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I like calculus and I've always liked it more than other branches of mathematics. That's not to say that I consider myself a true expert in the subject as I'm n

I like calculus and I've always liked it more than other branches of mathematics. That's not to say that I consider myself a true expert in the subject as I'm not (I still find some aspects of the subject difficult). I like calculus because it allows us to put a measure on how things in our world change, it greatly simplifies our calculations of the rate at which these changes occur than we would otherwise be without it. Things in this universe are rarely fixed, unchanging and immutable and the fact that they're not is why the universe works as it does, and it's why we exist and are around to actually study it. Calculus is a principal tool in helping us make observations about how the universe works. From my experience, probably the single biggest problem with calculus is its frightening reputation. There's likely nothing better to give someone a lifelong phobia of calculus than for a teacher to say to him or her as a kid 'you must be very good at arithmetic if you are ever to be good at that advanced subject calculus'. When I was a kid calculus had a frightening and awesome reputation for being difficult so by the time we'd reached highschool and were confronted with learning it many kids already had preconceived notions that they weren't going to do well in the subject. This notion about calculus being difficult isn't new and it goes back a long way. Only a few days ago I read a HN story about physicist Richard Feynman learning calculus as a teenager and that he learned it from an early type of 'teach yourself' book titled Calculus Made Easy (1914) by Silvanus Thompson. I'd not seen this book previously so out of curiosity about how the great man came to learn the subject I downloaded a copy and it was an eye-opener. Its Chapter I titled To Deliver You from the Preliminary Terrors, only confirmed the fact that not only my generation of kids had been forewarned of calculus' 'terrors' but also so had previous generations of kids - even those long before the author's time. Thompson, an electrical engineer, professor of physics and educator, was well aware of its 'terror' factor ipso facto the chapter's title. In only one half pages of the most delightfully written prose for a mathematics text he attempts to alleviate the reader's fears with simple straightforward explanations. Therein he explains the dreaded 'd' as simply meaning 'a little bit of' so dx means a little but of x. (In my opinion all educators of the subject would do well to read this chapter.) Despite Thompson's best efforts to dispel fears of calculus they still remain with many of us today. Perhaps the reason why the students referred to in the article failed calculus is that they've carried this preconceived notion of its difficulty with them from their earliest days and that the level of (or the subject material) wasn't appropriate to their courses. I'm not in favor of removing calculus from courses because I believe it is necessary to understand what it teaches us about the world, as it explains the underpinnings of almost everything we do, especially so the physical world, and we need to have some knowledge of how that happens. If I could I would go even further than Thompson and teach calculus to primary school kids. Get in early enough and kids wouldn't have time to develop a fear of the subject. For skeptics who say we can't teach anything meaningful about calculus at that age then they ought to think again. For instance, getting kids to learn Simpson's Rule by showing them how to work out the area under a curve by having them cut out rectangles of cardboard and best-fitting them to the area is well within their capabilities. Couple this with a stress on how important this is in real world siutations and we'd be well on the way. Then there's the old fable about the frog on a stone in the middle of a pool who wants to jump to the outside bank. At first he jumps half way and then a quarter and then an eighth and so on as he tires. Every kid knows this story and that the poor hapless frog never makes it. Now teach kids the frog really doesn't perish after all - because our hero Calculus says his feet are too big. Happy ending. (Apologies to any mathematicians whose sensibilities I've offended.) ___ Edit: I should have stressed that in professions that have a less rigorous dependence on mathematical calculations, their calculus courses should place more emphasis on the meaning of what it teaches us rather than the manipulation of symbols and equations per se. (Humans have always had difficulties in recognizing small and large rates of change - exponential growth etc. - especially in their early stages (when often still manageable). We should always emphasize to students why calculus is an essential tool for processing the mathematics of rates of change. In many cases understanding the underlying reasons is more important than doing calculations (if one actually understands the problem then one can always call on someone with better mathematical knowledge - after all, at times even the best did this - Einstein for instance).

