understand how those outside the gender binary fit into all this. ^↩

3 Theoretical computer science. ^↩

4 Or other underrepresented groups—my focus here is on gender, but I expect this part is more broadly applicable. ^↩

Contra Rowena

 “Wit beyond measure is man’s greatest treasure.”

-Rowena Ravenclaw

When I was a kid, I loved Harry Potter. And when I say “loved,” I mean that one time my parents confiscated my Harry Potter books for six months because I kept rereading them when I was supposed to be getting ready for school in the morning, and I couldn’t possibly survive six months without Harry Potter, so I secretly checked them out of the school library instead.

So the question of “What Hogwarts House are you?” is a very familiar one. You can guess what my short answer is.

But I’ve always had a longer answer to the question, too. I have Ravenclaw hobbies. I fit in with Ravenclaw communities. But the merits I prize most highly aren’t Ravenclaw merits.

As a young teenager, I used to try to keep my academic accomplishments quiet, so my classmates wouldn’t make a big deal about them. It wasn’t that I thought there was anything wrong with being smart. It was that I didn’t want people’s first gloss on who I was as a person to be “that smart girl” (or worse, “that know-it-all”). I wanted to be perceived as kind, or helpful, or hardworking, not to be judged by a metric that had much to do with luck and little to do with my values. 

Of course, there is a middle ground between “arrogant” and “hiding your light under a barrel,” and my teenage self certainly worried an unnecessary amount about other people’s opinions of her. But I still agree with her on the basic point that wit beyond measure is not man’s greatest treasure.

In academic cultures, intelligence is often idolized in unhealthy ways. Perhaps you, too, have heard the cruel little voice in the back of your head telling you that that you’re not smart enough, that you don’t understand things as quickly as such-and-such classmate, that if you don’t solve this homework problem or if you don’t get this fellowship or whatever then you’re not good enough. I can’t tell you whether that voice will ever shut up permanently. Maybe someday. But in the meantime, don’t listen to it! If you constantly compare yourself to others, you will never be satisfied -- or at least, this is true for (100-epsilon)% of mathematicians. Sure, yeah, there is someone smarter than you, or more academically successful than you, or whatever. But who cares? Intelligence is a useful tool, but it isn’t a virtue, the way compassion or perseverance is; and it certainly doesn’t determine your worth as a person.

I want a good academic job so that I can provide for my family and do interesting research. But what I want most from my career isn’t academic success. I care more about teaching cool math to students who wouldn’t otherwise get access to good math education than I do about proving famous theorems. I care more about helping the mathematical community become more diverse than I do about winning fancy awards. Alternatively, I could focus on doing research beneficial to society, or take high-paid jobs so as to be able to donate significant amounts to charity. The thing that matters most is not whether you climb to the very top; it is whether you lift up others as you go.

How Do You Get Into [Princeton]?

Or alternatively, how do you help your kid get into [Princeton] (by which I mean MIT, or Yale, or Stanford, or...)

This is truly the most frequent of FAQs. So here are my answers.

• To parents: start young -- but keep it low stakes.

From the time your children are learning to read, encourage intellectual curiosity. Talk about books, science, math, history, and so on at home. You may want to give them extra lessons, beyond their schoolwork; when I was a kid, I was lightly homeschooled during the first few summers of elementary school.

However, don't go overboard with this. Elementary and middle school kids shouldn't be doing work for hours every day in the summer. Your goal is not to teach your kids as much as you possibly can; they'll have plenty of time for intensive learning later. Your real goal here is to inspire a love of learning that will last them throughout their lives. Make it fun, and cater to your kids' interests as much as you can.

Schoolwork is not the only important thing: encourage your kids to pursue whatever creative/exploratory things they want. Try to keep lots of books in the house, or make regular trips to a library. Reading lots of books will improve your kids' vocabularies and writing abilities. I also think it's good for kids to do martial arts, sports, or another kind of athletic activity. Tae kwon do helped me learn self-discipline and perseverance. You can also suggest to your kids that they sign up for the science fair, or robotics team, or quiz bowl, etc, or write stories, or make art projects, or do theatre, etc. That said, it's important that academic and creative activities should be fun, low-pressure, and at least somewhat self-directed -- if a kid tries some activity and consistently dislikes it, then it's not for them. Kids will be much more motivated and passionate about their activities if they're doing things they actually like, rather than things they're being forced to do.¹

• A strong academic record is basically a must.

