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I quite like math, but I hated the college-level calc series. I had to struggle to just pull of a B average, even though I thoroughly understood the concepts. I went on to apply calculus and diff eqs in higher level classes quite successfully, where the rote memorization isn't the point.

But those weed-out classes test a bunch of arcane mechanics and memorized formulas and transformations in the most intentionally obtuse exam questions possible. If you don't know the "one weird trick" you're kind of screwed.

Math weed-out classes are a lot like the tech interview problem, but for STEM majors.

It's funny because while I don't consider myself "good at math", I'd always learned math by learning the fundamentals, and deriving what I need at exam time, since I find memorizing leads me to (a) go insane with boredom and (b) if I don't understand what I'm memorizing, I might apply it incorrectly without it "looking wrong".

Calc II was a class where it just wasn't possible because there was too much to derive on any given exam. It took me 1/2 way through the course to course-correct and make flashcards and such nonsense. Unfortunately that was the last (non-discrete) math class I took, so I never discovered what happens next.

It's interesting that these classes have a particular character to them, and they are also coincidentally the core math "service" courses for engineering, the hard sciences, and pre-med students. The fun isn't allowed to begin until those kids are gone. Also, the accreditation requirements for those disciplines makes it very hard to change the lower level math curriculum.

To help my kids stay interested in math, I offered them the following promise: "Suspend your judgement until you get a chance to do proofs, because proofs are when math comes alive." One of my kids became a math major.

Ha, I became a math major accidentally when I happened to take a proof based course and math finally came alive for me.

And the worst is that when you apply the "one weird trick" unexpectedly, you thoroughly confuse the graders. (In this particular case, it was an integration problem in which I used a substitution that wasn't taught by the professor.)

I am guessing that people with an aptitude for math ,such as top physicists and mathematicians, just breeze through calc 2. They don't study more than everyone else but rather it just clicks faster.

Yes, I think you’re right. As a mediocre mathematician, I don’t know of anyone in my good-but-not-top PhD program that struggled with computational math. This is the feeling that I got from their relaxed attitude towards TAing those classes. My classmates also had close to 4.0 gpas in undergrad.

i always found it funny how much time was spent drilling techniques for different integral types or doing transforms for derivatives, yet the most important idea: the continuous nature of the reals and why this is the rug that pulls it all together, and the delta-epsilon definition of the limit, only saw about a grand total of 3 minutes of hand waving and an optional problem on one homework.

ug calculus was about having algebraic/trig manipulations memorized along with a table of transforms and a handful of tricks; where the actually beautiful ideas that if we use an infinitely "elastic" representation of numbers, we can solve hard approximation problems both correctly and easily- get totally glossed over.

physics has the same problem. basic physics without calculus is just a bunch of rote memorization. the idea that such a thing is taught and that is somehow "easier" is nuts. they should be taught together, as they were developed, as many of the expressions given to undergrads in physics are simply definitions of integrals and derivatives applied a few times.

no student in calculus should ever be wondering what the constant is for in a computed integral and no student in physics should wonder where the constants come from in equations of motion. (there should be no equations of motion, just definitions in integral/derivative form and definitions of integrals and derivatives)

As someone who has taught intro calculus a few times, one reason for the emphasis on computational techniques is simply that 90% of the students in such a class do not care/are not capable of grasping the epsilon-delta definition of the limit.

Solution: spend more time on epsilon-delta so that students have time to wrap their minds around the idea. But I think the engineering departments would complain that the students who we send on to them cannot do basic computations. Also students would complain that we spend too much time on theory and not enough on application. There are probably other reasons that someone more experienced would know about.

I tutored calc too, but I had an unanswered complaint about the epsilon/delta definition of a limit - I found it to be circular. There was no impact on the computations required by the class, but the definition seemed rather circular, as it presupposes the reality of the infinity implicit in induction. Over time I recognized that my intuition was rebelling against unphysical math concepts. I was a physics major and so I was content to hand-wave my own concerns away as a merely philosophical problem, since you could always pick a sufficiently large number of induction iterations to satisfy any practical need. But yeah, it left a bad taste in my mouth that no-one seemed to care about this issue. (Oddly the use of induction for similar things, like Cantor's diagonal argument, didn't bother me at all. It was specifically epsilon delta that did, and still does, bother me.)