jqgatsby 4y ago
that's funny, I ended up buying the book based on exactly the same HN comment and received it today. The idea that calculus could be learned based on the dual numbers is really intriguing to me, especially because I've been studying them in conjunction with their cousins the double (or split-complex) numbers, and trying to understand the whole Cayley-Klein geometry framework, for some years now. It's sort of a never-ending spiral of research, where I never seem to be able to get to the core of the subject (geometry). Today I am looking at Stillwell's "The Four Pillars of Geometry", in yet another attempt to understand the cross-ratio, and in particular, Von Staudt's reconstruction of arithmetic based on projective geometry. Of course one could go deeper, to Klein and Russell's criticisms of Von Staudt and so on... Each time I think to close my hand on the golden bird, she flits from my grasp.
- hilbert42 4y ago
  Yeah right. I'm not a mathematician by profession so much of my time has been spent just plugging values into equations. The trouble with that is twofold, first one can forget those less frequently used esoteric tricks one was taught, and second, one can overlook whole swathes of the subject - those parts outside one's normal orbit. I never seem to have enough time to sit down and do nothing but mathematics for its own sake.
fuzzfactor 4y ago
>not only my generation of kids had been forewarned of calculus' 'terrors' I think the real terror as a youngster is that you could fail a timed exam and have to repeat the whole course.
- hilbert42 4y ago
  That, I agree, is very true. I recall the horrible stress and anxiety I experienced early on in high school when I ran out of time in a mathematics exam and thus couldn't complete all of it. Panic started to set in because I faced the dilemma of having to decide which of the questions I could best answer (and that was far from being clear). The alternative option was to sketch out parts of more of the questions and leave each partially unanswered (on grounds of obtaining some marks fór at least having some knowledge of the material). I cannot remember exactly what I did but I think it was a mixture of those two approaches. What I vividly do recall however was that I wrote a somewhat sarcastic comment on top of my examination paper with words to the effect that the examination was too long to do within the allotted time. Under the circumstance, that wasn't the wisest of moves, it didn't win me any Brownie points with the teacher. Also, I was ridiculed and humiliated by my classmates as they laughed and crackled at me when the teacher stopped to read my comment out during his handing out of the marked papers. I can't remember what marks I received except that it was borderline fail (and I can't even remember what side of the line the marks were on). I do recall the only consolation I had was that some kids received lower marks in the examination than I did. There are two reasons for why that exam remains particularly memorable, the first is that it was first [but not the last] time I'd done badly in what I considered an important and key subject (unlike languages which back then I considered of little importance—much to my later chagrin and regret); and second, I knew that I could have done much better if I had had more time to compete it. Still that's hardly a reasonable excuse for being poorly prepared (which was a fact). Here, I'd also make a distinction between examinations where one simply doesn't know answers to questions and those where one cannot provide satisfactory or adequate answers in the time available (for whatever the reasons), as marks won't necessarily provide an accurate assessment of a student's actual knowledge of a subject. Again, that's unlikely to be a problem if one is sufficiently prepared. Nevertheless, there are some students who are particularly adept at doing well in exams and I'm quite envious of them or their ability. Especially, so those who've less interest in a subject than me and or who have demonstrated by other means that they've less knowledge of it than I have. So be it, the world's not fair, and neither are examinations the ideal way to determine one's knowledge—but likely they're still the best method we have for said purpose.

jqgatsby 4y ago

that's funny, I ended up buying the book based on exactly the same HN comment and received it today. The idea that calculus could be learned based on the dual numbers is really intriguing to me, especially because I've been studying them in conjunction with their cousins the double (or split-complex) numbers, and trying to understand the whole Cayley-Klein geometry framework, for some years now. It's sort of a never-ending spiral of research, where I never seem to be able to get to the core of the subject (geometry). Today I am looking at Stillwell's "The Four Pillars of Geometry", in yet another attempt to understand the cross-ratio, and in particular, Von Staudt's reconstruction of arithmetic based on projective geometry. Of course one could go deeper, to Klein and Russell's criticisms of Von Staudt and so on... Each time I think to close my hand on the golden bird, she flits from my grasp.

hilbert42 4y ago
Yeah right. I'm not a mathematician by profession so much of my time has been spent just plugging values into equations. The trouble with that is twofold, first one can forget those less frequently used esoteric tricks one was taught, and second, one can overlook whole swathes of the subject - those parts outside one's normal orbit. I never seem to have enough time to sit down and do nothing but mathematics for its own sake.

fuzzfactor 4y ago

>not only my generation of kids had been forewarned of calculus' 'terrors' I think the real terror as a youngster is that you could fail a timed exam and have to repeat the whole course.

hilbert42 4y ago
That, I agree, is very true. I recall the horrible stress and anxiety I experienced early on in high school when I ran out of time in a mathematics exam and thus couldn't complete all of it. Panic started to set in because I faced the dilemma of having to decide which of the questions I could best answer (and that was far from being clear). The alternative option was to sketch out parts of more of the questions and leave each partially unanswered (on grounds of obtaining some marks fór at least having some knowledge of the material). I cannot remember exactly what I did but I think it was a mixture of those two approaches. What I vividly do recall however was that I wrote a somewhat sarcastic comment on top of my examination paper with words to the effect that the examination was too long to do within the allotted time. Under the circumstance, that wasn't the wisest of moves, it didn't win me any Brownie points with the teacher. Also, I was ridiculed and humiliated by my classmates as they laughed and crackled at me when the teacher stopped to read my comment out during his handing out of the marked papers. I can't remember what marks I received except that it was borderline fail (and I can't even remember what side of the line the marks were on). I do recall the only consolation I had was that some kids received lower marks in the examination than I did. There are two reasons for why that exam remains particularly memorable, the first is that it was first [but not the last] time I'd done badly in what I considered an important and key subject (unlike languages which back then I considered of little importance—much to my later chagrin and regret); and second, I knew that I could have done much better if I had had more time to compete it. Still that's hardly a reasonable excuse for being poorly prepared (which was a fact). Here, I'd also make a distinction between examinations where one simply doesn't know answers to questions and those where one cannot provide satisfactory or adequate answers in the time available (for whatever the reasons), as marks won't necessarily provide an accurate assessment of a student's actual knowledge of a subject. Again, that's unlikely to be a problem if one is sufficiently prepared. Nevertheless, there are some students who are particularly adept at doing well in exams and I'm quite envious of them or their ability. Especially, so those who've less interest in a subject than me and or who have demonstrated by other means that they've less knowledge of it than I have. So be it, the world's not fair, and neither are examinations the ideal way to determine one's knowledge—but likely they're still the best method we have for said purpose.