Take lots of honors and AP classes. College classes too, if you can. Use prep books and practice tests to prepare for AP exams and the SATs. It's not important to have perfect scores, you don't need ten APs, and one bad test score won't hurt you -- everyone knows people have bad days sometimes. But you need to have very good grades/test scores overall, especially in 10th, 11th, and the beginning of 12th grade, because colleges want to be sure that you are academically prepared to go there.

Regarding SATs specifically, colleges publish the 25th and 75th percentile scores of their admits. This is a decent way to figure out if your academic record is approximately in range for the school. If you beat 75th percentile that's good, and it doesn't really matter by how much. If you're below 25th percentile (and you've already done a reasonable amount of SAT prep), then you should pay extra attention to the next bullet point.

• Do what you can to stand out, at a state or even national level.

Highly-selective colleges receive huge numbers of applications and can't admit even half of the ones who are academically qualified, so they have to choose somehow. It is always at least something of a lottery who gets in, but you can help increase your chances by having accomplishments that stand out.  

This can mean doing well in an academically-related competition (e.g., math, science, CS, debate team, quiz bowl, etc), or publishing an essay in a well-regarded student journal, or being captain of a high-performing team of some variety, or doing something impressive in creative writing or the arts, or being a top athlete, or taking a leadership position in a major volunteering/service initiative -- or lots of other things. Don't try and juggle six serious extracurriculars; rather, pick a small number that you really care about, and then put your heart into those things. (Although this is not to say you shouldn't ever be in low-time-commitment extracurriculars that you aren't as serious about, you can be if you want, it's just not necessary for college applications.)

• Make good use of your summers.

Apply to selective academic enrichment programs, e.g. your state Governor's School or a program in your subject of interest. Many of these programs offer financial aid of some sort. Selectiveness/prestige matters, because if an admissions officer sees you have been vetted and deemed impressive by respected authorities, they're more likely to admit you.

MIT lists a number of STEM-focused summer programs here, and the AMS has a longer list of programs in math specifically.

Alternatively, see if you can work in a lab at your local university, or find a mentor to do research in your subject of interest, or get an internship with a state congressperson, or so on.

• Don't procrastinate on your college essays.

There's tons of college essay advice out there, but the main thing I have to add is that you should write your essays months ahead of time and send the drafts to an English teacher, a professor, or a good adult writer you know, with a request for feedback. Nobody's college essay is ever good on the first try -- get other people to criticize your essay and help you revise it into something that doesn't sound exactly the same as the other twenty essays your admissions officer will read that day.

• Make use of your counselor rec letter.

When I applied to college, I realized that by default my guidance counselor letter was going to be basically useless, because my counselor barely knew me. This is a common situation in large high schools. So, I asked a couple of my teachers -- ones who weren't going to be my recommendation letter writers, but who knew me well -- to write a paragraph about me and email it to my counselor, so she could get a better sense of what I was like. She told me later that she quoted my teachers when she wrote her letter. I don't know whether this made a difference to my application or not but I'm sure it didn't hurt.

• Don't add extra materials to your application, unless you've got a good reason.

If it's not really, actually going to make a big difference to your application, leave it out. You don't want your admissions officer to start grumbling about how you are making them read even more stuff than they already have to read. This goes for extra rec letters too -- if you've got three teachers who could write you strong letters, just pick your top 2 (and tell the third to email your guidance counselor!)

• Take care of your mental health and get enough sleep.

