Annals of math: how integral calculus did me in
[NOTE: This fond reminiscence was sparked by a comment on the Periodic Table thread.]
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 test paper and then at the 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.
I earned a degree in chemical engineering with honors from a first class Chem. E. department, but I found calculus hard to understand at first. Fortunately, my prof. spoke excellent English but he was VERY intent on emphasizing elegant proofs as opposed to using the math to solve problems.
I memorized the proofs with little understanding but memorized them well enough to get high scores on tests. FINALLY, with the help of Schaum’s Outline on Calculus and a paid seminar just before finals given by an upperclassman, it began to sink in.
After that, it was clear sailing and differential calculus, integral calculus, advanced calculus, and differential equations helped me to keep my GPA high.
That famous “Bridge of Fools” was differential calculus for me…not the one in plane geometry. I had to struggle to get across mine.
I was in a class where someone did that. And it was obvious it wasn’t because they were a genius who’d finished quickly. I think they even said some funny comment as they proceeded to walk out the door. If nothing else, they had style.
I took a Linear Algebra course where the instructor , while from Bosnia, spoke good English. Her spoken English was so good that I had her pegged not for an ESL person but as someone born in the US who had grown up among Middle /Eastern European immigrants, as I had. Such as “der” for “there” which I also do, if not all the time. One time I said “tree” instead of “three” to an elementary school class where I was subbing – which the class corrected. I tot to myself, dere’s my hometown showing up.
She was not a bad lecturer, at least for me. However, she did not have a good idea of how the problem sets she assigned correlated with the abilities of her students. Both the problem sets and the exams had a lot of proofs. Most students had little experience with proofs. Little of the math I had previously taken in college had much in the way of proofs, but I was very well practiced in math proofs from the New Math I had taken in high school years before – which had proofs starting in the 9th grade.
She also assigned too much homework. She would improvise an assignment at the end of class: “Do 3, 5, 6 … on page 157.” She didn’t have a clue about how long the problems took. I was one of the most proficient in the class, and I was taking way too much time for the homework. I imagine most of the students gave up from the overwhelming load.
While I got an A, a third of the class flunked.
Before I took this course, I had sat in on some other Liner Algebra courses. The Linear Algebra course for Engineers course had instructors who skipped steps, did not explain well, and had too much computation as opposed to proofs. I recall one instructor who RUSHED through the material, then when 20 minutes remained in class, would ask, ” Are there any questions?” Of course there are questions, you doofus. You rushed through things so fast, no one could follow what you did.
For all the faults of the instructor, she was better than the ones I had sat in on before deciding they were not my idea of good instructors.
The quality of math instruction in is quite variable.
My 9th grade math teacher, in addition to being a family friend, was a horrible teacher. I learned the course from the book, a New Math creation. I did well in the course and liked math a lot more at the end of the year than I did at the beginning of the year- because of the book and in spite of the teacher. Having a bad teacher can help you find out what your real strengths are, because you are on your own. Before that 9th grade math class, I would never had pegged myself as someone who liked math – which I did after the course.
Back in the late 60’s I took third semester Calc. There were 27 in the class all had succesfully completed two courses of Calc. One person got a “D”, the rest “F”. There was no language problem, just a new prof with no idea what he was doing.
Oh, Neo. You bring back memories. After twelve or so years as a Naval Aviator, the Navy decided I needed a degree (I had entered through the Cadet program which only required two years of college. A wonderful program that the USN, in its wisdom, later discontinued.)
First the Naval Post-Graduate school would not accept me because I was deficient in math. So, I took a quickie Algebra course at a local JC. Then, since I had a recent math course, the NPGS placed me in an introductory math course with the Engineering Science students, rather than with the International Relations students (two options only.) For some crazy reason I decided to continue, quite unnecessarily, into calculus along with the Engineering class, in a quest to erase my personal shame over being a math Klutz.
Your description of calculus rings so true. In my case the Professor was a geriatric English man, but his teaching skills compare. The second quarter was worse. The Professor was a young American who should never have been given a piece of chalk nor an eraser. To make it worse, he did not like the text so he did not use it. Everything came from the blackboard. As described above he spent the entire hour with his back to the class writing from his notes, and we spent the entire hour copying.
