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A lot of undergraduate math programs in the US start with unnecessarily hard calculus classes "weed-outs" which is unfortunate, since it discourages students wh

• cameldrv 5y ago
  My college roommate called this "Blue Collar Mathematics."
  • djmips 5y ago
    What in particular was being referred to as blue collar and what's the analogy?
• ptmcc 5y ago
  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.
  • kccqzy 5y ago
    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.)
  • Afton 5y ago
    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.
  • analog31 5y ago
    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.
    • hansvm 5y ago
      Ha, I became a math major accidentally when I happened to take a proof based course and math finally came alive for me.
  • paulpauper 5y ago
    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.
• paulpauper 5y ago
  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.
  • thebooktocome 5y ago
    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.
  • nwallin 5y ago
    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)
    • Retric 5y ago
      AP classes have caused most US collages to split the material in similar ways between in Calc I, II, and III.
  • pazimzadeh 5y ago
    What's between Calc 1 and 3??
• ska 5y ago
  > 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.
• knicholes 5y ago
  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.
  • a-dub 5y ago
    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)
• hammock 5y ago
  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.
• acchow 5y ago
  Integral problems is to math as Leetcode is to software engineering.
• gip 5y ago
  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).
  • commandlinefan 5y ago
    > 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.
• a-dub 5y ago
  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)
  • concreteblock 5y ago
    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.
    • javajosh 5y ago
      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.)
• Koshkin 5y ago
  > the very definition of what the real numbers are I wonder what that was. (In my world it was just a bunch of axioms.)
  • siraben 5y ago
    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 https://www.cs.umd.edu/~rrand/vqc/Real.html
    • subset 5y ago
      There are also set-theoretic constructions of the Reals that satisfy the axioms (e.g. Dedekind cuts)
• rsj_hn 5y ago
  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.
  • acchow 5y ago
    What university is this where a humanities major is required to learn and apply integration rules?
    • rsj_hn 5y ago
      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.
• randcraw 5y ago
  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.
• zwaps 5y ago
  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.
  • javajosh 5y ago
    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.)
    • zwaps 5y ago
      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.