Spooky season has come and gone, and UChicago celebrated as it always has. Students threw costume parties, Rockefeller screened its annual horror film, and we all ignored Eric Heath’s Halloween safety email. But behind this cheery facade, something truly dread-inducing was lurking—a demonic force that traumatized most of our first-year students, as it has for generations:

“For all , there exists a such that, for all x, if , then ."

You’ll find this phrase everywhere around Halloweekend, madly scrawled across whiteboards in dripping red Expo. For those of you not in the know, what you just read is the formal, so-called “delta-epsilon” definition of the limit lim f(x) (as x → a) being equal to L. Limits are standard fare for introductory calculus classes, of course. But at UChicago, as with everything, we’re proud to treat the subject at a higher level. You may have succeeded in high school calculus, but an intuitive grasp of the subject will not suffice here. Here, we ask for rigor. We ask you to prove that a function’s limit is what it is, using the above definition. The goal is to teach you what taking a limit “really means.” Unfortunately, as many students can attest, what often occurs is that you never really learn what a limit means. What you learn instead is that you never really cared. Discouraged and defeated, you lose whatever interest you once had in math. Needless to say, this is not what Core calculus is supposed to do.

If that sounds like you, it’s not your fault. Most high school calculus classes emphasize computation: take the derivative, evaluate the integral. Proofs, if they are ever introduced, are usually covered at a basic level in earlier classes. Learning to write good proofs takes time. In an ideal world, our math curriculum would slowly introduce proof-based math using simpler definitions. But you have only nine weeks of class, so instead you are whacked across the head with this behemoth:

“For all , there exists a such that, for all x, if , then ."

And told to sink or swim. 

This must be approaching, pun intended, the worst possible introduction to proofs. Most students have never parsed formal mathematical statements, and this one is especially opaque. The values of the variables in this definition (epsilon, delta , and x) depend on each other in subtle ways. And most importantly, writing proofs does not always come naturally to “computationally trained” students. Proof-writing is more about making arguments and using logic than solving equations. Since these skills are not explicitly taught in class, many first-years learn to argue by simply copying their instructor’s examples. As a result, students develop a shaky foundation in proof-writing, which they must apply to this especially tricky class of proofs. Perhaps with more time, all of this could be taught well. But asking students to pick these skills up by osmosis before exams, in four weeks, is a bit like asking someone to learn Chinese by watching Beijing’s nightly news – for an hour a night, three times a week.

Yet we insist on shoving these proofs down the throats of our students, to the point where more students can regurgitate the aforementioned definition than the school motto. Is all of this worth it? For most students in the MATH 130s and 150s, the formal definition of the limit, per se, is completely irrelevant to their future studies. And effective education is a resource management problem. Every second a student spends struggling with these proofs keeps them from building confidence and proficiency in other areas. Our current approach reminds me of a quip from the bard-mathematician Tom Lehrer: “The important thing is to understand what you’re doing, rather than to get the right answer.”

There is a noble sentiment at the heart of how we teach calculus. At UChicago, we insist that students do delta-epsilon proofs, take biology courses, and read Plato’s Republic because we believe in rigorous, broad-minded education. An ambition to study topics thoroughly is commendable. So is the exhortation for students to read widely and understand many disciplines. But it is pointless to insist on rigor without providing adequate resources. That is how you get zombified writing seminars.

Teaching delta-epsilon proofs is the Core at its most pathological. UChicago expects its first-years to understand these difficult proofs, but it only gives them a month to master them. To guide them, UChicago provides a cohort of PhD students, all busy with their own studies, many of them teaching for the very first time. One wonders how many students leave class dejected, convinced that math is just not for them, and eagerly anticipating the day they can forget about it for good. The Core should fuel students’ curiosity, not destroy it.

What’s to be done? If you’re a student struggling with delta-epsilon proofs right now, go talk to your professor. If they can’t help, talk to your classmates. Make friends with a math major (we don’t bite, and some of us shower), or seek out resources like Core Tutoring. But, longer-term, the university has a choice to make. It should either commit to giving subjects like delta-epsilon proofs the time they require, or it should remove them from the general education curriculum. Our school’s high standards are laudable, but there is little point in teaching something poorly.