The epsilon delta definition of limit is not circular and there is no infinity anywhere in it. Are you referring to the limit as something goes to +infinity on the real line? That's just bookeeping shorthand for some quantifiers over bounded numbers.

I'm genuinely curious as to what you were getting hung up on, because odds are you learned the wrong definitions or somehow changed the definition to something that is incorrect, and then discovered the error later on. It's a fairly straightforward definition if you can do the proper bookkeeping over the quantifiers, and is designed to not have anything to do with infinities. But many people struggle with quantifiers.

this is the point. they do not stress in beginning calculus that continuity is actually a profound property of a mathematical construction (the reals) that we use for modeling physical phenomena.

it's not the delta-epsilon definition that is slightly weird, it's what it means to be truly continuous that is. (uncountable infinity, or as i like to call it, endless zooming everywhere)

once that is clear, the delta-epsilon definition becomes obvious and clear.

Continuity is just a formal property. I'm not sure if it is "profound", that's more of an emotional response, which is very personal.

Mathematicians use words like "deep" to describe properties that are useful in many branches of math. Is continuity deep? I think what's deep is the notion of compactness and the essence of compactness is open covers and finite subcovers, so the deep idea really is about epsilons and deltas and not handwaving about "what it really means" to be continuous.

For ruminations on what things "truly mean", you are not going to find that in a math book. Perhaps a philosophy book would better scratch that itch for you. Math is self-contained and provides its own meaning in terms of the relationships between formal properties and precise answers to precise questions. Some people truly love exploring the relationships between formal properties and discovering the answers to difficult questions that reveal new logical structures. If that is not interesting enough for you, perhaps you will get more satisfaction studying a different field.

Any statement about infinity is not something you can check directly, by hypothesis. The powerful influence of social proof cannot and should not be ignored. There are too many smart people who don't know anything about the foundations of mathematics, and believe that there aren't any consistency issues.

My point is that as a practical matter, they're right. Calculus works really well. But consider for the moment the question of why you are convinced that induction is valid. There is no proof of induction's validity. The epsilon delta thing is a definition, not a proof, and serves, effectively, to invert infinity into an infinitesimal. I am on firm ground with these statements, like it or not.

I agree there's little to be gained examining what things really mean when it comes to meaning itself - it is always going to be circular. I believe that the foundations of math lay primarily in its utility, and secondarily in the social fabric of experts (which may or may not be related to utility). The eerie, beautiful way in which math describes the world is fundamentally a human phenomena, and it's based on aesthetics, not logic. I suppose my objection is simply that one shouldn't go around thinking that the epsilon delta definition removes the ambiguity and messiness that actually underlies mathematics.

> But consider for the moment the question of why you are convinced that induction is valid. There is no proof of induction's validity. The epsilon delta thing is a definition, not a proof, and serves, effectively, to invert infinity into an infinitesimal. I am on firm ground with these statements, like it or not.

I'm afraid these statements are a mishmash of some correct and incorrect statements, and a logical argument like that is considered incorrect.

* Yes, the definition of limit is a definition.

* The definition of limit has nothing to do with induction or infinity. I'm honestly baffled why these three distinct concepts are being conflated.

* For well ordered sets, induction is just reductio ad absurdum in which you assume the smallest element does not satisfy a condition and then show this to be incorrect because the next smallest element must meet the condition and it's satisfication means the next larger element must meet it as well. There is a valid question as to whether every set can be well-ordered, which is an axiom, but for countable sets, which is where virtually all induction arguments are used, no axiom of choice is needed to use induction-style arguments.

* The statement "to invert infinity into an infinitesimal." is gibberish.

In an effort to understand what you mean by 'induction' and 'infinity', can you explain where induction comes up in the following standard definition of a limit:

"f has a limit L at x" means exactly that

"forAll epsilon>0 thereExists delta>0 such that if |x-y|<delta then |f(y)-L|<epsilon."

"forAll"

Okay, I see a link to 'infinity' there, because there are infinitely many reals. Is that what you're going for?

If so, the link seems irrelevant and in particular I don't see how this leads to circularity in the definition. Can you explain this further?

Integral problems is to math as Leetcode is to software engineering.

