Annals of math: how calculus did me in
[NOTE: This is a repeat of a previous post I happened to run across the other day and enjoyed all over again. The original comments were entertaining, as well.]
The college I attended had a fairly rigorous set of science requirements for its liberal arts candidates: two (count em, two) year-long lab science sequences, or a year of a lab science and a year of math. Not business math either; math math.
Even though I’d been good at both science and math in high school, this was a daunting prospect. I started out thinking I’d do a year of biology and a year of math. The biology worked out, but not the math; not at all. I somehow managed to slide though the first course, differential calculus, although I can’t say my understanding of it was very deep. But with integral I hit the wall.
The professor was from Turkey and could not speak English in anything but the most rudimentary way. Each class meeting followed the same format: he turned his back on the students and his face to the blackboard, and then covered the board’s entire surface with a faint and squiggly scrawl of numbers and symbols that were virtually unreadable, all the while engaged in nearly-inaudible mumbling.
I tried to keep up by reading the book, but the text (although nominally in English) was impenetrable as well, at least to me. I don’t think I even knew that tutoring might be available—I’d never struggled with a subject before—and by the time I realized I was in deep do-do, I felt helpless to reverse the trend.
It was too late to drop the course, and I managed to pass the midterm. But the final was another matter. Finals at that particular university were three hours long, and this one consisted of five problems. I don’t remember anything about them except that I had no idea how to tackle a single one.
I spent about fifteen minutes staring at the questions and then at my empty bluebook, back and forth and back and forth, as though by sheer concentration and force of will I could figure out some way to respond. But absolutely nothing came to mind and—after looking around at the class of about eighty fellow students, most scratching away diligently, but some immobile and sweating, like me—I closed the pages, stood up, walked resolutely to the front desk, and placed my bluebook there.
The entire class gasped in unison. I never knew whether they were flabbergasted because they mistakenly believed I had completed the test in fifteen minutes, or whether they were stunned because they realized I had given up so very early and dramatically. But I walked out of the room knowing that my grade would be a big fat “F” and I would have to pick up either another entire year of a lab science to make up for it, or another math course that would almost undoubtedly be over my head.
I later discovered that about a third of the students in the class had failed along with me, although not as flamboyantly. I never knew what happened to the professor, because I transferred soon after, to a university that serendipitously required only a single year of a lab science and no math at all, a fact for which I was exceedingly grateful.
The late and very great blogger Neptunus Lex (Captain Carroll LeFon, USN)…an obviously brilliant man…had an interesting post about his difficulties with math and specifically with calculus:
https://thelexicans.wordpress.com/2016/10/07/on-maths/
I think your difficulty was not with calculus, but with a teacher who was utterly unable to explain the significance of the formulae and why anyone should even bother. I passed a calc class for my undergraduate degree, but I truly had no idea what it was about. I just regurgitated formulae. When I went into an MBA program, I had to pass a competence exam in the subject, so I turned to my physics major husband. He explained what it was about, what the differentials and integrals meant, and how the concepts could be applied to real life. “Oh,” I said, “is that what it’s for!” I aced the exam. It was simple once I understood why I was doing what I was doing.
As it happens, I’m re-taking the summer Calc 3 course after getting a D+ last year. Ten days to go to the final exam. This weekend I’m studying for the third exam.
It’s going much better, but calculus is tough. It’s a whole ‘nother level from high school algebra and trig. I was shocked when I took calculus back in 1970. I managed to pass Calc 1 and 2, just barely. So it ended up as unfinished business I wanted to settle.
A couple years ago I self-studied Calc 1 and 2 and thought I could parachute into a college Calc 3 class, but the pace was a chapter a day with a quiz or a test everyday. No homework which I could do at my own speed. My grade depended entirely on timed tests. I just didn’t have the fluency to execute at speed. So D+.
This time I’m studying harder and practicing to work fast. To do well I must know the material cold, so working the problems becomes pattern recognition. I think I can hit a B this time. I would have to drill a lot more for an A.
I’ve got the concepts and pattern recognition mostly, but calculus problems involve 20-50 steps and I’m getting nibbled to death on small mistakes — flipping a sign, confusing a variable, multiplying instead of dividing. So I don’t get the right answer and I don’t get full credit. I’m not sure how much of it is age or rustiness.
It reminds me of learning piano for performance. You can’t decide not to make mistakes. You have to practice until you can’t make a mistake. That’s a lot of practice.
1)Proof yet again that femmes do not do as well in STEM.
2)Integral Calculus is not about formulae. It is about concepts, as Kate’s husband pointed out to her.
3)As to turning in a blank exam, I did just that in my 1st week of college…a pop quiz written on the blackboard in my advanced chem class. I apparently muttered “I can’t do it” as I placed my paper with name on the lab bench, went to the prof and asked to be put into a lower-level freshman course, which was done. A future roommate was in the class, and he told me later that no one could solve the problem, but only I had publicly admitted it.
We both majored in chemistry.
The big wash out course for pre-meds at Creighton was organic chemistry. Lots of med school dreams dashed there.
Cicero:
Seems that you are far more STEM-challenged than I, since you are making the error of believing that my anecdotal experience can be “proof” of anything of the sort about “femmes” in general.
In fact, I’m very good at math, although no sort of math genius. But the teacher was abominable and plenty of men and women failed that course, as I note.
However, I do think—based on statistical evidence—that statistically speaking there are more men who are good at math than women who are good at math, and what’s more, there are a great many more men who are super-duper good at math than women at that level. Like Larry Summers, I think that phenomenon is worth pondering and studying. The differential may indeed have some genetic (brain or other) basis or it may be more largely environmental, but it seems very real.
However, my personal experience with this particular integral calculus class has nothing to do with any “proof” of what happens with women in general when they study math. This particular experiment has an “n” of one.
My summer class started with 16 students. We’re now down to 8 students — seven men, one woman.
My hardest course was thermodynamics, which I just barely managed to advance place (studying over the summer to take the test).
Entropy and enthalpy and Gibbs Free Energy were dancing in my brain.
Grand canonical ensembles.
Three laws of thermodynamics:
1) you cannot win
2) you cannot break even
3) you cannot get out of the game
I self-taught pretty much all the math I ever learned right through partial differential equations when I was in high school and college. Mathematics was always easy for me, but I majored in chemistry instead. I often think about picking up math study again. At this point, I would probably just start over with algebra again, even though I haven’t forgotten everything yet, over 30 years later- I still understand all the concepts, and still a lot of the procedures- just really, really rusty is all.
I was a poor calculus student the first time around. However, I don’t remember then the time pressure I experience now. I believe my ability to calculate has eroded some with age, but I’m not sure that’s the whole story.
The internet has made math homework problematic. You can look up fully worked-out solutions for any problem in a popular math calc text in five or six mouse clicks. My teacher doesn’t believe in homework. Maybe teachers spread the grade curve using time pressure.
Good math books are few and far between. Most of the math texts seem to have taken a page out of this book.
https://www.amazon.com/Mathematics-made-difficult-Carl-Linderholm/dp/0529045524
Also, that is why philosophy was invented. It allows one to cogitate about grand thoughts without knowing the language of universe, i.e. math.
I noticed the same thing in college- I found organic chemistry really easy and I loved the lab work, which is why I ended up with a PhD in organic chemistry. I was premed for a while, but I realized I didn’t like anatomy or physiology classes or the labs all that much, so moved to just a chemistry focus after my sophomore year in college.
Huxley,
When I was studying any kind of math, I worked every problem in every book. I did the same for chemistry and physics classes. I can’t imagine a teacher that doesn’t require homework for such a class- that just boggles my mind.
}}} I somehow managed to slide though the first course, differential calculus, although I can’t say my understanding of it was very deep. But with integral I hit the wall.
To be honest, Neo, this IS the really difficult point. I have a natural talent for math, but, when I was 15, and teaching it to myself — they would not let me take it in HS, as it was “independent study”, and they argued I was not “mature” enough to take it in my Jr. year — despite having been mature enough to get through the prereqs to take it in my Jr. year… :-/
So I checked out a college text from the public library that had the answers in the back, and began to teach myself. I had no problems all the way through derivatives, but def. bounced at integrals. I would have figured it out by asking for some help from teachers who weren’t assholes (there were some) but other things happened (I moved) and did not get back to it for another year, but at the local JuCo, where I did ok.
But yes, integral calculus is posdef a step up from what preceded it.
