The same thing happened with Galileo and the Pope (or some other priest.)
One calculus, two calculi. Literally, “pebble”, it was the standard name for what we would call “abacus beads”. (“Pebble” rather than “bead” because western calculating tables were not strung.) They were the normal way of doing math until “Arabic” numerals displaced them, not because numerals were better (expert abacus workers could beat even adding machines until the 1970’s), but because an abacus leaves no paper trail.
John W. Kennedy
“Compact is becoming contract; man only earns and pays.”
– Charles Williams
Snark,
I just wanted to share with you my experience that, unlike every other form of math I’d ever been subjected to, I found Calculus to be interesting and even fun at times.
You might be surprised if you give it a try.
Rmariamp
Rmariamp, I was thinking about refreshing my math skills, as I’m 33 and haven’t seen the inside of a math book for more than 10 years. But I think I’d have to start with algebra, trig and geometry in order to understand calculus, and those are big obstacles to me. But I’ll keep your advice in mind and hope you’re right, should I ever make it that far in math studies.
rmariamp, and others: My experiences have been very different than yours. My entire life, and most especially from kindergarten until early October of 12th grade (1971), I found all forms of mathematics to be fun and very intuitive. Geometry, algebra, trig, they all came very naturally to me, because I was always able to picture what was going on, and relate to it in a very practical way.
All that vanished two weeks into the Calculus course. We started out calculating the area under a curve, and after that, Calculus no longer had anything to do with the real world. It was all just rote memorization of arcane formulas. (OK, formulae.) Sure, they all derived from each other, and could be proven from each other, but nothing was intuitive any more.
Has anyone else had any similar experiences?
Does anyone follow what I’m talking about?
I hope that in the years since then, someone has found a way to make Calculus more interesting. I shudder to think of what the study of fractals must be like.
Keeves: I agree with you. Calculus is not usually as interesting as high school math because you can’t really figure it out on your own. But on the other hand, it’s used all the time in physics so it makes up for the boredom in usefulness.
There’s some really good books on physics problems, and a lot of the time they ask you to solve it without using calculus. That forces you to actually use your head and not just plug in different formulas or play with the algebra.
There are 2 approaches to teaching calculus in this country. Good schools prove every formula and don’t require you to memorize anything. Crappy schools make you memorize formulas. Folks who advocate the latter typically have math ed degrees (rather than math phds) and, I suspect, are too stupid to understand the proofs.
dlv: Thanks, but to me, a proof is no better than a memorization, unless there is some sort of picture image that I can use in order to relate to the concept.
Examples: The area of a triangle is half the length times height. I like this, because I can draw a rectangle around the triangle, and see the congruent triangles within it, and the rectangle contains both of the pair while the triangle contains on of each pair, so the triangle has half the area of the rectangle. What is particularly cool is how this can be done even with an obtuse triangle whose base is one of the shorter sides.
Another example is how the volume of a right cone is 1/3 the volume of the cylinder around it. This is nowhere near as intuitive as the triagle in my previous paragraph, but at least I can visualize it, and use that picture as a memory aid for the memorized formula.
But in calculus there are no such visual aids at all.
NanoByte wrote:
K-7th grade: Missouri
8-10th grades: South Dakota
11-12th grades: Utah
Don’t get the idea that these schools didn’t teach calculus to their students (well, maybe not K-7th grades). I was just a bit too stupid to comprehend higher mathematics, I guess, so I declined taking the classes.
My brother took calculus in the same high school that I didn’t.
Keeves, I have to confess that, after taking Calculus for several years in Engineering school, I graduated and haven’t used it since. And I had the same problems: if I can’t visualize what I’m working with, I have trouble understanding and remembering it.
If a way to do a job wrong exists, someone someday will do it that way.
- Capt. Edward Aloysius Murphy Jr. (yes, THAT Murphy)
Developmental Engineer, Edwards AFB, 1941