I was and am bad at integration problems like this. I find them scary. And mysterious. Somehow I associate these with engineering type classes, but that may not be generally true.

On the other hand, I really enjoyed and did well in classes like real analysis, functional analysis and measure theory, even though I am not a mathematician.

But those classes are fun. Say, the construction of measures and Lebesgue type integrals makes sense to me.

Integration by parts? And this stuff? I feel like a stupid monkey. It's weird.

For a real mathematician, both of these types of skills are probably important. I wish I could do integrals better.

They are just games with symbols. But yes, some moves are scary. The moves that scare me are those whacky out-of-nowhere algebraic additions that lead to a simplification later. They are scary because somehow someone anticipated that one addition out of an infinity of possibilities would be useful "down the road" in the computation. (This is done a couple of times in the OP's integral.)

I think it is interesting that some notions in this thread boil down to the idea that this ain't real math like, say, real analysis.

In my limited view, many proofs are easier. Especially easy ones that are (if that makes sense) more sequential in the required thinking.

I think there is a lot of cleverness in these integral calculations, a lot of creativity to, as you say, come up with some seemingly nonsensical transformation that down the road turns out useful.

Perhaps it's because I am so bad at it, but I find people who does this easily very impressive.

I feel the same way for differential equations, where, incidentally, I also suck.

I wonder whether anyone ever did a study on the cognitive skill "dimensions" in maths.

Very similar experience for a lot of math programs in France. For about a year we did a lot of repetitive and uninteresting stuff (integration by parts, compute Taylor series on the whiteboard, ...).

The only concept from that time I used in my day job is the binomial coefficients. Yet I don't regret taking the class, in some inexplicable way I feel it has made be better (at what I have no idea).

> some inexplicable way I feel it has made be better

The only thing I really "learned" from studying calculus that I actually apply to the real-world is the ability to slow down and take each part individually. When I first took calculus, I tried to rush through the problems and inevitably dropped something important. It wasn't until I started forcing myself to write down each step (even if I thought I could do it in my head) that I started actually getting the right answers, and knowing that I actually had.

At least when I went to undergrad, calculus was a university requirement, not a major requirement. Like English 101, except it is of course taught by math department faculty just as English 101 is taught by English faculty. Everyone had to take it or test out of it.

For actual math requirements, you started with real and complex analysis, abstract algebra, geometry and topology, and then some applied math classes such as partial differential equations, or numerical methods. There was also a requirement for probability and statistics.

One way you can tell the difference between a general requirement class and a major class is the size of the classroom and the majors taking the course. If you are in an auditorium with 300 freshmen taught by a TA and almost no one else in that course is a math major, then you are looking at a university requirement rather than a college requirement.

University requirements are not intended to weed anyone out, that would be contrary to the goals of the university. They should be doable by all who are admitted. When I went to grad school, I had the pleasure of teaching some of these calculus classes, and no one considered this to be a weed out class or a math major class. All the math majors we had tested out of calculus in high school, and most of our students had humanities majors (as the STEM students also tended to test out of it). Giving those humanity majors lots of tricky problems in order to try to weed them out from their own majors wouldn't make any sense.

Moreover the key skill in being a math major is the ability to do proofs. So the weed out classes tend to be real analysis or abstract algebra, as these are the classes where students first do proofs. As there are traditionally no proofs in calculus classes (the books may provide proofs, but you are not tested in your ability to prove theorems, but in your ability to calculate). Thus it wouldn't be a good weed out class for math majors even if it wasn't a general requirement class taken by all majors.

What university is this where a humanities major is required to learn and apply integration rules?

undergrad was Arizona State. Yes, having a college education requires knowing basic stuff like how to write a college essay or how to find the area under a curve. At that time, it was grouped by Numeracy or Literacy Requirements, so calculus met the N1 requirement and you could satisfy your L1 with English 101. Of course you could take more advanced classes as well if you wanted, but there was no credit for taking high school math classes to satisfy the university Numeracy requirements. So no trig or pre-calc would cut it. There were also social studies requirements, etc. The idea is that a "liberal arts education" requires these. And of course virtually all math majors would already have tested out of them, even in big state schools.

