Anyone with a basic understanding of algebra, can be taught to DO calculus... It's the UNDERSTANDING the calculus that's the hard part. I can tell you don't believe me… So let's look at some analogies.
Say I told you EVERY penguin has feathers. And we went to the Antarctic to see for ourselves. We wander up to a big flock of penguins. Now, how many penguins are you, personally, going to need to look at, before you believe me that it's true?
Five penguins? Ten penguins? A hundred penguins? OK, so say you look at 100 penguins, and it takes you about 15 seconds to inspect them thoroughly. You just spent 25 minutes to completely internalize the fact that all penguins have feathers.
So, let's take it a step further, and use a math analogy. You remember Pythagorean Theorem from geometry? You know, when you have a right triangle, a^2 + b^2 = c^2? Well, I bet your geometry teacher showed you 10 examples in class.
But being the skeptic you ARE, you still weren't convinced. So, you went home and you did some homework. Another 10 problems. And if you are REALLY hard to convince, you went and drew a few of your very OWN triangles, to try to make the Pythagorean Theorem untrue.
How long did those 20 or 25 problems take you to complete? Two or three minutes each? Well, you just spent over an hour there, to INTERNALIZE that Pythagorean Theorem actually works, is true every time.
So what does that mean for the student of Calculus? A really well-constructed Calculus problem, one that addresses something you can actually relate to -- say, the relationship between total distance traveled, time of travel, speed and acceleration? This sort of really THOROUGH problem isn't going to take you two or three minutes. It's going to take you 45-60 minutes!
So, if you are going to get the most basic idea of Calculus, and work that enough to truly INTERNALIZE it and understand that it works, every single time… Well, you are going to put in about 20 hours just to understand the first little bit.
If that sort of thing is the first in 12 chapters, you had better plan on spending HUNDREDS OF HOURS to actually be able to not only DO the Calculus, but to UNDERSTAND what the heck you are doing and WHY you are doing it.
The funny thing is, there was a group of students in high school who took Calculus, and they broke down into three distinct groups.
The first group always had to work hard at math. They did well, and got good grades, but they were never really sure WHY they were doing what they were doing… Other than because they wanted a good grade, their parents wanted them to take the class, or they thought they needed more math to get into college.
The second group maybe had to start actually practicing the math at the Trig part of Algebra 2 Trig. Oh, they really GOT the math before that. And they totally understood WHY, usually without being told. But they learned to work, and in Pre-Calc they learned that the ante was upped.
The third group breezed through Algebra 2 Trig with their eyes closed and one hand tied behind their back. They could have taken the final exam on the first day of class, and passed. And Pre-Calc? Piece of cake. There were fewer problems these guys needed explained to them than fingers on a hand. Sure, they didn't get EVERYTHING on the first try. But they got everything on the second try.
Now, I stop and ask you this: If you work at the Ford plant, and I train you to assemble the fuel injector for a Mustang… How many times do I need to show you this, before you can make a working fuel injector 99% of the time? Once? Time for you to get a promotion. Ten times? You'll be supervisor by the end of the year. 20 or 30 times? Yeah, this is more like it.
BUT… Once you make all your fuel injectors perfectly… Does this mean you actually understand how it WORKS? And, really, do you NEED to understand that, to make them right?
So, in my earlier groups -- who passes Calculus. Only group two. And right now, you are asking the question that truly vexes us all: WHY?
Well, group one got in there, and try as they might, they just didn't see the connection (probably because it wasn't shown to them) between Calculus and Physics. They gave it a good go, and put in 100 hours of real, brain-wracking work… But they just couldn't get the fuel injector to fire up right. At least not all the time. And they got discouraged. Then they gave up.
Group three, on the other hand, went in thinking it was going to be like Pre-Calc. Ask a few questions, do all the homework, and it's going to be easy. It never occurred to them that they would have to absolutely WRACK THEIR BRAINS (something they probably hadn't tried before, anyway!) for 200 or 300 hours, to get really good at this stuff. So they figured their teacher sucks. They got discouraged. Then they gave up.
Group two, on the other hand, actually understood the difference between doing and understanding -- and didn't take for granted that with the doing comes the understanding. They realized that, when they put in enough effort, some time this stuff was going to click. They didn't get discouraged. They didn't give up.
The only problem I have with Calculus, at this exact moment -- is that my husband wants me to teach him Calculus. And the bugger of it is, he wants to UNDERSTAND the damn Calculus!
Which brings me full circle… Anyone with a basic understanding of algebra, can be taught to DO calculus... It's the UNDERSTANDING the calculus that's the hard part.
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