The precious time spent on calculus also detracted from my other studies. But, it did pay off very slightly in an Economics course when I finally made it to graduate school. Believe it or not, the ability to compute the area under a curve does have practical application in the sense that it can significantly reduce the verbiage needed to explain some arcane concepts.
In a bit of irony; many years later I undertook as Adjunct Faculty at a nameless college, to teach remedial algebra to Sailors. What a discouraging undertaking.
Calculus is just like algebra, except really, really different.
The memories above of bad math teachers belie the old aphorism “Those who can, do. Those who can’t teach.”
There is an art to teaching and there is a special art to teaching math. I, personally, am sorry to say I never had a great math teacher. Most were mediocre, a few were just awful. As a result it took me an additional 30 years to understand, respect and admire the intricacy and beauty of mathematics.
It’s a condemnation of the entire educational system which (at the university level) disdains teaching for research. A poor teacher can make one hate English Lit, Biology, Art History or Math even more so than a good teacher can attract one to the same subject. Unfortunately, there are many more opportunities for the former than the latter.
The more you understand the concept of Algebra and Geometry Proving, which requires a lot of practicing, the better you are in Calculus.
Neo,
Calculus is all about finding the slope of the line, the derivative, or the area under the line, the integral. Of course it looks complicated if you don’t start simple and work up to the complicated cases gradually. Most college instructors aren’t very good teachers. In grad school I was forced to teach a course in partial differential equations. It was an interesting experience and sometimes I was baffeled by the students inability to understand what was obvious to me. You would have to explain something a number of different ways before everybody caught on. When you see the puzeled look on the students faces, you know you haven’t done a good job. The students had the math prerequisites to take the course so it wasn’t like teaching students with no math background.
I took algebra, geometry, trigonometry and calculus as an undergraduate majoring in geology. I could do the work and got good grades, but never understood it. Fast forward from 1954 to 1962. The Navy sent me to a year of Post Graduate School. Except, instead of Monterey where Oldflyer attended, they sent me to San Diego State University. The summer term was spent brushing up on algebra, trig, and calculus in succession. They were my only courses and were intense. Like drinking from a firehose. Fortunately, the Math Department offered a three hour evening tutoring class for we Navy types. It was taught by a grad student who was a natural teacher. Without his ability to help me refresh my knowledge it would have been much harder – maybe even a disaster.
The result was gratifying. I went on to complete advanced calculus, theory of numbers, solid geometry, and statistics during the regular academic year. This time I could not only do the work, but understood it as well. Unfortunately, I never had to use all that hard won knowledge in the real world. Today, it is all just a pleasant memory. However, when you understand how it works, it provides an appreciation for all the many uses of math in science and engineering. It has enriched our lives immensely even though few of us have mastered it.
I feel sorry for everyone who mentioned that their difficulties in math stemmed from a teacher (probably an elementary school teacher) who did not teach the subject well.
Up until May 2011, I taught developmental mathematics at a community college in my home town of Memphis. That term refers to a sequence of three courses in basic mathematics, elementary algebra, and intermediate algebra. The students in my courses were students who were deemed to need more help in bringing their math skills up to the level of college math. There were a number of students who had not been in a math class for years.
My job was to make the math understandable for them. I think I succeeded in many cases (although in some cases students did not really want to do the work, and subsequently did not pass). I was “old-school,” but I understood the responsiblity of not making the experiences of my students worse because of my teaching. So I was sensitive to their concerns.
I say “until” because due to not having my tenure-track appointment renewed due to the economic situation at the time, according to the rules I had to leave. Too bad, I loved that job.
Keep in mind, though, that calculus, in my opinion, is in the “big leagues.” Not everyone needs calculus in order to be math-literate; algebra is in use even if we don’t recognize it, but some topics admittedly may not be as useful as others. Maybe a good finite math course would be good for most, especially if there was a good financial math component to it. But a lot of the relative benefits of the math curriculum is another topic for other forums.