I can’t imagine a teacher that doesn’t require homework for such a class- that just boggles my mind.
Yancey Ward: Oh, she believes in working problems as homework, but not turning in one’s work for a grade.
Yancey Ward: I worked just about all the calculus problems when self-studying the online MIT course, but this summer course is usually a chapter a day and I don’t have the stamina to work 30-50 problems a night.
}}} A couple years ago I self-studied Calc 1 and 2 and thought I could parachute into a college Calc 3 class
Calc 3 is also another step up.
One thing to look at math teaching is that they tend to teach stuff in the order it was developed — that is all the stuff you know before calculus was mostly developed before 1650. The calc stuff is 1650-1700, and “DiffEq” (Differential Equations) is post 1700. (Note: Not hard-and-fast, just “rule of thumb”).
Calc three starts with the stuff they worked on to solve some more complex problems, usually in three-dimensions (e.g., the volume of an elliptical plate with a tapering thickness), before some Bright Boy worked out the idea of multiple integrals.
Needless to say, these early methods are tortuous and overly complex and require a lot of head-wrappings to actually figure them out, while Multiple Integrals make the same problem so freaking easy it’s not even funny.
(It works likewise-sorta, with DiffEq — the problems they teach you to deal with are remarkably simplistic and utterly useless, but the techniques are used to develop more advanced ideas used later on, so teaching them is still a Good Thing, even though the specific techniques you learn are defacto useless. They are simple building blocks for the next story…)
BTW, speaking of time pressure, you very rarely have in-class exams past Calculus. I can only recall ONE course which was only in-class exams, and I have about 60-odd semester credits calc and above (including 4 graduate level courses, 2×5000, and 2×6000), all taken as an undergrad.
After that point, it tends to be more “take home” exams.
The reasoning is obvious — yeah, on a take-home, you can cheat, and have others do your work for you, but the point is not to merely memorize stuff, but to learn to THINK in mathematical manipulations. If someone does the work for you, will they be available next term for you to cheat off when you take another course? Will you be able to answer any questions put to you in a class that should be easy for someone “at this point”, if you have cheated your way this far?
No, of course not. If you cheat in Math, it will be obvious, if not in this course, within a couple more courses, because you will not have developed the MENTAL PROCESSES needed to understand and ask smart questions, much less solve, the problem.
I was quite pleased in a couple of my classes (with my favorite professor, one of the three best teachers I ever had in college or HS), when he introduced a concept and demonstrated some basic uses of it, and I came up to him after class and talked/asked about its application to another sort of problem, and he told me that that was exactly the topic for the next class…
When I started high school all the math in our small independent school was taught by a married couple. After my first year, one of them left to work as a programmer. My second, the other one did. They were replaced by a teacher with the qualifications but not the experience of teaching math, and by another who (I wish I were making this up) was about a week ahead of us in the book. I was woefully unprepared for college math courses but my college advisor looked at my transcript and told me I should take Calculus I rather than Pre-Calculus.
What a mistake. The teacher wasn’t as bad as most of the horror stories here but he also rushed through things and rarely allowed time for questions. I got behind and never caught up. Stopped going to class the last two weeks of the semester and didn’t even bother showing up for the final.
Took it again the next semester with a young teacher with a soft Irish brogue. I managed a soft B, and never looked back.
I had an interesting first semester Calc experience. Math was my easy A in high school. I was usually first or second in the class. I loved the Illinois Math (UICSM) version of New Math, with its emphasis on proofs. Incidentally, math before 9th grade bored me.
In my freshman year first semester I enrolled in Honors Calc 1, as my SAT score was comfortably above the cutoff for the class. Unfortunately, the bookstore had run out of textbooks (Lipman Beers), so I had limited access to the textbook- when I could borrow it. The first exam, I got a 67. I suspect that lack of regular access to the textbook may have had something to do with that.
I decided to transfer to the regular Calc 1 class. This was 4-6 weeks into the semester. Unfortunately, Calc 1 and Honors Calc 1 had by this time covered entirely different material.Which meant that I got a failing grade for my first Calc 1 exam. After all, the tests are cumulative, and I was 4-6 weeks behind. With a lot of work, I was able to pull my grade up to a C by the end of the semester. Coincidentally, the prof teaching Calc 1 had written our textbook.
Several decades later, into a second career, I took an undergrad Linear Algebra course. The prof was a Postdoc from Bosnia. Her English was fine, her lectures were OK, but she had no idea what constituted a reasonable homework assignment. She would decide what homework examples to assign at the end of class. Which turned out to be very long homework assignments. What the heck, I sucked it up and did them. An additional difficulty for most of the class was that a lot of the problems involved math proofs. Because of my exposure to Illinois Math, where I wrote proofs from the beginning of 9th grade, this wasn’t an issue for me. I suspect this was an issue for most of the class, as most high school math these days doesn’t emphasize proofs, and most Calc 1 -3 courses don’t either.
I got an A for the course, but over half the class got a D or an F. Though my memory is probably fuzzy on details.
Before enrolling in that particular Linear Algebra section, I had sat in on some other sections. One prof would go so fast that there wasn’t time to write down what he wrote/said. (IIRC, his lecture notes were pre-written, and slapped on the overhead.) Why did he go so fast? So he could have 15 minutes of questions from the class. If he had lectured at a slower rate, there wouldn’t have been so many questions. In addition, my experience was that a STEM lecture was too early to ask questions- it was only for first exposure. Questions would arise for me when I went over the material myself.
I once had a Korean teacher for a Math class. His accent was at times difficult, but his lecture notes on the board were logical and well-organized.
BTW, speaking of time pressure, you very rarely have in-class exams past Calculus
OBloodyHell: That’s good to hear. I’m surprised at the emphasis on speed. In my career as a programmer, it goes entirely against my grain. Of course, I always had deadlines, but never to bang out a problem in the next seven minutes.
My programming approach is to get quiet inside, get a feel for the problem, write down what I know on paper clearly and cleanly, then put slow, careful thoughts together into code until I solve the problem.
I’m dead in the water if I take a calculus test that way.
Does anyone have some good info or links on how seniors (65+) manage in college STEM courses?
I know I’ve lost a step or two in being able to retain information and calculate quickly, but I’m not sure how much. I like to think I can make it up by being better organized and more aware in how best I learn, but I don’t know.
huxley:
My hat is off to you for even attempting calculus at this point.
“However, I do think—based on statistical evidence—that statistically speaking there are more men who are good at math than women who are good at math, and what’s more, there are a great many more men who are super-duper good at math than women at that level.”
That is correct. Look at the winners of the Putnam Math competition.
https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition
My hat is off to you for even attempting calculus at this point.
neo: Thank you.
I’ve worked about as hard as I can on this course and it’s galling I can’t get an A. OTOH, hey, I’m 67. I self-studied Calc 1 and 2, and now I’m holding my own against 20 year-olds. Not too shabby.
I still want an A, but it doesn’t look like it’s in the cards.
BTW, any seniors out there interested in returning to school ought to check the senior discount available at many state schools.
At UNM I am only paying $5 per credit hour, so essentially it’s free! There are limitations — only six hours per term and I can’t register for classes until the first day. Still, such a deal. It’s worth it just for the pool and weight room…
In New Mexico almost no one seems aware of this. I’m half-inclined to keep my shut lest I queer the deal. I see almost no seniors taking advantage of the discount.
Ha, ha. Flashbacks.
I swear that college math teachers ( I don’t think tenured Professors stoop to teaching Calculus, or lower levels) are screened for several attributes. First, they should have no personality. Second, they should have no teaching experience, and not a clue about how to teach. Third, bonus points if they speak English (poorly) as a second language.
I stumbled into calculus. It was not required for my major, but I had been somewhat deficient in math and had set a course to catch up. What a miserable experience.
huxley, have you tried supplementing your class lectures with the video lectures at Khan Academy? Sometimes watching a presentation by a different teacher could make it click. I’m almost 60 and am thinking about taking some college classes. Instead of STEM however I would be learning Spanish.
I often think about picking up math study again. At this point, I would probably just start over with algebra again, even though I haven’t forgotten everything yet, over 30 years later- I still understand all the concepts, and still a lot of the procedures- just really, really rusty is all.
Me, too. I am starting to review again like some people do NY Times crossword puzzles to keep the brain active. I had a similar Indian professor of Calculus in college but still managed a C, but that cost me my scholarship.