I mean it. The college application process is really stressful, and it's damaging to lots of students' mental health, especially students who go to highly competitive high schools or students whose parents put an excessive amount of emphasis on their academic achievements. It is easy, when you're dealing with all this stuff, to feel like one small mistake is the end of the world, to catastrophize and feel like everything you do carries a huge, inescapable weight. It is (unfortunately) normal to be stressed about this and none of it means you are a bad person or a failure or whatever, no matter what the doom-and-gloom voice in your head might tell you. Take time off, eat enough, sleep enough, do things just for fun. Ultimately, you do not have control over whether you get into a highly selective college, and it's not worth driving yourself crazy trying to guarantee it. You honestly will be just fine at whatever college you end up at.



A final aside: None of my three closest friends from childhood went to highly selective colleges. One went to a large public school to train as a pharmacist; she's also contemplating going into real estate, and I'm sure she's going to make more money than I ever do. One is applying to psych master's programs so she can become a social worker, and I'm sure she's going to make a bigger difference in the lives of vulnerable people than I ever will. And the third went to community college, is now married with a toddler and another little one on the way, and has spent the past few years living their dreams of traveling all over the world. You should only try to get into a top college if that's genuinely part of what you want your life to look like -- happiness and fulfillment in life are not contingent on it.

---

1 As an example, here's my own experience. In elementary and middle school, I didn't really learn any advanced math or science. School was easy, and my creative/academic energy was mostly directed elsewhere. I think it was very good for me to have lots of free time and no internet, TV, XBox, Wii, etc -- my brother and I were limited to 1 hour of internet time per day. So I filled my time with reading books, writing stories, learning miscellaneous bits of physics and chemistry from my parents, working on art and photography projects, taking tae kwon do classes, playing with Scratch (a programming language for kids), and inventing silly games with my friends. I played saxophone for a few years, but I really didn't like practicing, so my parents let me quit. I was in stage crew for two years. I went to MathCounts once, in 8th grade, with no preparation. I didn't do well at all, but I didn't care because it was just for fun.
I think I would have benefited from getting to learn more challenging math in 7th-8th grade, e.g. by working through some AoPS books, or taking AoPS classes online slightly earlier than I actually did (the summer after 9th grade). I think I also would have enjoyed doing more science fair projects -- I didn't do any until 8th grade. However, I don't think I would have benefited from a serious focus on math contests at that age; I think the competitive aspect would have made me unhappy. ^↩

The Genius Myth

Once upon a time, there was a student named Carol, who was taking a class called Systems Programming. Although Carol thought of herself as smart, she didn't have much background in computer science, and she struggled long, emotionally exhausting hours working on the assignments. Her class was filled mostly with men, many of whom liked to throw around technical jargon she didn't know. One of the other students, a young man named "Joe" with a strong background in CS, seemed to find the class downright easy. At the end of the semester, she mentioned to her professor that she wasn't sure she was cut out for computer science. After all, it seemed like she didn't know nearly as much as students like "Joe." Her professor said: "You got a better final grade than he did."

---

I'm telling you this story because Carol is my mother, and she made sure to drill it into my head (love you Mom!) because she wanted me not to fall prey to the same lack of confidence and anxiety which she felt.

There are these myths that people who are good at quantitative subjects are geniuses, that they achieve "effortless perfection." But the reality is, everyone who does math has to work hard at it. Yeah, even the ones who act like it's a breeze.

Math is an open playground. If you see someone swinging along the monkey bars, don't be afraid she possesses some magical "monkey bar genius" that you lack. She learned how to do it through practice, and so can you.

(And furthermore, while you weren't looking, she fell off. And picked herself back up and tried again.)

Closing the Gender Gap

I have been the only female student in a math class at Princeton, not once but twice. These weren’t unusually small classes: one had ten students, the other fourteen. I’ve taken math classes with eight professors, of whom one was female.

This is all par for the course in the Princeton Math Department.

We’ve come a long way since the days when women were first being admitted to the university and the math building (so I’ve heard) had no women’s restrooms. Equal treatment is the expectation; the department chair, David Gabai, has assured me that he is “committed to have the Math Department be a welcoming place where students and faculty are treated respectfully, fairly and professionally.” His view is shared by all the other faculty I’ve spoken to about this.