Neo,
There could have been an inernational conspiracy here. Yes, it was calculus, my Prof. a korean, same M.O.. First day 10 second intro then the filling of white grease boards front and sides of the room with what could have been Minoan Linear B for all I know. He couldn’t speak english I couldn’t speak his math, I ran for my life. Oh yeah, it was a chem. major too.
I had a graduate math course taught by a Frenchman who had just spent 20 years in South Africa. OY!
The first time around, at a small but excellent engineering school in the midwest, my Calculus class was taught by a small but excellent, straight-from-central-casting professor named R. E. Doubt. Having missed most of his classes because I was hopelessly immature, I asked him the day before if it would be worth my while to take the final. He got out his grade book, found my name, pondered, and said “Son, you don’t have a snowball’s chance in Hell of passing.” I was quite relieved, because I could go partying instead of studying that night.
A year later, I had switched schools, and majors (!?). to industrial design, but re-took Calc just to prove to myself I could pass it. I was 20. The teacher was female, 23, and HOT! Having no other plan to get next to her, I studied like mad. The Calc “light bulb” went on for me, and I aced the course.
Motivation, that’s the key to learning calculus.
I think math is a bit like music, as much a talent as a consequence of IQ. I don’t have a particularly high IQ, but I got an A in my multivariable calculus course by splitting the text in half and reading the first half the day before the final; I had skipped most of the classes and didn’t know how far they gotten. Likewise, I tested out of high school trigonometry by reading the text over the weekend. OTOH, I’ve known people with IQ’s up around 170 who just didn’t ‘get’ math and struggled horribly to manage the simplest things.
All very good stories, even if some were bad memories.
I speak conversational math — my vocabulary is better than a tourist’s, but I still have a strong accent and am guilty of bad math malapropisms. I certainly don’t dream in differential equations.
The problem of math is that it veers between “so what” and “huh?”, between “that’s too obvious to mention” and ‘how in the h#ll did you get there?”
You need a good teacher, a clear understanding of your goal (e.g., how do I find the slope if it’s changing?), and the willingness to make mistakes. Lots of mistakes and false starts. If you are curious, the motivation will power you through and the why will drive the how. Some people are born with that drive, the intrinsic need to do math, the same way that some people are born with the need to dance or sing. The rest of us need to be nurtured until the math becomes second nature, until the gap between our desire and ability is not too great.
Too often teaching is a display of dominance. Good teachers are humble before their subjects. Math has the additional twin burdens of precision and perfectionism. There is a lot of tension between the necessary willingness to make mistakes along the way and the need to get it right in the end. It’s also so easy to forget that what seems obvious now might once have been considered insoluble, if it was even perceived as problematic in the first place! And that beautiful equation didn’t emerge full-blown like Athena from Zeus’ forehead. It was probably a wrinkled baby and the product of a laborious birth.
I am awful at math, simply awful. Always have been, always will be. Even good teachers – and I have had some – are only able to effect marginal improvements in my performance.
In high school, I studied overtime every day and my dad hired a personal tutor to help me on weekends for Algebra I. I got a C (my only C).
Somehow, I squeaked by with the bare minimum numerical grade for a B- in Algebra II, and did the same in pre-Calculus. A few lucky guesses on the exams, I suppose.
I took the SAT four times, three of them with a paid tutor, and my absolute ceiling on the math section was 650 (I got that grade, exactly, 3 times).
In college, I was an economics major at first, and I did well – until the math became necessary. Statistics practically drove me to an early grave (it didn’t help that my professor was abominably bad – I think half the class failed). I would have failed Calculus if my professor didn’t allow the people who failed to re-take the final exam.
I had realized by that point that this was a constitutional thing – no amount of teaching could help me attain anything like proficiency in math. Or rather, maybe if I dedicated my life, 24/7, to rudimentary algebra and calculus, I could be average – but no more. Thus it was pointless for me to keep going with economics, since everything from the intermediate sequence onwards requires calculus/econometrics. I switched majors to what I was good at – and truly loved – anyway, and never looked back.
The one feeling I can recall about all of this personal history is a persistent sense of humiliation, sometimes even shame. I watched tutor after tutor damn near throw their hands up in exasperation at my seeming inability to progress. Quite simply, I felt borderline retarded. It didn’t help matters when everyone around me, teachers and tutors, insisted good-naturedly that idiocy in math is a result, for the most part, of bad teachers. It was not pleasant to serve as a falsifying test case for that hypothesis.