I feel you pain. When I was in college and learning different methods of integration, someone in my recitation section (I forget if it was me or someone else) asked the section leader, “Is it possible to tell, by inspection, what integration method will yield the best results with a particular equation?”
The answer was, “No.”
That kind of frosted me on calculus.
I can still integrate e to the x, though.
I am starting to review again like some people do NY Times crossword puzzles to keep the brain active.
Mike K: I’ve wondered why more people don’t do math for fun like crosswords or sudoku. Why not have math puzzle books with a very gradual ascent so people don’t get frustrated but still get somewhere over time?
Of course the problem there is that most people, from what I can tell, did not enjoy math at all in school and it’s about the last place they would look for enjoyment or diversion.
Math is no fun when you’re scrambling to keep your head above water.
huxley, have you tried supplementing your class lectures with the video lectures at Khan Academy?
big al: I haven’t tried Khan Academy, but I’ve checked other youtube lectures. I’m not proud. If I get stuck I’ll try lots of things. I’ve got six other calculus textbooks. I’ve watched all of the MIT 1801 lectures and some of the 1802. There’s a Professor Leonard who gives great, slow 3 1/2 hour lectures where he doesn’t gloss over anything. However, like neo, I get impatient with video presentations.
Here you go Neo, I just stumbled over this story a week ago, short but cool. The background is there was controversy over who invented (or discovered) calculus – Leibniz or Newton, so I think that was the basis of this. I believe it’s been decided that they each came up with calculus independently.
https://gordma.wordpress.com/2014/03/26/we-know-the-lion-by-his-claw/
Neo:
“Proof”, as in Quod erat demonstrandum ,aka Q.E.D.
Used in math!
What are your math skills and interests? At which you are “pretty good”.
How fluent are you in statistical methodologies?
I had difficulty with calculus as an undergraduate – passing with a then “gentleman’s C”, but not really confident that I understood it and not willing to pursue further mathematics courses. It was only several years later – when I was a graduate student – that a friend doing a doctorate in mathematics suggested that I should approach learning mathematics as I would a foreign language. That insight made all the difference for me. YMMV, of course. A year or so later, when I became interested in doing graduate work in economics, I was able to re-learn calculus by myself over a Summer, and moved on from there to successfully do theoretical work in mathematical economics requiring topology and measure theory.
Learning college level mathematics (calculus +) is highly individual – most of the books are more or less “cookbooks” and this style suits many students, especially those in engineering who will use primarily applied mathematics, but for others (and I was one) it is a disaster. Mathematics majors are (or were in my day some 50 years ago) taught very differently with a highly theoretical and analytical approach like that in Tom Apostle’s two volume calculus book. When I re-taught myself calculus, I found the Apostle books exceptionally intuitive.
Interesting stories, Neo’s and others. My own calculus tale: I was never a math person but did ok till 2nd year algebra when I got behind and never caught up. Skated past math in college because I could substitute an extra lab science. Ten years later I started working on a computer science degree. I was/am a competent programmer despite my math limitations. Had to go back and take math through calculus. Survived without really understanding it because my wife was a math major and honor student who was able to coach me enough to get me through the exams. Barely.
Now at 70 I’d sort of like to really understand the concepts with no obligation to work problems and pass tests. Probably won’t though.
Kate had it right. It is about learning the concepts. Once you learn to understand the concepts and language of calculus, you will have a powerful tool that can never be taken away.
BTW: The time to take your stand was the very first day when you didn’t understand the material. As a paying student, you have the right to ask that the subject be explained until you do understand it.
Cicero:
Your sentence is straightforward English, not a mathematics equation. You wrote: “Proof yet again that femmes do not do as well in STEM.” It’s meaning is clear, and it is incorrect. If you meant something more specifically mathematical by the word “proof,” it would require you to indicate that. If you meant “Q.E.D.” all you had to do was write it.
One person not being extremely good at something indicates nothing about the class of people to which that person belongs and whether that class is as good at that thing as another class (you didn’t specify the other class, but I am assuming you meant to contrast “femmes” with men).
And to answer your question, since you’re so interested in my math background: I took a course in statistics in college and did well (don’t remember my grade), and then later I took one in advanced statistics at the Master’s level and received an A. I also had to understand statistics to read and evaluate research on human subjects for my Master’s degree. I write about statistics on this blog quite often, when I discuss research, and I understand the basic concepts quite well.
Cornhead and Yancy Ward:
Ditto on organic chemistry. I washed out of pre-med because of it while having no difficulty with anatomy and physiology. I decided to switch to my real love, music. That may seem strange, or the cop out of a lazy hard science student, but it hasn’t failed me as an avocation these many years. I don’t think you can truly appreciate Bach without a grounding in and love of math. Music is math made real, touchable, sensual. The best of it bridges reason and emotion.
In my last year of university I took a graduate level class called Structural Analysis II. This class was about utilizing Matrix Algebra methods to analyze indeterminate structures which do not yield to simple analysis. I did well in this class and got my A, after which I graduated and never again had to use such mathematics.
Many years later, while cleaning out my belongings, I ran across my file folder from this class. Perusing the contents, I realized that, although the writing was recognizably in my hand and I still had a recollection of doing the work, I had absolutely no understanding of the contents!
“especially those in engineering who will use primarily applied mathematics”
That is correct. I am an electrical engineer and to be an engineer you have to be a good applied mathematician. I had lots of mathematics courses such as differential equations, vector analysis, complex variables, statistics, partial differential equations, numerical analysis, tensor analysis. I’m not sure if tensor analysis is math or physics. It was a prerequisite for the relativity course which I never took.
Freshman year at an engineering college in the Black Hills. Not doing well in calc, I asked my 80 year old professor (Dr. R. E. Doubt) if it was worth it for me to take the final. He got his little black grade book, mumbled a “hmmm,” and said “Son, you don’t stand a snowball’s chance in hell.”
Four months later, after transferring to a large midwestern university, and switching majors to Art, I took it again. The professor was only a few years older than me, and she was “hot.” Figuring she would be impressed, I aced it.
A guy just has to have the right motivation.
(Found out she was married.)
This post and the comments remind me of so many stories from my college days. I was hard core STEM, and I wish I could share them.
Carried a post versalog slide rule on my belt, and had a pocket protector.
We Badgers use the One, Two, Three, Many school of Math.
OK, one calculus story from my school.
Our text for two semesters (differential, then integral) was by Thomas.
We had three lectures per week and two recitation classes per week.
Recitation classes were for doing the homework and then working out the problems in class, usually taught by graduate students.
But sometimes senior faculty would fill in when a graduate student was absent. One such time, Norbert Wiener did so. It was a three-dimensional integral problem.
After working out the problem in the manner that was provided by the Thomas text, one of us asked Prof Wiener if there was another way to solve the problem.
Norbie (as we referred to him among ourselves) stared at the blackboard for about 30 seconds and then wrote the correct answer, turned to the class and said “you are right, there is another way.” He had solved the problem in his head, without writing the steps down.
The good professor’s eyesight was quite poor, so he wore bifocals. However, he read so much that his bifocals were reversed — the reading prescription was the main lens and the distance prescription was the little area at the bottom of the lens. Thus he had to walk carefully with his head raised upward so he could peer through the distance prescription.
Hi David Foster,
I am a fellow Lexican — I followed Lex for years and was heartbroken when that bright light was extinguished. I sorely miss his bright wit and spirit,
All here would do well to go to this URL and dive through the archives preserved and made available by Bill Brandt. Drop Bill a line, for he is a good soul as well.
The stories by Lex are wonderful. Here is a GREAT sea story that will raise the hairs on the back of your neck:
https://thelexicans.wordpress.com/2016/05/06/time-to-get-up/
https://thelexicans.wordpress.com/
https://thelexicans.wordpress.com/category/best-of-neptunus-lex/
https://thelexicans.wordpress.com/2018/02/21/index-the-rest-of-neptunus-lex/
I have almost the exact same story about flunking out of Calculus. I wanted to be a veterinarian so Calculus was required. I wasn’t to worried about Calculus as I had done well at college-level Algebra, Geometry, and Trig. In my case the Teaching Assistant who did most of the instructing was from Taiwan. But he also could only barely speak English. And he, too, would face the chalkboard, his face just inches away from the board, and start writing equations. Every once in a while he’d mumble something barely audible and always unintelligible.