I think your claim that "calculus was a university requirement" doesn't align with what I'm reading in their degree requirements, which is "MA and CS: Mathematical Studies (combined six credit hours)"

> This core area has two categories. Mathematics (MA) is the acquisition of essential skills in basic mathematics. Computer/statistics/quantitative applications (CS) applies mathematical reasoning and requires students to complete a course in either the use of statistics and quantitative analysis or the use of a computer to assist in serious analytical math work.

> This requirement has two parts: At least three credit hours must be selected from courses designated MA and at least three credit hours must be selected from courses designated CS, and all students are expected to fulfill the MA requirement by the time they accumulate 30 credit hours in residence at ASU.

Sounds like you could take calculus to satisfy the Math requirement, but you could decide to do other Maths instead?

https://catalog.asu.edu/ug_gsr

> I finally understood calculus after taking introduction to real analysis,

Real analysis is the version of these things taught to math students, rather than the (often mostly service) version that is taught for other programs (engineering, physics, etc.) that need calculus. It is unfortunate that many programs are structured so you can't even see this before surviving the standard 1st year calc progression, especially at large universities.

How math-oriented or not a particular program is varies obviously, but it's pretty common to see this distinction. When I was an undergraduate, entry to the honors math program ignored all calculus classes and results entirely (if I recall correctly, it was based on having a 1st class standing in linear and modern algebra courses). I think was entirely possible to complete a math major with little or no calculus at all.

Since many other countries fund their universities differently, I imagine weeder courses would be less unnecessary there. As such, is calculus taught differently there? If intuition were valued more in math instruction, I think many US students would take more math courses, as well as more technical courses that employ higher math and math-based insights.

raises hand

I started freshman year intending to major in math, started with Calc 3. When the average grade among my classmates on the first test was a 56/100, curved of course, I knew something had to give. This was not the fun math I knew from before. A+ student up until this moment.

What finally made calculus click for me was my class in Numerical Mathematics where I actually wrote programs to take derivatives. Also because of that course, functions became familiar, fun, and easy to rearrange. It was life changing.

going between discrete and continuous really helped me too. i think such a thing, using computers even, could do wonders for ug math education.

some kind of awesome integrated class where kids work with robot toy cars comes to mind as an interesting way it could be presented. (start by measuring their behavior and collecting data, computing crude integrals on the computer, moving into analysis using the reals)

> the very definition of what the real numbers are

I wonder what that was. (In my world it was just a bunch of axioms.)

It was given axiomatically, with classical set theory as the foundation. Define the reals as a complete, ordered field. In fact it can also be proven that any two complete ordered fields are isomorphic. From this you can prove various cancellative, associative and distributive laws and obvious things like 0 < 1. See[0] for a readable Coq formalization.

[0] https://www.cs.umd.edu/~rrand/vqc/Real.html

There are also set-theoretic constructions of the Reals that satisfy the axioms (e.g. Dedekind cuts)

My college roommate called this "Blue Collar Mathematics."

What in particular was being referred to as blue collar and what's the analogy?

Calculus 3 is probably what you mean. calculus 1 does not cover integrals that much beyond some of the basic techniques. The handwaving typically makes the class easier instead of harder. I find it hard to belive that someone who struggles with intro calculus will underderstand it by starting with elliptic functions.

Not sure if the author's edit changed which calculus he's talking about, but at the time of this posting, they say Calculus 2, which meshes with my experience.

Lots of schools break up calculus in different ways, and that's fine. My school (and the schools of lots of people I know) break calculus up into calc 1, which is limits, differentials, and a toe dipping into integrals. Calc 2 is the 8 or so different tools for integrating progressively more difficult integrals. Calc 3 is multivariate calculus.

Lots of people have a very difficult time with calc 2. It feels very plodding- calc 1 and calc 3 (and diff eq and lin alg...) felt like I was learning new insight every week, calc 2 just felt like memorizing new vocabulary words. It wasn't just that it was hard, it was that it was hard and boring.

(obviously if a school breaks calc up differently your experiences will probably be different)

AP classes have caused most US collages to split the material in similar ways between in Calc I, II, and III.

There are different course names in common use for the various divisions of the calculus curriculum. There's no standard, so quibbling over course titles is kind of empty.

What's between Calc 1 and 3??