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Why make a big stink about calculus? To a smart person, after learning about it, calculus is simple and self-evident. The rate of change of something is the sl

Why make a big stink about calculus? To a smart person, after learning about it, calculus is simple and self-evident. The rate of change of something is the slope, and to find it at a point you take two points arbitrarily close together. The rate of change of the area under a curve IS the curve. That's calculus, and the details of the rest follow. Why not give a bright young person those insights and let them play? They never teach the really interesting and cool parts about calculus. How many people know that the rate of change of volume of a sphere _is_ the surface area, for instance? It makes wonderful sense.

kragen 17y ago
There are a lot of useful tricks for finding formulas for the rate of change and its inverse, though. You could probably teach them in a more socratic, leading-the-student way than is traditional.
Retric 17y ago
the standard curriculum is not designed for the top students it’s not a question of learning calculus that's easy the question is how to keep up with your potential. If you understand calculus and diff EQ you can do 95% of the useful math for most walks of life. But getting to that point at 16 is not much more useful than getting there at 20. What’s often harder and more interesting is to get into topology and number theory so you can start to explore higher math. PS: High level math completions in the US are also focused on a wider range of math skills. With a little effort you can start to step out of the "normal mold" and compete at that level.
- WilliamLP 17y ago
  > What’s often harder and more interesting is to get into topology and number theory so you can start to explore higher math. If you learn the basic ideas of calculus when you're 12, this is not going to dissuade you from learning about topology and number theory too. Prime numbers pretty much sell themselves.
sokoloff 17y ago
I still vividly remember an "Aha!" moment in AP Calc in high-school where a problem was stated as (paraphrasing very roughly): Suppose a spherical snowball melts according to the formula: dV/dt = K * pi * r^2 What is the shape of r plotted vs t for a snowball with initial radius of R1? It was astounding to me at first that it was linear, and then when I thought it through and realized that one might naturally expect it to be linear (in real life), and that the change in volume per unit time to accomplish that MUST be proportional to the surface area just snapped into focus all at once. It's so vivid, I can recall what the room looked like, where that problem was on the page, and what desk I was sitting at 23 years ago. This from someone who can't remember what I went upstairs to get just 23 seconds earlier. Both parents were high school math teachers, and they lamented even back then that the focus of high school education was much more about teaching the minimums to pass the exams and frankly, babysitting students than it was instilling a lifelong love of learning or deep understanding of the subject matter. Famous quote from a math teacher at my high-school (neither of my parents :)): "The square root of two is an irrational number because it says so on page 169."

kragen 17y ago

There are a lot of useful tricks for finding formulas for the rate of change and its inverse, though. You could probably teach them in a more socratic, leading-the-student way than is traditional.

Retric 17y ago

the standard curriculum is not designed for the top students it’s not a question of learning calculus that's easy the question is how to keep up with your potential. If you understand calculus and diff EQ you can do 95% of the useful math for most walks of life. But getting to that point at 16 is not much more useful than getting there at 20. What’s often harder and more interesting is to get into topology and number theory so you can start to explore higher math. PS: High level math completions in the US are also focused on a wider range of math skills. With a little effort you can start to step out of the "normal mold" and compete at that level.

WilliamLP 17y ago
> What’s often harder and more interesting is to get into topology and number theory so you can start to explore higher math. If you learn the basic ideas of calculus when you're 12, this is not going to dissuade you from learning about topology and number theory too. Prime numbers pretty much sell themselves.

sokoloff 17y ago

I still vividly remember an "Aha!" moment in AP Calc in high-school where a problem was stated as (paraphrasing very roughly): Suppose a spherical snowball melts according to the formula: dV/dt = K * pi * r^2 What is the shape of r plotted vs t for a snowball with initial radius of R1? It was astounding to me at first that it was linear, and then when I thought it through and realized that one might naturally expect it to be linear (in real life), and that the change in volume per unit time to accomplish that MUST be proportional to the surface area just snapped into focus all at once. It's so vivid, I can recall what the room looked like, where that problem was on the page, and what desk I was sitting at 23 years ago. This from someone who can't remember what I went upstairs to get just 23 seconds earlier. Both parents were high school math teachers, and they lamented even back then that the focus of high school education was much more about teaching the minimums to pass the exams and frankly, babysitting students than it was instilling a lifelong love of learning or deep understanding of the subject matter. Famous quote from a math teacher at my high-school (neither of my parents :)): "The square root of two is an irrational number because it says so on page 169."