Fortunately it was early enough in the term that I could still drop the class. Then I changed my major to Poli Sci and I never looked back.
Actually since I dropped the course, technically I didn’t flunk.
Wait until you hit Newton’s 3 body problem in calculus.
(To the battle hymn of the republic…)
Free energy and entropy were whirling in his brain,
With partial differentials that you never have to name,
Sigma, delta, gamma, theta, epsilon and pi,
Were driving him distracted as they danced before his eyes;
They asked him on his final if a mole of any gas,
In a vessel with a membrane through which Hydrogen could pass,
Were subjected to a pressure what the entropy would be,
If on half of delta sigma equaled half of delta p;
He said he thought the entropy would have to equal four,
Unless the second law would raise it up a couple more,
Or it almost might be seven if the Carnot law applied,
Or maybe even zero if the delta p should slide;
His professor read his paper with a corrugated brow,
For he knew he had to grade it, though he didn’t quite know how,
When an inspiration in his cerebellum suddenly smote,
He raised his trusty fountain pen and this is what he wrote:
Just as you guessed the entropy I’ll have to guess your grade,
But the second law won’t raise it to the mark you might have made,
For it might have been one hundred if your guesses all were good,
But I think it must be zero ’til they’re rightly understood.
refrain:
Glory glory dear old thermo…we’ll try again next year.
You may be interested in the book “The Language God Talks” by Herman Wouk (2010). While writing “The Winds of War” and “War and Remembrance,” Wouk met the physicist Richard Feynman to learn about the Manhattan Project. The title of the book comes from Wouk’s first meeting with Feynman: “…[Feynman] said as we were parting, ‘Do you know calculus?’ I admitted that I didn’t. ‘You had better learn it,’ he said. ‘It’s the language God talks.’ ”
It’s a great book, about the meaning of life. Wouk tried but never did succeed in learning calculus.
New Math
https://www.youtube.com/watch?v=W6OaYPVueW4
I took calculus for about for weeks. The lecturer was okay. She was German and her English was good. But we were expected to get our real learning with our TA’s. Mine was Chinese, barely spoke English. Essentially repeated the lecture on the board, couldn’t answer or questions. If you didn’t do about twenty to fifty problems every day, you couldn’t keep up. Twenty was doable; fifty pretty much meant you couldn’t do any other homework. I gave up.
Interesting thread. My calculus adventures are uninteresting — but I always loved math and after first an engineering major and then business I wasn’t really feeling it and decided my next major should be math and my career should be high school math teacher.
[Narrator] He would *not* become a high school math teacher.
Linear algebra bored me and I hated theory of probability and statistics and both were too abstract for me, but I also took FORTRAN (computer programming) and loved it. Got an A. Switched to Comp Sci major immediately, graduated two years later.
Heh. And for math majors there is the famous Putnam competition.
The highest score on the six-hour exam was 89 out of a possible 120 points. Only 20% of participants earned a score above 13.
The median score is zero. Looking through the list of past Putnam scholars, one notices some who became well known (Feynman), but many just disappear. One of my boyhood friends became a Putnam scholar, but I don’t think he was motivated to be a mathematician, he was just *very* good at math and enjoyed winning.
“Each class meeting followed the same format: he turned his back on the students and his face to the blackboard, and then covered the board’s entire surface with a faint and squiggly scrawl of numbers and symbols that were virtually unreadable, all the while engaged in nearly-inaudible mumbling.”
This! I had a math class in college in which the professor did almost the same thing; except that his writing was legible. However, we couldn’t see what he wrote on the board as he would stand directly in front of it, facing the board with his back to the class blocking our view of the board. Also, since he was facing the board we could barely hear him!
Within a couple of sessions we had all learned that in order to see the board we had to sit in rows near the edge of the classroom.
The professor was actually a nice guy; and he was the head of the Math Dept! He said that he taught one undergraduate class each semester so that he could “keep in touch” with students.
It is just a shame that he didn’t know how to teach to a class; which sadly seems to happen with a lot of college professors – they might be excellent in their chosen field; but, teaching isn’t something they all are good at.
Guess I should add my calculus story. I took multivariable calculus as a freshman and cut almost all the classes. Come finals, I realized I didn’t even know how far in the text the class had gotten. It was a two semester class, so I divided the text in half, read the first half, and got an A. Had a similar experience in a biology class where I cut all the classes, missed all the quizzes, but got the high score on the final. The professor called me in, and after a bit of negotiation, I ended up with a C.
I was a *terrible* student 🙂
Your story brings back memories …
… as a freshman engineering student back in the 70’s, I had a similar experience. I went to a small rural high school and was not nearly as prepared for college math as I thought I was. My first calculus instructor was from India and he thought he spoke fluent English; many disagreed. I will never forget that man’s name, but will not speak it here. He covered the blackboard and chattered away at mach speed. I had no idea what he was saying or doing. I also did not realize how badly I was drowning in time to drop the class with dignity; I got the F.
The next semester, I registered for the same class and got an American instructor and a much smaller size class. The A was easy that time around.
Later, the university instituted a mandatory training requirement for all foreign instructors, including a test of oral fluency. That Indian man would never have passed it. I guess I can take comfort that the university saw the problem and took action to reduce it.
Then there was the physics class I took before I even had the math to be able to do it (stupid academic counselor!) … that was my introduction to the concept of Purgatory. I got a C in that class, along the way teaching myself the math I hadn’t yet had in math class … fun times …
Hi WGraves,
I believe “Glory, Glory Dear Old Thermo” was written by AR Lewis and published by him in the Songs of Phciance
Dear The Switched Philosopher,
If you have not read this book, go to Amazon and get it:
Surely You’re Joking, Mr. Feynman! (Adventures of a Curious Character)
by Feynman
One interesting tidbit in the book is how he broke into a secure safe at Los Alamos.
The combination had been set by a scientist, and thus the combination had been set to digits of Pi or e or to their birthday. He also used the mechanical slop of the 3-dial safe combination system to whittle down the number of combinations that had to be tried.
3.1415626
My Calc 3 textbook is James Stewart’s “Multivariable Calculus,” 7th Edition. I’m not sure how Stewart won the calculus textbook wars, but he did, the market was lucrative, and he became a multimillionaire in the process.
He funneled much of that money into building the home of his dreams. He called it “Integral House,” and the design does indeed suggest the look of a calculus integral. The house is gorgeous. I recommend taking a quick look via this link:
https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/oct/05/maths-palace-built-by-calculus-rock-star-on-sale-for-14m
If only we all had worked harder on our calculus…
Nobody at all – I had an Indian econometrics professor in graduate school who was completely unintelligible. Nice guy, but hopeless as a teacher. OTOH, I had a Chinese professor for an advanced mathematical programming course who was an outstanding (if demanding) teacher despite a heavy French accent in English (which will give the professor’s identity away to older mathematicians in the commentariat).
As badly as it is often taught, I think I come down on the side of thinking that mastery of calculus at the level of the old freshman year course covering elementary differential and integral calculus should be a requirement for college graduation.
There used to be a core of general knowledge that a college graduate was expected to have mastered (even if he or she forgot a fair bit over time): year courses in Western civilization and American history, calculus, English literature (with a remedial composition and grammar course if testing showed it was required), two of chemistry, physics and biology, and a couple of years of a modern (usually European, most often French or German) language, and an introductory philosophy or logic course.
I’ve long taken the view that the elimination of this core has been one of the greatest tragedies of the past fifty years, and that much of the idiocy we see today could have been avoided had the key liberals in the universities of the 1960s and 1970s held the line against the New Left instead of trying to cater to them.
Neo, the main difference between men in women in possessing very high skills in almost anything is neither genetic nor environmental, but epigenetic. The term reflects development patterns specific for each sex. There is such thing as a norm of reaction: how far an individual can diverge away from genetically prescribed norm. It is rather narrow for men, but quite wide for women. (The same is true for the most vertebrates.) So, lots of regulations in the course of development compensate for deviations induced by genetic mutations or environmental factors, bringing the developing organism to some standard norm. These mechanisms are more effective in females than in males, so the statistical dispersal is more narrow in females. That means that the bell curve for males has heavy tales, on the both sides of the curve. There are more talented men and true geniuses among men than among women, but also more fools, deviants and idiots, too. Women in general tend to be mediocre – not too dumb and not too smart. And the higher standard of performance we set, the more prevalence of men can be observed. Among the top 1% of IQ distribution the ratio men to women is about 20 to 1. Among Nobel prize winners it is 100 to 1. (Alas, the same is true for criminals and idiots: they almost always are male.) This conclusion is simply mathematically inevitable, if you superimpose two Gaussian curves, one with a small dispersal and another with a big dispersal.