Nonetheless, we’ve definitely got a ways to go. The proportion of female students and faculty remains pretty low, both at Princeton and comparable universities. The 2018 class of math majors were 1/3 women, a department all-time record. In my year, the fraction is 1/5.

What causes this gender gap?

There isn’t one single answer to this question, of course. But one of the biggest factors -- and one that isn’t often discussed -- is underconfidence.

Math has a pernicious reputation for being accessible only to geniuses. If you're a student in a challenging math class, it is easy to get intimidated and think, "Man, this is really hard, I must not be smart enough."

It is especially easy to fall victim to this genius myth at places like Princeton, where some students enter already having a background in proof-based math. Freshmen who do not have this background may compare themselves to the ones that do, and ascribe the difference to talent instead of to a temporary head start.

Underconfidence doesn't exclusively afflict women. However, research has found that it does so disproportionately: on average, women consider themselves lower-performing than men at the same performance level. This lines up with my experience. I have heard from more than a few female math majors -- no less talented than their male counterparts -- who at some point doubted their ability to succeed in math. I also know of two female students in my year who would have liked to major in math, but didn't because they didn't believe they were smart enough.

To close the gender gap, we have to close the confidence gap. We need supportive mentors and female role models for the younger students. We need female math majors and grad students and professors to talk to freshmen, and say, "I've struggled too. Everyone does. It's not just you."

Accomplishing this is hard! If it weren’t, everybody probably would have done it already.

But I’m part of the undergraduate Noetherian Ring at Princeton, and we’re doing what we can -- and now I’ll describe what exactly it is that we’re doing, in the hope that our ideas are useful to other people who care about this issue.

A year ago, math major Aria Wong spearheaded an initiative to encourage more women to major in math. She made a website with resources and advice from older students, and she sent personal welcome emails to every single one of the dozens of incoming freshman women who indicated an interest in math on their applications.

On top of that, Aria held a dinner for freshman women interested in majoring in math. She especially encouraged students who weren't sure about it to come to the dinner. She invited all the female junior and senior math majors, along with a handful of grad students and professors; we were there to give advice and answer questions for the freshmen. At one point during the dinner, we all went around one by one and talked about our experiences: the struggles we'd faced, the friends we'd found in the department who had helped us through. Two of the then-seniors, who became friends during their first proof-based math class, said they would not have gotten through that class without each other. (They’re both now getting PhDs at top-15 math grad schools.)

In Fall 2018, inspired by Aria, I started organizing weekly study sessions for female math students. These are designed to be casual: we just chat and work on homework. These study sessions give younger female students a way to meet juniors and seniors, who can serve as an academic resource and a source of encouragement. Honestly, I think all of us -- older students too -- benefit from having a better support network. The department is helping out with these study sessions by funding a surefire method of attracting students, namely free food.

The Noetherian Ring also started holding once-a-semester dinners around course selection time, where students give and receive advice about choosing courses, applying to summer research programs and internships, applying to graduate school, and so on. This semester, the Noetherian Ring and the Math Club jointly hosted a play on issues of gender and race in math, and afterwards held a dinner discussion with the playwright.

What else could we do? We could add a "Women and Minorities" page to the Math Department website, with links to websites of groups supporting women and under-represented minorities in math, both within and without Princeton. Perhaps this webpage could also include profiles of female professors or grad students, along with their advice to young female math students. We could host a once-a-semester math lecture by a female mathematician, perhaps followed by a dinner discussion in which she talks about her experience and advice with gender issues. We could have free one-on-one coffee chats between volunteer mentors and younger female students. We could also work on advocating and designing more academic support for students in the introductory proof sequences, MAT 215-217 and 216-218.

It is too soon to tell how successful our efforts will be at encouraging more female math majors. But with serious thought and persistent effort, I believe we can bring about a time when, if the gender ratio in a graduating class of math majors is 1/3, then that’s a local minimum -- not a global maximum.




For more guidance on supporting female students, see the "Guidelines on Best Practices" published by an NSF-funded initiative called WATCH US: https://www.womendomath.org/watch-us/

A Drizzle of Puzzles

Today's forecast will include a light rain of puzzles. We are expecting at least 1", but it's harder to determine the upper bound.