I can, however, balance a check book. Baby steps.
A lot of math horror stories in the comments. Nothing is hard once you understand it. The challenge must be explaining it well enough that it is easy to understand. That is ‘good’ teaching. As online teaching becomes more popular I expect we’ll find some really good teachers and teaching methods that make ‘hard’ math easy (and fun) to understand.
My grandfather described our high school math teacher as the brightest kid he ever taught. I always assumed that I was just bad at math, until my younger brother was in teachers’ college, and told us all at Thanksgiving dinner that Mrs. A. had used a method of teaching math that came into style every few years, and was quickly discarded, until the next generation experienced the curse. Finding out I was not actually stupid was a great encouragement to me. I did much better in math, thereafter.
I’m not sure how to react to this post to be honest. I mean on the one hand I guess I can relate to be stuck taking a sequence of courses you’re terrible at because some dirt bag admin decided that we just have to shove some topic down the undergrads’ figurative throats with a hypothetical broomstick. On the other hand I’m a little pissed you got off so easily. I guess I’m going to sound like the 4 yorkshiremen but the thing that got me was the foreign language requirement, aka the undergrad torture requirement. You had to look with dred to another year because you failed a 2nd semester math course? I was looking at 2 years because I kept failing the 4th semester of French.(God how I now despise French because of those courses.) Hey, here’s an interesting thing to note. You know how they say foreign language will make you smarter and you’ll do better in your other courses? In my case that is demonstrably not true, lucky me.
(For those that care I only passed because I got to switch from the regular series to the reading only but I had to go back to 3rd semester reading French. All in all how many semesters of foreign language claptrap did I get stuck taking? 9 semesters of it.)
I later discovered that about a third of the students in the class had failed along with me, although not as flamboyantly.
The first class I had in grad school was in thermodynamics, a subject I knew well, but taught by a self-admitted appallingly bad teacher. On the final exam I worked one question, then spent the rest of the three hours struggling with the remaining questions, wandering down mathematical blind alleys and getting mugged. I left totally dejected, convinced I was toast.
Turns out that everybody in the course had done exactly the same thing (including one of our number who subsequently won the Nobel Prize), that we all had solved the same problem, and miserably failed to solve any of the others. The “curve” was a delta function, and so we were all given a grade of B+.
I guess I’m going to sound like the 4 yorkshiremen but the thing that got me was the foreign language requirement
A friend of mine in college was a math genius, who taught math courses at age 19. That’s the good news. The bad news is that he was essentially pre-verbal. He struggled with the concept of “subject verb object,” and that was in English (of which he was a native speaker. Sort of.)
The foreign language requirement – which included holding a conversation with two native speakers for 20 minutes – for him was equivalent to asking Stephen Hawking to juggle running chainsaws. He’d have graduated early but for that requirement. Come to think of it, I don’t that he ever graduated, but I hope that somebody finally had the sense to waive that requirement for him.
I suspect that poor verbal skills is one of the major reasons that math instruction is often so dire.
Gotta confess. I loved math on all levels and did well in all the courses I took.
Starting with high school, I had good teachers up to and including analytical geometry. College (Caltech) was the same with some great teachers, up to and including Feynman’s Principles of Math Physics (can’t remember the formal title: it was grad school level).
So OB – that explains my love affair with QM and MO theory.
Math always came easy for me. I still study it both formally and informally. Want to try Number Theory some day before I die. I keep checking Coursera hopefully…
Hope you all hang in there with math…it’s truly beautiful.
I had a similar experience with calculus. I failed Integral, but kept the book, and wrestled with it on my own for the next several years. Got it, finally. Yay, I win!
On a similar note, a friend of mine flunked out of college for gross immaturity. Now, thirty years later, he’s back and just made the dean’s list in his first semester. He, too, feels that he has something to prove, even if only to himself.
I had thought maybe I’d be able to pass because I had a rudimentary ability to do a few things in integral calculus, but to the best of my recollection the final was all proofs.