As badly as it is often taught, I think I come down on the side of thinking that mastery of calculus at the level of the old freshman year course covering elementary differential and integral calculus should be a requirement for college graduation.
CatoRenasci: In other words the percentage of college graduates in the US should return to where it was in 1965 — around 10%.
I’m not against the idea.
I view calculus as one of the crown jewels of Western civilization. I like to know stuff but I also feel a weird reverence and patriotism when studying calculus.
Western Civ used to be a requirement, and I think it still should be, although not if it’s just going to be the litany of the evils and horrors of Western Civ, which is almost certainly the way it would be taught today.
Latin used to be a requirement, for that matter.
But I don’t think there was a time when calculus was ever a universal requirement. And there are plenty of brilliant people who are unable to do math at an even lower level than that, much less calculus.
It might surprise non-math-types that there has been substantial controversy on how calculus and undergraduate math in general should be taught.
When I took Calc 1 and 2 in 1970, much of the early curriculum involved the tedious delta-epsilon method for proving limits. Historically, delta-epsilon was key to putting calculus on a rigorous mathematical footing, but for engineering types who just want the ideas and the cookbook for using calculus, it’s pretty pointless.
Now that calculus classes have been overrun by engineers, compared to math majors, they no longer teach delta-epsilon beyond “Oh yeah, there’s D-E if you really wanna know.”
> And there are plenty of brilliant people who are unable to do math at an even lower level
It is almost like music, some folks have the knack, some don’t. It isn’t directly connected to intelligence, it’s more like a talent.
neo – a fair number of the better colleges required calculus in the ’50s and ’60s, though you’re probably quite correct that it wasn’t a universal requirement. I do think it ought to be, however.
And, of course you’re right that Western Civ should be taught along the lines it used to be…. I’d recommend R.R. Palmer, et. al. The Making of the Modern World as a textbook, preferably one of the editions from the late ’60s or early ’70s. And, Morison, et. al. The Growth of the American Republic for a good mainstream American history course text.
But I don’t think there was a time when calculus was ever a universal requirement. And there are plenty of brilliant people who are unable to do math at an even lower level than that, much less calculus.
neo: True. I’m something of a STEM chauvinist, but I know too many really bright people who just don’t get math. I’m not sure what that’s about or what might be done about it.
huxley: I view calculus as one of the crown jewels of Western civilization.
The amount time spent on teaching statistics is woefully inadequate even though it is much more useful. If you want to learn for the sake of learning then sure calculus is fine but stats is much more useful in everyday life and lot more time should be spent on teaching and learning it.
As for “crown jewels” argument, that though is very similar to the argument that led physics astray and is explored in the following book.
Lost in Math: How Beauty Leads Physics Astray by Sabine Hossenfelder
A contrarian argues that modern physicists’ obsession with beauty has given us wonderful math but bad science
Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades. The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these “too good to not be true” theories are actually untestable and they have left the field in a cul-de-sac. To escape, physicists must rethink their methods. Only by embracing reality as it is can science discover the truth.
Andy: That’s surely a glass half-empty argument. I take your point on statistics for personal usefulness. But calculus changed the world full stop.
Calculus upped science’s game for modeling the world to an astonishing degree. Modern technology would simply be impossible without calculus. Feynman was probably exaggerating when he said calculus was the language God talks. But he was saying something important.
Calculus is beautiful, but that’s the glory, mystery and fault of mathematics in general. I think the problems of today’s physics is more that of groupthink and politics than seduction by mathematical beauty.
Wow Neo. I hadn’t read this post before, but is mirrors my experience somewhat. As a pre-med student, I was required to take a year of Calculus. I had always thought myself a poor math student, and so dreaded taking this class. At Berkeley there were several hundred students in Math A,B,C all attending the same lecture by the professor who wrote the textbook. Fortunately he lectured in English. The homework problem sets were addressed by TA’s ( Math Grad Students) who graded them, and explained the solutions in small classes. My TA was Pakistani, and as you experienced a rudimentary English speaker. The first midterm, the entire section failed the exam. The TA was replaced by Mr. Feldman, a sharp young grad student from Brooklyn. He was hands down, the best teacher I’ve ever had, and spent many office hours one on one with me explaining the “tricks,” as he called them required to solve each type of Calculus problem. I got straight A’s, and the rest is history. Mr. Feldman is one of the very few teachers I had in college whose name I remember. I thank him.
CatoRenasci:
I’d be very curious which colleges required calculus in the 50s and 60s. I have never heard of a single one, much less a “fair number,” although my guess is that perhaps one like MIT required it. But that’s a special type of college. (I was in college during the 60s, by the way.)
Calculus was (and still is) required for certain majors, of course. And there were general science requirements, as there were in my school that had quite rigorous ones for liberal arts majors. But math was not required, much less calculus. High school math was considered sufficient.
Scott Robinson:
Fascinating. Your story indicates that teachers can be extremely important. I’ve sometimes wondered whether I could have done pretty well in the course if I’d actually had a teacher worth his or her salt.
But I haven’t wondered enough to actually try again 🙂 .
I’ve sometimes wondered whether I could have done pretty well in the course if I’d actually had a teacher worth his or her salt.
neo: My bet is yes.
It’s easy to lose traction in calculus and the next thing you know, you’re in trouble. It’s happened to me twice.
The problem of the modern physics is that it came to the limit of applicability of a scientific method as such. (Yes, there are limits for everything, and a naive belief that the science can expand infinitely is simply wrong.) Now, it came to energies that can not be reproduced in any lab, so it increasingly is a Natural Philosophy as it was understood by Ancient Greeks and medieval scholars of Aristotelian school, and here logic and beauty are the only criteria of truth. All the low-hanging fruits were already harvested, and only really hard problems left. Most of them would hardly be ever solved. It now takes a prophet and a divine revelation to go beyond what can be achieved by mere humans.
I second the recommendation for Surely You’re Joking. Feynman was not just extremely curious and a near genius, he had a great sense of humor and lived life to the hilt all the time.
Very entertaining. A fascinating character, he was.
huxley:
Calculus is a slippery slope.
A glass hill.
I guess Einstein had a “knack.” For the original Chuck:
https://experimentalmath.info/blog/2016/04/why-are-so-many-mathematicians-also-musicians/
When I took, Diff Calc in the 60’s as part of my Engineering course, I was uneasy the way the derivative was defined by our teacher. It seemed forced and arbitrary and to my young mind comes close to include division by zero.
Then I came across a quote by Bishop Berkeley criticizing Calculus as formulated by Newton, which make me smile every time I read it:
“And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?”
As it turned out due in part to criticisms like these, beginning in the late 60’s, after I have finished my Calc courses, the definition of the derivative was made more rigorous by using the concept of limits.
Third vote for Feynman’s book, and the others he wrote for quasi-lay readers.
From my own experience the teacher is Everything. Someone mentioned the Khan Academy, someone else Professor Leonard’s YouTube video courses, and while I deeply appreciate the former, the latter is absolutely superb. For some, any level of his various courses (pre-algebra, algebra, trig, calculus) can become monotonous because he has endless patience and explains over and over again every element of his problems and the operations necessary to solve them, but this is essential for those who, like me, are math challenged. The Khan Academy supplies endless levels of practice problems and short explanations of any specific point, but Leonard is peerless at both explaining both underlying concepts and solution techniques. If you have even the most minimal interest in going back to any level of math I cannot recommend Leonard (especially) and Khan too highly.
“Proof yet again that femmes do not do as well in STEM.”
Umm, where does a lack of M have anything to do with S, T and E ?
You can be a brilliant biologist or surgeon with minimal maths. And much of actual engineering doesn’t require any more than competence in maths.
Even whole spheres of Maths don’t require Calculus. My daughter gained a first class degree in Statistics without doing any calculus at all (although, to be fair, she’s actually really good at it — she just doesn’t like it).
There’s this thing where people who are good at Maths too often equate it with intelligence. That they then treat us to that opinion then instantly makes people think the opposite is true. (And no I’m not jealous, I’m quite good at Maths.)