Purplified Chessboard (source, with solution)

At time t = 0, some of the squares on my 8-by-8 chessboard inexplicably caught a disease which turned them purple. This disease is contagious: if at time t, a square is adjacent (either vertically or horizontally) to at least two purple squares, then at time t+1, the square becomes overwhelmed by purple-germs and turns purple itself. Once a square has been purplified, it never recovers.

I can tell that my chessboard will eventually become entirely purple. What's the largest possible number of non-purple squares my chessboard can have at time t = 0?




Guessing Game (source)

Aliens have invaded the planet again and chosen you and two other people to determine its fate. Your task is as follows:

• Each of you is given a hat, red or blue, with a 1/2 chance of each color. You can see the colors of the other two people's hats, but not your own.

• Without communicating anything to each other, each of you privately writes down either "my hat is red," "my hat is blue," or "I will not guess."

• If at least one person guesses, and everyone who guesses is correct, the aliens will leave you in peace. Otherwise, they will annihilate humanity.

You can discuss your strategy with the other two people beforehand. If you choose an optimal strategy, what's the probability that the world is saved?

Cutting a Cross (source, with lower bound solution)

You may make two straight-line cuts into the following cross. What is the largest number of pieces you can get?

A Guide to College Math Classes

Hey there! Are you a freshman, just starting out, heart set on majoring in math? A junior who thought she wanted to do computer science but had an unexpected change of heart? A geosciences major who wants to try a math class or two? All are welcome. Sit down, have a cup of tea with me, and I shall bestow my wisdom upon you.

• Find the best professors. Ask older students which ones are good teachers and which ones aren't. Having a good professor can make a huge difference in whether a class is enjoyable or painful.

• If you're allowed to collaborate on p-sets, do that. Having p-set buddies is incredibly helpful, especially in challenging classes. When you get stuck on a problem, they'll help un-stick you, and vice versa. This is a perfectly normal thing to do: mathematicians have an extremely collaborative culture, despite the stereotype of the recluse working alone in an ivory tower. Indeed, in proof-based math classes, working together with your classmates is not only normal, but often essential to doing well in the class.

• Go to office hours if you need help. You've probably heard this before, but that's for a reason. Any decent professors or TAs will be happy to re-explain stuff that you didn't get the first time around. They will not think you are dumb for asking; on the contrary, they'll respect you for putting in the effort to learn. It's much better to be the kid who goes to office hours and asks for help than the kid who doesn't and is eternally confused. If you have a schedule conflict that prevents you from going to office hours, it's totally normal to email the professor or TA to ask to meet at an alternate time.

• Ask questions in class. Admittedly, this depends on your class size; if you're in a 100-person calculus lecture where asking questions feels super awkward, then, well, no pressure. But in smaller classes, definitely go for it. In my experience, professors appreciate being asked questions, and see above re: "what if they think I'm dumb." Plus, if you're confused by something, other students probably are too.

• Sometimes, even if your professor is nice and means well, their style of explanation might not mesh with your style of learning. Or sometimes, it's Sunday and they're not available. Or sometimes, it's Tuesday and you could totally go to office hours, but it's raining and you'd rather stay in your nice cozy dorm. The point is, if you want, you can just ask your classmates for help instead of asking the professor. In my own experience, every time I've asked a classmate to explain something I didn't get, they've been happy to.

• If you've exhausted all other options (or even if you haven't), and there's still a topic you don't really understand, the internet is your friend (as long as you're not using it to look up answers to homework problems). For example, check out my page of math resources. A caveat: the internet has a lot more instructional material for lower-level classes like calculus and linear algebra than for advanced proof-based classes.

• Finally, don't blame yourself if you get lost. It can be really disheartening when you find that your first college-level math class, or your first proof-based math class, is a lot harder than you expected -- but don't beat yourself up. Most college-level math classes are designed to be hard! Be proud: You have taken on this challenge.

And this is the part where I stop, because I have completely forgotten to drink my hypothetical tea.