He’d have graduated early but for that requirement. Come to think of it, I don’t that he ever graduated, but I hope that somebody finally had the sense to waive that requirement for him.
Well there’s your problem. “Sense” and “College Administration” are 2 phrases that don’t belong in the same sentence. If his school was anything like mine then there’s basically no way they’d waive a requirement. (Even if they got sued over it which mine did.)
NotSoHeavyD: oh, but I’ve got another story about language requirements. Some day maybe I’ll tell it.
As far as my getting off easy goes on the science/math, let me just say that, at the time I went to school, most colleges required a year of math or science. My school had an unusually rigorous set of requirements in that area: two years instead of one, and the sciences had to be regular one-year sequences of lab science each, rather than a semester here and there of this and that in courses not geared for majors. So when I transferred to a school with only a one-year requirement, I was leaving an unusual situation and going to the standard one.
And by the way, we used to have to get up out of the shoebox at twelve o’clock at night, and lick the road clean with our tongues. Just sayin’.
I too had this experience, but in grad school. And it was a physics class (Electricity & Magnetism. E&M, otherwise known as Elves and Magicians…) I swear, it was the same guy!
It’s not an accident that so many of us had that experience. THIS WAS NO ACCIDENT!
We all experienced something we used to call “weed-out classes”. The point was precisely to discourage the weaker students to reconsider the path they were on. Sometimes that was their major, and sometimes that was college itself.
I am current calculus & stats teacher, who enjoys your posts on ballet immensely (as that world had been a closed book to me), and composes & plays jazz (woodwinds).
I had a class or two like that as an undergrad at U.C.__.; exams that completely baffled me when I was a young engineering student. I remember the crushing flop-sweat pretty vividly. Also the recurrent nightmares of taking a final but wearing no pants.
All that to say this: you had the wrong teacher, more of a supervisor than a teacher.
Cheers,
Wry Mouth: you just reminded me—I had nightmares about math exams for many many decades afterward.
My stumbling blocks in college were calculus and organic chem. I think the problem was that I never had to study science or math in HS. They just clicked for me, and I never studied for trig, analytic geometry or anything in 2 years of algebra. When I hitcalculus, I didn’t know how to study it, and honestly I don’t think I was motivated when math reached a level beyond game-like problem solving. I felt bad about getting befuddled by organic chem when the benzene ring was introduced, but I now know several 1st-rate scientists who say that that was the hardest science course they ever took. I also know a first-rate biologist who started out as a physics major at MIT and changed to bio when he realized that some in his class understood instantly what it took him a weekend of work to grasp. He said he knew he could never compete with them.
I think to overcome this instant understanding barrier, you have to be highly motivated to working in a field for the rest of your life. I certainly wasn’t.
It had always stuck in my craw that I was so utterly stupid when it came to Math, but I went into a field in which spells of unemployment aren’t unusual. During one of my involuntary sabbaticals, I picked up an old calculus book in a 2nd hand book store. The Practical Man’s Course in Calculus, or something like that. I think it was published around the turn of the last century, when the tradition of the self-educated man was still strong. Anyway, wisely I think, it eschewed self-important mathematical proofs and theorems in favor of things like “Given a muzzle velocity of 400 feet-per-second, at what elevation should a soldier point a cannon in order to hit a target 1200 yards away?”
I could get my teeth into that!
No, I did not discover my inner sleeping mathematical genius. It wasn’t home. I think beyond a supple brain able to flip concepts upside down and turn them inside out, you need plain common sense, which I lacked and lack. But with dogged effort you can attain competence, and more importantly, the realization that math and the sciences do not exclusively belong to the immeasurably brilliant.
My first two semesters of calculus was taught by a Chinese graduate student with a strong accent. He would write on the blackboard with his right hand while erasing the previous blackboard panel with his left and talked to us over his right shoulder. Luckily the book was great and I was able to understand it by just reading. The next four semesters was taught by a High School Math teacher (moonlighting) who just might have been the world’s best teacher.
‘The college I attended had a fairly rigorous set of science requirements for its liberal arts candidates’
Well, good I say. One of the major problems with modern society is the lack of scientific understanding amongst non scientists.