I have a friend who has an ABD (all but dissertation) in physics from MIT. I asked him why he didn’t do his thesis and his response was “I realized I wasn’t smart enough”. Sure he could have obtained a PhD but do to what – teach at some mid-level university?
Lot of people here are bemoaning the profs and the TA but if you can’t read through the textbook on your own and learn it all by yourself then you are certainly not going to to be able to appreciate the deeper insights. So what if you don’t know calculus, it isn’t as if you are going to lead a fuller life if you learned to solve diffy-q.
FWIW, there are endless comments on Leonard’s videos from current students bewailing the vile quality of their teachers and stating that Leonard’s videos have given them a lifeline for surviving their classes. My Spousal Unit, who is currently forging through his calculus courses, agrees wholeheartedly and adds that at the higher levels things become more difficult simply because of the increasingly complex analyses that have to be applied to solve problems, but unless you have some need to pass an actual exam you in fact have all the time you need to take notes, review and solve without the compulsion to absorb and remember everything. Without a grade or a potential career at stake, taking these classes for the sake of the intellectual exercise becomes a matter of interest, not penance for your sins
Interesting work is being done at Marshall University on the use of a mechanical differential analyzer…long obsolete for practical computing problems…to aid in the teaching of calculus. Vannever Bush, who constructed the original differential analyzer in 1931, wrote of the machinist who had helped him build the machine:
“I never consciously taught this man any part of the subject of differential equations; but in building that machine, managing it, he learned what differential equations were himself … it was interesting to discuss the subject with him because he had learned the calculus in mechanical terms — a strange approach, and yet he understood it. That is, he did not understand it in any formal sense, he understood the fundamentals; he had it under his skin.”
The objective of the Marshall project is to assist in achieving this type of intuitive understanding.
https://pdfs.semanticscholar.org/b4ec/e9c0a982013a335fe12aedc4dbc9cd946758.pdf?_ga=2.186344361.1067441178.1563140788-1987828002.1544545292
Mike H:
In my case, because of the requirements at my school, if I had passed that one course in integral calculus (which was a quarter, by the way) I would have then had the credit sequence to skip one full year of a lab science sequence. I didn’t care if I knew calculus or not, but you better believe I cared about not having to do another full year of a lab science.
I already had to take one full year of a lab science, which I did. But I didn’t want to have to take two, which my school required. By taking differential calculus (which I took and got something like a B or B+ in) and passing integral as well, it would have stood as an equivalent of my second year of lab science.
In the end, I transferred (not for that reason, although it may have been a factor somewhere in the back of my mind) to a school that only required one year of lab science, so it turned out I never had to deal with the problem.
And these weren’t dumbed-down lab science either. They were the basic sequences, three quarters each.
“Calculus was (and still is) required for certain majors, of course.”
I went to New Mexico State and anybody that wanted into the engineering school had to take and pass the 2 semester freshman calculus course. The reason was very simple. Many engineering courses used calculus so if you couldn’t do calculus you weren’t going to pass the course.
While I didn’t think about it at the time, my hat is off to the Berkeley math department for caring enough about its students to quickly replace a TA who wasn’t getting through to us. He was probably a pretty good mathematician, but unfortunately, not a very good teacher
Neo:
My experience with calculus was more positive than yours. During my freshman year at Rice, I had to take beginning calculus. The course consisted of a lot of proofs at least the way my professor taught it. I just memorized the proofs so I was making good grades although I didn’t understand them. Finally, about Christmas vacation, the light began to dawn on me. (I think studying “Shaum’s Outline” helped me understand Calculus…does anyone remember those books?). Anyway I never had any trouble understanding college math from that time on. I graduated with a degree in Chem. Engineering.
Later, I earned an Outstanding Contribution Award at IBM for my work in monitoring Gas Chromatographs which made extensive use of first and second derivatives. So I had a practical use of my calculus.
I agree with all the Feynman recommendations! I have at least a dozen books by or about him, for non-scientists. All very good.
I have been a fan of Richard Feynman and Herman Wouk for decades, and I was delighted to learn when I read Wouk’s 2010 book “The Language God Talks” that they knew each other. They were both geniuses, in different ways. Feynman was gifted with math. Wouk could never learn calculus although he tried.
Wouk dedicated “The Language God Talks” as follows:
“To the memory of our fathers, Abraham Isaac Wouk and Melville Feynman, who emigrated from Minsk and gave us our lives in America.”
You can no more be an engineer without calculus than you could be a carpenter without a hammer.
That should be “Schaum’s Outlines”. I was never any good at spelling.
🙂 And it’s gotten worse since my stroke.
Switched Philosopher,
If you have not read “When Einstein Walked with Godel: Excursions to the Edge of Thought” by Jim Holt, I suggest you investigate it. I believe you will like it.
Lot of people here are bemoaning the profs and the TA but if you can’t read through the textbook on your own and learn it all by yourself then you are certainly not going to to be able to appreciate the deeper insights.
Mike H: I think it’s perfectly fine, admirable even, for people to learn things in whatever way they can and to whatever depth they wish, deeper insights or no.
There was a movie back in the 1980s, “Stand and Deliver,” in which a retired engineer (played by Edward Olmos) chose to teach high school math to working-class Hispanics in the East LA. It’s based on a true story and faithful to the essentials.
To challenge the kids and blow away their minimal expectations of themselves, Olmos resolves to teach them math with the goal of taking the Advanced Placement Calculus exam when they are seniors. Son of a gun, he succeeds.
I found the film inspiring. People have limits, true, I’m up against some of mine now in Calc 3, but with good teaching I do believe calculus is within range of more people than conventionally expected.
I’ve often wondered why so many people are convinced they can’t do math or they don’t want to do math. With appropriate teaching I truly believe high school algebra and trig and even into calculus are possible for people with normal IQs and higher.
It seems many people are burned by their experiences of math when younger and never recover. The writer Robert Anton Wilson, as I recall, said mathphobia is a contagion passed on by female elementary school teachers. Perhaps.
Something happens that shuts people down. I see it with poetry too. We have more pressing problems than worrying about people fearful of math or poetry, but it still seems a shame. As far as I’m concerned, math and poetry are high points in our human heritage.
> With appropriate teaching
One problem is that teaching tends to concentrate on details. I tell people that the important thing is to have a general idea first so you know where you are trying to go. Students tend to get lost wandering around unorganized piles of memorized material. Math isn’t like organic chemistry, you really don’t need to memorize that much if you have a feeling for how things work.
As to calculus texts, I recommend Calculus Made Easy by Silvanus P. Thompson. The first edition was published in 1910. I learned calculus from it at 14, and some hints make me think that Feynman may have learned from the same book.
Lot of people here are bemoaning the profs and the TA but if you can’t read through the textbook on your own and learn it all by yourself then you are certainly not going to to be able to appreciate the deeper insights.
I taught myself 9th grade math from the textbook- Illinois Math (UICSM) version of New Math. My teacher, a family friend of sorts, was bright and knowledgeable about Math, but a horrible classroom manager. The class was a zoo. I ignored the classroom chaos and taught myself from the book. At the beginning of the year, I was indifferent to Math. By the end of the year, Math was my favorite subject. The textbook was that good.
The mid-late 1600’s saw the invention of the calculus. Another century of work by some of the most brilliant minds (Euler, and many others) expanded this new field, laying the groundwork for a revolution in science that continues to this day.
What about calculus that made all of this possible? Our universe is not a static entity, like a bridge. It is in constant motion and in constant flux. This new tool calculus allowed for explaining change mathematically. It very cleverly uses the concept of the ‘limit’ to split whatever it is being described into infinitesimally small segments. It takes a complex thing and breaks it into manageable parts.
So I think it’s the conceptual part that is often what’s missing in learning this tool. I took the Univ of Calif first two years of math for the sciences back in the early 70’s, did pretty good with it, but never felt I really ‘got’ it. I believe a teacher, or a book, that can give a historical/conceptual explanation instead of just plowing through the proofs and problems would be a great improvement.
The world plodded along for millennia until this breakthrough happened in the West. The West soon dominated the world. And anywhere else in the world has had to emulated the West’s science to even stand a chance. That’s because the calculus does model much of the real world very closely – much more than the old pre-calc math.
So from this point of view, I believe it to be one of the most interesting of all human developments, right up there with religion. It has had a profound effect on human progress in the last couple of centuries. I would think, at a minimum, an understanding of how it came to be and how it changed things should be on every educated person’s list of things to better understand. IMO, it’s that important.
But that’s just my opinion!
}}} That is correct. Look at the winners of the Putnam Math competition.
https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition
I believe that in most things involving men and women vs. a Normal Curve, you find that women have a much more narrow standard deviation while a slightly higher mean, if it’s not something where physiology is a clear blatant advantage (the mean for men having babies, for example, is vanishingly close to zero, whereas men seem to excel above women for power lifting… go figure).
So men tend to have more variation — more people who excel, as well as more people who suffer failure — e.g., more male geniuses AND more male dunces.
Which means women do equal or better at the centralized tasks, but don’t tend towards excelling enough to dominate in the top end. I don’t believe we should have a system which says woman cannot be top dog, but I also don’t believe that the system should push women to be top dogs when they are not better equipped for it.
Yes, I tend to believe in meritocracy. That’s what equality of opportunity, not equality of results, is all about.
As to calculus texts, I recommend Calculus Made Easy by Silvanus P. Thompson.
Chuck: Dear old Silvanus! I still have my copy.
I have fallen into collecting calculus textbooks. I have an old Thomas, 3rd Edition, which some early 70s STEM types still rave about like a fine vintage wine. Though T3 lacks color illustrations and historical photographs, it remains an admirable and concise text.
The basic ideas of calculus are not that tough to get, really. Nor are the basic tricks. If that’s all you want. But the devil is very much in the details. If you want to use calculus in situations beyond toy demonstrations, the more thorough-going university approach is the way to go.
Kleppner & Ramsey’s “Quick Calculus: A Self-Teaching Guide” is an excellent compromise between Silvanus and a university text. That’s where I would tell a non-STEM person to start if they were inclined to get the overview of calculus’s grand edifice.
The world plodded along for millennia until this breakthrough [calculus] happened in the West. The West soon dominated the world. And anywhere else in the world has had to emulated the West’s science to even stand a chance. That’s because the calculus does model much of the real world very closely – much more than the old pre-calc math.
So from this point of view, I believe [calculus] to be one of the most interesting of all human developments, right up there with religion.
M Williams: Yes!
I think an international holiday celebrating calculus is in order. But I’m not waiting up nights for it.
OBloodyHell,
I found your comment about women having a lower deviation from the mean in abilities than men interesting. Do you have a reference or link for that? I have never heard that before. It would provide a reasonable explanation for why men would dominate the top tiers in nearly every human endeavor.
I found your comment about women having a lower deviation from the mean in abilities than men interesting. Do you have a reference or link for that?
Bing Search: men women intelligence distribution.
As anecdotal support to the variability hypothesis, during my student days I worked at an institution for the mentally retarded. As I recall, there were more males than females living at the institution.
huxley, thanks for pointing out that seniors get a tuition break in New Mexico. Your pointing that out prompted me to find out that Texas has a similar tuition break for seniors.
I also have collected a bunch of calculus texts.
> I have an old Thomas, 3rd Edition
That was my next text 🙂 I skipped trig and went directly to “official” calculus. My most memorable classes in high school were geometry (for proofs), chemistry (for bonds, acids/bases, and the periodic table), and calculus.
The stuff of nightmares. Had a similar class with Professor Lofti Zadeh, the father of fuzzy logic, a brilliant man but an unintelligible lecturer from Azerbaijan. I can’t remember a single thing I was actually able to glean from his class.
Socratease:
The stuff of nightmares. Had a similar class with Professor Lofti Zadeh, the father of fuzzy logic, a brilliant man but an unintelligible lecturer from Azerbaijan. I can’t remember a single thing I was actually able to glean from his class.
Brilliant researchers do not necessarily make good teachers. A cousin of mine had Nobel Prize winning Melvin Calvin, for one of his grad courses at Berkeley. Calvin was not a good teacher, my cousin informed me. IIRC, “horrible” was the way my cousin described Calvin’s teaching abilities.
I suspect that the main part of the problem is that material which to appears intuitively obvious to the brilliant researcher, does not appear so to “ordinary” students. The brilliant researcher leaves out some steps which he automatically did in his brain, leaving “ordinary” students at a loss.
If you ever want to see how bad math instruction is, I recommend checking out
The Calculus Direct: An intuitively Obvious Approach to a Basic Understanding of the Calculus for the Casual Observer
Weiss, John
The author, a lowly community college professor, starts at number lines and in less than 100 pages, step by step takes you through Calculus.
Of course, what is really valuable from calculus is not the formulaic application, but the underlying thought process. Summing incremental pieces for Integral calculus and examining the impact of how the equation changes when one variable is changing, such as varying with time or location.
Once I got through my summer-term partial differential calculus class that was much like your experience, we were informed that only a few equations could actually be solved. One of my professors in the Physics department explained not to worry about it, there were only five or so forms of equations and you just learn to recognize them and their solutions.
“The world plodded along for millennia until this breakthrough [calculus] happened in the West. The West soon dominated the world.”
Not to downplay the importance of calculus, but I think that’s an overstatement. Did either Newcomen or Watt (steam engine inventors) or Boulton (Watt’s manufacturing partner) use calculus in their work? I don’t think so. Ditto for the early textile equipment such as the water frame (for mechanized spinning) and the power loom.
And the oceangoing sailing ships were developed pretty much empirically. Celestial navigation did require a lot of math, but that was trigonometry at the level of the individual navigator. Not sure exactly how the celestial-body position tables were created; possibly calculus was involved there.
I still remember my summer-term partial differential equations class. Only 3 students signed up for the class so the professor was going to cancel the class. We persuaded the professor to hold the class, but he made us students teach the class and he sat in the audience and asked questions. Each student was assigned sections of the book to teach and you had to write everything on the blackboard and explain it to the other students and the professor. Even worse, you were not allowed to use the book or notes, you had to memorize everything. After you do that a dozen times, you get pretty good at it. We didn’t have any tests, you were graded on your presentation of the assigned material. What I really dreaded was the questions from the professor. He asked questions to see if you really understood what you were talking about and sometimes you had to present detailed explanations. You couldn’t get away with rote memorization. That was a really formidable math course, you had to teach yourself the math, then teach other people.
David Foster
You could also cite the development of the marine chronometer and the problem of longitude for navigation.
I still remember my summer-term partial differential equations class. Only 3 students signed up for the class so the professor was going to cancel the class. We persuaded the professor to hold the class, but he made us students teach the class and he sat in the audience and asked questions. Each student was assigned sections of the book to teach and you had to write everything on the blackboard and explain it to the other students and the professor.
That’s pretty much how the seminars in mathematical economics were taught in the ’70s – we had an important math econ book each quarter, in which the mathematics was usually outlined, but rarely proved in depth. The seminar participants were each assigned the principal theorems to explain to the group, with full mathematical proofs of the underlying material as well as the main theorem. Very intense, very difficult for me (as the only member of the seminar without either an undergraduate or graduate degree in mathematics), but exhilarating.
It is said that you haven’t truly mastered a subject until you can explain it to a layman.
This is an interesting discussion, especially the professions and vocations that are heavily math oriented and their curriculum, where many of the applications of math courses aren’t used later once one is in the field and practicing. This sort of reminds me of the debate on who can choose, administer, score and report certain assessments and tests in the education field. Currently it’s all those who are masters level clinicians vs Ph.Ds and Psy. Ds. It’s almost a consensus of the former that a vast majority of the assessments should be done by those with training. I would agree – the purely psychological ones. But when a school district offers training to administer particular assessments (non-cognitive, psychoeducational) relating to SEL (social-emotional) or adaptive + behavior, to masters level clinicians (MSW) it makes you wonder if all the talk of training on the doctoral level is just self-preservation. Even if you have proper training, the annoying question of “you can but should you” tends to show its head, always casting doubt on the masters level clinicians’ abilities, and always asked by the doctoral level clinicians.
I think a (large?) reason more people have problems with integral calculus than differential calculus is that it involves a lot more memorization of seemingly arbitrary formulas. I have taken a lot of advanced math courses but I still can’t remember the sign of the indefinite integral of cos(x). Is it -sin(x) or just sin(x)?
Actually once you passed the course, most people relied on big fat books like Gradshteyn and Ryzhik with tables of thousands of integrals
A big change since most of us took calculus is the introduction of symbolic math programs like the Maxima (http://maxima.sourceforge.net/). These things are amazing. They can do both indefinite and definite integrals, simplify algebraic formulas, solve systems of equations, plus a lot more. An interesting anecdote is that when they were first introduced people found lot of mistakes in the integral tables in books.
Here is an example of the program’s output:
(%i1) integrate(cos(x),x);
(%o1) sin(x)
(%i2) integrate(sin(x),x);
(%o2) -cos(x)
Maxima is free but there are many other symbolic math programs out there like Sagemath (free) and very expensive ones like Mathematica and Maple. The company that makes Mathematica provides a free online interface to it called Wolfram Alpha.
One problem is that their interface can be quirky so you are trading memorizing math formulas for memorizing the interface. But most people nowadays have gotten used to memorizing the use of software. The process Neo uses to post on and maintain this blog is also very complex but she has mastered it.
In high school I was very interested in learning chemistry; how things were put together and interacted with each other fascinated me.
That was until I took my first chemistry class with a teacher who, it was said, had worked for DuPont and, then, something happened, and here he was, teaching basic chemistry to a bunch of kids in Philly.
Well, it seemed to me that he took all of his resentment out on us.
He would start each class by facing the board and starting to write formulas which—as I recall—is what he did for practically the whole period.
Moreover, his explanations for what he was writing made no accommodations for our complete ignorance; he was explaining things on what I imagined was the level on which he had discussed things with his colleagues at DuPont.
A totally unsympathetic and unapproachable guy.
That was it for chemistry for me!
“he was explaining things on what I imagined was the level on which he had discussed things with his colleagues at DuPont.”
Which might explain why he was no longer working at DuPont.
Snow On Pine: Too bad you’re chem teacher didn’t catch the meth wave like Walter White in “Breaking Bad.”
I do hate confusing “you’re” and “your.”
neo: Any ETA on a fix for comment edits?
Not to downplay the importance of calculus, but I think that’s an overstatement. Did either Newcomen or Watt (steam engine inventors) or Boulton (Watt’s manufacturing partner) use calculus in their work? I don’t think so. Ditto for the early textile equipment such as the water frame (for mechanized spinning) and the power loom.
David Foster: To be sure, Newton and Leibniz didn’t come down from their respective mountains with tablets of integrals, and lo, the modern age began. It took a century or for calculus to really kick in. Fair enough.
However, no tech we see today in the urban 21st Century would be possible without calculus.
It is said that you haven’t truly mastered a subject until you can explain it to a layman.
Roy Nathanson: I’d agree, mostly.
My impression is no one, not even math majors, get calculus until they’ve TA’d or taught it a few times. Leaving aside freaks like Feynman and Freeman Dyson, of course.
I have no wisdom of my own; I merely point.
E.O. Wilson, preeminent biologist, was mathematically “semiliterate”. He too aspired to learn calculus, and so as a full professor at Harvard, sat in on a basic undergrad calculus class. He earned a C.
A quote from the incomparable one:
“I had a feeling once about Mathematics – that I saw it all. Depth beyond depth was revealed to me – the Byss and Abyss. I saw – as one might see the transit of Venus or even the Lord Mayor’s Show – a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go.”
Do you recognize the lion by his paw?
P.S.
https://winstonchurchill.org/publications/finest-hour/finest-hour-159/wit-and-wisdom-churchill-on-tergiversation/
Read the whole thing, as they say!
Mathematics has always been a struggle for me.
I’ve sensed that there was a universe there, but one that I somehow could not get the key to; couldn’t visualize and understand.
That universe of mathematics has always been “terra incognita,” except for one brief semester in college, in my “Physics for Dummies” class.
The professor somehow uttered the magic words, patiently explained things in a certain way and, voila, I could suddenly look at the Laurence-Fitzgerald equations, and see them as not just a jumble of meaningless symbols, but as something I understood; they created a picture in my mind.
I understood what they were trying to say.
Then, of course, that all too brief moment of clarity and understanding fled, and it was back to facing an enormous cloud of things and relationships that I could not comprehend; barriers that would not yield.
Today marks the 50th anniversary of the Apollo 11 launch to the moon. Archival footage (CBS) is running on YouTube now, with liftoff approx. 20 mins from now. Link: https://youtu.be/zYnF31el-ik
Highly recommended.
For the Eagle has landed
Tell your children when
Time won’t drive us down to dust again
–Leslie Fish, “Hope Eyrie”
https://www.youtube.com/watch?v=aGk5I6Fejh4
Hard to believe that was 50 years ago. I was bussing tables at a Sheraton Hotel when it happened. I assumed we would have a moonbase and much more by now.
Extraordinarily disappointing how the U.S dropped the ball after we landed on the Moon.
What do we have now?
No functional “Space Program,” some satellites in orbit, a few fading “Mission Patches” for sale on Ebey, and some private ventures trying to get things going again.
What a colossal failure of U.S. national will, nerve, and imagination!
Some “Space Age.”
We may not have our own U.S. space station in orbit, or a permanent presence Moon, but we have 53 sexes–or is it 72?
Yay!
As a software engineer I’m similarly disappointed. We have an astonishing number of bright, well-trained tech people, but most of their work these days seems to be about sucking clicks and life force out of people.
“Actually once you passed the course, most people relied on big fat books like Gradshteyn and Ryzhik with tables of thousands of integrals” — Bob
I once had a long open-book physics exam where we were expected to set up and solve a few integrals. Usually I used a nice little compendium of integral solutions put out by CRC.
But on this occasion I stupidly brought the gigantic Russian tome (G&R) listed above. I’m sure a lost at least 20 exam points on the time wasted searching that book. You should approach exam taking a little like game theory. The little CRC book wasn’t comprehensive, but probably good enough and very efficient.
_____
It’s a sin to spend substantial time on calculus and not really understand what it is about. Some of my math textbooks had a fair bit of physics in them, purely because a good example can really help with comprehension.
I love that movie “Stand and Deliver.” Part way through the film I wondered if the students truly understood the calc. concepts. Sure enough, the scenes covering that were included. The film, if accurate, shines a light on what’s wrong with leftist public education.
_____
So many thoughts and memories.
As a sophomore, I once took a junior/senior level math class intended for the math majors, on differential geometry. Three of us sophomore physics students had to get permission from the prof. to get into the large class.
There was a 1 hour mid-term exam. The young physics guys crushed the curve. I got a 78, my good friend got a 90 or 95, and the resident physics genius Shinichi, got 100 percent. The median (average?) score was 23%. The excellent math prof., who ran a very conversational classroom, was so chagrined.
An interesting book on that period of time when calculus was developed is ‘The Clockwork Universe’. The thing that I remember most from it is how strong a hold religion had on people at that time. It was thought – actually, taken totally for truth – that the heavens were perfect and things down on earth were debased and sinful, ie., imperfect, and that they were two different realms that God death with accordingly. The fact that they obeyed the same rules was blasphemy.
I don’t think we can imagine the cultural hurdles that faced these thinkers of 400 years ago. And Newton didn’t think this up out of the blue. There was a lot of activity (Brahe, Kepler and others) that laid the groundwork for his new math.
I don’t know, though, how Leibniz came around to the same concepts. The friction between these two is another story that I’m not sure has ever clearly settled.
huxley—After we landed on the Moon, the U.S. had the “momentum”—public enthusiasm and support, some of the infrastructure built, the experience, and—for those with some vision–the pretty clear promise of new industries and techniques, resources, and raw minerals “out there,” in orbit and further out.
It was no lie that this truly was the “New Frontier.”
Thus, if we’d have played our cards right, we would be the ones “out there” now, somewhat established–perhaps a lot established—with a lot more experience and hardware, and leading the way.
As it stands now, it looks like the Russians, the Chinese, the Israelis, the Indians, or who knows who else will be the leaders, and the beneficiaries of this New Frontier.
Snow on Pine: Agreed. With sorrow.
huxley–An abundance of science fiction writers who do have lots of imagination and vision, but no one in power able to lift their eyes from the ground, look up at the stars, make a bold choice, and place their bet on a new future.
We’re stuck with unimaginative, scared, risk-averse plodders, fighting, down in the pig trough, all their focus and efforts on grabbing scraps of garbage, instead of on changing the game, on climbing out, and seeking a far wider and more productive and rewarding environment and much, much better and more worthwhile prizes.
huxley–speaking of our Space/Apollo program, see this commentary about the anti-male/woke take on that Herculean effort today–
here at https://pjmedia.com/instapundit/